# Stream function

The **stream function** is a scalar function used in fluid dynamics to describe a two-dimensional incompressible flow: its partial derivatives give the velocity components, so that the continuity equation (conservation of mass) is satisfied automatically, and its contours are the streamlines of the flow.<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/fluiddynamics.html)</sup> It exists whenever a flow is two-dimensional and incompressible, whether or not the flow is irrotational.<sup>[2](https://web1.eng.famu.fsu.edu/~dommelen/courses/flm/09/topics/pot/node3.html)</sup>

| Key fact | Detail |
|---|---|
| Definition (planar, Cartesian) | \( u = \partial \psi/\partial y \), \( v = -\partial \psi/\partial x \); substituting into \( \partial u/\partial x + \partial v/\partial y = 0 \) gives an identity, so continuity is automatically satisfied<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/fluiddynamics.html)</sup><sup> • </sup><sup>[3](https://courses.washington.edu/me431/handouts/Smits_text_part1.pdf)</sup> |
| Streamlines | Lines of constant \( \psi \) are exactly the streamlines; no flow crosses a streamline<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/fluiddynamics.html)</sup><sup> • </sup><sup>[4](https://warwick.ac.uk/fac/sci/eng/staff/ymc/members/former/azimi/project/references/white_frank_m._-_fluid_mechanics_4th_ed_mcgraw_hill.pdf)</sup> |
| Flow rate | The volume flow rate between two points equals the difference in stream function, \( Q_{1\to 2} = \psi_2 - \psi_1 \)<sup>[4](https://warwick.ac.uk/fac/sci/eng/staff/ymc/members/former/azimi/project/references/white_frank_m._-_fluid_mechanics_4th_ed_mcgraw_hill.pdf)</sup><sup> • </sup><sup>[5](https://eng.libretexts.org/Bookshelves/Civil_Engineering/Fluid_Mechanics_(Bar-Meir)/10:_Inviscid_Flow_or_Potential_Flow/10.2/10.2.1:_Streamline_and_Stream_function)</sup> |
| Speed relation | \( d\psi/dn = U \), where \( n \) is the shortest distance between two streamlines: closely spaced streamlines mean fast flow<sup>[5](https://eng.libretexts.org/Bookshelves/Civil_Engineering/Fluid_Mechanics_(Bar-Meir)/10:_Inviscid_Flow_or_Potential_Flow/10.2/10.2.1:_Streamline_and_Stream_function)</sup> |
| Compressible form | For steady 2-D compressible flow, \( \rho u = \partial \Psi/\partial y \), \( \rho v = -\partial \Psi/\partial x \)<sup>[4](https://warwick.ac.uk/fac/sci/eng/staff/ymc/members/former/azimi/project/references/white_frank_m._-_fluid_mechanics_4th_ed_mcgraw_hill.pdf)</sup> |
| Axisymmetric form | The Stokes stream function gives \( u_r = -(1/r)\,\partial \psi/\partial z \), \( u_z = (1/r)\,\partial \psi/\partial r \) in cylindrical coordinates<sup>[6](https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/20b3e1b7b85203a798bb3e995591a312_potential_flow.pdf)</sup> |
| Governing equations | Irrotational 2-D flow: \( \psi \) satisfies Laplace's equation; Stokes (creeping) flow: the biharmonic equation \( \nabla^4 \psi = 0 \)<sup>[3](https://courses.washington.edu/me431/handouts/Smits_text_part1.pdf)</sup><sup> • </sup><sup>[7](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/6be103e9-015c-447b-8104-c758ff61c666/content)</sup> |

