A fluid has density field and velocity field

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Conservation of mass, momentum and energy

Problem

A fluid has density field ρ(x,y,z,t)\rho(x,y,z,t) and velocity field u=(u,v,w)\mathbf{u} = (u,v,w). Consider a fixed rectangular box whose corner of least coordinates is at (x,y,z)(x,y,z) and whose sides are δx\delta x, δy\delta y, δz\delta z, small enough that ρ\rho and u\mathbf{u} may be expanded to first order across the box. The outward normal is used on every face. ((a)) Write the mass contained in the box and the mass flux through each of the six faces, giving the sign of u⋅n\mathbf{u}\cdot\mathbf{n} on each face, and hence show that ∂ρ∂t+∇⋅(ρu)=0.\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf{u}) = 0. ((b)) Hence show that if the density is a single uniform constant, the velocity field must satisfy ∇⋅u=0\nabla\cdot\mathbf{u} = 0, and state in one sentence what that says about the volume of a fluid parcel.

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