## PHY454H1S Continuum mechanics. Problem Set 3. Velocity scaling, non-dimensionalisation, boundary layers.

Posted by peeterjoot on March 30, 2012

# Disclaimer.

This problem set is as yet ungraded.

# Problem Q1.

## Background.

In fluid convection problems one can make several choices for characteristic velocity scales. Some choices are given below for example:

where is the acceleration due to gravity, is the coefficient of volume expansion, length scale associated with the problem, is the applied temperature difference, is the kinematic viscosity and is the thermal diffusivity.

- For :
Observing that

we must have

We also find

so that

- For :
- For
- For
According to http://scienceworld.wolfram.com/physics/ThermalDiffusivity.html, the thermal diffusivity is defined by

so that

That gives us

We’ve verified that all of these have dimensions of velocity.

## Part 1. Statement. Check whether the dimensions match in each case above.

## Solution.

## Part 2. Statement. Pure liquid.

For pure liquid, say pure water at room temperature, one has the following estimates in cgs units:

For a layer depth and a ten degree temperature drop convective velocities have been experimentally measured of about .

With , calculate the values of , , , and . Which ones of the characteristic velocities , , do you think are suitable for nondimensionalising the velocity in Navier-Stokes/Energy equation describing the water convection problem?

We have

Use of gives the closest match to the measured characteristic velocity of .

## Solution.

## Part 3. Statement. Mantle convection.

For mantle convection, we have

and the actual convective mantle velocity is . Which of the characteristic velocities should we use to nondimensionalise Navier-Stokes/Energy equations describing mantle convection?

## Solution

Let’s compute the characteristic velocities again with the mantle numbers

Both and come close to the actual convective mantle velocity of . Use of to nondimensionalise is probably best, since it has more degrees of freedom, and includes the gravity term that is probably important for such large masses.

# Problem Q2.

## Statement

Nondimensionalise N-S equation

where is the unit vector in the direction. You may scale:

- velocity with the characteristic velocity ,
- time with , where is the characteristic length scale,
- pressure with ,

Reynolds number and Froude number .

## Solution

Let’s start by dividing by , to make all terms (most obviously the term) dimensionless.

Our suggested replacements are

Plugging these in we have

Making a replacement, using the Froude number, we have

Scaling by we tidy things up a bit, and also allow for insertion of the Reynold’s number

Observe that the dimensions of Froude’s number is that of velocity

so that the end result is dimensionless as desired. We also see that Froude’s number, characterizes the significance of the body force for fluid flow at the characteristic velocity. This is consistent with [1] where it was stated that the Froude number is used to determine the resistance of a partially submerged object moving through water, and permits the comparison of objects of different sizes (complete with pictures of canoes of various sizes that Froude built for such study).

# Problem Q3.

## Statement

In case of Stokes’ boundary layer problem (see class note) calculate shear stress on the plate . What is the phase difference between the velocity of the plate and the shear stress on the plate?

## Solution

We found in class that the velocity of the fluid was given by

where

Calculating our shear stress we find

and on the plate () this is just

We’ve got a constant term, plus one that is sinusoidal. Observing that

The phase difference between the non-constant portion of the shear stress at the plate, and the plate velocity is just . The shear stress at the plate lags the driving velocity by 90 degrees.

# References

[1] Wikipedia. Froude number — wikipedia, the free encyclopedia [online]. 2012. [Online; accessed 27-March-2012]. http://en.wikipedia.org/w/index.php?title=Froude_number&oldid=479498080.

## Leave a Reply