## Definition and relation to continuity

For an incompressible fluid the continuity equation reduces to \( \nabla \cdot \mathbf{v} = 0 \).<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/fluiddynamics.html)</sup> In two dimensions this is \( \partial_x v_x + \partial_y v_y = 0 \), which is satisfied identically by taking

\[ v_x = \frac{\partial \psi}{\partial y}, \qquad v_y = -\frac{\partial \psi}{\partial x} \]

for any twice-differentiable function \( \psi(x, y) \), because the mixed partial derivatives \( \partial^2 \psi/\partial x \partial y \) and \( \partial^2 \psi/\partial y \partial x \) cancel.<sup>[1](https://interactivetextbooks.tudelft.nl/nb1140/content/fluiddynamics.html)</sup><sup> • </sup><sup>[5](https://eng.libretexts.org/Bookshelves/Civil_Engineering/Fluid_Mechanics_(Bar-Meir)/10:_Inviscid_Flow_or_Potential_Flow/10.2/10.2.1:_Streamline_and_Stream_function)</sup> The existence of the stream function is therefore a direct consequence of the incompressibility assumption, and it replaces the two velocity components with a single scalar unknown even when the flow is rotational.<sup>[2](https://web1.eng.famu.fsu.edu/~dommelen/courses/flm/09/topics/pot/node3.html)</sup>

**Streamlines and speed.** Because \( \mathbf{v} \cdot \nabla \psi = 0 \), the velocity vectors are everywhere parallel to the contours of \( \psi \); these contours are the streamlines.<sup>[8](https://solar.physics.montana.edu/dana/incompressible.pdf)</sup> Where streamlines are close together, \( |\nabla \psi| \) is large and the flow is fast; closed contours around local maxima or minima show circular flow, called eddies.<sup>[8](https://solar.physics.montana.edu/dana/incompressible.pdf)</sup> Quantitatively, the change of \( \psi \) across the shortest distance \( dn \) between two streamlines is \( d\psi = U\,dn \), so \( d\psi/dn = U \).<sup>[5](https://eng.libretexts.org/Bookshelves/Civil_Engineering/Fluid_Mechanics_(Bar-Meir)/10:_Inviscid_Flow_or_Potential_Flow/10.2/10.2.1:_Streamline_and_Stream_function)</sup>

## Flow rate between streamlines

Since no flow crosses a streamline, the volume flow rate between any two points in the flow is the integral of the velocity component normal to a connecting line, and this integral equals the difference in stream function:

\[ Q_{1\to 2} = \int_1^2 (\mathbf{V} \cdot \mathbf{n})\,dA = \int_1^2 d\psi = \psi_2 - \psi_1. \]

<sup>[4](https://warwick.ac.uk/fac/sci/eng/staff/ymc/members/former/azimi/project/references/white_frank_m._-_fluid_mechanics_4th_ed_mcgraw_hill.pdf)</sup><sup> • </sup><sup>[5](https://eng.libretexts.org/Bookshelves/Civil_Engineering/Fluid_Mechanics_(Bar-Meir)/10:_Inviscid_Flow_or_Potential_Flow/10.2/10.2.1:_Streamline_and_Stream_function)</sup> The absolute value of \( \psi \) carries no physical meaning; only differences between stream-function values matter, which is why the additive constant is chosen arbitrarily, consistently with the boundary conditions of the problem.<sup>[9](https://innovationspace.ansys.com/courses/wp-content/uploads/sites/5/2020/08/Simple-Approximations-of-Fluid-Flows-Lesson-3-Handout.pdf)</sup>

## Compressible and axisymmetric forms

**Compressible flow.** For steady, two-dimensional compressible flow the continuity equation is \( \partial(\rho u)/\partial x + \partial(\rho v)/\partial y = 0 \), and a stream function can be defined by \( \rho u = \partial \Psi/\partial y \) and \( \rho v = -\partial \Psi/\partial x \); the density is absorbed into the definition.<sup>[4](https://warwick.ac.uk/fac/sci/eng/staff/ymc/members/former/azimi/project/references/white_frank_m._-_fluid_mechanics_4th_ed_mcgraw_hill.pdf)</sup>

