Discussion
A two-dimensional incompressible flow field is defined as,
\[ \vec{V}(x, y) = (Axy)\,\hat{i} + (By^{2})\,\hat{j} \]
where, \(A\) and \(B\) are constants. The dynamic viscosity of the Newtonian fluid is \(\mu\). In the absence of body force, which among the following expressions represents the pressure gradient at the location (5, 0) in the concerned flow field?
Answer: C
The pressure gradient is \(\mu A(-\hat{j})\).
Incompressibility fixes \(B\):
\[ \begin{aligned} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &= Ay + 2By = 0 \\ \Rightarrow B &= -\frac{A}{2} \end{aligned} \]
The steady Navier-Stokes equation gives \(\nabla p = -\rho(\vec{V}\cdot\nabla)\vec{V} + \mu\nabla^{2}\vec{V}\). At (5, 0) both velocity components vanish (\(u = 5A \cdot 0 = 0\), \(v = 0\)), so the convective term is zero there. The viscous term:
\[ \begin{aligned} \nabla^{2}u &= \nabla^{2}(Axy) = 0 \\ \nabla^{2}v &= \nabla^{2}(By^{2}) = 2B = -A \end{aligned} \]
So \(\nabla p = \mu(0\,\hat{i} - A\,\hat{j}) = \mu A(-\hat{j})\).
Options with an \(\hat{i}\) component wrongly take \(\nabla^{2}(Axy)\) as non-zero, or keep \(B\) without applying continuity.
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