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7/30/2019 Transport Processes Lec
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Chapter 1. Viscosity and the Mechanisms ofMomentum Transport
Newtons law of viscosity (molecular momentumtransport)
Generalisation of Newtons law of viscosity
Pressure and temperature dependence of viscosity Molecular theory of the viscosity of gases at lowdensity, of liquids, and of suspensions andemulsions
Convective momentum transport
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Newtons law of viscosity(molecular transport of momentum)
At steady-state
Newtons law ofviscosity
F V
A Y
xyx
dv
dy
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Newtons law of viscosity
Molecular transport of momentum
= Viscosity,yx = Shear stress in the x-direction at constant y
Velocity gradient is the driving force formomentum transport
xyx
dv
dy
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Newtons law of viscosity
Another interpretation x-momentum is transmitted through the fluid in
the y-direction
yx = viscous flux of x-momentum in thepositive ydirection.
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Newtonian fluids
All gases and most simple liquids Viscosity is independent of -> =(T) Units: Pa.s
Non-Newtonian fluids Viscosity is dependent of Polymeric liquids, pastes, slurries
Kinematic viscosity, Units: m2/s
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Viscosity Definition: the measure of a fluids resistance to flow under applied
stress
Newtonian: origin, constant slope( = viscosity), not time dependent
Non-Newtonian: all others
The apparent viscosity is the ratio of the shearstress to the shear rate. It is not the slope.
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Example: Calculation of momentum flux Fluid between two parallel plates
Details: plate velocity V= 1 ft/s, fluid viscosity = 0.7 cP, Y=0.001 ft
Required: calculate steady state momentum flux yx
2
5 2 2/ 0 1.0 /
(0.7 )(2.0886 10 ) 1.46 10 /0.001 0
fx xyx f
lb s ftdv v ft scp lb ft
dy y cp ft
Solution
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Generalization of Newtons lawof viscosity
Velocity is a vector
Stress is a tensor with 9 components
, ,x y zv v v v
xx xy xz
yx yy yz
zx zy zz
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Generalization of Newtons lawof viscosity
A half of the cube is removed. Forces onthe free surface:
x x xx xy xzpe and
Associated to the pressure(perpendicular to the surface)
Associated to the viscous forces.For a plane perpendicularto x-direction
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Generalization of Newtons lawof viscosity
Pressure and viscous forces;
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Molecular stress
Molecular Stress : Normal stress :
Shear stress :
Interpretationsij force in the j-direction on a unit areaperpendicular to the i-direction or
ij flux of j-momentum in the positive ofi-direction
ij ij ijp ( )ij i j
( )ij i j
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= viscosity
= dilatational viscosity (commonly not used)
Stress tensor
2
3
2
3
j yi x zij ij
i j
T
v vv v v
x x x y z
v v v
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Viscosity of gases and liquids
Viscosity is a function of =(T,p) Viscosity may be obtained from Molecular theories
Empirical correlations
Correlation based in molecular theories
Reference The Properties of Gases and Liquids by Poling, Prausnitz,
and OConnell, 5th Ed.
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Pressure and temperaturedependence of viscosity
Reduced viscosity infunction of thereduced temperatureand reduced pressure
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Critical pressure andtemperature
The critical temperatureof a liquid is thattemperature beyond which the liquid cannot exist,no matter how much pressure is placed on it.
The pressure that is needed to cause the gas to
condense at the critical temperature is thecritical pressure.
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Pressure and temperaturedependence of viscosity
Viscosity of a gas at low density increases withincreasing temperature
Viscosity of a liquid decreases with increasingtemperature
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Molecular theory of the viscosityof gases at low density
At low pressure, average velocity (random) andmean free path
Viscosity
2
8 1
2
Tu
md n
2
2
3
m T
d
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Rigorous kinetic theory.Monoatomic gases Lennard-Jones potential
Viscosity for a pure mono-atomic gas (it can be usedfor poly-atomic gases aswell) in term of Lennard-
Jones parameters
12 6
( ) 4rr r
2
5
16
m T
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Convective momentumtransport
Momentum may be transported by the bulk flow ofthe fluid. Convective transport Momentum flux across the shaded area are
Convective momentum-flux tensor (2nd order)
Combined momentum flux
( ) ( ) ( )x y zv v v v v v
i j i ji j
vv vv ee
vv p vv
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Convective momentum transport.Convective momentum flux tensor