Name: stream function for incompressible flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

In two-dimensional incompressible flow, there are horizontal and vertical velocity components.
The continuity equation ensures mass conservation by requiring that the sum of the rates of change of these components is zero, meaning the net inflow and outflow in any region are balanced.
To satisfy this condition, the stream function is introduced.
The horizontal velocity is the partial derivative of the stream function with respect to the vertical direction, while the vertical velocity is the negative partial derivative with respect to the horizontal direction. This ensures mass conservation is automatically maintained.
A key feature of the stream function is that it remains constant along streamlines, which represent the paths of fluid particles and are tangent to the velocities in the flow field.
Streamlines, often depicted as contour lines, allow intuitive interpretation of fluid behavior.
The stream function simplifies fluid motion calculations. The difference in stream function values between two streamlines represents the volume flow rate between them.
This eliminates the need to solve for velocity components separately, making it an efficient tool for analyzing two-dimensional flow.

stream function

Stream Function

In two-dimensional incompressible fluid flow, the continuity equation is essential for ensuring mass conservation, meaning that any change in fluid entering or exiting a region is balanced by a corresponding change elsewhere. For incompressible flow, where density remains constant, this requirement simplifies to the condition that the divergence of the velocity field must be zero. Mathematically, this is expressed as,
<img alt="Continuity equation, &#8706;u/&#8706;x + &#8706;v/&#8706;y = 0, partial differential equation diagram" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
where u and v represent the horizontal and vertical velocity components, respectively.
To inherently satisfy this equation, we introduce the stream function &#968;, which allows the velocity components to be defined in terms of &#968; as:
<img alt="Dynamic equilibrium; phase diagram showing vapor-liquid equilibrium of hexane and ethanol mixture." src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
These definitions automatically satisfy the continuity equation, as the mixed partial derivatives of &#968; cancel out.
The stream function &#968; is constant along streamlines, which are the paths that fluid particles follow within the flow. Streamlines can therefore be visualized as contour lines of &#968;, providing a clear representation of fluid motion. Each streamline is tangent to the velocity vector at any point along its path, illustrating the direction of flow without the need for calculating velocities at each point individually.
Moreover, the difference in &#968;\psi&#968; values between two streamlines represents the volumetric flow rate per unit depth between them, enabling direct calculation of flow rates. This property significantly simplifies the analysis of two-dimensional incompressible flows by removing the need to solve separate equations for u and v. As a result, the stream function &#968; is a powerful tool for modeling and interpreting fluid behavior in applications involving incompressible, steady flow.

In two-dimensional incompressible flow, there are horizontal and vertical velocity components.
The continuity equation ensures mass conservation by ...

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Differential Analysis of Fluid Flow

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stream function for incompressible flow

The stream function helps describe two-dimensional incompressible flow. In this type of flow, the continuity equation keeps mass conserved. That means fluid entering a region must be balanced by fluid leaving it.

For incompressible flow, the density stays constant. The continuity requirement then becomes a zero-divergence condition for the velocity field. In two dimensions, the horizontal and vertical velocity components are usually written as u and v.

A stream function, written as ψ, is used to satisfy this equation automatically. The velocity components can be defined in terms of ψ. When they are written this way, the mixed partial derivatives of ψ cancel, so the continuity equation is already met.

The stream function is also linked to streamlines. A streamline is the path a fluid particle follows in the flow, and ψ stays constant along each one. Because of this, streamlines can be viewed as contour lines of ψ. Each streamline is tangent to the velocity vector at every point, so it shows flow direction without finding the velocity at every location.

Differences in ψ between two streamlines also have meaning. They represent the volumetric flow rate per unit depth between those lines. This makes it easier to analyze steady, two-dimensional incompressible flows. In many cases, the stream function reduces the need to solve separate equations for u and v, so it is a useful tool for modeling and interpreting fluid motion.

Stream Function for Incompressible Flow

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The stream function helps describe two-dimensional incompressible flow. In this type of flow, the continuity equation keeps mass conserved. That means fluid entering a region must be balanced by fluid leaving it. For incompressible flow, the density stays constant. The continuity requirement then becomes a zero-divergence condition for the velocity field. In two dimensions, the horizontal and vertical velocity components are usually written as u and v. A stream function, written as ψ, is used...

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key terms and concepts

Learn how the stream function describes two-dimensional incompressible flow, keeps continuity satisfied, and shows streamlines and flow rate in fluid motion.

Stream Function for Incompressible Flow | JoVE K12

JoVE Core Civil Engineering offers a foundational exploration of essential construction materials such as bricks, stones, steel, wood, and concrete, along with fluid mechanics topics like flow over immersed bodies, open channel flow, and dimensional modeling. It also covers key surveying concepts, including leveling, traverse adjustments, curve design, topographic mapping, and geospatial technologies (GPS & GIS). Science-in-action videos showcase laboratory research on materials, fluids, and surveying practices in real-world engineering systems.

civil engineering

Stream Function for Incompressible Flow

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