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 net mass flow across the fluid element is given by ______

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

(b) \([\frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} + \frac{\partial(\rho w)}{\partial z}]\)

(c) [ρ]dx dy dz

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

This question was posed to me in semester exam.

This is a very interesting question from Continuity Equation in chapter Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct answer is:

(a) [∂(ρu)∂x+∂(ρv)∂y+∂(ρw)∂z]dx dy dz[∂x∂(ρu)​+∂y∂(ρv)​+∂z∂(ρw)​]dxdydz

Explanation: This question refers to the net mass flow across a fluid element in the context of the continuity equation in fluid dynamics.

The term ρρ represents the density of the fluid, and the velocity components uu, vv, and ww represent the velocity in the xx, yy, and zz directions, respectively. The net mass flow is determined by the rate at which mass is flowing across the boundaries of the infinitesimal fluid element.

The expression for mass flow is derived from the conservation of mass, and it is related to the divergence of the mass flux vector ρV⃗ρV, where V⃗=ui^+vj^+wk^V=ui^+vj^​+wk^ is the velocity vector. The divergence of the mass flux is:

∂(ρu)∂x+∂(ρv)∂y+∂(ρw)∂z∂x∂(ρu)​+∂y∂(ρv)​+∂z∂(ρw)​

This term gives the rate of change of mass per unit volume. To find the net mass flow across the fluid element, you multiply the divergence of the mass flux by the volume element dx dy dzdxdydz.

Thus, the correct answer is (a).