A flow field is represented by the stream function ψ=x^2-y^2 . Find the corresponding velocity f

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A flow field is represented by the stream function ψ=x^2-y^2...

$ \mathrm{A}$ flow field is represented by the stream function $\psi=x^{2}-y^{2} .$ Find the corresponding velocity field. Show that this flow field is irrotational. Plot several streamlines and illustrate the velocity field.

Key Concepts

Stream Function

A stream function is a scalar function whose contours represent streamlines in an incompressible two?dimensional flow field. The stream function is defined such that its partial derivatives with respect to the spatial coordinates yield the velocity components, ensuring continuity automatically. In this problem, the stream function is given by ? = x² - y², which uniquely defines the flow field via its gradients.

Velocity Field

The velocity field in a two-dimensional flow can be obtained from the stream function through the relations u = ??/?y and v = -??/?x. Taking the derivatives for ? = x² - y², we find u = -2y and v = -2x. Thus, the velocity field is represented by the vector (u, v) = (-2y, -2x), which describes the fluid particle velocities at any point in space.

Irrotational Flow

A flow is said to be irrotational if the curl of its velocity field is zero. For two-dimensional flows, this condition is checked by computing ?v/?x - ?u/?y. Applying this to the obtained velocity components, we have ?v/?x = -2 and ?u/?y = -2, so their difference is zero. This confirms that the flow is irrotational, meaning that there is no local rotation of fluid elements.

Streamlines

Streamlines are curves that are tangent to the velocity field at every point and represent the paths a fluid element would follow. They coincide with the constant values of the stream function. For the stream function ? = x² - y², setting ? = C (a constant) yields the equation x² - y² = C, which represents a family of hyperbolas. Plotting these hyperbolic curves along with vectors representing the velocity field gives a clear visualization of the flow structure.

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