Fluid Mechanics and Energy Transport
BIEN 301
Lecture 3
Viscosity, Flow Visualization, Flow Analysis Methods
Juan M. Lopez, E.I.T.
Research Consultant
LeTourneau University
Adjunct Lecturer
Louisiana Tech University
Viscosity

Definition (White 1.7)
Quantitative measure
of a fluid’s resistance
to flow.



t
 
u
, In generalize d coordinate s
g 2
 
u
, In cartesian coordinate s, flat plane
y
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BIEN 301 – Winter 2006-2007
Viscosity

Newtonian vs. Non-Newtonian Fluids
• Newtonian Fluids: Linear Viscosity Equation

Examples?
• Non-Newtonian Fluids: Non-linear Viscosity Eq.

Examples?
• Usually this difference is established as constant viscosity vs.
non-constant viscosity. In ALL real fluids, however, viscosity
can never be a true constant and varies with T and P.

No-Slip Condition
• For real fluids, it is assumed that at the boundaries, zero slip
occurs.
• This is due to jump balance across the interface.


Temperature Balance
Momentum Balance
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BIEN 301 – Winter 2006-2007
Viscosity Effects

The principal effects of viscosity are divided into two readily
visible areas:
• The boundary layer formation
• Turbulence


We’ll cover boundary layer in greater depth later on. For now,
we’ll focus on the second effect: generation of turbulence.
We use a dimensionless number to correlate the effects of
viscosity on a flow regime: the Reynolds Number.
Re 
VL VL


v
where
v
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

BIEN 301 – Winter 2006-2007
Viscosity Effects
 Dimensionless


Numbers
Many dimensionless numbers in fluid mechanics. We will cover
many, many of them throughout the quarter.
Reynolds number relates the kinetic energy to the viscosity.
• It helps us identify transition into turbulent region, where the
viscosity is too great relative to the fluid motion, and orderly
movement can no longer occur.

Why dimensionless?
• These are numbers that apply to a flow regime, related to a
characteristic dimension.
• Pick the characteristic dimension that better describes the region
you are interested in studying, and apply the equation.
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BIEN 301 – Winter 2006-2007
Viscosity Effects
 Identify
the characteristic dimension for
each flow.
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BIEN 301 – Winter 2006-2007
Viscosity Effects

It’s important to think carefully about the characteristic
dimension, however it’s not absolutely a perfect science.

You can pick an alternative characteristic dimension as long as
you remain consistent as you compare systems
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BIEN 301 – Winter 2006-2007
Viscosity Effects

Understanding the meaning of dimensionless numbers


Even though they are quite an alien concept to people outside of
fluid mechanics, it is important that you grasp these numbers
and learn to obtain a “feel” for what these numbers are telling
you.
Reynolds number:
• High Values, mean a high kinetic energy relative to its ability to flow.
This should cause instability, therefore, we would expect turbulence.
• Medium values, mean that the kinetic energy is smoothly related to
its ability to flow. This should cause smoothly changing laminar flow.
• Low values, mean a low kinetic energy related to its ability to flow.
This would be a creeping flow where inertial effects are almost, if
not entirely, negligible.
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Viscosity Effects - Example

A standard example a moving plate with viscous fluid
between the plate and a fixed surfaceVplate2
Vplate1
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Viscosity Changes


Viscosity is dependent on temperature and pressure. For
water, we can ignore the effects of pressure.
Viscosity changes as a result of temperature are evident
in all real fluids. For gases, we have two representations:
Power Law and Sutherland Law:
n

T 
 

Power Law
 T0 
 
3/ 2

 0   T  T  S 
0
T0 


Sutherland Law

T S
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Viscosity Changes

For Liquids, the approximation is as follows:
 
T0   T0 

ln    a  b   c 
T  T 
 0 

2
You can see how this gets much uglier very
quickly.

We really like idealized fluids. It’s unfortunate we don’t
have more of them around.
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BIEN 301 – Winter 2006-2007
Thermal Effects
 Similar
to viscosity, thermal conductivity
changes the way heat transfer occurs
within the fluid.
 Thermal conductivity acts on the
temperature gradient present in the fluid to
obtain a vector form of heat transfer
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BIEN 301 – Winter 2006-2007
Thermal Effects

The gradient of a property is the partial
differential of that property with respect to each
of the principal dimensions of the field.

Note the Del Operator is a vector operator.

X X X
X g1 , g 2 , g 3  


, In generalize d coordinate s
g1 g 2 g 3

T
T
T
 kT x, y, z   k
 k
 k
, for the thermal conductivi ty problem
x
y
z
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BIEN 301 – Winter 2006-2007
Surface Tension
 Surface
tension arises at the interface
between a fluid and another system.


This system can be another fluid, a solid, or
nothing at all.
Surface tension is a much more complicated
subject than it will appear to be here. We will
cover a simplified method of studying surface
tension.
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BIEN 301 – Winter 2006-2007
Surface Tension

The coefficient of surface tension, Υ, is a
measure of force per unit length or energy per
unit area present everywhere tangent to the
surface, at the surface.


