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16 March, 12:27

In an incompressible three-dimensional flow field, the velocity components are given by u = ax + byz; υ = cy + dxz. Determine the form of the z component of velocity. If the z component were not a function of x or y what would be the form be?

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  1. 16 March, 13:52
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    An incompressible flow field F in a 3D cartesian grid with components u, v, w:

    F = u + v + w

    where u, v, w are functions of x, y, z

    Must satisfy:

    ∇·F = du/dx + dv/dy + dw/dz = 0

    We have a field F defined:

    F = u+v+w, u = ax+byz, v = cy+dxz

    du/dx = a, dv/dy = c

    Recall ∇·F = 0:

    ∇·F = du/dx + dv/dy + dw/dz = 0

    a + c + dw/dz = 0

    dw/dz = - a-c

    Solve for w by separation of variables:

    w = ∫ (-a-c) dz

    w = - az - cz + f (x, y)

    f (x, y) is some undetermined function of x and y

    The question states that w is not a function of x and y, therefore f (x, y) = 0 ...

    w = - az - cz
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