- #1
Peeter
- 305
- 3
Homework Statement
Consider the steady flow between two long cylinders of radii [itex]R_1[/itex] and [itex]R_2[/itex], [itex]R_1 > R_1[/itex], rotating about their axes with angular velocities [itex]\Omega_1[/itex], [itex]\Omega_2[/itex]. Look for a solution of the form, where [itex]\hat{\boldsymbol{\phi}}[/itex] is a unit vector along the azimuthal direction:
[tex]\mathbf{u} = v(r) \hat{\boldsymbol{\phi}}[/tex]
[tex]p = p(r).[/tex]
Write out the Navier-Stokes equations and find differential equations for [itex]v(r)[/itex] and [itex]p(r)[/itex].
Fix the constants [itex]a[/itex] and [itex]b[/itex] from the boundary conditions. Determine the velocity [itex]v(r)[/itex] and pressure [itex]p(r)[/itex].
Homework Equations
Navier-Stokes equations for steady state incompressible flow are
[tex](\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} = -\frac{1}{{\rho}} \boldsymbol{\nabla} p + \nu \boldsymbol{\nabla}^2 \mathbf{u}[/tex]
[tex]\boldsymbol{\nabla} \cdot \mathbf{u} = 0.[/tex]
We'll have no-slip boundary value conditions on the surfaces of each cylinder, so
[tex]\begin{align}v(R_1) &= R_1 \Omega_1 \\ v(R_2) &= R_2 \Omega_2.\end{align}[/tex]
The Attempt at a Solution
Working in cylindrical coordinates where the gradient is
[tex]\boldsymbol{\nabla} = \hat{\mathbf{r}} \partial_r + \frac{\hat{\boldsymbol{\phi}}}{r} \partial_\phi.[/tex]
I find a pair of differential equations to solve
[tex]r^2 v'' + r v' - v = 0[/tex]
[tex]p' = \frac{\rho v^2}{r}[/tex]
solving these and applying the boundary value conditions I find
[tex]v(r) = \frac{1}{{R_2^2 - R_1^2}}\left(\left( R_2^2 \Omega_2 - R_1^2 \Omega_1\right) r+\frac{R_1^2 R_2^2}{r} (\Omega_1 -\Omega_2)\right)[/tex]
[tex]p(r) - p_0 = \frac{\rho }{(R_2^2 - R_1^2)^2} \left( \frac{1}{{2}} \left( R_2^2 \Omega_2 - R_1^2 \Omega_1\right)^2r^2 -\frac{R_1^4 R_2^4}{2 r^2} (\Omega_1 - \Omega_2)^2+ 2 \left( R_2^2 \Omega_2 - R_1^2 \Omega_1\right) R_1^4 R_2^4 (\Omega_1 - \Omega_2)^2 \ln r\right)[/tex]
That's almost the whole solution, but the part that I am unsure of is what can we use to determine the integration constant for the pressure (I've called it [itex]p_0[/itex] above)? Is there another boundary value constraint that I am missing?
Last edited: