The velocity components of a two-dimensional plane motion of a fluid with constant density are u = 2x - x2y and v = xy² - 2y. Which of the following is the correct statement?

Concept:

Continutiy equation in Three-Dimension

\(\frac{{\partial {\rm{\rho }}}}{{\partial {\rm{t}}}} + \frac{{\partial \left( {{\rm{\rho U}}} \right)}}{{\partial {\rm{X}}}} + \frac{{\partial \left( {{\rm{\rho V}}} \right)}}{{\partial {\rm{Y}}}} + \frac{{\partial \left( {{\rm{\rho W}}} \right)}}{{\partial {\rm{Z}}}} = 0{\rm{\;}}\)

Where,

U, V, and W are components of velocity in X, Y and Z direction respectively

When the flow is steady\(,\;\;\frac{{\partial {\bf{\rho }}}}{{\partial {\bf{t}}}} = 0\)

\(\therefore \frac{{{\rm{\partial }}\left( {{\rm{\rho U}}} \right)}}{{{\rm{\partial X}}}} + \frac{{{\rm{\partial }}\left( {{\rm{\rho V}}} \right)}}{{{\rm{\partial Y}}}} + \frac{{{\rm{\partial }}\left( {{\rm{\rho W}}} \right)}}{{{\rm{\partial Z}}}} = 0\)

When flow is steady and incompressible, ρ = constant

\(\therefore \frac{{\partial {\rm{U}}}}{{\partial {\rm{X}}}} + \frac{{\partial {\rm{V}}}}{{\partial {\rm{Y}}}} + \frac{{\partial {\rm{W}}}}{{\partial {\rm{Z}}}} = 0{\rm{\;}}\)

When the flow is steady, incompressible and 2-D, \(\frac{{\partial {\bf{W}}}}{{\partial {\bf{Z}}}} = 0\)

\(\therefore \frac{{\partial {\rm{U}}}}{{\partial {\rm{X}}}} + \frac{{\partial {\rm{V}}}}{{\partial {\rm{Y}}}} = 0{\rm{\;}}\)

Calculation:

Given:

\(U = 2x - x^2 y \)

\(V = xy^2 - 2y \)

As per continuity equation, for flow to occur

\(\therefore \frac{{\partial {\rm{U}}}}{{\partial {\rm{X}}}} + \frac{{\partial {\rm{V}}}}{{\partial {\rm{Y}}}} = 0{\rm{\;}}\)

\(\frac{\partial U}{\partial x} + \frac{\partial V}{\partial y} = 2 - 2xy + 2xy - 2 = 0 \)

Since, \(\frac{{\partial {\rm{U}}}}{{\partial {\rm{X}}}} + \frac{{\partial {\rm{V}}}}{{\partial {\rm{Y}}}} = 0{\rm{\;}}\), then flow is steady, incompressible.