GATE 2022 ME (ME2) – Question 44
The steady velocity field in an inviscid fluid of density 1.5 is given to be $\mathbf V=(y^2-x^2)\hat i+(2xy)\hat j$. Neglecting body forces, the pressure gradient at $(x=1,y=1)$ is _________.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (C) $-6\hat i-6\hat j$
Explanation
**Step 1: velocity at (1, 1).** With $u=y^2-x^2$ and $v=2xy$, at $(1,1)$: $u=0$, $v=2$.
**Step 2: velocity gradients.**
- $u_x=-2x=-2$, $u_y=2y=2$
- $v_x=2y=2$, $v_y=2x=2$
**Step 3: acceleration** (steady flow, so convective only):
$$a_x=u\,u_x+v\,u_y=0+2(2)=4,\qquad a_y=u\,v_x+v\,v_y=0+2(2)=4 .$$
**Step 4: Euler's equation** (no body force): $\rho\mathbf a=-\nabla p$, so
$$\nabla p=-1.5\,(4\hat i+4\hat j)=-6\hat i-6\hat j .$$
Answer **C**.