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CP502
Advanced Fluid Mechanics
Flow of Viscous Fluids
Set 01
What do we mean by ‘Fluid’?

Physically: liquids or gases

Mathematically:

A vector field u (represents the fluid velocity)

A scalar field p (represents the fluid pressure)

fluid density (d) and fluid viscosity (v)
R. Shanthini
15 Mar 2012
Recalling vector operations

Del Operator:

Laplacian Operator:

Gradient:

Vector Gradient:

Divergence:
R. Shanthini
Directional
15 Mar 2012
Derivative:
Continuity equation
for incompressible
(constant density) flow
- derived from conservation of mass
where u is the velocity vector
u, v, w are velocities in x, y, and z directions
R. Shanthini
15 Mar 2012
Continuity equation
derivation
 IF v I
F
y J
v  y J
G
G
H y KH y Kxz
Mass flux out of differential volume  
Rate of change of mass in 
 y x z
differential volume
t
y
x
R. Shanthini
15 Mar 2012
z
Mass flux into differential volume vxz
Continuity equation
derivation
Mass flux into differential volume
= Mass flux out of differential volume
+ Rate of change of mass in differential volume
vxz
R. Shanthini
15 Mar 2012
=
 IF v I
F
  y J
v  y J
G
G
H y KH y Kxz
+

 y x z
t
Continuity equation
in 1-dimension
v


 v  
y
y
t
v 

0
y t
known as one
dimensional Continuity
equation
R. Shanthini
15 Mar 2012
Continuity equation
in 3-dimension
af af a f
  u  v  w



0
t
x
y
z
where u, v, w are velocities in x, y, and z directions

   V  0
t
divergence
R. Shanthini
15 Mar 2012
Continuity equation
for
incompressible flow
Density is constant for incompressible flow:
u v w
 
0
x y z
or
V  0
R. Shanthini
15 Mar 2012
Navier-Stokes equation for incompressible
flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum
υ
kinematic
viscosity
(constant)
ρ
density
(constant)
pressure
external force
(such as
gravity)
R. Shanthini
15 Mar 2012
Navier-Stokes equation for incompressible
flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum
υ
υ
R. Shanthini
15 Mar 2012
ρ
ρ
Navier-Stokes equation for incompressible
flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum
υ
Acceleration term:
change of velocity
with time
R. Shanthini
15 Mar 2012
ρ
Navier-Stokes equation for incompressible
flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum
υ
Advection term:
force exerted on a
particle of fluid by the
other particles of fluid
surrounding it
R. Shanthini
15 Mar 2012
ρ
Navier-Stokes equation for incompressible
flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum
υ
ρ
viscosity (constant) controlled
velocity diffusion term:
(this term describes how fluid motion is
damped)

R. Shanthini
15 Mar 2012
Highly viscous fluids stick together (honey)
 Low-viscosity fluids flow freely (air)
Navier-Stokes equation for incompressible
flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum
υ
ρ
Pressure term:
Fluid flows in the
direction of
largest change
in pressure
R. Shanthini
15 Mar 2012
Navier-Stokes equation for incompressible
flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum
υ
ρ
Body force term:
external forces that
act on the fluid
(such as gravity,
electromagnetic,
etc.)
R. Shanthini
15 Mar 2012
Navier-Stokes equation for incompressible
flow of Newtonian (constant viscosity) fluid
- derived from conservation of momentum
υ
ρ
change
body
in
= advection + diffusion + pressure + force
velocity
with time
R. Shanthini
15 Mar 2012
Continuity and Navier-Stokes equations
for incompressible flow of Newtonian fluid
υ
R. Shanthini
15 Mar 2012
ρ
Continuity and Navier-Stokes equations
for incompressible flow of Newtonian fluid
in Cartesian coordinates
Continuity:
Navier-Stokes:
x - component:
y - component:
z - component:
R. Shanthini
15 Mar 2012
Steady, incompressible flow of Newtonian fluid in an
infinite channel with stationery plates
- fully developed plane Poiseuille flow
Fixed plate
Fluid flow direction
y
x
h
Fixed plate
Steady, incompressible flow of Newtonian fluid in an
infinite channel with one plate moving at uniform velocity
- fully developed plane Couette flow
Moving plate
Fluid flow direction
y
R. Shanthini
15 Mar 2012
x
Fixed plate
h
Continuity and Navier-Stokes equations
for incompressible flow of Newtonian fluid
in cylindrical coordinates
Continuity:
Navier-Stokes:
Radial component:
Tangential component:
Axial component:
R. Shanthini
15 Mar 2012
Steady, incompressible flow of Newtonian fluid in a pipe
- fully developed pipe Poisuille flow
φ
Fixed pipe
r
z
R. Shanthini
15 Mar 2012
Fluid flow direction
2a
2a
Steady, incompressible flow of Newtonian fluid between
a stationary outer cylinder and a rotating inner cylinder
- fully developed pipe Couette flow
a
r
b
aΩ
R. Shanthini
15 Mar 2012