Respuesta :
Answer:
The relationship is only between the coefficients A, E and J which is:
[tex]A + E + J = 0[/tex]. The remaining coefficients can be anything without any constraints.
Explanation:
Given:
The three components of velocity is a velocity field are given as:
[tex]u = Ax + By + Cz\\\\v = Dx + Ey + Fz\\\\w = Gx + Hy + Jz[/tex]
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,
[tex]\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0[/tex]
Now, let us find the partial derivative of each component.
[tex]\frac{\partial u}{\partial x}=\frac{\partial }{\partial x}(Ax+By+Cz)\\\\\frac{\partial u}{\partial x}=A+0+0=A\\\\\frac{\partial v}{\partial y}=\frac{\partial }{\partial y}(Dx+Ey+Fz)\\\\\frac{\partial v}{\partial y}=0+E+0=E\\\\\frac{\partial w}{\partial z}=\frac{\partial }{\partial z}(Gx+Hy+Jz)\\\\\frac{\partial w}{\partial z}=0+0+J=J[/tex]
Hence, the relationship between the coefficients is:
[tex]A+E+J=0[/tex]
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.
