The stream function for an incompressible flow field is...

The stream function for an incompressible flow field is given by the equation $$\psi=3 x^{2} y-y^{3}$$ where the stream function has the units of $m^{2} / s$ with $x$ and $y$ in meters. (a) Sketch the streamline(s) passing through the origin. (b) Determine the rate of flow across the straight path $A B$ shown in Fig. $P 6.20$.

Key Concepts

Incompressible Flow

In incompressible flows the fluid density remains constant, and mathematically this is expressed by a divergence-free velocity field. This fundamental property leads to the introduction of stream functions in two dimensions, which inherently satisfy continuity. It is essential for understanding how the flow behaves and for applying potential flow theory to solve problems without directly tackling the full Navier–Stokes equations.

Stream Function

The stream function is a scalar function whose contours represent streamlines of the flow. In two-dimensional incompressible flow, the stream function automatically satisfies the continuity equation. Each level curve of the stream function corresponds to a streamline, meaning that fluid particles move along these curves, making it a powerful tool for visualizing and analyzing flow patterns.

Streamlines

Streamlines are curves in the flow field that are everywhere tangent to the local velocity vector. They provide a snapshot of the flow geometry at a given instant and are obtained by setting the stream function equal to a constant. This concept is essential because it helps in visualizing the flow, determining circulation, and analyzing patterns of fluid movement.

Velocity Components Derived from the Stream Function

In a two-dimensional incompressible flow, the velocity components can be obtained from the stream function using partial derivatives; typically, the horizontal component is given by the partial derivative of the stream function with respect to the vertical coordinate, and the vertical component is the negative partial derivative with respect to the horizontal coordinate. This relationship not only simplifies the calculation of velocities but also provides a deeper physical interpretation of the stream function in the context of the flow field.

Flow Rate Calculation

The difference in the stream function values between two streamlines represents the volumetric flow rate per unit depth between them. This property is exploited to calculate the flow crossing a curve, even if it is not a streamline, by integrating the velocity components along the given path. Understanding this concept is crucial for determining the net flow in regions of interest within the fluid domain.

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