Mechanical Engineering

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Mechanical EngineeringFluid MechanicsMultiple Select (MSQ)2 Marks
Q1.

Let ρ(x,y,z,t)\rho(x, y, z, t) and u(x,y,z,t)u(x, y, z, t) represent density and velocity, respectively, at a point (x,y,z)(x, y, z) and time tt. Assume ∂ρ∂t\frac{\partial \rho}{\partial t} is continuous. Let VV be an arbitrary volume in space enclosed by the closed surface SS and n^\hat{n} be the outward unit normal of SS. Which of the following equations is/are equivalent to ∂ρ∂t+∇⋅(ρu)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho u) = 0?

A
∫V∂ρ∂tdv=−∮Sρu⋅n^ds\int_V \frac{\partial \rho}{\partial t} dv = -\oint_S \rho u \cdot \hat{n} ds
B
∫V∂ρ∂tdv=∮Sρu⋅n^ds\int_V \frac{\partial \rho}{\partial t} dv = \oint_S \rho u \cdot \hat{n} ds
C
∫V∂ρ∂tdv=−∫V∇⋅(ρu)dv\int_V \frac{\partial \rho}{\partial t} dv = -\int_V \nabla \cdot (\rho u) dv
D
∫V∂ρ∂tdv=∫V∇⋅(ρu)dv\int_V \frac{\partial \rho}{\partial t} dv = \int_V \nabla \cdot (\rho u) dv