Question

State whether following flow field is physically possible?

U = 3 xy² + 2x + y² and V = x² - 2y - y³
1. Possible for steady, incompressible flow
2. Possible for unsteady, incompressible flow
3. Possible for steady, compressible flow
4. Not possible

Answer 1

Correct Answer - Option 1 : Possible for steady, incompressible flow

Concept:

Steady flow: A flow is said to be steady if the properties do not change with respect to time. Also known as time invariant flow.

Incompressible flow: A flow is said to be incompressible if the density does not change during the flow else it is called a compressible flow.

For an incompressible flow:

\(\frac{{\partial u}}{{\partial x}} + \;\frac{{\partial v}}{{\;\partial y}} + \;\frac{{\partial w}}{{\partial z}} = 0\) 

Calculation:

Given, U = 3 xy² + 2x + y² and V = x² - 2y - y³

Clearly they are not function of time, hence, the flow is steady

Checking for compressibility:

\(\frac{{\partial u}}{{\partial x}} + \;\frac{{\partial v}}{{\;\partial y}} + \;\frac{{\partial w}}{{\partial z}}\) Should be equal to zero for a compressible flow

\(\frac{{\partial u}}{{\partial x}} = \frac{{\partial \left( {3\;x{y^2}\; + 2x\; + \;{y^2}} \right)}}{{\partial x}} = 3{y^2} + 2\) 

\(\frac{{\partial v}}{{\;\partial y}} = \;\frac{{\partial \left( {{x^2}\; - 2y\; - \;{y^3}} \right)}}{{\;\partial y}} = - 2 - 3{y^2}\) 

\(\frac{{\partial u}}{{\partial x}} + \;\frac{{\partial v}}{{\;\partial y}} = 3{y^2} + 2 + \left( { - 2 - 3{y^2}} \right) = 0\)

Hence, it is incompressible

∴ the flow is steady and incompressible.