ME 3350 – Spring 17
handout 6.4
SCALING: EXAMPLE OF APPLICATION
SAE 30 oil (   912 kg/m3;   3.8 101 Ns/m2) is contained between wide, parallel plates
spaced a distance h  2 cm apart.
The upper plate is fixed, and the bottom plate oscillates harmonically with a velocity amplitude U
and a frequency  . The differential equation for the velocity distribution between the plates is:

u
 2u
 2 ,
t
y
where u is the velocity, t is the time, and  and  are the fluid density and viscosity,
respectively.
1- Rewrite this equation in a suitable non-dimensional form using h, U ,  as reference
parameters.
2- Discuss the solution of this problem when   1 rad/s and   0.01 rad/s.
Scaling the variables:
Calculation of the derivatives:


t
2

y 2


y
ME 3350 – Spring 17
handout 6.4
Governing equation in scaled form:
Divide all terms by
U
h2
:
Scaled equation:
The dimensionless coefficient in front of the unsteady term (on the left side of the equation) is the
12
  
square of a well-known dimensionless number called the Womersley number   h 
 ,
  
which represents the ratio of pulsatile effects to viscous effects.

When   1 rad/s,
 h 2
 0.96 . Therefore the coefficients in front of the unsteady term

(on the left of the equation) and the viscous term (on the right of the equation) are of the
same order of magnitude. In dimensional form, the governing equation for this flow is:


u
 2u
 2
t
y
When   0.01 rad/s,
 h 2
 0.0096 . Therefore the coefficient in front of the unsteady

term is two orders of magnitude smaller than that in front of the viscous term. Therefore,
unsteady effects can be neglected and the flow is dominated by viscous effects. In
 2u
dimensional form, the governing equation for this flow is:  2  0
y