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Condition for irrotationality of a 2-d ideal fluid flow

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Working from first principles show that the condition for irrotationality of a two-dimensional ideal fluid flow is given by:

delta(u)/delta(y) = delta(v)/delta(x)

Hence, define in mathematical terms the velocity potential psi, and show that potential lines of constant psi are perpendicular to streamlines of constant streamfunction phi. You may assume that the gradient of the tangent to a streamline is given by dy/dx = v/u.

By deriving the condition for continuity for the flow,

delta(u)/delta(x) + delta(v)/delta(y) = 0,

Show that the velocity potential psi satisfies Laplace's equation.

Show that the flow given by psi = x^2 - y^2 satisfies Laplace's equation.

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Solution Preview

1. What first principles is the problem referring to? They should be in your notes or in your textbook. For instance if you were using that irrotationality means that Del cross velocity = 0. You would set up the determinant matrix for del cross velocity putting the components of del (partial x, partial y, partial z) in row 2 and the components of velocity (u, v, 0) in row 3. After calculating the cross product, you should ...

Solution Summary

This solution helps show the velocity potential psi satisfies Laplace's equation.