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Chapter 6: Problem 2

Construct an alternative derivation of the equation of continuity based on Gauss' theorem $$ f \mathrm{~V} \cdot d \mathrm{~S}=\int \nabla \cdot \mathrm{V} d r $$ Hinf: Let \(\mathbf{V}=\rho \mathbf{v}\) and interpret what Gauss' theorem is telling us.

Short Answer

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Question: Derive the equation of continuity using Gauss' theorem. Answer: The equation of continuity derived using Gauss' theorem is given by: $$ \frac{\partial \rho}{\partial t} = \rho (\nabla \cdot \mathbf{v}) + \mathbf{v} \cdot (\nabla \rho) $$ This equation represents the conservation of mass for a fluid element, where \(\rho\) is the fluid density, \(\mathbf{v}\) is the fluid velocity, and \(t\) is time.

Step by step solution

01

Review Gauss' theorem

Note that Gauss' theorem relates the flux of a vector field, \(\mathbf{V}\), through a closed surface \(S\) to the volume integral of the divergence of the vector field over the volume \(V\), enclosed by the surface \(S\). Gauss' theorem is mathematically represented as: $$ \oint_{S} \mathbf{V} \cdot \mathrm{d}\mathbf{S} = \int_{V} (\nabla \cdot \mathbf{V})\,\mathrm{d}V $$
02

Interpret Gauss' theorem for the given vector field

As per the hint, let \(\mathbf{V} = \rho \mathbf{v}\). Now, we need to interpret Gauss' theorem in terms of the given vector field. The left-hand side of Gauss' theorem represents the flux of mass through the closed surface \(S\). The right-hand side represents the net mass generation rate inside the volume \(V\).
03

Calculate the divergence of the vector field

To find the net mass generation rate inside volume \(V\), first, calculate the divergence of the vector field \(\mathbf{V}=\rho\mathbf{v}\): $$ \nabla \cdot \mathbf{V} = \nabla \cdot (\rho \mathbf{v}) = \rho (\nabla \cdot \mathbf{v}) + \mathbf{v} \cdot (\nabla \rho) $$
04

Apply Gauss' theorem

Now, apply Gauss' theorem using the calculated divergence of the vector field: $$ \oint_{S} (\rho \mathbf{v}) \cdot \mathrm{d}\mathbf{S} = \int_{V} \left(\rho (\nabla \cdot \mathbf{v}) + \mathbf{v} \cdot (\nabla \rho)\right) \mathrm{d}V $$
05

Relate the flux term to the mass balance equation

The left-hand side of the equation can be written as the rate of change of mass inside the volume \(V\) (mass conservation principle): $$ \int_{V} \frac{\partial \rho}{\partial t} \mathrm{d}V = \int_{V} \left(\rho (\nabla \cdot \mathbf{v}) + \mathbf{v} \cdot (\nabla \rho)\right) \mathrm{d}V $$
06

Simplify the equation for a differential volume element

Now, assume the differential volume element \(\mathrm{d}V\) is small enough so that we can bring the integrals inside and simplify the equation: $$ \frac{\partial \rho}{\partial t} = \rho (\nabla \cdot \mathbf{v}) + \mathbf{v} \cdot (\nabla \rho) $$ This equation represents the alternative derivation of the equation of continuity, derived using Gauss' theorem.

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