Question

Consider an infinitesimally small fluid element with density ρ (of dimensions dx, dy and dz) fixed in space and fluid is moving across this element with a velocity \(\vec{V}=u\vec{i}+v\vec{j}+w\vec{k}\). What is the final reduced form of net mass flow across the fluid element?

(a) \(\frac{\partial\rho}{\partial t}\)

(b) \(\rho\vec{V} dx \,dy \,dz\)

(c) \(\nabla.(\rho\vec{V})\)

(d) \(\nabla.(\rho\vec{V})\)dx dy dz

I had been asked this question during an interview.

My question comes from Continuity Equation topic in portion Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct answer is (d) \(\nabla.(\rho\vec{V})\)dx dy dz

The best I can explain: Net mass flow across the element = change in mass flow in x direction + change in mass flow in y direction + change in mass flow in z direction

= \(\frac{\partial(\rho u)}{\partial x} dx \,dy \,dz + \frac{\partial(\rho v)}{\partial y} dx \,dy \,dz + \frac{\partial(\rho w)}{\partial z} dx \,dy \,dz \)

=\(\left[(\frac{\partial}{\partial x}\vec{i}+\frac{\partial}{\partial y}\vec{j}+\frac{\partial}{\partial z}\vec{k})dx \,dy \,dz.((\rho u)\vec{i} + (\rho v)\vec{j} + (\rho w)\vec{k})\right] \)

Net mass flow across the element = \([\nabla.(\rho \vec{V})]dx \,dy \,dz\).