Transcript SESM3004 Fluid Mechanics

Lecture 4: Isothermal Flow.
Fundamental Equations
•
•
•
•
•
Continuity equation
Navier-Stokes equation
Viscous stress tensor
Incompressible flow
Initial and boundary conditions
Continuity equation

t

 div   v   0
V
expresses the mass balance in a control volume,
one scalar equation
No gain or loss of mass in a fixed volume V,
The change of mass is solely due to
inlet/outlet fluxes.
This statement leads to the above-written
continuity equation.
Navier-Stokes equation




 v

 v   v       f
 t

expresses conservation of
momentum (second Newton’s law
applied to a fluid particle).
Vector equation = three scalar
equations
 ij   p  ij   ij -- stress tensor divided into pressure (short-range intermolecular forces dependent the relative position of
molecules) and viscous stress tensor (short-range
forces dependent on the relative motion of molecules)

f -- long-range forces – volume (or body) forces, e.g.
gravity force
Substitution gives




 v

 v   v    p      f
 t

Viscous stress tensor
 ij
 v i v j 2






  ij div v   ij div v
 x


x
3
j
i


This expression is obtained based on the following reasoning
(i) the viscous force is due the relative motion of molecules, defined by
the velocity gradients, i.e. the viscous stress should be proportional to
velocity gradients
(ii) velocity gradients are assumed to be small: no higher derivatives
and non-linear terms
 

(iii) we must exclude the case of uniform rotation ( v    r ), when
the viscous stress should also vanish (only symmetric combinations of
velocity gradients)
(iv) summing up (i)-(iii) to write the most general tensor (of rank 2) from
the velocity derivatives
,
 -- first and second coefficients of viscosity (phenomenological
coefficients to be determined experimentally)
Incompressible flow
  constant
Criterion: slow motion (v<<c) where v is the
typical fluid velocity and c is the speed of
sound (cair=343m/s and cwater=1560m/s).
Comment: for a single-phase fluid, density is function of temperature
and pressure. We consider isothermal motion, when there are no
temperature variations. The incompressibility assumption is equivalent
to saying that the pressure-related variations in fluid density are
negligibly small.

The continuity equation is simplified to: div v  0
The viscous force can be also simplified as follows:
 i ij   i   iv j   j v i     i  iv j
Or, the Navier-Stokes
equation becomes:
it is assumed that coefficient of
viscosity  is constant



 
 v

 v   v    p    v  f
 t

Initial and boundary conditions
Initial conditions: field of velocity at the initial moment
Boundary conditions:
 
a) rigid wall v  v wall (‘no-slip condition’: result of molecular attraction
between a fluid and the surface of a solid body), illustration:
http://www.youtube.com/watch?v=cUTkqZeiMow (
)
b) interface between two immiscible fluids:


v1  v 2
( 1)
(2)
 ik n k   ik n k : force exerted by the first fluid acted on the
second fluid equals to the force exerted by
the second fluid acted on the first one
n k -- unit normal vector to the interface
c)free interface (e.g. air/liquid):
 ik n k  0
Navier-Stokes
Claude-Louis Navier
(10 February 1785 in
Dijon – 21 August 1836
in Paris) was a French
engineer and physicist
specialized in
mechanics.
Sir George Stokes, 1st Baronet ,
(13 August 1819 in Skreen, Ireland
– 1February 1903, Cambridge,
England) was a mathematician and
physicist, who at Cambridge made
important contributions to fluid
dynamics, optics, and mathematical
physics. He was secretary, then
president, of the Royal Society.
Lecture 5: Circular Poiseuille Flow
• Governing equations (isothermal,
incompressible flow):

div v  0

 

 v

  v  v    p    v
 t

• Configuration:
L
R
z
r
p
1
p
2
Assumptions

• Steady flow:  0, no time-dependence
t

• Plane-parallel flow: v  0 ,0 , v r 
z
Consequences:
a) Zero non-steady term
b) Continuity equation is
automatically satisfied for any v z r 
c) Non-linear term vanishes
Resultant equations

 p  v  0
r- and z- projections: (r ) :
(z ) :
Pressure:


p
r
p
z
0
 p  p (z )
 v z  0
differentiation of the second equation (z-projection) in respect to z gives
 p
2
0
 p   Az  B
A and B are the constants of
z
integration
Use of the boundary conditions at the left and right end of the pipe
gives
2
p 0   p 1 
  p  p 1  Az ;
p L   p 2 
A 
p1  p 2
L
0
Velocity profile
z-projection of the Navier-Stokes equation:
1   v z 
A 
r
0
r r  r 
 vz  
Ar
2
 4
1. Velocity is limited, v z  
2. At the wall, v z r  R   0
Finally,
vz 
A
R
4
2
r
2

parabolic velocity
profile:
Volumetric flow flux:
R
Q   v z d S   v z r 2  r d r 
S
0
A
8
R
4
 c 1 ln r  c 2
Jean Louis Marie Poiseuille \pwä-'zəi\
(22 April 1797 - 26 December 1869) was a
French physician and physiologist
Blood circulation
What happens to the blood flow
as blood viscosity changes
(increased cholesterol level)?
What happens to the flow as the
capillary radius changes
(artery blockage due to
cholesterol deposits)?