ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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Stream function

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</style><div role="note" class="hatnote navigation-not-searchable">For three-dimensional flows with <a href="Axisymmetry" class="mw-redirect" title="Axisymmetry">axisymmetry</a>, see <a href="Stokes_stream_function" title="Stokes stream function">Stokes stream function</a>.</div>

In <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, two types of stream function (or streamfunction) are defined:

<ul><li>The two-dimensional (or Lagrange) stream function, introduced by <a href="Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a> in 1781,<a href="#cite_note-1">[1]</a> is defined for <a href="Incompressible_flow" title="Incompressible flow">incompressible</a> (<a href="Divergence-free" class="mw-redirect" title="Divergence-free">divergence-free</a>), two-dimensional <a href="Fluid_flow" class="mw-redirect" title="Fluid flow">flows</a>.</li>

<li>The <a href="Stokes_stream_function" title="Stokes stream function">Stokes stream function</a>, named after <a href="George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">George Gabriel Stokes</a>,<a href="#cite_note-2">[2]</a> is defined for incompressible, three-dimensional flows with <a href="Axisymmetry" class="mw-redirect" title="Axisymmetry">axisymmetry</a>.</li></ul>

The properties of stream functions make them useful for analyzing and graphically illustrating flows.

The remainder of this article describes the two-dimensional stream function.

<div class="mw-heading mw-heading2"><h2 id="Two-dimensional_stream_function">Two-dimensional stream function</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Assumptions">Assumptions</h3></div>

The two-dimensional stream function is based on the following assumptions:

<ul><li>The flow field can be described as two-dimensional plane flow, with velocity vector</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \quad \mathbf {u} ={\begin{bmatrix}u(x,y,t)\\v(x,y,t)\\0\end{bmatrix}}.}</annotation>

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</math><img src="./8cc4c80d67aa1e2e6dc6d19982fe2c3ab387c1e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:19.937ex; height:9.509ex;" alt="{\displaystyle \quad \mathbf {u} ={\begin{bmatrix}u(x,y,t)\\v(x,y,t)\\0\end{bmatrix}}.}" loading="lazy"></dd></dl>

<ul><li>The velocity satisfies the <a href="Continuity_equation" title="Continuity equation">continuity equation</a> for incompressible flow:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \quad \nabla \cdot \mathbf {u} =0.}</annotation>

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</math><img src="./f7cbface40db50e3ec5b1465d3bd381bdcf0d8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.331ex; height:2.176ex;" alt="{\displaystyle \quad \nabla \cdot \mathbf {u} =0.}" loading="lazy"></dd></dl>

<ul><li>The domain has no holes, or only has holes that have no net flux inwards or outwards.</li></ul>

Although in principle the stream function doesn't require the use of a particular coordinate system, for convenience the description presented here uses a right-handed <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> with coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}">

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<annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation>

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</math><img src="./22a8c93372e8f8b6e24d523bd5545aed3430baf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3></div>

<div class="mw-heading mw-heading4"><h4 id="The_test_surface">The test surface</h4></div>

Consider two points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

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<annotation encoding="application/x-tex">{\displaystyle xy}</annotation>

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</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy"> plane, and a continuous curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AP}">

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<annotation encoding="application/x-tex">{\displaystyle AP}</annotation>

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</math><img src="./0c5ad98caec3c0a2b988d988be2c78cd0dbfe442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.489ex; height:2.176ex;" alt="{\displaystyle AP}" loading="lazy">, also in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

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<annotation encoding="application/x-tex">{\displaystyle xy}</annotation>

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</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy"> plane, that connects them. Then every point on the curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AP}">

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<annotation encoding="application/x-tex">{\displaystyle AP}</annotation>

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</math><img src="./0c5ad98caec3c0a2b988d988be2c78cd0dbfe442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.489ex; height:2.176ex;" alt="{\displaystyle AP}" loading="lazy"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> coordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}">

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<annotation encoding="application/x-tex">{\displaystyle z=0}</annotation>

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</math><img src="./b92bfc06485cc90286474b14a516a68d8bfdd7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}" loading="lazy">. Let the total length of the curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AP}">

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<annotation encoding="application/x-tex">{\displaystyle AP}</annotation>

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</math><img src="./0c5ad98caec3c0a2b988d988be2c78cd0dbfe442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.489ex; height:2.176ex;" alt="{\displaystyle AP}" loading="lazy"> be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">.

