Discussion
\(u = ax + by\), \(v = cx + dy\) where \(a, b, c, d\) are non-zero constants are velocity components of a possible incompressible fluid motion only, if :
Answer: B
The motion is possible only if \(a + d = 0\).
Incompressibility requires the divergence of the velocity to vanish:
\[ \begin{aligned} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &= a + d \\ &= 0 \end{aligned} \]
The constants \(b\) and \(c\) do not enter the divergence; they decide instead whether the flow is irrotational (\(c - b = 0\), i.e. \(b = c\)). So \(b = c\) and \(b + c = 0\) concern vorticity, not incompressibility, and \(a = d\) would give divergence \(2a \neq 0\).
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