# Time scale for steady flow to occur in parallel disk viscometer

Hey to whomever is reading this!

I'm currently trying to solve a problem given to the class in a hydrodynamics course. I have to main questions. The following describes the problem:

We are considering a parallel disk viscometer (type of rheometer with plate-plate geometry) with disk radius $$R = 10mm$$ and disk spacing $$H = 0.1mm$$ (z-axis). An incompressible Newtonian fluid is placed between the disks. The top disk is rotating at a constant angular speed $$\Omega = 10 rad/s$$, generating a velocity gradient where the velocity at $$z=0$$ is $$u = 0$$ and at $$z=H$$ is $$u = R\Omega$$.

We consider the fluid in question to be silicone oil with a kinematic viscosity of $$\nu = 8.3\ \cdot 10^{-5} m^2/s$$. We want to calculate the time scale $$\tau$$ over which the motion induced by the upper plate will be transmitted to the bottom plate (time necessary to generate steady flow) based on the given parameters.

In another subquestion of this problem (later stage of the exercise) we are asked to derive the expression for the velocity $$u_\theta (r,z)$$. Based on the result of that question, namely $$u_\theta (r,z)=\frac{\Omega}{H}zr$$, I can calculate the time scale as the reciprocal of the stear strain rate $$\dot{\gamma} = \frac{du_\theta}{dz}$$ giving me a $$t = 10^{-3} s$$.

1st question:

How can I calculate (estimate?!) this time scale based on the parameters given (radius, disk spacing, angular velocity, kinematic viscosity), without knowing the expression for $$u_\theta$$?

I just found an answer purely by looking at the units and trying to get seconds. So I calculated this: $$t = \frac{\Omega R^2 H^2}{\nu^2} = 1.4\ \cdot 10^{-3} s$$, which is very close to the value I get from the reciprocal of the shear strain rate. However, we are supposed to explain our calculations and I do not know how to justify what I did in a physical sense.

2nd question:

If I want to calculate the Reynolds number for my problem, what do I choose as my characteristic length?

I thought of using the form $$Re = \frac{\Omega L^2}{\nu}$$. As the characteristic length, I calculated the geometric mean of the disk spacing and two times the radius. Is this correct? I would not know what else to use as the characteristic length for this kind of geometry.

Any help is greatly appreciated!

Thanks :)

• There is a different time scale $t_0=R^2/\nu=1.20482 s$. – Alex Trounev Feb 22 '19 at 14:53
• How do you motivate that calculation? I don't understand what it physically means. I can also see by looking at the units, that it works out to be in seconds – alonoid Feb 22 '19 at 15:13
• This is the characteristic time for a change in viscous flow parameters when moving from a state of rest to a state of motion. – Alex Trounev Feb 22 '19 at 15:20
• Okay but I do not understand why it is the radius squared divided by the kinematic viscosity. How can you explain that calculation physically? – alonoid Feb 22 '19 at 15:23
• See my answer, please. – Alex Trounev Feb 22 '19 at 17:14

To clarify the physical meaning of the characteristic time scale $$t_0=R^2/\nu$$, we consider the equations of fluid motion in the gap between the disks. In a cylindrical coordinate system $$(r,\phi,z)$$, the laminar flow velocity has one non-zero component which depends on $$t,r,z$$, thus $$\vec {v}=(0,u(t,r,z),0)$$ and

$$\frac {\partial u}{\partial t}=\nu \nabla^2u-\frac {u}{r^2}$$

$$\rho\frac {u^2}{r}=\frac{\partial P}{\partial r}$$

$$\nabla^2 =\frac {1}{r}\frac{\partial }{\partial r}(r\frac{\partial }{\partial r})+\frac{\partial^2 }{\partial z^2}$$

Choose time scales and lengths $$t_0=R^2/\nu, R$$, we introduce new coordinates and time $$r'=r/R, z'=z/R, t'=t/t_0$$. Then the first equation is reduced to the form (strokes omitted)

$$\frac {\partial u}{\partial t}= \nabla^2u-\frac {u}{r^2}$$

Set $$u(0,r,z)=0$$ and $$u(t,0,z)=0, u(t,r,0)=0, u(t,r,H/R)=u_0r$$, here $$u_0=\Omega t_0$$. We solve the problem numerically for different values $$u_0=1;10;100$$. The pictures show $$u(t,r,H/2R)$$. We see that the speed reaches the expected value $$u_0/2$$ at $$r=1$$ in a time $$t=1$$. Hence in dimensional variables $$t=t_0$$ is a characteristic time of transition from a state of rest to a state of motion. • @alonoid You're welcome! – Alex Trounev Feb 25 '19 at 14:04

The characteristic time for establishing steady state flow is $$H^2/\nu$$. This is because in the approximation you are using, the flow is being treated locally as rectilinear flow between two parallel plates, with the upper plate moving and the lower plate stationary. And, for this flow, the smallest eigenvalue of the transient solution is a constant times $$H^2/\nu$$. You can find the solution to this rectilinear flow problem in Transport Phenomena by Bird, Stewart, and Lightfoot.

• Thank you, I will look into the source you gave! :) – alonoid Feb 25 '19 at 13:30