Chapter 5, Problem 1P

Which of the following sets of equations represent possible three-dimensional incompressible flow cases?

1. (a) u = 2y² + 2xz; υ = −2xy + 6x²yz; w = 3x²z² + x³y⁴
2. (b) u = xyzt; υ = −xyzt²; w = z² (xt² − yt)
3. (c) u = x² + 2y + z²; υ = x − 2y + z; w = −2xz + y² + 2z

To determine

The set of equation that represent three-dimensional incompressible flow.

Explanation of Solution

Given:

The given set of equations are:

1. a) $u=2y^2+2xz;v=-2yz+6x^{2}yz;w=3x^2{z}^2+x^3{y}^4$
2. b) $u=xyzt;v=-xyz{t}^2;w=z^2\left(x{t}^2-yt\right)$
3. c) $u=x^2+2y+z^2;v=x-2y+z;w=-2xy+y^2+2z$

Calculation:

The criteria that must be satisfied for a flow to be incompressible is,

  $\frac{\partial }{\partial x}u+\frac{\partial }{\partial y}v+\frac{\partial }{\partial z}w=0$

Check the given sets to satisfy the above criteria.

For set (a),

  $\begin{array}{c}\frac{\partial }{\partial x}u+\frac{\partial }{\partial y}v+\frac{\partial }{\partial z}w=\frac{\partial }{\partial x}\left(2y^2+2xz\right)+\frac{\partial }{\partial y}\left(-2yz+6x^{2}yz\right)+\frac{\partial }{\partial z}\left(3x^2{z}^2+x^3{y}^4\right)\\ =2z+6x^{2}z-2z+6x^{2}z\\ =12x^{2}z\ne 0\end{array}$

For set (b),

  $\begin{array}{c}\frac{\partial }{\partial x}u+\frac{\partial }{\partial y}v+\frac{\partial }{\partial z}w=\frac{\partial }{\partial x}\left(xyzt\right)+\frac{\partial }{\partial y}\left(-xyz{t}^2\right)+\frac{\partial }{\partial z}\left(z^2\left(x{t}^2-yt\right)\right)\\ =yzt-xz{t}^2+2z\left(x{t}^2-yt\right)\ne 0\end{array}$

For set (c),

  $\begin{array}{c}\frac{\partial }{\partial x}u+\frac{\partial }{\partial y}v+\frac{\partial }{\partial z}w=\frac{\partial }{\partial x}\left(x^2+2y+z^2\right)+\frac{\partial }{\partial y}\left(x-2y+z\right)+\frac{\partial }{\partial z}\left(-2xy+y^2+2z\right)\\ =2x-2+2+2x\\ =0\end{array}$

Hence, of the three only set (c) satisfies the criteria for the incompressible flow.

Thus, the set of equations that represent three-dimensional incompressible flow is $u=x^2+2y+z^2;v=x-2y+z;w=-2xy+y^2+2z$