Similarity Transformations in Fluid Mechanics
Similarity Transformations in Fluid Mechanics
Similarity Transformations in Fluid Mechanics
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<strong>Fluid</strong> <strong>Mechanics</strong> General<br />
Perspective and Application<br />
Department of Mathematics<br />
COMSATS Institute of Information Technology<br />
Islamabad Pakistan
• Archimedes(287-217 B.C.) observed float<strong>in</strong>g objects<br />
on water and reasoned out the pr<strong>in</strong>ciple of bouncy.<br />
• Da V<strong>in</strong>ci(1452-1519) built the first chamber canal lock.<br />
• Castelli(1577-1644) stated the cont<strong>in</strong>uity pr<strong>in</strong>ciple for<br />
the river flow.<br />
• Torricelli(1608-1647) perfected the barometer.<br />
• Pascal(1623-1662) discover the scalar nature of<br />
pressure<br />
• Newton(1642-1727) developed the resistance law.<br />
• Bernoulli(1700-1782)developed developed energy<br />
equation for <strong>in</strong>viscid fluids
Physical science deal<strong>in</strong>g with the action of fluids at rest (fluid<br />
statics) or <strong>in</strong> motion (fluid dynamics), and their <strong>in</strong>teraction with<br />
flow devices and applications <strong>in</strong> eng<strong>in</strong>eer<strong>in</strong>g.<br />
The subject branches out <strong>in</strong>to sub-discipl<strong>in</strong>es such as:<br />
Aeronautics/Astronautics: Aircraft and missile aerodynamics,<br />
control hydraulics, gas-bear<strong>in</strong>g gyros, propeller, turbojet and<br />
rocket, satellites and cool<strong>in</strong>g system.<br />
Civil eng<strong>in</strong>eer<strong>in</strong>g: Pipe and channel flows, surface and ground<br />
water hydrology, w<strong>in</strong>d and water structure loads, lake and<br />
harbor tides, coastl<strong>in</strong>e flows, sediment transport, river flood<strong>in</strong>g<br />
and meander<strong>in</strong>g and water and water-water treatment.<br />
Physics: Magneto hydrodynamics, fusion devices, cryogenics<br />
and superconductivity.<br />
Astrophysics: Star and galaxy formation, and evolution,<br />
<strong>in</strong>terstellar gas dynamics, solar w<strong>in</strong>d and comet tails.
Mathematics: Solution of differential equations,<br />
boundary conditions, nonl<strong>in</strong>ear differential equations,<br />
dynamic analogies and computational fluid dynamics.<br />
Mechanical/ Nuclear eng<strong>in</strong>eer<strong>in</strong>g: Pumps and<br />
compressor, impulse and reaction turb<strong>in</strong>es, bear<strong>in</strong>g<br />
lubrication, heat exchangers, process control, fluid<br />
controls, cool<strong>in</strong>g system, electrochemical devices,<br />
Two-Phase flows and heat<strong>in</strong>g ventilation and air<br />
condition<strong>in</strong>g.<br />
Chemical, Petroleum eng<strong>in</strong>eer<strong>in</strong>g: Material transport,<br />
filter<strong>in</strong>g, heat transfer, mix<strong>in</strong>g and multiphase flow.<br />
Biophysics: Blood flow, artificial organs, breath<strong>in</strong>g<br />
aids, heart-lung mach<strong>in</strong>es, and artificial hearts, cellular<br />
mass transport, heat transfer, locomotion.<br />
Geophysics: Meteorology, oceanography, upper<br />
atmosphere, space, planetary atmospheres,<br />
geomagnetism, cont<strong>in</strong>ental drift, mantel convection<br />
and Glacier flows.
• Aerodynamics: deals with the motion of air and other<br />
gases, and their <strong>in</strong>teractions with bodies <strong>in</strong> motion<br />
such as lift and drag.<br />
• Hydraulics: application of fluid mechanics to<br />
eng<strong>in</strong>eer<strong>in</strong>g devices <strong>in</strong>volv<strong>in</strong>g liquids such as flow<br />
through pipes, weir and dam design<br />
• Geophysical fluid dynamics: fluid phenomena<br />
associated with the dynamics of the atmosphere and<br />
the oceans such as hurricane and weather systems<br />
• Bio-fluid mechanics: fluid mechanics <strong>in</strong>volved <strong>in</strong><br />
biophysical processes such as blood flow <strong>in</strong> arteries,<br />
and many others<br />
• Astrophysical <strong>Fluid</strong> Dynamics :the fluid mechanics<br />
of the sun, stars and other astrophysical objects
The mass of a fluid <strong>in</strong> a control volume rema<strong>in</strong>s<br />
conserved. This fact leads to establish a relation<br />
between the fluid density and fluid velocity at any<br />
po<strong>in</strong>t. Mathematical form of this relation is called<br />
equation of cont<strong>in</strong>uity<br />
Dρ<br />
∂ρ<br />
+ ρdiv( V ) = 0 or + div( ρV<br />
) = 0<br />
Dt<br />
∂t<br />
ρ<br />
