An incompressible velocity field is given by u = a(x^2 - y^2) v unknown w = b where a and b are constants. What must the form of the velocity component v be? ^3An exception occurs in geophysical flows, where a density change is imposed thermally or mechanically rather than by the flow conditions

Question

An incompressible velocity field is given by

[latex]u = a(x^2 - y^2) v[/latex] unknown w = b

where a and b are constants. What must the form of the velocity component [latex]v[/latex] be?

Accepted Answer

Again Eq. (4.12a) applies:

[latex]\frac{\partial}{\partial x}(ax^2 - ay^2)+\frac{\partial v}{\partial y}+\frac{\partial b}{\partial z}=0[/latex]

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

or [latex]\frac{\partial v}{\partial y}=-2ax[/latex] (1)

This is easily integrated partially with respect to y:

[latex]v[/latex](x, y, z, t) = -2axy + f(x, z, t)

This is the only possible form for [latex]v[/latex] that satisfies the incompressible continuity equation. The function of integration f is entirely arbitrary since it vanishes when [latex]v[/latex] is differentiated with respect to [latex]y^4[/latex].

Question 4.3: An incompressible velocity field is given by u = a(x^2 - y^2...

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