Alternative Derivation of the Continuity Equation There are other, equivalent ways of deriving the continuity equation. To illustrate, derive the continuity equation in vector form by following the motion of a fluid element of constant mass, whose volume may be changing, as shown in Fig. E5.7, alon

Question

Alternative Derivation of the Continuity Equation

There are other, equivalent ways of deriving the continuity equation. To illustrate, derive the continuity equation in vector form by following the motion of a fluid element of constant mass, whose volume may be changing, as shown in Fig. E5.7, along the path A-B-C.

Accepted Answer

Focus attention when the element is at location B, where its volume is [latex]d V[/latex]. By definition of the divergence of the velocity vector as the net rate of outflow of volume per unit volume, the fractional rate at which the element is increasing in volume is [latex] abla \cdot \mathbf{v} .[/latex] (A positive value of the divergence means that the constant mass is now occupying a larger volume, because this consists of the original volume plus the outflow or expansion.) But since the mass of the element is constant, its density must be decreasing at the same fractional rate. Since we are following the path taken by the fluid, the latter is given by [latex](\mathcal{D} ho / \mathcal{D} t) / ho[/latex]. The relationship between the two quantities is therefore:

[latex]\begin{array}{l}\frac{1}{ ho} \frac{\mathcal{D} ho}{\mathcal{D} t}=- abla \cdot \mathbf{v}, \ \end{array}[/latex] (E5.7.1)

or,

[latex]\begin{array}{l}\frac{\mathcal{D} ho}{\mathcal{D} t}+ ho abla \cdot \mathbf{v}=0 .\end{array}[/latex] (E5.7.2)

The reader can show (see Problem 5.6) that Eqns. (5.47) and (E5.7.2) are equivalent. As a further example, Problem [latex]5.7[/latex] illustrates yet another way of deriving the continuity equation.

[latex]\frac{\partial ho}{\partial t}+ abla \cdot ho \mathbf{v}=0[/latex] (5.47)

Question 5.7: Alternative Derivation of the Continuity Equation There are ...

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