Suppose a ribbon-shaped surface is created by extending the curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AP}">

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<annotation encoding="application/x-tex">{\displaystyle AP}</annotation>

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</math><img src="./0c5ad98caec3c0a2b988d988be2c78cd0dbfe442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.489ex; height:2.176ex;" alt="{\displaystyle AP}" loading="lazy"> upward to the horizontal plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=b}">

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<annotation encoding="application/x-tex">{\displaystyle z=b}</annotation>

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</math><img src="./b655a5698d722586f63fc8062ef3206439685eb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.184ex; height:2.176ex;" alt="{\displaystyle z=b}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b>0)}">

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<annotation encoding="application/x-tex">{\displaystyle (b>0)}</annotation>

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</math><img src="./d89ccb2dd7e8f9cef657765f262523845b31258e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.068ex; height:2.843ex;" alt="{\displaystyle (b>0)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is the thickness of the flow. Then the surface has length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">, width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">, and area <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\,L}">

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<annotation encoding="application/x-tex">{\displaystyle b\,L}</annotation>

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</math><img src="./51d8ebc9a093b10177a94ba10c4639e10eb69e05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.967ex; height:2.176ex;" alt="{\displaystyle b\,L}" loading="lazy">. Call this the test surface.

<div class="mw-heading mw-heading4"><h4 id="Flux_through_the_test_surface">Flux through the test surface</h4></div>

The total <a href="Flux" title="Flux">volumetric flux</a> through the test surface is

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<annotation encoding="application/x-tex">{\displaystyle Q(x,y,t)=\int _{0}^{b}\int _{0}^{L}\mathbf {u} \cdot {\hat {\mathbf {n} }}\,\mathrm {d} s\,\mathrm {d} z}</annotation>

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</math><img src="./10a683f4f0df7e3b33c32020c41f823255e71e7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.318ex; height:6.343ex;" alt="{\displaystyle Q(x,y,t)=\int _{0}^{b}\int _{0}^{L}\mathbf {u} \cdot {\hat {\mathbf {n} }}\,\mathrm {d} s\,\mathrm {d} z}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> is an arc-length parameter defined on the curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AP}">

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<annotation encoding="application/x-tex">{\displaystyle AP}</annotation>

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</math><img src="./0c5ad98caec3c0a2b988d988be2c78cd0dbfe442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.489ex; height:2.176ex;" alt="{\displaystyle AP}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=0}">

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<annotation encoding="application/x-tex">{\displaystyle s=0}</annotation>

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</math><img src="./7903b8069a44c70f6f96511675bdd9a4ff200ed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=0}" loading="lazy"> at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=L}">

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<annotation encoding="application/x-tex">{\displaystyle s=L}</annotation>

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</math><img src="./d27c075e88f70d757acc08ac86e8200189e04baf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.772ex; height:2.176ex;" alt="{\displaystyle s=L}" loading="lazy"> at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">.

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation>

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</math><img src="./aae87b164ba005e99b51066c46d1eacc7f56564a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}" loading="lazy"> is the unit vector perpendicular to the test surface, i.e.,

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}\,\mathrm {d} s=-R\,\mathrm {d} \mathbf {r} ={\begin{bmatrix}\mathrm {d} y\\-\mathrm {d} x\\0\end{bmatrix}}}</annotation>

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</math><img src="./767cb3cb24e360a8135f4f547113ad249ad2903e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:25.089ex; height:9.509ex;" alt="{\displaystyle {\hat {\mathbf {n} }}\,\mathrm {d} s=-R\,\mathrm {d} \mathbf {r} ={\begin{bmatrix}\mathrm {d} y\\-\mathrm {d} x\\0\end{bmatrix}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}">

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<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>

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</math><img src="./ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" loading="lazy"> <a href="Rotation_matrix" title="Rotation matrix">rotation matrix</a> corresponding to a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}">

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<annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation>

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</math><img src="./c326d317eddef3ad3e6625e018a708e290a039f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}" loading="lazy"> anticlockwise rotation about the positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> axis:

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<annotation encoding="application/x-tex">{\displaystyle R=R_{z}(90^{\circ })={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}}.}</annotation>

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</math><img src="./dd4b82195370a714667b24c945130c6cdaca6cf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:30.355ex; height:9.176ex;" alt="{\displaystyle R=R_{z}(90^{\circ })={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}}.}" loading="lazy"></dd></dl>

The integrand in the expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> is independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">, so the outer integral can be evaluated to yield

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x,y,t)=b\,\int _{A}^{P}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right)}</annotation>

</semantics>

</math><img src="./ccb309cbd4ca026e4e99a43bd456ca9a53a23603.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.18ex; height:6.176ex;" alt="{\displaystyle Q(x,y,t)=b\,\int _{A}^{P}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right)}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Classical_definition">Classical definition</h4></div>

<a href="Horace_Lamb" title="Horace Lamb">Lamb</a> and <a href="George_Batchelor" title="George Batchelor">Batchelor</a> define the stream function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> as follows.<a href="#cite_note-3">[3]</a>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (x,y,t)=\int _{A}^{P}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right)}</annotation>

