Under what conditions does the velocity field
$V = (a_1x + b_1y + c_1z)i + (a_2x + b_2y + c_2z)j + (a_3x + b_3y + c_3z)k$
where $a_1, b_1$ , etc. = const, represent an incompressible flow that conserves mass?

Question

Under what conditions does the velocity field

[latex]V = (a_1x + b_1y + c_1z)i + (a_2x + b_2y + c_2z)j + (a_3x + b_3y + c_3z)k[/latex]

where [latex]a_1, b_1[/latex], etc. = const, represent an incompressible flow that conserves mass?

Accepted Answer

Recalling that V = ui + vj + wk, we see that [latex]u = (a_1x + b_1y + c_1z)[/latex], etc. Substituting into Eq. (4.12a) for incompressible continuity, we obtain

[latex]\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} =0[/latex] (4.12a)

[latex]\frac{\partial}{\partial x} (a_1x + b_1y + c_1z) + \frac{\partial}{\partial y} (a_2x + b_2y + c_2z) + \frac{\partial}{\partial z} (a_3x + b_3y + c_3z)=0[/latex]

or [latex]a_1+b_2+c_3=0[/latex]

At least two of constants [latex]a_1, b_2, and c_3[/latex] must have opposite signs. Continuity imposes no restrictions whatever on constants [latex]b_1, c_1, a_2, c_2, a_3, and b_3[/latex], which do not contribute to a volume increase or decrease of a differential element.

Question 4.2: Under what conditions does the velocity field V = (a1x + b1y...

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