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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 dzI 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»

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 = \([\nabla.(\rho \vec{V})]dx \,dy \,dz\).


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