Consider the incompressible, two-dimensional flow of a...

Consider the incompressible, two-dimensional flow of a nonviscous fluid between the boundaries shown in Fig. P6.22. The velocity potential for this flow field is $$\phi=x^{2}-y^{2}$$ (a) Determine the corresponding stream function. (b) What is the relationship between the discharge, $q$ (per unit width normal to plane of paper) passing between the walls and the coordinates $x_{i}, y_{i}$ of any point on the curved wall? Neglect body forces.

Key Concepts

Stream Function

In two-dimensional incompressible flow, a stream function is defined so that its contours represent the streamlines of the flow. The stream function is related to the velocity potential through the Cauchy-Riemann equations, ensuring that the flow is both irrotational and divergence?free. In other words, the partial derivatives of the stream function with respect to the spatial coordinates yield the orthogonal components of the fluid velocity, and its constant values identify the paths followed by fluid particles.

Discharge

Discharge in the context of fluid mechanics refers to the volumetric flow rate per unit width in a two-dimensional flow. It is calculated as the difference in stream function values between two streamlines. In problems with boundaries, such as a curved wall, the relationship between the discharge and the coordinates is determined by equating the stream function difference (which remains constant along a given streamline) to the integrated normal component of velocity over the distance along the boundary, ensuring conservation of mass.

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