Question

Let \(\hat{u}\) be the specific internal energy of a system moving along with the flow with a velocity \(\vec{V}\). What is the time rate of change of the total energy of the system per unit mass?

(a) \(\hat{u}+\frac{1}{2}\vec{V}.\vec{V}\)

(b) \(\frac{D}{Dt}(\hat{u}+\frac{1}{2}\vec{V}.\vec{V})\)

(c) \(\frac{\partial}{\partial t}(\hat{u}+\frac{1}{2}\vec{V}.\vec{V})\)

(d) \(\frac{D}{Dt}(\hat{u}+\vec{V}.\vec{V})\)

The question was asked during an internship interview.

My query is from Energy Equation topic in portion Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

Right answer is (b) \(\frac{D}{Dt}(\hat{u}+\frac{1}{2}\vec{V}.\vec{V})\)

The best explanation: The total energy of the system is

\(E=\hat{u}+\frac{1}{2}\vec{V}.\vec{V}\)

Take its substantial derivative as the model is not stationary.

rate of change of total energy=\(\frac{DE}{Dt}=\frac{D}{Dt}(\hat{u}+\frac{1}{2} \vec{V}.\vec{V})\).