Similarity Transformations in Fluid Mechanics

Fluid Mechanics General
Perspective and Application
Department of Mathematics
COMSATS Institute of Information Technology
Islamabad Pakistan

• Archimedes(287-217 B.C.) observed floating objects
on water and reasoned out the principle of bouncy.
• Da Vinci(1452-1519) built the first chamber canal lock.
• Castelli(1577-1644) stated the continuity principle for
the river flow.
• Torricelli(1608-1647) perfected the barometer.
• Pascal(1623-1662) discover the scalar nature of
pressure
• Newton(1642-1727) developed the resistance law.
• Bernoulli(1700-1782)developed developed energy
equation for inviscid fluids

Physical science dealing with the action of fluids at rest (fluid
statics) or in motion (fluid dynamics), and their interaction with
flow devices and applications in engineering.
The subject branches out into sub-disciplines such as:
Aeronautics/Astronautics: Aircraft and missile aerodynamics,
control hydraulics, gas-bearing gyros, propeller, turbojet and
rocket, satellites and cooling system.
Civil engineering: Pipe and channel flows, surface and ground
water hydrology, wind and water structure loads, lake and
harbor tides, coastline flows, sediment transport, river flooding
and meandering and water and water-water treatment.
Physics: Magneto hydrodynamics, fusion devices, cryogenics
and superconductivity.
Astrophysics: Star and galaxy formation, and evolution,
interstellar gas dynamics, solar wind and comet tails.

Mathematics: Solution of differential equations,
boundary conditions, nonlinear differential equations,
dynamic analogies and computational fluid dynamics.
Mechanical/ Nuclear engineering: Pumps and
compressor, impulse and reaction turbines, bearing
lubrication, heat exchangers, process control, fluid
controls, cooling system, electrochemical devices,
Two-Phase flows and heating ventilation and air
conditioning.
Chemical, Petroleum engineering: Material transport,
filtering, heat transfer, mixing and multiphase flow.
Biophysics: Blood flow, artificial organs, breathing
aids, heart-lung machines, and artificial hearts, cellular
mass transport, heat transfer, locomotion.
Geophysics: Meteorology, oceanography, upper
atmosphere, space, planetary atmospheres,
geomagnetism, continental drift, mantel convection
and Glacier flows.

• Aerodynamics: deals with the motion of air and other
gases, and their interactions with bodies in motion
such as lift and drag.
• Hydraulics: application of fluid mechanics to
engineering devices involving liquids such as flow
through pipes, weir and dam design
• Geophysical fluid dynamics: fluid phenomena
associated with the dynamics of the atmosphere and
the oceans such as hurricane and weather systems
• Bio-fluid mechanics: fluid mechanics involved in
biophysical processes such as blood flow in arteries,
and many others
• Astrophysical Fluid Dynamics :the fluid mechanics
of the sun, stars and other astrophysical objects

The mass of a fluid in a control volume remains
conserved. This fact leads to establish a relation
between the fluid density and fluid velocity at any
point. Mathematical form of this relation is called
equation of continuity
Dρ
∂ρ
+ ρdiv( V ) = 0 or + div( ρV
) = 0
Dt
∂t
ρ
If the density ( ) is constant (incompressible flow), Eq.
(1) reduce to the simple equation:
(1)
divV = 0 (2)

The well known Navier-Stokes equations
(momentum Equation) for unsteady, incompressible
viscous fluid in rectangular coordinate system are
given by
∂ ∂ ∂ ∂ ⎛∂ ∂
+ u + v = − + ν ⎜ +
∂t ∂x ∂y ρ ∂x ⎝∂x ∂y
2 2
u u u 1 p u u
2 2
∂ ∂ ∂ ∂ ⎛∂ ∂
+ u + v =− + ν ⎜ +
∂t ∂x ∂y ρ ∂x ⎝∂x ∂y
2 2
v v v 1 p v v
D
Dt
2 2
where is the material time derivative, ρ is the
density, p is pressure, µ is the viscosity of fluid, g is
acceleration due to gravity and v is the velocity vector.
⎞
⎟
⎠
⎞
⎟
⎠
(3)
(4)

( uvw , , )andT
If are velocity components and the
temperature of the fluid respectively, then the energy
equation is given by:
Φ
DT
Dt
2
= α∇ + Φ
where is the dissipation function and is thermal
diffusivity of the fluid.
Φ =
∂ ∂
τ μ δ λ λ μ
u u
' = (
i
+
i) + div v where 3 +2 =0
ij
ij
∂x
∂x
(7)
j
j
τ
T
'
ij
∂
∂
u
x
α
i
j
(5)
(6)

