CP502
Advanced Fluid Mechanics
Flow of Viscous Fluids
and Boundary Layer Flow
[ 10 Lectures + 3 Tutorials ]
Computational Fluid dynamics (CFD) project
Midsemester (open book) examination
R. Shanthini
18 Aug 2010
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
18 Aug 2010
Recalling vector operations
Del Operator:
Laplacian Operator:
Gradient:
Vector Gradient:
Divergence:
Directional Derivative:
R. Shanthini
18 Aug 2010
Continuity equation
for incompressible
(constant density) flow
where
u
is the velocity vector
u, v, w
are velocities in
x, y,
and
z
directions

derived from conservation of mass
R. Shanthini
18 Aug 2010
ρ
υ
Navier

Stokes equation
for incompressible
flow of Newtonian (constant viscosity) fluid

derived from conservation of momentum
kinematic
viscosity
(constant)
density
(constant)
pressure
external force
(such as
gravity)
R. Shanthini
18 Aug 2010
Navier

Stokes equation
for incompressible
flow of Newtonian (constant viscosity) fluid

derived from conservation of momentum
ρ
υ
ρ
υ
R. Shanthini
18 Aug 2010
Navier

Stokes equation
for incompressible
flow of Newtonian (constant viscosity) fluid

derived from conservation of momentum
ρ
υ
Acceleration term:
change of velocity
with time
R. Shanthini
18 Aug 2010
Navier

Stokes equation
for incompressible
flow of Newtonian (constant viscosity) fluid

derived from conservation of momentum
ρ
υ
Advection term:
force exerted on a
particle of fluid by the
other particles of fluid
surrounding it
R. Shanthini
18 Aug 2010
Navier

Stokes equation
for incompressible
flow of Newtonian (constant viscosity) fluid

derived from conservation of momentum
ρ
υ
viscosity (constant) controlled
velocity diffusion term:
(this term describes how fluid motion is
damped)
Highly viscous fluids stick together (honey)
Low

viscosity fluids flow freely (air)
R. Shanthini
18 Aug 2010
Navier

Stokes equation
for incompressible
flow of Newtonian (constant viscosity) fluid

derived from conservation of momentum
ρ
υ
Pressure term:
Fluid flows in the
direction of
largest change
in pressure
R. Shanthini
18 Aug 2010
Navier

Stokes equation
for incompressible
flow of Newtonian (constant viscosity) fluid

derived from conservation of momentum
ρ
υ
Body force term:
external forces that
act on the fluid
(such as gravity,
electromagnetic,
etc.)
R. Shanthini
18 Aug 2010
Navier

Stokes equation
for incompressible
flow of Newtonian (constant viscosity) fluid

derived from conservation of momentum
ρ
υ
change
in
velocity
with time
advection
diffusion
pressure
body
force
=
+
+
+
R. Shanthini
18 Aug 2010
Continuity and Navier

Stokes equations
for incompressible flow of Newtonian fluid
ρ
υ
R. Shanthini
18 Aug 2010
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
18 Aug 2010
Steady, incompressible flow of Newtonian fluid in an
infinite channel with stationery plates

fully developed plane Poiseuille flow
Fixed plate
Fixed plate
Fluid flow direction
h
x
y
Steady, incompressible flow of Newtonian fluid in an
infinite channel with one plate moving at uniform velocity

fully developed plane Couette flow
Fixed plate
Moving plate
h
x
y
Fluid flow direction
R. Shanthini
18 Aug 2010
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
18 Aug 2010
Steady, incompressible flow of Newtonian fluid in a pipe

fully developed pipe Poisuille flow
Fixed pipe
z
r
Fluid flow direction
2a
2a
φ
R. Shanthini
18 Aug 2010
Steady, incompressible flow of Newtonian fluid between
a stationary outer cylinder and a rotating inner cylinder

fully developed pipe Couette flow
a
Ω
a
b
r
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