'Fluid'? - Rshanthini.com

stickshrivelMechanics

Oct 24, 2013 (3 years and 8 months ago)

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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