</semantics>

</math><img src="./af25496de879840473ef9adb24cba6a24a2d4e92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.083ex; height:6.176ex;" alt="{\displaystyle \psi (x,y,t)=\int _{A}^{P}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right)}" loading="lazy"></dd></dl>

Using the expression derived above for the total volumetric flux, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

</semantics>

</math><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy">, this can be written as

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (x,y,t)={\frac {Q(x,y,t)}{b}}}</annotation>

</semantics>

</math><img src="./42b381c3a8ccf177404eab897d04b6c99944805c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.69ex; height:5.843ex;" alt="{\displaystyle \psi (x,y,t)={\frac {Q(x,y,t)}{b}}}" loading="lazy">.</dd></dl>

In words, the stream function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> is the volumetric flux through the test surface per unit thickness, where thickness is measured perpendicular to the plane of flow.

The point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a reference point that defines where the stream function is identically zero. Its position is chosen more or less arbitrarily and, once chosen, typically remains fixed.

An <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> shift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} P=(\mathrm {d} x,\mathrm {d} y)}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} P=(\mathrm {d} x,\mathrm {d} y)}</annotation>

</semantics>

</math><img src="./8fccea817b193820543d7bba480c984b39219b4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.05ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} P=(\mathrm {d} x,\mathrm {d} y)}" loading="lazy"> in the position of point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> results in the following change of the stream function:

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \psi =u\,\mathrm {d} y-v\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./bfd52dc67677e5798e349975f857932517a811d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.046ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \psi =u\,\mathrm {d} y-v\,\mathrm {d} x}" loading="lazy">.</dd></dl>

From the <a href="Exact_differential" title="Exact differential">exact differential</a>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \psi ={\frac {\partial \psi }{\partial x}}\,\mathrm {d} x+{\frac {\partial \psi }{\partial y}}\,\mathrm {d} y,}</annotation>

</semantics>

</math><img src="./560b6c1ad0e23427b25483eb3d62153330050547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.571ex; height:6.176ex;" alt="{\displaystyle \mathrm {d} \psi ={\frac {\partial \psi }{\partial x}}\,\mathrm {d} x+{\frac {\partial \psi }{\partial y}}\,\mathrm {d} y,}" loading="lazy"></dd></dl>

so the flow velocity components in relation to the stream function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> must be

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u={\frac {\partial \psi }{\partial y}},\qquad v=-{\frac {\partial \psi }{\partial x}}.}</annotation>

</semantics>

</math><img src="./287f5e8834582821fedf5338eda84b84b79d52fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.123ex; height:6.176ex;" alt="{\displaystyle u={\frac {\partial \psi }{\partial y}},\qquad v=-{\frac {\partial \psi }{\partial x}}.}" loading="lazy"></dd></dl>

Notice that the stream function is <a href="Linear" class="mw-redirect" title="Linear">linear</a> in the velocity. Consequently if two incompressible flow fields are superimposed, then the stream function of the resultant flow field is the algebraic sum of the stream functions of the two original fields.

<div class="mw-heading mw-heading3"><h3 id="Effect_of_shift_in_position_of_reference_point">Effect of shift in position of reference point</h3></div>

Consider a shift in the position of the reference point, say from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'}</annotation>

</semantics>

</math><img src="./98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi '}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi '}</annotation>

</semantics>

</math><img src="./3e4792d3361b67c2f3eb7a3394932d28a6ef2e7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.198ex; height:2.843ex;" alt="{\displaystyle \psi '}" loading="lazy"> denote the stream function relative to the shifted reference point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'}</annotation>

</semantics>

</math><img src="./98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy">:

<msup>

</msup>

<msup>

</msup>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi '(x,y,t)=\int _{A'}^{P}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right).}</annotation>

</semantics>

</math><img src="./efc51bf7e09a2bc75b339471aae7eacb8ce9a85e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.801ex; height:6.176ex;" alt="{\displaystyle \psi '(x,y,t)=\int _{A'}^{P}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right).}" loading="lazy"></dd></dl>

Then the stream function is shifted by

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</msup>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta \psi (t)&=\psi '(x,y,t)-\psi (x,y,t)\\&=\int _{A'}^{A}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right),\end{aligned}}}</annotation>

</semantics>

</math><img src="./4aee8e63693515fc71830f6f8d2bee31e1cf403d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:30.903ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\Delta \psi (t)&=\psi '(x,y,t)-\psi (x,y,t)\\&=\int _{A'}^{A}\left(u\,\mathrm {d} y-v\,\mathrm {d} x\right),\end{aligned}}}" loading="lazy"></dd></dl>

which implies the following:

<ul><li>A shift in the position of the reference point effectively adds a constant (for steady flow) or a function solely of time (for nonsteady flow) to the stream function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> at every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">.</li>

<li>The shift in the stream function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \psi }</annotation>