**Axisymmetric flow.** For flow that is the same at every azimuth about an axis and has no azimuthal component, a Stokes stream function \( \psi(r, z) \) exists in cylindrical coordinates with

\[ u_r = -\frac{1}{r}\frac{\partial \psi}{\partial z}, \qquad u_z = \frac{1}{r}\frac{\partial \psi}{\partial r}. \]

<sup>[10](https://www.damtp.cam.ac.uk/user/tong/books/examples/f1.pdf)</sup> In spherical coordinates the corresponding components are \( u_r = (1/(r^2 \sin\theta))\,\partial \psi/\partial \theta \) and \( u_\theta = -(1/(r \sin\theta))\,\partial \psi/\partial r \).<sup>[6](https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/20b3e1b7b85203a798bb3e995591a312_potential_flow.pdf)</sup> Planar flow in cylindrical coordinates instead uses \( u_r = (1/r)\,\partial \psi/\partial \theta \), \( u_\theta = -\partial \psi/\partial r \).<sup>[3](https://courses.washington.edu/me431/handouts/Smits_text_part1.pdf)</sup>

## Relation to vorticity and the governing equations

Taking the curl of the stream-function representation gives the vorticity. In two dimensions the vorticity has a single nonzero component, perpendicular to the flow plane, and it is the (negative) Laplacian of the stream function, \( \omega = -\nabla^2 \psi \) under the \( \mathbf{u} = -\nabla \times (\psi \hat{z}) \) convention.<sup>[11](https://math.nyu.edu/faculty/greengar/inteq_ns_2d.pdf)</sup><sup> • </sup><sup>[12](https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=kent1500818090455148&disposition=inline)</sup> This kinematic relation \( \nabla^2 \psi = \omega \) is a constraint, not a time-evolution equation.<sup>[13](https://www.cfm.brown.edu/faculty/gk/AM258/Reading/chapter1/)</sup>

**Irrotational flow.** If the vorticity vanishes, \( \nabla^2 \psi = 0 \): the stream function satisfies [Laplace's equation](https://www.edgepedia.org/laplaces-equation), as does the velocity potential for the same flow.<sup>[3](https://courses.washington.edu/me431/handouts/Smits_text_part1.pdf)</sup><sup> • </sup><sup>[14](https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node69.html)</sup> For two-dimensional, irrotational, incompressible flow both functions exist, and either one determines the velocity field; for steady flows the pressure then follows from Bernoulli's equation.<sup>[3](https://courses.washington.edu/me431/handouts/Smits_text_part1.pdf)</sup>

**Viscous flows.** The stream function is also the basis of the stream function-vorticity formulation of the two-dimensional incompressible Navier-Stokes equations, in which the velocity and pressure fields are replaced by \( \psi \) and the scalar vorticity; the governing equation becomes a single fourth-order equation in \( \psi \).<sup>[11](https://math.nyu.edu/faculty/greengar/inteq_ns_2d.pdf)</sup><sup> • </sup><sup>[15](https://arxiv.org/abs/1810.04222)</sup> In the [Stokes flow](https://www.edgepedia.org/stokes-flow) regime, where viscous terms dominate and inertia is negligible, the stream function satisfies the two-dimensional [biharmonic equation](https://www.edgepedia.org/biharmonic-equation) \( \nabla^2(\nabla^2 \psi) = 0 \).<sup>[7](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/6be103e9-015c-447b-8104-c758ff61c666/content)</sup>

## Use in potential flow and superposition

In two-dimensional potential flow the velocity potential \( \phi \) and the stream function \( \psi \) combine into a complex potential \( \Phi(z) = \phi + i\psi \), and elementary flows have simple stream functions: a uniform stream of components \( (U, V) \) has \( \psi = -Vx + Uy \); a source or sink of strength \( Q \) has \( \psi = (Q/2\pi)\theta \); a free vortex of circulation \( \Gamma \) has \( \psi = -(\Gamma/2\pi)\ln r \).<sup>[6](https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/20b3e1b7b85203a798bb3e995591a312_potential_flow.pdf)</sup> Because the governing equation for \( \psi \) is linear (Laplace's equation), combinations of these elementary stream functions model more complicated flows such as flow around a cylinder.<sup>[16](https://www.open.edu/openlearn/science-maths-technology/kinematics-fluids/content-section-0?active-tab=description-tab)</sup><sup> • </sup><sup>[17](https://open.oregonstate.education/intermediate-fluid-mechanics/chapter/potential-flows/)</sup>