• Modeling of pulsating diaphragms
• Sweet cooling or heating
• Isotope separation
• Filtration
• Paper manufacturing
• Irrigation
• Grain regression during solid propellant
combustion

Diagram of a simple
filtration
Filtration process in
kidneys

Cross-flow Microfiltration (MF) is a low pressure process for
separation of larger size solutes from aqueous solutions using a
semi-permeable membrane. This process is carried out by having a
process solution flow along a membrane surface under pressure.
Particulate matter circulates through the membrane tube, cleaning the
membrane tube surface while filtrate flows through the membrane

The channel has a width in the y-direction of h, a length
in the z-direction of l, and a length in the x-direction,
the direction of flow. There is a pressure drop along
the length of the channel, so that the pressure
gradient is constant
From the continuity equation we have
u=
u( y)
And the momentum equation takes the following form
2
1 dp ⎛ d u ⎞
0 = − + ν ⎜ ,
2 ⎟
ρ dx ⎝ dy ⎠
u = at y = and y = h
The boundary conditions are: 0 0
(9)

y-axis
a
Porous wall
y=a
Geometry
x-axis
Porous wall
y=-a
A channel of rectangular cross section, one side of the
cross section, representing the distance between the
porous walls, is taken to be much smaller than the other.
Both channel walls are taken to have equal permeability.
Furthermore he considered steady state,
incompressible, laminar, no external forces on the fluid
and the suction/injection velocity is independent of
position.

Under the above assumptions the continuity and
momentum equations reduce to the following form
∂u
∂x
∂v
+ =
∂y
0,
∂ ∂ ∂ ⎛∂ ∂
u
∂x ∂y ρ ∂x ⎝∂x ∂y
2 2
u u 1 p u u
+ v =− + ν ⎜ +
2 2
∂ ∂ ∂ ⎛∂ ∂
u
∂x ∂y ρ ∂y ⎝∂x ∂y
The appropriate boundary conditions are
2 2
v v 1 p v v
+ v =− + ν ⎜ +
2 2
⎞
⎟,
⎠
⎞
⎟,
⎠
(10)
(11)
(12)
uxy ( , ) = 0, vxy ( , ) = vw
at y=± a,
(13)
∂ u = 0, v = 0 at y = 0,
∂ y
(14)
u=0 at x=0. (15)

Let us consider the two-dimensional, unsteady,
incompressible viscous fluid in an elongated
rectangular channel bounded by two porous walls.
The mass and momentum equations give
∂u
∂x
∂v
+ =0,
∂y
∂ ∂ ∂ ∂ ⎛∂ ∂
⎜
∂t ∂x ∂y ρ ∂x ⎝∂x ∂y
2 2
u u u 1 p u u
+ u + v =− + ν +
2 2
2 2
∂v ∂v ∂v 1 ∂p ⎛∂ v ∂ v⎞
+ u + v =− + ν ⎜ + ,
2 2 ⎟
∂t ∂x ∂y ρ ∂y ⎝∂x ∂y
⎠
The appropriate boundary conditions are
⎞
⎟,
⎠
(16)
(17)
(18)
u=0, v=-v
w
at y=a(t), (19)

∂ u = 0, v = 0 at y = 0,
∂y
u
= 0 at x = 0. (20)
y-axis
Porous wall
a(t)
X-axis
Porous wall
y=a(t)
y=a(t)
Geometry of the bulk fluid motion

The governing equations are
∂u
∂x
∂v
+ =0,
∂y
∂ ∂ ∂ ∂ ⎛∂ ∂
∂ ∂ ∂ ∂ ⎝∂ ∂
2 2
u u u 1 p u u νεu
+ u + v =− + ν ⎜ +
2 2⎟−
t x y ρ x x y k
∂ ∂ ∂ ∂ ⎛∂ ∂
∂ ∂ ∂ ∂ ⎝∂ ∂
2 2
v v v 1 p v v νεv
+ u + v =− + ν ⎜ +
2 2 ⎟−
t x y ρ y x y k
Where ɛ is the porosity and k is the permeability.
The appropriate boundary conditions are
u=0,
v=-v at y=a(t),
w
∂ u = 0, v = 0 at y = 0,
∂y
⎞
⎠
⎞
⎠
,
,
(21)
(22)
(23)
(24)

References
1. Exact solutions using symmetry methods and conservation laws for the
viscous flow through expanding–contracting channels.
S. Asghar, M. Mushtaq, A.H. Kara
Applied Mathematical Modelling,Volume 32, Issue 12, 2008.
2. Application of Homotopy perturbation method to deformable channel
with wall suction and injection in a porous medium.
M. Mahmood, M. A. Hussain, S. Asgar, T. Hayat
International Journal of Nonlinear Sciences and Numerical Simulation.
3. Application of Homotopy perturbation method to deformable channel with
wall suction and injection in a porous medium.
M. Mahmood, M. A. Hussain, S. Asgar, T. Hayat
International Journal of Nonlinear Sciences and Numerical Simulation.

