FLUID MECHANICS  THEORY



Another important fluid property will be introduced in
this section. Viscosity is a fluid property that measures the resistance
of
the fluid
due to
an applied force.




Fluid between Two Parallel Plates

Viscosity, μ 
20 C (68 F) 
Ns/m^{2} 
lbs/ft^{2} 
Water, pure 
1.12e3 
2.34e5 
Water, sea 
1.20e3 
2.51e5 
Carbon
Tetrachloride 
9.58e4 
2.00e5 
Gasoline 
3.1e4 
6.5e6 
Glycerin 
1.50e0 
3.13e2 
Mercury 
1.57e3 
3.28e5 
SAE 30W Oil 
3.8e1 
8.0e3 


Viscosity


To illustrate the concept of viscosity, consider a fluid between
two parallel plates, as shown in the figure. If the top plate is moved
at a velocity U while the bottom plate is fixed, the fluid
is subjected to deformation. The fluid in contact with the top
plate moves with the plate velocity U and noslip condition is applied
at the bottom plate (i.e.,
the fluid is stuck to the bottom plate, u = 0). The velocity profile
of the fluid motion between the plates is assumed to be linear and is
given by
u = U y/h
Note that the velocity gradient (also known as the rate of shear
strain) in this case is a constant (du/dy = U/h). Experiments
have shown that shear
stress (τ) is directly
proportional
to the rate of shear strain:




Shearing Stress
versus Rate of
Shearing Strain for Various Fluids 

Most common fluids, such as water, air and oil, are
called Newtonian fluids in which the shear stress is related to the
rate of shear strain in a linear fashion. That is,
The above equation is referred to as Newton's law of viscosity.
The proportionality constant (μ) is called
the absolute viscosity, dynamic viscosity or simply the viscosity. It
has
units of Ns/m^{2 }in
SI units (lbs/ft^{2} in US units). Sometimes it
is also expressed in the CGS system as dynes/cm^{2 }and
this unit is called a poise (P).
Note that the shear stress can also be determined by dividing the shear
force with the surface area. 



Shear Thinning Fluid: Latex Paint
Click
to view movie (29k) 

For nonNewtonian fluids, the shear stress is not a linear
function of the rate of shear strain. Some common types of
nonNewtonian fluids are shear thinning fluids, shear thickening
fluids and Bingham plastic. To describe these
nonNewtonian fluids, an apparent
viscosity is introduced and it represents the slope (not constant) of
the shear stress versus the rate of shear strain. It is obvious that
for
Newtonian
fluids,
the apparent viscosity is the same as the viscosity and is not a function
of the shear rate. 



Shear Thickening Fluid: Quicksand
Click
to view movie (41k) 

For shear thinning fluids, the apparent viscosity decreases with shear
rate, whereas for shear thickening fluids, the apparent viscosity increases
with shear rate. An example of a shear thinning fluid is latex paint. When
brushing paint on a wall, note that the larger the applied shear rate,
the less resistance that (viscosity) is encountered.
Examples for shear thickening
fluids are quicksand and a watercorn starch mixture. The larger the
applied shear rate trying to mix water with corn starch, more resistance
will be encountered. 



Bingham Plastic: Toothpaste
Click to view movie (22k) 

Another nonNewtonian fluid is Bingham plastic,
which is neither a fluid nor a solid. Bingham plastic, such as toothpaste,
can
withstand a finite shear stress without any motion, however it moves
like a fluid once this yield stress is exceeded. Note that only Newtonian
fluids will be considered in the future discussion; nonNewtonian
effects are beyond the scope of this eBook.
Viscosity is not a strong function of pressure, hence the
effects of pressure on viscosity can be neglected. However,
viscosity depends
greatly on temperature. For liquids, the viscosity
decreases with temperature, whereas for gases, the viscosity increases
with temperature. For example,
crude oil is often heated to a higher temperature to reduce
the viscosity
for
transport. 





Kinematic Viscosity



The kinematic viscosity is the ratio of absolute viscosity and density.
That is,
ν = μ
/ ρ
The SI unit for the kinematic viscosity is m^{2}/s. The unit in
the CGS system is cm^{2}/s and this is referred
as a stoke (St).



