The GATE Grind

GATE 2021 CH – Question 15

Fluid Mechanics and Mechanical Operations · Equations of continuity, motion and mechanical energy, Euler and Bernoulli equations · 1 mark · Multiple choice

A three-dimensional velocity field is given by $\mathbf V=5x^2y\,\mathbf i+Cy\,\mathbf j-10xyz\,\mathbf k$, where $\mathbf i,\mathbf j,\mathbf k$ are the unit vectors in $x,y,z$ directions, respectively, describing a cartesian coordinate system. The coefficient $C$ is a constant. If $\mathbf V$ describes an incompressible fluid flow, the value of $C$ is

  1. −1
  2. 0
  3. 1
  4. 5

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) 0

Explanation

Incompressibility requires zero velocity divergence:
$$\nabla\cdot\mathbf V=\frac{\partial(5x^2y)}{\partial x}+\frac{\partial(Cy)}{\partial y}+\frac{\partial(-10xyz)}{\partial z}=10xy+C-10xy=C.$$

Thus $C=0$, independent of position. **Answer: B.**