Download Transport Phenomena 3

Transport Phenomena 3
Unit 2: Momentum Transport
Topics
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Definitions
Properties at a point
Molecular Transport of Momentum
Convective Momentum Transport
Definitions
• Fluid: as a substance which deforms
continuously under the action of a shear
stress.
• Then when a fluid is at rest, there can be
no shear stress
Concept of a continuum
• Fluids, like matter , are composed of
molecules whose numbers stagger the
imagination.
• In a cubic inch of air at room conditions
there are some 1020 molecules. Any theory
to predict the individual motions of this
many molecules would be extremely
complex, far beyond out present abilities
Concept of a continuum (Cont.)
• Most engineering work is concerned with the
macroscopic or bulk behavior of a fluid rather
than with the microscopic or molecular behavior.
• In most cases is convenient to think as a
continuous distribution of matter or a continuum.
• Validity of this concept is seem to be dependent
upon the type of information desired rather than
the nature of the fluid.
Definitions
• Steady state flow – physical properties of
the fluid does not change.
• Unsteady state flow – the properties
change with time in any point of the
system
Definitions
• Uniform flow – when the velocity at any point,
across transversal section in the direction that
motion occurs is constant. In other words when
the velocity vector are parallel and preserved
their intensity
• Non uniform – when the velocity vectors are not
parallel and did not maintain their intensity
constant. Example: The flow fluid through a
tube of constant diameter is uniform, but the flow
through a tube of variable diameter (conic form)
is not uniform.
Properties At a Point
• Density: It is defined as the mass per unit
volume. Under flow conditions, particularly in
gases, the density may vary greatly throughout
the fluid.
• The density, ρ, at a particular point in the fluid is
defined as
m
  lim
V  V V
• Where Δm: mass contained in the volume ΔV
• δV is the smallest volume.
Properties At a Point (Cont.)
• Stress:
• Consider the force ΔF acting on an
element ΔA of the body shown in the
figure.
• The force ΔF is resolved into components
normal and parallel to the surface of the
element.
• The force per unit area or stress ata point
is defined as the limit of ΔF/ΔA as ΔA→δA
Force on an element of fluid
Fn
lim
  ii
A A A
Fs
lim
  ij
A A A
ΔF
ΔFn
ΔFs
ΔA
• Where σii is called the normal stress and
the shear stress (Flux tensor)
 ij is
Molecular Transport Momentum
• The buildup to the steady, laminar velocity
profile for a fluid contained between two
plates.
• Each of the plates have an area A,
separated by a distance Y.
• In the space let’s consider there is a fluid
(gas or liquid).
Build-up to the steady-state laminar
velocity profile
Y
t<0
vx(y,t)
Small t
vx(y)
t=0
V
Large t
• In this Figure 1:
• First: The system is at rest, but a time t= 0
• Second: At t= 0 the lower plate is set in
motion in the direction at a constant
velocity
• Third: At time proceeds (Small t), the fluid
gain momentum
• Fourth: Ultimately the linear steady-state
velocity profile shown in the figure is
established.
• At state-steady motion, a constant force F is
required to maintain the motion of the lower
plate.
F
V
(1)

A
Y
• The force is proportional to the area and to the
velocity and inversely proportional to the
distance between plates.
• Where F/A is , which is the force in the
direction x per unit area perpendicular to the y
direction.
• It is the force exerted by the fluid of lesser y on
the fluid of greater y, therefore V/Y can be
replaced by –dvx/dy.

