Question

Problem 2: The vorticity-streamfunction formulation is to be used to solve the Navier- Stokes equations in a stationary circular cylinder of radius $R$. Obtain the expression for the Dirichlet boundary condition for vorticity at the surface using Thom's method. In cylindrical coordinates, the streamfunction equation is $\frac{\partial^2 \psi}{\partial r^2} + \frac{1}{r} \frac{\partial \psi}{\partial r} + \frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2} = -\omega$, and the velocity components are defined by $v_r = \frac{1}{r} \frac{\partial \psi}{\partial \theta}$, $v_\theta = -\frac{\partial \psi}{\partial r}$.

          Problem 2:
The vorticity-streamfunction formulation is to be used to solve the Navier-
Stokes equations in a stationary circular cylinder of radius $R$. Obtain the
expression for the Dirichlet boundary condition for vorticity at the surface
using Thom's method. In cylindrical coordinates, the streamfunction equation
is
$\frac{\partial^2 \psi}{\partial r^2} + \frac{1}{r} \frac{\partial \psi}{\partial r} + \frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2} = -\omega$,
and the velocity components are defined by
$v_r = \frac{1}{r} \frac{\partial \psi}{\partial \theta}$, $v_\theta = -\frac{\partial \psi}{\partial r}$.

Problem 2:
The vorticity-streamfunction formulation is to be used to solve the Navier-
Stokes equations in a stationary circular cylinder of radius R. Obtain the
expression for the Dirichlet boundary condition for vorticity at the surface
using Thom's method. In cylindrical coordinates, the streamfunction equation
is
(∂^2 ψ)/(∂ r^2) + (1)/(r)(∂ψ)/(∂ r) + (1)/(r^2)(∂^2 ψ)/(∂θ^2) = -ω,
and the velocity components are defined by
vr = (1)/(r)(∂ψ)/(∂θ), vθ = -(∂ψ)/(∂ r).

Added by Vicki R.

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