The velocity components are given as follows: u = (2x + y + z)t;

Question

The velocity components are given as follows:

u = (2x + y + z)t; v= (x — 2y + z)t; w = (x + y)t

Show that they satisfy continuity equation in 3-D.

Accepted Answer

Continuity equation in 3-D is given by Eq. (4.6) as:

[latex]\frac{\pmb{\delta}u }{\pmb{\delta} x}+\frac{\pmb{\delta} v}{\pmb{\delta} y}+\frac{\pmb{\delta} w}{\pmb{\delta} z}=0[/latex] (i)

Now,

[latex]\frac{\pmb{\delta}u }{\pmb{\delta} x}=2t,\frac{\pmb{\delta} v}{\pmb{\delta} y}=-2t, and \frac{\pmb{\delta} w}{\pmb{\delta} z}=0[/latex]

Substituting these values in Eq. (i), we get

[latex]\frac{\pmb{\delta}u }{\pmb{\delta} x}+\frac{\pmb{\delta} v}{\pmb{\delta} y}+\frac{\pmb{\delta} w}{\pmb{\delta} z}=2t-2t+0=0[/latex]

Hence the three components satisfy the continuity equation.

Question 4.4: The velocity components are given as follows: u = (2x + y + ......