Question

3 Problem Consider an incompressible flow in a periodic 2D domain of side $2\pi$ in each direction. Assume the initial conditions for velocity are of the form $u(x, y, t = 0) = \sin(x)\cos(y)$, $v(x, y, t = 0) = -\cos(x)\sin(y)$. (3) • Using these initial conditions and the Navier-Stokes equations find the pressure and velocity at all time assuming that $u = u(x,y,t = 0)F(t)$ and $v = v(x, y, t = 0)F(t)$ by finding $F(t)$.

          3 Problem
Consider an incompressible flow in a periodic 2D domain of side $2\pi$ in each
direction. Assume the initial conditions for velocity are of the form
$u(x, y, t = 0) = \sin(x)\cos(y)$, $v(x, y, t = 0) = -\cos(x)\sin(y)$. (3)
• Using these initial conditions and the Navier-Stokes equations find the
pressure and velocity at all time assuming that $u = u(x,y,t = 0)F(t)$
and $v = v(x, y, t = 0)F(t)$ by finding $F(t)$.

3 Problem
Consider an incompressible flow in a periodic 2D domain of side 2π in each
direction. Assume the initial conditions for velocity are of the form
u(x, y, t = 0) = sin(x)cos(y), v(x, y, t = 0) = -cos(x)sin(y). (3)
• Using these initial conditions and the Navier-Stokes equations find the
pressure and velocity at all time assuming that u = u(x,y,t = 0)F(t)
and v = v(x, y, t = 0)F(t) by finding F(t).

Added by Michael W.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Mukesh Devi

Step 1

Step 1: Write down the Navier-Stokes equations for an incompressible flow in 2D: ∂u/∂t + u∂u/∂x + v∂u/∂y = -1/ρ ∂p/∂x + ν (∂²u/∂x² + ∂²u/∂y²) ∂v/∂t + u∂v/∂x + v∂v/∂y = -1/ρ ∂p/∂y + ν (∂²v/∂x² + ∂²v/∂y²) ∂u/∂x + ∂v/∂y = 0 where u and v are the velocity components Show more…

Show all steps

Thanks for your feedback!

Problem Consider an incompressible flow in a periodic 2D domain of side 2t in each direction. Assume the initial conditions for velocity are of the form u(x,y,t=0) = sin(x)cos(y), v(x,y,t=0) = -cos(x)sin(y). (3) Using these initial conditions and the Navier-Stokes equations, find the pressure and velocity at all times, assuming that u = u(x,y,t = 0)F(t) and v = v(x,y,t = 0)F(t) by finding F(t).