## References

1. [11. Fluid dynamics, Introduction to particle and continuum physics, TU Delft](https://interactivetextbooks.tudelft.nl/nb1140/content/fluiddynamics.html)
2. [2D Incompressible, Fluid Mechanics course notes, FAMU-FSU College of Engineering](https://web1.eng.famu.fsu.edu/~dommelen/courses/flm/09/topics/pot/node3.html)
3. [Laplace's Equation (Smits, Fluid Mechanics text excerpt), University of Washington ME 431](https://courses.washington.edu/me431/handouts/Smits_text_part1.pdf)
4. [Frank M. White, Fluid Mechanics, 4th ed., McGraw-Hill (excerpt)](https://warwick.ac.uk/fac/sci/eng/staff/ymc/members/former/azimi/project/references/white_frank_m._-_fluid_mechanics_4th_ed_mcgraw_hill.pdf)
5. [10.2.1: Streamline and Stream function, Engineering LibreTexts (Bar-Meir, Fluid Mechanics)](https://eng.libretexts.org/Bookshelves/Civil_Engineering/Fluid_Mechanics_(Bar-Meir)/10:_Inviscid_Flow_or_Potential_Flow/10.2/10.2.1:_Streamline_and_Stream_function)
6. [Stream Functions for planar and axisymmetric flow, 2.016 Hydrodynamics, MIT OpenCourseWare](https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/20b3e1b7b85203a798bb3e995591a312_potential_flow.pdf)
7. [Jie Chen, dissertation chapter on lid-driven cavity flow, Virginia Tech](https://vtechworks.lib.vt.edu/server/api/core/bitstreams/6be103e9-015c-447b-8104-c758ff61c666/content)
8. [Dana Longcope, Incompressible Hydrodynamics notes, Montana State University](https://solar.physics.montana.edu/dana/incompressible.pdf)
9. [Wall Boundary Conditions for Stream Function, Ansys Innovations Space handout](https://innovationspace.ansys.com/courses/wp-content/uploads/sites/5/2020/08/Simple-Approximations-of-Fluid-Flows-Lesson-3-Handout.pdf)
10. [Example problems on stream functions, David Tong, University of Cambridge DAMTP](https://www.damtp.cam.ac.uk/user/tong/books/examples/f1.pdf)
11. [Greengard, L. and Kropinski, M. C., SIAM Journal on Scientific Computing (2D Navier-Stokes in polar coordinates)](https://math.nyu.edu/faculty/greengar/inteq_ns_2d.pdf)
12. [Numerical Simulation of 2D Incompressible Navier-Stokes Flow, OhioLINK ETD](https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=kent1500818090455148&disposition=inline)
13. [Applied Mathematics AM 258 reading notes, Brown University](https://www.cfm.brown.edu/faculty/gk/AM258/Reading/chapter1/)
14. [Velocity Potentials and Stream Functions, Farside Fluid Mechanics, University of Texas](https://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node69.html)
15. [Vorticity and stream function formulations for the 2D Navier-Stokes equations in a bounded domain, arXiv](https://arxiv.org/abs/1810.04222)
16. [Kinematics of fluids, OpenLearn, The Open University](https://www.open.edu/openlearn/science-maths-technology/kinematics-fluids/content-section-0?active-tab=description-tab)
17. [Potential Flows, Intermediate Fluid Mechanics, Oregon State University](https://open.oregonstate.education/intermediate-fluid-mechanics/chapter/potential-flows/)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Continuum, solid, and fluid mechanics › Fluid mechanics › Inviscid and potential flow › Potential flow theory*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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