Question

The velocity field of incompressible flow in a Cartesian system is represented by

\(\vec V = 2\left( {{x^2} - {y^2}} \right)\hat i + v\hat j + 3\hat k\) 

Which one of the following expressions for v is valid?
1. -4xz + 6xy
2. -4xy – 4xz
3. 4xz – 6xy
4. 4xy + 4xz

Answer 1

Correct Answer - Option 2 : -4xy – 4xz

Concept:

If velocity is given as \({\rm{\vec V}} = {\rm{u\hat i}} + {\rm{v\hat j}} + {\rm{w\hat k}}\)

Then for incompressible flow continuity equation has to be satisfied,

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

Calculation:

\(\vec V = 2\left( {{x^2} - {y^2}} \right)\hat i + v\hat j + 3\hat k\)

u = 2(x² – y²)

v = v

w = 3

Putting the values in continuity equation

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

⇒ \(4{\rm{x}} + {\rm{\;}}\frac{{\partial {\rm{v}}}}{{\partial {\rm{y}}}} + 0 = 0\)

⇒ \(\frac{{\partial {\rm{v}}}}{{\partial {\rm{y}}}} = {\rm{\;}} - 4{\rm{x}}\)

⇒ v = -4xy + f(x,z)

∴ Out of the given options, option 2 matches our answer.