Question

If p and τ are the net pressure and net shear stress acting on an infinitesimally small element (volume dx dy dz) moving along with the flow (velocity \(\vec{V}\)), what is the net work done on the system?

(a) \(\rho (\nabla .(p\vec{V} )+\nabla .(τ.\vec{V}))\)

(b) \(((p\vec{V})+(\tau.\vec{V}))dx \,dy \,dz\)

(c) \(\rho(\nabla.(p\vec{V})+\nabla.(\tau.\vec{V})) dx \,dy \,dz\)

(d) \((\nabla .(p)+\nabla.(\tau))dx \,dy \,dz\)

I got this question during an online exam.

Origin of the question is Energy Equation topic in section Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct answer is (c) \(\rho(\nabla.(p\vec{V})+\nabla.(\tau.\vec{V})) dx \,dy \,dz\)

To elaborate: The rate of work done is power which is the product of force and velocity. This can be represented by \((\nabla.(p\vec{V})+\nabla.(\tau.\vec{V})) dx \,dy \,dz\).