</semantics>

</math><img src="./56cd77c82e3bef469ed0c406ca046620a4d8d2fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.449ex; height:2.509ex;" alt="{\displaystyle \Delta \psi }" loading="lazy">, is equal to the total volumetric flux, per unit thickness, through the continuous surface that extends from point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'}</annotation>

</semantics>

</math><img src="./98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy"> to point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. Consequently <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \psi =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta \psi =0}</annotation>

</semantics>

</math><img src="./a2f563d12213dca7719e4cebb8d23b859955b505.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.71ex; height:2.509ex;" alt="{\displaystyle \Delta \psi =0}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A'}</annotation>

</semantics>

</math><img src="./98a12527148d6ed68adc91d9b419eb4b92d58ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}" loading="lazy"> lie on the same streamline.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="In_terms_of_vector_rotation">In terms of vector rotation</h3></div>

The velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation>

</semantics>

</math><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"> can be expressed in terms of the stream function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> as

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =-R\,\nabla \psi }</annotation>

</semantics>

</math><img src="./3e12f3f1e1db678e8fa4041ce04c31849780ec8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.992ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} =-R\,\nabla \psi }" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>

</semantics>

</math><img src="./ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" loading="lazy"> rotation matrix corresponding to a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation>

</semantics>

</math><img src="./c326d317eddef3ad3e6625e018a708e290a039f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}" loading="lazy"> anticlockwise rotation about the positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> axis. Solving the above equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla \psi }</annotation>

</semantics>

</math><img src="./1c0efa8fbd06a61d2e56e1fdf036f63639bd2a83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.449ex; height:2.509ex;" alt="{\displaystyle \nabla \psi }" loading="lazy"> produces the equivalent form

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla \psi =R\,\mathbf {u} .}</annotation>

</semantics>

</math><img src="./c4598e4d9095a1d2649e700e1ab6f02d93392b8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.831ex; height:2.509ex;" alt="{\displaystyle \nabla \psi =R\,\mathbf {u} .}" loading="lazy"></dd></dl>

From these forms it is immediately evident that the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation>

</semantics>

</math><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla \psi }</annotation>

</semantics>

</math><img src="./1c0efa8fbd06a61d2e56e1fdf036f63639bd2a83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.449ex; height:2.509ex;" alt="{\displaystyle \nabla \psi }" loading="lazy"> are

<ul><li>perpendicular: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot \nabla \psi =0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot \nabla \psi =0}</annotation>

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</math><img src="./4b5c96fcda1985ce33de9da1a97d51cc051b54fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.874ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} \cdot \nabla \psi =0}" loading="lazy"></li>

<li>of the same length: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {u} |=|\nabla \psi |}">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {u} |=|\nabla \psi |}</annotation>

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</math><img src="./fd4e183a5a7cbd531bfff35af827127acb212704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.62ex; height:2.843ex;" alt="{\displaystyle |\mathbf {u} |=|\nabla \psi |}" loading="lazy">.</li></ul>

Additionally, the compactness of the rotation form facilitates manipulations (e.g., see <a class="mw-selflink-fragment" href="#Condition_of_existence">Condition of existence</a>).

<div class="mw-heading mw-heading3"><h3 id="In_terms_of_vector_potential_and_stream_surfaces">In terms of vector potential and stream surfaces</h3></div>

In general, a divergence-free field like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation>

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</math><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy">, also known as a <a href="Solenoidal_vector_field" title="Solenoidal vector field">solenoidal vector field</a>, can always be represented as the curl of some <a href="Vector_potential" title="Vector potential">vector potential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}}</annotation>

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</math><img src="./5a8b5a6d1dbadead8b1dc48719c888e6cac5f861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {A}}}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =\nabla \times {\boldsymbol {A}}.}</annotation>

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</math><img src="./b9912f3b1e43e099987030bddba4619015a7ec59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.026ex; height:2.176ex;" alt="{\displaystyle \mathbf {u} =\nabla \times {\boldsymbol {A}}.}" loading="lazy"></dd></dl>

The stream function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> can be understood as providing the strength of a vector potential that is directed perpendicular to the plane:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}(x,y,t)={\begin{bmatrix}0\\0\\\psi (x,y,t)\end{bmatrix}},}</annotation>

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</math><img src="./d680107333e200f377c6ebd2e6d496c0e45be556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:25.534ex; height:9.509ex;" alt="{\displaystyle {\boldsymbol {A}}(x,y,t)={\begin{bmatrix}0\\0\\\psi (x,y,t)\end{bmatrix}},}" loading="lazy"></dd></dl>

in other words <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {A}}=\psi {\hat {\mathbf {z} }}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {A}}=\psi {\hat {\mathbf {z} }}}</annotation>

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</math><img src="./bb783954323add28e021399e29ae166263ef61ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.819ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {A}}=\psi {\hat {\mathbf {z} }}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {z} }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {z} }}}</annotation>

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</math><img src="./2d4717aa1fda3698bf5489269af7a5904c39c07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {z} }}}" loading="lazy"> is the unit vector pointing in the positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> direction.

