Discussion
In a two-dimensional, incompressible and irrotational flow, fluid velocity (v) in the y-direction is given by \(v = 2x - 5y\). The velocity (u) in the x-direction is
Answer: C
The x-velocity is \(u = 5x + 2y\).
Incompressibility requires \(\partial u/\partial x + \partial v/\partial y = 0\). With \(v = 2x - 5y\), \(\partial v/\partial y = -5\), so \(\partial u/\partial x = 5\) and \(u = 5x + f(y)\).
Irrotationality requires \(\partial v/\partial x - \partial u/\partial y = 0\). Here \(\partial v/\partial x = 2\), so \(f'(y) = 2\) and \(f(y) = 2y\) (up to a constant). Hence \(u = 5x + 2y\).
\(u = 5x - 2y\) satisfies continuity but gives vorticity \(2 - (-2) = 4\). \(u = 2x \pm 5y\) gives \(\partial u/\partial x = 2\), which violates continuity.
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