# Consider the following steady, two-dimensional, incompressible velocity field:V=(u,v) = (-ax²)i + (2axy)j, where a is a constant. Calculate the pressure as a function of x and y.

Question-AnswerCategory: Fluid MechanicsConsider the following steady, two-dimensional, incompressible velocity field:V=(u,v) = (-ax²)i + (2axy)j, where a is a constant. Calculate the pressure as a function of x and y.

Consider the following steady, two-dimensional, incompressible velocity field:, where a is a constant. Calculate the pressure as a function of x and y.

Step: 1

Write the expression for steady, two dimensional, incompressible velocity field.

Write the velocity components u and v.

Find the differentiation of velocity components.

Find the differentiation of velocity components with respect to time.

Step: 2

Check the velocity field satisfies the two dimensional, incompressible continuity equation or not.
Write the two dimensional continuity equation.

Substitute for and for

Hence continuity equation satisfied.

Step: 3

Write the x-component of the two dimensional incompressible Navier-Stokes equation.

Here,  is density, is gravity along x direction, and is dynamic viscosity.
Substitute 0 for , 0 for , 0 for  for for for u, for v,  for , 0 for , and 0 for

Now, differentiate the above equation with respect to y.

Now, take the y-component of the two dimensional incompressible Navier-Stokes equation.

Substitute 0 for , 0 for , 0 for  for for , 2ax for for u, for v, 0 for , 0 for , and 0 for

Now, differentiate the above equation with respect to x.

Since Hence P is not a smooth function of x and y.
Therefore, the pressure as a function of x and y cannot be calculated.