The three components of velocity in a velocity field are given by u = Ax + By + Cz, v = Dx + Ey + Fz, and w = Gx + Hy + Jz. Determine the relationship among the coefficients A through J that is necessary if this is to be a possible incompressible flow field.

The relationship is only between the coefficients A, E and J which is:

. The remaining coefficients can be anything without any constraints.

Explanation:

Given:

The three components of velocity is a velocity field are given as:

The fluid is incompressible.

We know that, for an incompressible fluid flow, the sum of the partial derivatives of each component relative to its direction is always 0. Therefore,

Now, let us find the partial derivative of each component.

Hence, the relationship between the coefficients is:

There is no such constraints on other coefficients. So, we can choose any value for the remaining coefficients B, C, D, F, G and H.