GATE 2023 ME – Question 63
Consider a unidirectional fluid flow with the velocity field given by $\mathbf{V}(x, y, z, t) = u(x, t)\hat{i}$ where $u(0, t) = 1$. If the spatially homogeneous density field varies with time $t$ as $\rho(t) = 1 + 0.2e^{-t}$ the value of $u(2, 1)$ is ______________. (Rounded off to two decimal places)
Assume all quantities to be dimensionless.
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Correct answer: 1.13 to 1.15
Explanation
The continuity equation is $\frac{\partial\rho}{\partial t} + \frac{\partial(\rho u)}{\partial x} = 0$. The density does not depend on $x$, so $\rho\frac{\partial u}{\partial x} = -\frac{d\rho}{dt} = 0.2e^{-t}$ and $\frac{\partial u}{\partial x} = \frac{0.2e^{-t}}{1 + 0.2e^{-t}}$, which does not depend on $x$. Integrating from $x = 0$: $u(x, t) = 1 + x\,\frac{0.2e^{-t}}{1 + 0.2e^{-t}}$. At $x = 2$ and $t = 1$: $u = 1 + 2 \times \frac{0.0736}{1.0736} = 1 + 0.137 = 1.14$.