A three-dimensional velocity field is given by $V = 5x^2y\,\mathbf{i} + Cy\,\mathbf{j} - 10xyz\,\mathbf{k}$, where $\mathbf{i},\mathbf{j},\mathbf{k}$ are unit vectors in $x,y,z$ directions. If $V$ describes an incompressible fluid flow, the value of $C$ is
A three-dimensional velocity field is given by $V = 5x^2y\,\mathbf{i} + Cy\,\mathbf{j} - 10xyz\,\mathbf{k}$, where $\mathbf{i},\mathbf{j},\mathbf{k}$ are unit vectors in $x,y,z$ directions. If $V$ describes an incompressible fluid flow, the value of $C$ is
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- $-1$
- $0$
- $1$
- $5$
The Correct Option is B
Solution and Explanation
[3pt] Compute divergence:
$\frac{\partial}{\partial x}(5x^2y) = 10xy$,
$\frac{\partial}{\partial y}(Cy) = C$,
$\frac{\partial}{\partial z}(-10xyz) = -10xy$.
[3pt] So, $\nabla \cdot V = 10xy + C - 10xy = C$.
[3pt] For incompressible flow: $C = 0$.
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