This can also be written as the vector cross product

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =\nabla \psi \times {\hat {\mathbf {z} }}}</annotation>

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</math><img src="./cb76197512a957241af907bab8b696c7e257139d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.061ex; height:2.676ex;" alt="{\displaystyle \mathbf {u} =\nabla \psi \times {\hat {\mathbf {z} }}}" loading="lazy"></dd></dl>

where we've used the <a href="Vector_calculus_identity" class="mw-redirect" title="Vector calculus identity">vector calculus identity</a>

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times \left(\psi {\hat {\mathbf {z} }}\right)=\psi \nabla \times {\hat {\mathbf {z} }}+\nabla \psi \times {\hat {\mathbf {z} }}.}</annotation>

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</math><img src="./26e9fc8514de7747846f418e6fef12e001d7a5ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.827ex; height:2.843ex;" alt="{\displaystyle \nabla \times \left(\psi {\hat {\mathbf {z} }}\right)=\psi \nabla \times {\hat {\mathbf {z} }}+\nabla \psi \times {\hat {\mathbf {z} }}.}" loading="lazy"></dd></dl>

Noting that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {z} }}=\nabla z}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {z} }}=\nabla z}</annotation>

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</math><img src="./b7c48db6cc81dbd2df39c318b3ae370389bae263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.31ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {z} }}=\nabla z}" loading="lazy">, and defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =z}">

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<annotation encoding="application/x-tex">{\displaystyle \phi =z}</annotation>

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</math><img src="./1fb03a5e8f95be029e28cf6969b4eec35547599c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.572ex; height:2.509ex;" alt="{\displaystyle \phi =z}" loading="lazy">, one can express the velocity field as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =\nabla \psi \times \nabla \phi .}</annotation>

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</math><img src="./abab2b8fa0de2403561c35aa81d6222cb6872298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.841ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} =\nabla \psi \times \nabla \phi .}" loading="lazy"></dd></dl>

This form shows that the level surfaces of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> and the level surfaces of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> (i.e., horizontal planes) form a system of orthogonal <a href="Streamsurface" title="Streamsurface">stream surfaces</a>.

<div class="mw-heading mw-heading3"><h3 id="Alternative_(opposite_sign)_definition">Alternative (opposite sign) definition</h3></div>

An alternative definition, sometimes used in <a href="Meteorology" title="Meteorology">meteorology</a> and <a href="Oceanography" title="Oceanography">oceanography</a>, is

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<annotation encoding="application/x-tex">{\displaystyle \psi '=-\psi .}</annotation>

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</math><img src="./cf127050b392c95331c88f38c9515ccc127617f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.264ex; height:2.843ex;" alt="{\displaystyle \psi '=-\psi .}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Relation_to_vorticity">Relation to vorticity</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Vorticity" title="Vorticity">Vorticity</a></div>

In two-dimensional plane flow, the <a href="Vorticity" title="Vorticity">vorticity</a> vector, defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {u} }">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {u} }</annotation>

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</math><img src="./7710b172dd06dc78db845ce4710126781860f037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.029ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {u} }" loading="lazy">, reduces to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \,{\hat {\mathbf {z} }}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega \,{\hat {\mathbf {z} }}}</annotation>

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</math><img src="./aa068e396792c6743ca107585b78f0c5b7269498.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.021ex; height:2.343ex;" alt="{\displaystyle \omega \,{\hat {\mathbf {z} }}}" loading="lazy">, where

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<annotation encoding="application/x-tex">{\displaystyle \omega =-\nabla ^{2}\psi }</annotation>

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</math><img src="./e3140dae2c8ebc060582bf622431780a0422f4b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.856ex; height:3.009ex;" alt="{\displaystyle \omega =-\nabla ^{2}\psi }" loading="lazy"></dd></dl>

or

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \omega =+\nabla ^{2}\psi '}</annotation>

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</math><img src="./a296da9d10766f3ba6fbe8b4092e09add4446df0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.54ex; height:3.009ex;" alt="{\displaystyle \omega =+\nabla ^{2}\psi '}" loading="lazy"></dd></dl>

These are forms of <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a>.