Generally, Υ, changes with the same sign as
temperature, reaching zero at a critical point.
When the surface curves, the tension generates a
pressure difference across the interface.
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BIEN 301 – Winter 2006-2007
Surface Tension
 The
pressure across the surface interface
can be expressed as follows:

p  , Pressure balance across a cylinder
R
2
p 
, Pressure balance across a sphere
R
4
p 
, Pressure balance across a hollow bubble
R
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BIEN 301 – Winter 2006-2007
Surface Tension

Wetting

When a liquid interacts with a solid surface, there is
an angle at that interaction. The magnitude of this
angle defines whether an item wets the solid or not.
• Water wets clean glass

Water tends to sheet off clean glass, beads only with
imperfections.
• Water does not wet wax

Water beading on the wax job of your car.
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BIEN 301 – Winter 2006-2007
Surface Tension

Surface tension also causes the capillary effect.

For a cylindrical capillary tube, the height of the fluid
column that rises due to the surface tension and the
material wetting the capillary can be found to be as
follows:
2 cos 
h
R
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Cavitation

Cavitation is a function of vapor pressure. It can
be highly damaging.
 Cavitation is described by a non-dimensional
parameter, the cavitation number.

Which is a function of Pa, ambient pressure, Pv,
vapor pressure, and V, a characteristic velocity.
p a  pv
Ca 
1
2
V
2
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BIEN 301 – Winter 2006-2007
Cavitation
 For
most fluids, there is a critical point for
Ca below which the fluid will begin to
cavitate.


The spontaneous generation and rapid
implosion of gas bubbles in the liquid can
be a highly destructive force.
If not monitored for appropriately, this can
internally destroy a mechanical system.
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Speed of Sound

Aside from cavitation, we can cause damage to
pipes, ductwork, and cells when we start
approaching flows that are a significant fraction of
the speed of sound in that fluid.



This produces damaging shock waves and erosion.
This comes from the compressibility effects present in the
fluid.
For an ideal gas, the speed of sound is:
a  kRT 
1/ 2
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Complicated Enough?
The answer is…no. Not nearly enough. We have too many
assumptions and simplifications. But this will allow us to begin
working with fluids from a theoretical perspective.
 Our problems continue to complicate themselves…viscosity,
then turbulence, now cavitation and compressibility-induced
shock waves…it is apparent that a careful and engineered
approach must be taken each and every time we deal with a
flow system.


Enough of the basics for now…how do we DEAL with a flow
system? How do we visualize the results? How do we make sense
of all of this stuff?
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BIEN 301 – Winter 2006-2007
Flow Visualization

One of the most important tools in understanding
fluid mechanics is being able to visualize what is
going on in our system.
 When we visualize these flows, we often use
one of a few standard visualization methods:



1) Streamline
3) Streakline
2) Pathline
4)Timeline
Note that these are IDENTICAL in steady flow.
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BIEN 301 – Winter 2006-2007
Flow Visualizations
 Streamlines

Everywhere tangent to the velocity vector at a
given instant in time.
dr dx dy dz



V
u
v
w
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BIEN 301 – Winter 2006-2007
Flow Visualizations
 Pathline

The displacement of a particle over a defined
period of time, defined by integrating its
velocity vectors in time to obtain a path.
t2
t2
t2
t1
t1
t1
x   udt , y   vdt, and z   wdt
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BIEN 301 – Winter 2006-2007
Flow Visualizations
 Streaklines


A streak line is a set of particles that has gone
through a particular point in space.
This is the way experimental work is generally
collected.
• Injecting hydrogen bubbles, smoke, etc.
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Flow Visualizations
 Timelines


The set of particles that form a line in any
given instant in time.
Note that for all of these visualizations,
streamlines are the “simple” ones to generate
analytically. The rest are a result of
experimentation.
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Flow Visualizations
 Check
out the external links on blackboard
to see some external references on flow
visualizations.
 Whole careers are dedicated to the
generation of adequate flow visualizations.
 Flow Visualization Example
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Flow Analysis Techniques
 Three
primary methods of approaching a
flow problem



Integral Analysis (Control-Volume Approach)
Differential Analysis (Infinitessimal Approach)
Dimensional Analysis (Experimental Study)
 For ANY
of these approaches, we must
satisfy the basic laws of mechanics.
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Flow Analysis Techniques

Flow Analysis Checklist:






Conservation of Mass (Continuity, can also include
conservation of species for mixtures)
Conservation of Momentum (Newton’s second law)
Conservation of Energy (First Law of
Thermodynamics)
A state relationship
Adequate boundary conditions, initial conditions.
Appropriate assumptions about our flow.
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BIEN 301 – Winter 2006-2007
Flow Analysis Techniques
 Assumptions



Our principal assumptions sets come in pairs, to aid in
describing our flow condition:
• Steady or Unsteady?
• Inviscid or Viscous?
• Incompressible or Compressible?
• Gas or Liquid?
Once we’ve made our assumptions and followed the analysis
checklist, we are more certain that our results will be meaningful.
Coupled with White’s 1.13, this is a powerful approach to fluid
mechanics.
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BIEN 301 – Winter 2006-2007
Assignment
 HW
3 has been posted on blackboard
 Project Proposals due next time!
 Individual project schedules are available
on blackboard.
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BIEN 301 – Winter 2006-2007
Questions?
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BIEN 301 – Winter 2006-2007