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 \(\vec{V} = u\vec{i} + v\vec{j} + w\vec{k}\). The rate of change in mass of the fluid element is given by ____________

(a) \(\frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} + \frac{\partial(\rho w)}{\partial z}\)

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

(c) \(\frac{\partial\rho}{\partial t}(dx \,dy \,dz) \)

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

I got this question in homework.

My enquiry is from Continuity Equation topic in chapter Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

Right option is (c) \(\frac{\partial\rho}{\partial t}(dx \,dy \,dz) \)

To explain: Mass=density × volume

mass of fluid element=ρ×dx dy dz

rate of change in mass of fluid element=\(\frac{\partial\rho}{\partial t}dx \,dy \,dz \)