Question

Consider three-dimensional Euler equations. What will you do to get the value of \((\frac{\partial^2 ρ}{\partial t^2})_{i,j}^t\)?

(a) Differentiate \(\rho_{i,j}^t\) with respect to time twice

(b) Differentiate the continuity equation with respect to time

(c) Differentiate the value of \((\frac{\partial \rho}{\partial t})_{i,j}^t\) with respect to time

(d) Differentiate the value of \(\rho_{i,j}^t\) with respect to time twice

This question was addressed to me in examination.

This intriguing question originated from Finite Difference Methods topic in chapter Finite Difference Methods of Computational Fluid Dynamics

Answer 1

Correct answer is (b) Differentiate the continuity equation with respect to time

The explanation: Differentiating the value of any variable or the value of its derivative have no sense as it will result in zero. To differentiateup \(\rho_{i,j}^t\) with respect to time twice, the equation for \(\rho_{i,j}^t\) should be known. But, it is not. So, differentiating the continuity equation with respect to time is the only way. Remember the continuity equation gives \((\frac{\partial \rho}{\partial t})\).