Consider an incompressible fluid, with viscosity $\mu$ and density $\rho$, flowing vertically in an annulus, described in cylindrical coordinates $$\left\{ r,\theta,z\right\}, \; \text{with } r=a, r=b, \text{ and } b>a$$ at the boundaries and the boundary walls moving in a time-dependent fashion.
Simplifying the Navier-Stokes Equation where fluid velocity $\vec{u}=u_{z}(r,t)$ gives a one-dimensional form similar to the diffusion equation:
$$\frac{\partial u_{z}}{\partial t}=\frac{\mu}{\rho}\left[\frac{1}{r}\frac{\delta u_{z}}{\delta r}+\frac{\partial^{2}u_{z}}{\delta r^{2}}\right],$$
with boundary conditions where $U,\omega$ are constants:
$$u_{z}(a,t)=Ua\sin(\omega t), u_{z}(b,t)=Ub\sin(\omega t), \text{and }u_{z}(r,0)=0.$$
Can this be solved by Separation of Variables? Could the solution for $u_{z}$ give a sigmoidal relation between z and t for $0\leq\omega t\leq\pi$
Many thanks,
Tiernan
Let $u_z(r,t)=v_z(r,t)+Ur\sin\omega t$ ,
Then $\dfrac{\partial u_z(r,t)}{\partial t}=\dfrac{\partial v_z(r,t)}{\partial t}+U\omega r\cos\omega t$
$\dfrac{\partial u_z(r,t)}{\partial r}=\dfrac{\partial v_z(r,t)}{\partial r}+U\sin\omega t$
$\dfrac{\partial^2u_z(r,t)}{\partial r^2}=\dfrac{\partial^2v_z(r,t)}{\partial r^2}$
$\therefore\dfrac{\partial v_z(r,t)}{\partial t}+U\omega r\cos\omega t=\dfrac{\mu}{\rho}\left(\dfrac{1}{r}\left(\dfrac{\partial v_z(r,t)}{\partial r}+U\sin\omega t\right)+\dfrac{\partial^2v_z(r,t)}{\partial r^2}\right)$
$\dfrac{\partial v_z(r,t)}{\partial t}=\dfrac{\mu}{\rho}\left(\dfrac{\partial^2v_z(r,t)}{\partial r^2}+\dfrac{1}{r}\dfrac{\partial v_z(r,t)}{\partial r}\right)+\dfrac{U\mu\sin\omega t}{\rho r}-U\omega r\cos\omega t$
with $v_z(a,t)=0$ and $v_z(b,t)=0$