Question

A plane flow has velocity components \({\rm{u}} = \frac{{\rm{x}}}{{{{\rm{T}}_1}}}\), \({\rm{v}} = - \frac{{\rm{y}}}{{{{\rm{T}}_2}}}\)and w = 0 along x, y and z directions respectively, where T1 (≠ )0 and T2 ≠ )0( are constants having the dimension of time. The given flow is incompressible if 
1. T₁ = - T₂
2. \({{\rm{T}}_1} = - \frac{{{{\rm{T}}_2}}}{2}\)
3. \({{\rm{T}}_1} = \frac{{{{\rm{T}}_2}}}{2}\)
4. T₁ = T₂

Answer 1

Correct Answer - Option 4 : T₁ = T₂

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\(,\;\;\frac{{\partial {\bf{\rho }}}}{{\partial {\bf{t}}}} = 0\)

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

When flow is steady and incompressible, ρ = constant

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

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

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

Calculation:

Given,

\({\rm{u}} = \frac{{\rm{x}}}{{{{\rm{T}}_1}}}\), \({\rm{v}} = - \frac{{\rm{y}}}{{{{\rm{T}}_2}}}\) and w = 0

As per continuity equation, for two dimensional incompressible flow,

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

\(∴ \frac{{\partial {\rm{(\frac{{\rm{x}}}{{{{\rm{T}}_1}}})}}}}{{\partial {\rm{X}}}} + \frac{{\partial {\rm{(- \frac{{\rm{y}}}{{{{\rm{T}}_2}}})}}}}{{\partial {\rm{Y}}}} = 0{\rm{\;}}\)

\(\frac{1}{{{{\rm{T}}_1}}} - \frac{1}{{{{\rm{T}}_2}}} = 0\)

∴T₁ = T₂

Hence The given flow is incompressible if T1 = T2