• The flow of urine from the kidneys into the
bladder through tubular organs
• Bile from the gallbladder into the duodenum
• Peristalsis pushes ingested food through the
digestive tract towards its release at the
anus
• Worms propel themselves through peristaltic
movement
• Spermatic flow is also due to the peristalsis
motion
• Peristaltic pump

Bolus move into the
esophagus
360 Degree Peristaltic Pump

The equations that govern the flow are the
continuity equation and Navier Stokes equations
The wall motion is described by:
2
hxt ( , ) = a+ bsin π ( x−ct)
λ
(25)
b
λ
Coordinate system and the channel under consideration

The governing equations are the continuity equation
and Navier Stokes equations
The wall motion is described by:
2
hxt ( , ) = at ( ) + bsin π ( x−ct)
λ
where the distance between the walls is changing in (26)
time
b
λ
Coordinate system and the channel under consideration

References
1. Peristaltic flow in a deformable channel.
D. N. Khan Marwat, S. Asghar
2. Application of Homotopy perturbation method to deformable
channel with wall suction and injection in a porous medium.
M. Mahmood, M. A. Hussain, S. Asgar, T. Hayat
International Journal of Nonlinear Sciences and Numerical
Simulation.
3. Application of Homotopy perturbation method to deformable
channel with wall suction and injection in a porous medium.
M. Mahmood, M. A. Hussain, S. Asgar, T. Hayat
International Journal of Nonlinear Sciences and Numerical
Simulation.

• A continuously moving surface through a
quiescent medium:
• Hot rolling, wire drawing, spinning of laments,
metal extrusion, crystal growing, continuous
casting, glass fiber production, and paper
production.
• The flow over a continuous material moving
through a quiescent fluid is induced by the
movement of the solid material and by thermal
buoyancy.
• Cooling of electronic devices

Cooling of electronic devices
CPU heat sink with fan
attached
Radial isotherm and swirling
forced convection flow
trajectories

Mixed convection flow along a vertical
stretching plate with variable plate
temperature
x
Mixed convection flow
along a heated
continuously moving
surface subject to nonuniform
surface
temperature assuming
that the surface velocity
is u 0 x and the wall
temperature is T 0 x 2 .
u=U(x)
v =0
T w
O
u
g
v
δ T
T ∞
δ
u=0
y

Boundary-layer equations
Making the usual boundary-layer approximations
∂
∂
u
x
∂v
+ =
∂y
0
2
u u u
∂ ∂ ∂
u + v = ν + g β ( T − T )
2
∞
∂x ∂y ∂ y
u
2 2 3
⎡ ∂ ⎛ ∂ u ⎞ ∂u ∂ v ∂ u
+ K ⎢ ⎜ u
2 ⎟ + + v
2 3
⎣ ∂x
⎝ ∂y ⎠ ∂y
∂y ∂y
2
T T T
∂ ∂ ∂
+ v = α
∂x ∂y ∂ y
2
⎤
⎥
⎦
(27)
(28)
(29)

Boundary conditions
We have assumed that the flow is caused by the
stretching of the wall and the buoyancy effect
due to variable surface temperature
u= U(), x v = 0, T= T +Δ T() x at y=
0
u→ 0, T= T as y→∞
∞
∞
where T ∞ is the temperature of the ambient fluid.
Here we consider the following form of the
surface temperature and the stretching velocity
of the surface
2
0 0
(30)
Δ Tx () = Tx, Ux () = ux
(31)

References
1. Mixed convection flow of second grade fluid along a vertical stretching flat
surface with variable surface temperature.
M. Mushtaq, S. Asghar and M. A. Hossain
Heat and Mass Transfer, Volume 43, Number 10 / August, 2007.
2. Squeezed flow and heat transfer over a porous surface for viscous fluid.
M. Mahmood, M.A. Hussain, S.Asgar
Heat and mass transfer .
3. Hydro-magnetic squeezed flow of a viscous incompressible fluid past a
wedge with permeable surface.
M. Mahmood, M.A. Hussain, S. Asgar,
ZAMM.
4. www.google.com.pk/images