GATE 2024 Electronics & Communication Q25

Question

Let $ho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial ho}{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$ and $\hat{n}$ be the outward unit normal of $S$. Which of the following equations is/are equivalent to $\frac{\partial ho}{\partial t} + 
abla \cdot (ho u) = 0$?

Answer

$\int_V \frac{\partial \rho}{\partial t} dv = -\oint_S \rho u \cdot \hat{n} ds$

Answer

$\int_V \frac{\partial \rho}{\partial t} dv = \oint_S \rho u \cdot \hat{n} ds$

Answer

$\int_V \frac{\partial ho}{\partial t} dv = -\int_V 
abla \cdot (ho u) dv$

Answer

$\int_V \frac{\partial ho}{\partial t} dv = \int_V 
abla \cdot (ho u) dv$