• The equation can also be written as
 yx
dvx

dy
Newton’s Law of viscosity
(2)
• Where:
• μ: viscosity of fluid, (Pa.s)

• yx : Flux tensor (shear stress) in the positive y
direction , [(N/m2)= Pa]
• vx: velocity in x direction, (m/s)
• y: distance, (m)

Generalization of Newton’s Law
of viscosity
• The equation (2) was defined only in terms
of a simple steady state shearing flowing
in which vx as a function of y alone, and vx
and vz are zero.
• This situation is not really so common.
Usually the system is composed by a flow
in which the three velocity components
may depend on all three coordinates and
possible on time.
Consider
• A general flow pattern in which the velocity
shows in various directions depends on
the time.
• The velocity components are given by
vx  vx ( x, y, z, t );
v y  v y ( x, y, z, t ),
vz  v2 ( x, y, z, t )

• There will be then, 9 stress components ij
(where i and j may be taken on the
designations (x,y,z), instead of the  yx
component that appears in equation (2)
Pressure and viscous forces acting
on planes
z
x, y, z
y
x
• The volume element can be cut to each of
three coordinates in turn.
• Find the forces that have been exerted on
that surface by the fluid that was removed.
• There will be two forces that contribute
– Associated with the pressure
– Associated with the viscous forces
• Pressure force
– Always perpendicular to the exposed surface
• For example in x direction the force will be a vector
pδx that is the pressure (a scalar) multiplied by the
unit vector δx in the x direction. Similarly for the
other sections.
– Pressure forces will be exerted when the fluid
is stationary as well as when it is in motion
• Viscous forces
• They exist only when the are velocity gradients
within the fluid.
• They are neither perpendicular nor parallel to the

surface element, rather at some angle
to the
surface.
• Those forces represent in the Figure are vectors
with scalar components.
x
– For example,
x
has components
 xx , xy , xz
• Pressure and Viscous
Forces acting on
planes
y
x
x
pδy
pδx
z
pδz
• For convenient a new variable is defined
that include both
 ij  p ij   ij
• Where δij is the kronecher delta, which is 1
if i = j and zero if i ≠ j.
Two Ways to Interpret the definition
 ij  p ij   ij
• force in the j direction on a unit area to the i
direction, where it is understood that the fluid in
the region of lesser xi is exerting the force on
the fluid of a greater xi.
• flux of j-momentum in the positive (+) i direction,
It is understood that is, from the region of lesser
xi to that of greater xi.
Generalizing Newtons’ Law
(Equation 2)
• How are the stresses related to the
velocity gradients in the fluid?
• Restrictions
– The viscous stresses may be linear
combinations of all velocity gradients
 x
 ij   k  ijkl
where i,j,k and l may be 1,2,3
x1
– Time derivatives or time integrals should not
appear in the expression.
– Do no expect viscous forces of the fluid is in a state
of pure rotation. No viscous forces present are only
those one that are symmetric linear combinations of
velocity gradients
  j i


xi
 xi

  x  y  z


 and 
y
z
 x


  ij

– If the fluid is isotropic (not preferred direction) the
coefficient in front of the two expressions must be
scalar so that
  j i 
  x  y  z 
 ij  A 



  B
  ij
y
z 
 x
 xi xi 
– Common agreement among fluid dynamicists
the scalar constant B is set equal (3/2μ – k),
where k is the identically zero for monoatomic
gases at low density.
  j  i
 ij    

  xi  xi
  x  y  z 
 2

 ij
     k   k


 3
 




x
y
z



– These relations for the stresses in a
Newtonian fluid are associated with the
names of Navier, Poisson and Stokes
• Writing these set of equations more concise in
vector-tensor notation
2

    v  (v)       k  .v 
3
 
• Where δ is the unit tensor with components δij,
v is the velocity gradient tensor with components
  / xi  j ,(v) is the transpose of the velocity
gradient tensor with components   / xi  j , and (.v)
is the divergence of the velocity vector.
Importance of this generalization
• This equation involve two coefficients
characterizing the fluid:
– The viscosity μ
– The dilatational viscosity k.
Convective Momentum Transport
• The momentum can be transported by bulk flow
of the fluid. This process is called convective
transport. Use the same figure on three planes
• The fluid vector at the centre is v
• The volume rate of flow across the area in the
first figure is vx. The fluid carries with it
momentum ρv per unit volume.
• Hence the momentum flux across this area is
then vxρv (vectors)
• Convective
momentum fluxes
through planes of unit
area perpendicular to
the coordinates
direction
x
pvyv
pvxv
pvzv
END