Discussion
The velocity components in x- and y-directions of a two-dimensional, incompressible flow field are \(u(x,y) = 2x^{2}+y^{3}\) and \(v(x,y) = x^{3}-2xy+f(x,y)\), respectively. Here, f(x, y) is a polynomial function and g(x) is a polynomial function of x only. Which one of the following options for f(x, y) is correct?
Answer: C
The function is \(f(x,y) = -2xy+g(x)\).
For incompressible two-dimensional flow, continuity requires \(\partial u/\partial x + \partial v/\partial y = 0\):
\[ \begin{aligned} \frac{\partial u}{\partial x} &= 4x \\ \frac{\partial v}{\partial y} &= -2x+\frac{\partial f}{\partial y} \\ 4x-2x+\frac{\partial f}{\partial y} &= 0 \end{aligned} \]
So \(\partial f/\partial y = -2x\), and integrating in y gives \(f = -2xy + g(x)\), the function of integration depending on x only.
Check the others: \(-xy^{2}\) gives \(\partial f/\partial y = -2xy\), \(-2x\) gives 0, and \(-2x^{2}y\) gives \(-2x^{2}\); none cancels the remaining 2x.
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