<div class="mw-heading mw-heading3"><h3 id="Relation_to_streamlines">Relation to streamlines</h3></div>

Consider two-dimensional plane flow with two infinitesimally close points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(x,y,z)}">

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<annotation encoding="application/x-tex">{\displaystyle P=(x,y,z)}</annotation>

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</math><img src="./7a1320b61607963c1f2ae3598093f6167d7e0ed6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.294ex; height:2.843ex;" alt="{\displaystyle P=(x,y,z)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'=(x+dx,y+dy,z)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle P'=(x+dx,y+dy,z)}</annotation>

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</math><img src="./2228c24f671cc542a506ad488ae3628b1f161fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.653ex; height:3.009ex;" alt="{\displaystyle P'=(x+dx,y+dy,z)}" loading="lazy"> lying in the same horizontal plane. From calculus, the corresponding infinitesimal difference between the values of the stream function at the two points is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {d} \psi (x,y,t)&=\psi (x+\mathrm {d} x,y+\mathrm {d} y,t)-\psi (x,y,t)\\&={\partial \psi \over \partial x}\mathrm {d} x+{\partial \psi \over \partial y}\mathrm {d} y\\&=\nabla \psi \cdot \mathrm {d} \mathbf {r} \end{aligned}}}</annotation>

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</math><img src="./1f732533d94b43e3701a0df0f86345d70cd4dd67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:44.879ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\mathrm {d} \psi (x,y,t)&=\psi (x+\mathrm {d} x,y+\mathrm {d} y,t)-\psi (x,y,t)\\&={\partial \psi \over \partial x}\mathrm {d} x+{\partial \psi \over \partial y}\mathrm {d} y\\&=\nabla \psi \cdot \mathrm {d} \mathbf {r} \end{aligned}}}" loading="lazy"></dd></dl>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> takes the same value, say <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">, at the two points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'}">

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<annotation encoding="application/x-tex">{\displaystyle P'}</annotation>

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</math><img src="./ef4efa52c8f47f06136f6ebfd1d68c1249aaca39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.506ex; height:2.509ex;" alt="{\displaystyle P'}" loading="lazy">. Then this gives

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<annotation encoding="application/x-tex">{\displaystyle 0=\nabla \psi \cdot \mathrm {d} \mathbf {r} ,}</annotation>

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</math><img src="./17ee737105e60a8305efdff3945edf6ad3be507b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.43ex; height:2.509ex;" alt="{\displaystyle 0=\nabla \psi \cdot \mathrm {d} \mathbf {r} ,}" loading="lazy"></dd></dl>

implying that the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \psi }">

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<annotation encoding="application/x-tex">{\displaystyle \nabla \psi }</annotation>

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</math><img src="./1c0efa8fbd06a61d2e56e1fdf036f63639bd2a83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.449ex; height:2.509ex;" alt="{\displaystyle \nabla \psi }" loading="lazy"> is normal to the surface <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =C}">

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<annotation encoding="application/x-tex">{\displaystyle \psi =C}</annotation>

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</math><img src="./822e4689821f9ff614071d42db5d3266a4b4914e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.378ex; height:2.509ex;" alt="{\displaystyle \psi =C}" loading="lazy">. Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \cdot \nabla \psi =0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \cdot \nabla \psi =0}</annotation>

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</math><img src="./4b5c96fcda1985ce33de9da1a97d51cc051b54fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.874ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} \cdot \nabla \psi =0}" loading="lazy"> everywhere (e.g., see <a class="mw-selflink-fragment" href="#Condition_of_existence">In terms of vector rotation</a>), each streamline corresponds to the intersection of a particular stream surface and a particular horizontal plane. Consequently, in three dimensions, unambiguous identification of any particular streamline requires that one specify corresponding values of both the stream function and the elevation (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> coordinate).

The development here assumes the space domain is three-dimensional. The concept of stream function can also be developed in the context of a two-dimensional space domain. In that case level sets of the stream function are curves rather than surfaces, and streamlines are level curves of the stream function. Consequently, in two dimensions, unambiguous identification of any particular streamline requires that one specify the corresponding value of the stream function only.

<div class="mw-heading mw-heading3"><h3 id="Condition_of_existence">Condition of existence</h3></div>

It's straightforward to show that for two-dimensional plane flow <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation>

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</math><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"> satisfies the curl-divergence equation

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\nabla \cdot \mathbf {u} )\,{\hat {\mathbf {z} }}=-\nabla \times (R\,\mathbf {u} )}</annotation>

</semantics>

</math><img src="./bf5d634ea72069afc68e533bc385f45b33c10620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.613ex; height:2.843ex;" alt="{\displaystyle (\nabla \cdot \mathbf {u} )\,{\hat {\mathbf {z} }}=-\nabla \times (R\,\mathbf {u} )}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation>

</semantics>

</math><img src="./ddc0d4d6106875f8006be1d898512ca5843bad8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}" loading="lazy"> <a href="Rotation_matrix" title="Rotation matrix">rotation matrix</a> corresponding to a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation>

</semantics>

</math><img src="./c326d317eddef3ad3e6625e018a708e290a039f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}" loading="lazy"> anticlockwise rotation about the positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> axis. This equation holds regardless of whether or not the flow is incompressible.

