Question

A two-dimensional incompressible flow field is defined as \[ \vec V(x,y)=(Axy)\,\hat{\imath}+(By^2)\,\hat{\jmath}, \] where $A$ and $B$ are constants. The dynamic viscosity is $\mu$. In the absence of body force, which expression represents the {pressure gradient} at the location $(5,0)$ in the concerned flow field?

• A. $\mu A(5\,\hat{\imath}+\hat{\jmath})$
• B. $\mu(-5B\,\hat{\imath}+A\,\hat{\jmath})$
• C. $\mu A(-\hat{\jmath})$
• D. $\mu A(5\,\hat{\imath})$

Hint:

At a stagnation point the convective term drops out, so with no body force: $\nabla p=\mu\nabla^2\vec V$. For incompressible prescribed fields, first impose $\nabla\! . \vec V=0$ to relate constants.

Answer 1

The Correct Option is C

Step 1: Use incompressibility to relate $A$ and $B$.
\[ \nabla\! . \vec V=\frac{\partial (Axy)}{\partial x}+\frac{\partial (By^2)}{\partial y} =Ay+2By=y(A+2B)=0\ \forall(x,y)\Rightarrow A+2B=0 \Rightarrow B=-\frac{A}{2}. \]
Step 2: Steady Stokes balance at the point $(5,0)$.
With no body force, the steady momentum equation is \[ -\nabla p+\mu\nabla^2\vec V+\rho(\vec V\! . \nabla)\vec V=0. \] At $(5,0)$, $\vec V=(0,0)$, so the convective term vanishes. Hence \[ \nabla p=\mu \nabla^2\vec V. \]
Step 3: Compute the vector Laplacian.
For $u=Axy$: $\nabla^2 u=u_{xx}+u_{yy}=0+0=0$. For $v=By^2$: $\nabla^2 v=v_{xx}+v_{yy}=0+2B=2B$. Thus \[ \nabla^2\vec V=(0)\,\hat{\imath}+(2B)\,\hat{\jmath}. \]
Step 4: Substitute $B=-A/2$.
\[ \nabla p=\mu(0\,\hat{\imath}+2B\,\hat{\jmath}) =\mu\,(0\,\hat{\imath}-A\,\hat{\jmath}) =\boxed{\mu A(-\hat{\jmath})}. \] Final Answer:\fbox{(C) $\mu A(-\hat{\jmath})$}