Question

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.

Answer 1

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

`A + E + J = 0`. The remaining coefficients can be anything without any constraints.

Step-by-step explanation:

Given:

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

`u = Ax + By + Cz\\\\v = Dx + Ey + Fz\\\\w = Gx + Hy + Jz`

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,

`(\partial u)/(\partial x)+(\partial v)/(\partial y)+(\partial w)/(\partial z)=0`

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

`(\partial u)/(\partial x)=(\partial )/(\partial x)(Ax+By+Cz)\\\\(\partial u)/(\partial x)=A+0+0=A\\\\(\partial v)/(\partial y)=(\partial )/(\partial y)(Dx+Ey+Fz)\\\\(\partial v)/(\partial y)=0+E+0=E\\\\(\partial w)/(\partial z)=(\partial )/(\partial z)(Gx+Hy+Jz)\\\\(\partial w)/(\partial z)=0+0+J=J`

Hence, the relationship between the coefficients is:

`A+E+J=0`

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.