If the flow is incompressible (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {u} =0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {u} =0}</annotation>

</semantics>

</math><img src="./8f70c85ec868f8dd21cf2662ebfa2ec2552df8be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.361ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {u} =0}" loading="lazy">), then the curl-divergence equation gives

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {0} =\nabla \times (R\,\mathbf {u} )}</annotation>

</semantics>

</math><img src="./51653b1e792fc52b1c5cb0b8c7e9c4317acac862.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.657ex; height:2.843ex;" alt="{\displaystyle \mathbf {0} =\nabla \times (R\,\mathbf {u} )}" loading="lazy">.</dd></dl>

Then by <a href="Stokes'_theorem" title="Stokes' theorem">Stokes' theorem</a> the line integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\,\mathbf {u} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\,\mathbf {u} }</annotation>

</semantics>

</math><img src="./812fb99e7e372987614ad65936a9fcccd9714356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.636ex; height:2.176ex;" alt="{\displaystyle R\,\mathbf {u} }" loading="lazy"> over every closed loop vanishes

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \oint _{\partial \Sigma }(R\,\mathbf {u} )\cdot \mathrm {d} \mathbf {\Gamma } =0.}</annotation>

</semantics>

</math><img src="./c9deee6b1b243a8e78ce035db18fe2c991139df8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.577ex; height:5.676ex;" alt="{\displaystyle \oint _{\partial \Sigma }(R\,\mathbf {u} )\cdot \mathrm {d} \mathbf {\Gamma } =0.}" loading="lazy"></dd></dl>

Hence, the line integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\,\mathbf {u} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\,\mathbf {u} }</annotation>

</semantics>

</math><img src="./812fb99e7e372987614ad65936a9fcccd9714356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.636ex; height:2.176ex;" alt="{\displaystyle R\,\mathbf {u} }" loading="lazy"> is path-independent. Finally, by the converse of the <a href="Gradient_theorem" title="Gradient theorem">gradient theorem</a>, a scalar function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,y,t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi (x,y,t)}</annotation>

</semantics>

</math><img src="./de2c12a27ede081d666d699690ee69be0ee6441d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.715ex; height:2.843ex;" alt="{\displaystyle \psi (x,y,t)}" loading="lazy"> exists such that

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\,\mathbf {u} =\nabla \psi }</annotation>

</semantics>

</math><img src="./fae95201d5fbed954b876794ee28d59767502517.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.184ex; height:2.509ex;" alt="{\displaystyle R\,\mathbf {u} =\nabla \psi }" loading="lazy">.</dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> represents the stream function.

Conversely, if the stream function exists, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\,\mathbf {u} =\nabla \psi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\,\mathbf {u} =\nabla \psi }</annotation>

</semantics>

</math><img src="./fae95201d5fbed954b876794ee28d59767502517.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.184ex; height:2.509ex;" alt="{\displaystyle R\,\mathbf {u} =\nabla \psi }" loading="lazy">. Substituting this result into the curl-divergence equation yields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {u} =0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {u} =0}</annotation>

</semantics>

</math><img src="./8f70c85ec868f8dd21cf2662ebfa2ec2552df8be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.361ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {u} =0}" loading="lazy"> (i.e., the flow is incompressible).

In summary, the stream function for two-dimensional plane flow exists if and only if the flow is incompressible.

<div class="mw-heading mw-heading3"><h3 id="Potential_flow">Potential flow</h3></div>

For two-dimensional <a href="Potential_flow" title="Potential flow">potential flow</a>, streamlines are perpendicular to <a href="Equipotential" title="Equipotential">equipotential</a> lines. Taken together with the <a href="Velocity_potential" title="Velocity potential">velocity potential</a>, the stream function may be used to derive a <a href="Complex_number" title="Complex number">complex</a> potential. In other words, the stream function accounts for the <a href="Solenoidal" class="mw-redirect" title="Solenoidal">solenoidal</a> part of a two-dimensional <a href="Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a>, while the velocity potential accounts for the <a href="Irrotational" class="mw-redirect" title="Irrotational">irrotational</a> part.

<div class="mw-heading mw-heading3"><h3 id="Summary_of_properties">Summary of properties</h3></div>

The basic properties of two-dimensional stream functions can be summarized as follows:

<ol><li>The x- and y-components of the flow velocity at a given point are given by the <a href="Partial_derivative" title="Partial derivative">partial derivatives</a> of the stream function at that point.</li>

<li>The value of the stream function is constant along every <a href="Streamlines%2C_streaklines%2C_and_pathlines" title="Streamlines, streaklines, and pathlines">streamline</a> (streamlines represent the trajectories of particles in steady flow). That is, in two dimensions each streamline is a <a href="Level_curve" class="mw-redirect" title="Level curve">level curve</a> of the stream function.</li>

