An Alternative to the Differential Element When performing operations similar to the above, some people prefer first to consider a finite volume, such as one of dimensions Δx × Δy × Δz, and then let this shrink to one of differential size. Investigate this alternative.

Question

An Alternative to the Differential Element

When performing operations similar to the above, some people prefer first to consider a finite volume, such as one of dimensions [latex]\Delta x imes \Delta y imes \Delta z[/latex], and then let this shrink to one of differential size. Investigate this alternative.

Accepted Answer

The net rate of outflow across the faces normal to the [latex]x[/latex] axis is now given by the difference of the velocities at locations [latex]x[/latex] and [latex]x+\Delta x[/latex], multiplied by the area [latex]\Delta y \Delta z[/latex] across which the transports occur:

[latex]\left(\left.v_{x} ight|_{x+\Delta x}-\left.v_{x} ight|_{x} ight) \Delta y \Delta z .[/latex] (E5.3.1)

By following similar arguments for the outflow across the other four faces, the total outflow rate for all six faces is:

[latex]\left(\left.v_{x} ight|_{x+\Delta x}-\left.v_{x} ight|_{x} ight) \Delta y \Delta z+\left(\left.v_{y} ight|_{y+\Delta y}-\left.v_{y} ight|_{y} ight) \Delta z \Delta x+\left(\left.v_{z} ight|_{z+\Delta z}-\left.v_{z} ight|_{z} ight) \Delta x \Delta y . (E5.3.2)[/latex]

Division by the volume [latex]\Delta x \Delta y \Delta z[/latex] then gives the rate of outflow per unit volume:

[latex]\frac{\left.v_{x} ight|_{x+\Delta x}-\left.v_{x} ight|_{x}}{\Delta x}+\frac{\left.v_{y} ight|_{y+\Delta y}-\left.v_{y} ight|_{y}}{\Delta y}+\frac{\left.v_{z} ight|_{z+\Delta z}-\left.v_{z} ight|_{z}}{\Delta z} .[/latex] (E5.3.3)

Each dimension is now allowed to shrink to a differential value. By definition of the derivative:

[latex]\underset{\Delta x ightarrow 0}{\lim } \frac{\left.v_{x} ight|_{x+\Delta x}-\left.v_{x} ight|_{x}}{\Delta x}=\frac{\partial v_{x}}{\partial x} .[/latex] (E5.3.4)

Likewise, the second and third terms in Eqn. (E5.3.3) yield [latex]\partial v_{y} / \partial y[/latex] and [latex]\partial v_{z} / \partial z[/latex], so that the total rate of outflow per unit volume, which by definition is the divergence of the velocity, becomes identical to that obtained previously:

[latex] abla \cdot \mathbf{v}=\frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{\partial v_{z}}{\partial z}.[/latex] (E5.3.5)

The preference of whether to start with a differential element, or with a finite element that is allowed to shrink to a differential one and then invoke the definition of a derivative, is a personal one. We have opted for the former (and have therefore generally followed this approach throughout), but recognize that the alternative may be preferred by others. The alternative approach may be adopted when performing both mass balances and momentum balances, the latter of which is discussed in Section 5.6.

Question 5.3: An Alternative to the Differential Element When performing o...

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