# 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.