<li>The difference between the stream function values at any two points gives the volumetric flux through the vertical surface that connects the two points.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Two-dimensional_stream_function_for_flows_with_time-invariant_density">Two-dimensional stream function for flows with time-invariant density</h2></div>

If the fluid density is time-invariant at all points within the flow, i.e.,

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \rho }{\partial t}}=0}</annotation>

</semantics>

</math><img src="./e0c9f617a0673ac5f483570f9f17d2e190330f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.617ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \rho }{\partial t}}=0}" loading="lazy">,</dd></dl>

then the <a href="Continuity_equation" title="Continuity equation">continuity equation</a> (e.g., see <a href="Continuity_equation#Fluid_dynamics" title="Continuity equation">Continuity equation#Fluid dynamics</a>) for two-dimensional plane flow becomes

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot (\rho \,\mathbf {u} )=0.}</annotation>

</semantics>

</math><img src="./08912a977ffc5fec679d0e384c3810d9d42e3d7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.406ex; height:2.843ex;" alt="{\displaystyle \nabla \cdot (\rho \,\mathbf {u} )=0.}" loading="lazy"></dd></dl>

In this case the stream function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> is defined such that

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho \,u={\frac {\partial \psi }{\partial y}},\quad \rho \,v=-{\frac {\partial \psi }{\partial x}}}</annotation>

</semantics>

</math><img src="./f250ddf0b1cd578f336e2e1129089bd2fadfe686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.332ex; height:6.176ex;" alt="{\displaystyle \rho \,u={\frac {\partial \psi }{\partial y}},\quad \rho \,v=-{\frac {\partial \psi }{\partial x}}}" loading="lazy"></dd></dl>

and represents the mass flux (rather than volumetric flux) per unit thickness through the test surface.

<ul><li><a href="Elementary_flow" title="Elementary flow">Elementary flow</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFLagrange1868" class="citation cs2 cs1-prop-long-vol"><a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange, J.-L.</a> (1868), "Mémoire sur la théorie du mouvement des fluides (in: Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, année 1781)", <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k229223s/f697.image">Oevres de Lagrange</a>, vol. Tome IV, pp. 695–748</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFStokes1842" class="citation cs2"><a href="George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">Stokes, G.G.</a> (1842), "On the steady motion of incompressible fluids", Transactions of the Cambridge Philosophical Society, 7: 439–453, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1848TCaPS...7..439S">1848TCaPS...7..439S</a></cite> Reprinted in: <cite id="CITEREFStokes1880" class="citation cs2">Stokes, G.G. (1880), <a rel="nofollow" class="external text" href="https://archive.org/details/mathphyspapers01stokrich">Mathematical and Physical Papers, Volume I</a>, Cambridge University Press, pp. 1–16</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFLamb1932">Lamb (1932</a>, pp. 62–63) and <a href="#CITEREFBatchelor1967">Batchelor (1967</a>, pp. 75–79)

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFKatopodes2019" class="citation book cs1">Katopodes, Nikolaos D. (2019). "Viscous Fluid Flow". Free-Surface Flow. pp. 324–426. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FB978-0-12-815489-2.00005-8">10.1016/B978-0-12-815489-2.00005-8</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-815489-2</bdi>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div>

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<ul><li><cite id="CITEREFBatchelor1967" class="citation cs2"><a href="George_Batchelor" title="George Batchelor">Batchelor, G. K.</a> (1967), An Introduction to Fluid Dynamics, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-09817-3</bdi></cite></li>

<li><cite id="CITEREFLamb1932" class="citation cs2"><a href="Horace_Lamb" title="Horace Lamb">Lamb, H.</a> (1932), Hydrodynamics (6th ed.), Cambridge University Press, republished by Dover Publications, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-60256-7</bdi></cite> <code class="cs1-code">{{citation}}</code>: ISBN / Date incompatibility (help)</li>

<li><cite id="CITEREFMasseyWard-Smith1998" class="citation cs2">Massey, B. S.; Ward-Smith, J. (1998), Mechanics of Fluids (7th ed.), UK: Nelson Thornes</cite></li>

<li><cite id="CITEREFWhite2003" class="citation cs2">White, F. M. (2003), Fluid Mechanics (5th ed.), New York: McGraw-Hill</cite></li>

<li><cite id="CITEREFGamelin2001" class="citation cs2">Gamelin, T. W. (2001), Complex Analysis, New York: Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95093-1</bdi></cite></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://glossary.ametsoc.org/wiki/Streamfunction">"Streamfunction"</a>, AMS Glossary of Meteorology, <a href="American_Meteorological_Society" title="American Meteorological Society">American Meteorological Society</a>, retrieved 2014-01-30</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://prj.dimanov.com/">Joukowsky Transform Interactive WebApp</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-14" href="https://en.wikipedia.org/wiki/?title=Stream_function&oldid=1285605432">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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