If the density ( ) is constant (<strong>in</strong>compressible flow), Eq.<br />
(1) reduce to the simple equation:<br />
(1)<br />
divV = 0 (2)
The well known Navier-Stokes equations<br />
(momentum Equation) for unsteady, <strong>in</strong>compressible<br />
viscous fluid <strong>in</strong> rectangular coord<strong>in</strong>ate system are<br />
given by<br />
∂ ∂ ∂ ∂ ⎛∂ ∂<br />
+ u + v = − + ν ⎜ +<br />
∂t ∂x ∂y ρ ∂x ⎝∂x ∂y<br />
2 2<br />
u u u 1 p u u<br />
2 2<br />
∂ ∂ ∂ ∂ ⎛∂ ∂<br />
+ u + v =− + ν ⎜ +<br />
∂t ∂x ∂y ρ ∂x ⎝∂x ∂y<br />
2 2<br />
v v v 1 p v v<br />
D<br />
Dt<br />
2 2<br />
where is the material time derivative, ρ is the<br />
density, p is pressure, µ is the viscosity of fluid, g is<br />
acceleration due to gravity and v is the velocity vector.<br />
⎞<br />
⎟<br />
⎠<br />
⎞<br />
⎟<br />
⎠<br />
(3)<br />
(4)
( uvw , , )andT<br />
If are velocity components and the<br />
temperature of the fluid respectively, then the energy<br />
equation is given by:<br />
Φ<br />
DT<br />
Dt<br />
2<br />
= α∇ + Φ<br />
where is the dissipation function and is thermal<br />
diffusivity of the fluid.<br />
Φ =<br />
∂ ∂<br />
τ μ δ λ λ μ<br />
u u<br />
' = (<br />
i<br />
+<br />
i) + div v where 3 +2 =0<br />
ij<br />
ij<br />
∂x<br />
∂x<br />
(7)<br />
j<br />
j<br />
τ<br />
T<br />
'<br />
ij<br />
∂<br />
∂<br />
u<br />
x<br />
α<br />
i<br />
j<br />
(5)<br />
(6)
• Model<strong>in</strong>g of pulsat<strong>in</strong>g diaphragms<br />
• Sweet cool<strong>in</strong>g or heat<strong>in</strong>g<br />
• Isotope separation<br />
• Filtration<br />
• Paper manufactur<strong>in</strong>g<br />
• Irrigation<br />
• Gra<strong>in</strong> regression dur<strong>in</strong>g solid propellant<br />
combustion
Diagram of a simple<br />
filtration<br />
Filtration process <strong>in</strong><br />
kidneys
Cross-flow Microfiltration (MF) is a low pressure process for<br />
separation of larger size solutes from aqueous solutions us<strong>in</strong>g a<br />
semi-permeable membrane. This process is carried out by hav<strong>in</strong>g a<br />
process solution flow along a membrane surface under pressure.<br />
Particulate matter circulates through the membrane tube, clean<strong>in</strong>g the<br />
membrane tube surface while filtrate flows through the membrane
The channel has a width <strong>in</strong> the y-direction of h, a length<br />
<strong>in</strong> the z-direction of l, and a length <strong>in</strong> the x-direction,<br />
the direction of flow. There is a pressure drop along<br />
the length of the channel, so that the pressure<br />
gradient is constant<br />
From the cont<strong>in</strong>uity equation we have<br />
u=<br />
u( y)<br />
And the momentum equation takes the follow<strong>in</strong>g form<br />
2<br />
1 dp ⎛ d u ⎞<br />
0 = − + ν ⎜ ,<br />
2 ⎟<br />
ρ dx ⎝ dy ⎠<br />
u = at y = and y = h<br />
The boundary conditions are: 0 0<br />
(9)
y-axis<br />
a<br />
Porous wall<br />
y=a<br />
Geometry<br />
x-axis<br />
Porous wall<br />
y=-a<br />
A channel of rectangular cross section, one side of the<br />
cross section, represent<strong>in</strong>g the distance between the<br />
porous walls, is taken to be much smaller than the other.<br />
Both channel walls are taken to have equal permeability.<br />
Furthermore he considered steady state,<br />
<strong>in</strong>compressible, lam<strong>in</strong>ar, no external forces on the fluid<br />
and the suction/<strong>in</strong>jection velocity is <strong>in</strong>dependent of<br />
position.
Under the above assumptions the cont<strong>in</strong>uity and<br />
momentum equations reduce to the follow<strong>in</strong>g form<br />
∂u<br />
∂x<br />
∂v<br />
+ =<br />
∂y<br />
0,<br />
∂ ∂ ∂ ⎛∂ ∂<br />
u<br />
∂x ∂y ρ ∂x ⎝∂x ∂y<br />
2 2<br />
u u 1 p u u<br />
+ v =− + ν ⎜ +<br />
2 2<br />
∂ ∂ ∂ ⎛∂ ∂<br />
u<br />
∂x ∂y ρ ∂y ⎝∂x ∂y<br />
The appropriate boundary conditions are<br />
2 2<br />
v v 1 p v v<br />
+ v =− + ν ⎜ +<br />
2 2<br />
⎞<br />
⎟,<br />
⎠<br />
⎞<br />
⎟,<br />
⎠<br />
(10)<br />
(11)<br />
(12)<br />
uxy ( , ) = 0, vxy ( , ) = vw<br />
at y=± a,<br />
(13)<br />
∂ u = 0, v = 0 at y = 0,<br />
∂ y<br />
(14)<br />
u=0 at x=0. (15)
Let us consider the two-dimensional, unsteady,<br />
<strong>in</strong>compressible viscous fluid <strong>in</strong> an elongated<br />
rectangular channel bounded by two porous walls.<br />
The mass and momentum equations give<br />
∂u<br />
∂x<br />
∂v<br />
+ =0,<br />
∂y<br />
∂ ∂ ∂ ∂ ⎛∂ ∂<br />
⎜<br />
∂t ∂x ∂y ρ ∂x ⎝∂x ∂y<br />
2 2<br />
u u u 1 p u u<br />
+ u + v =− + ν +<br />
2 2<br />
2 2<br />
∂v ∂v ∂v 1 ∂p ⎛∂ v ∂ v⎞<br />
+ u + v =− + ν ⎜ + ,<br />
2 2 ⎟<br />
∂t ∂x ∂y ρ ∂y ⎝∂x ∂y<br />
⎠<br />
The appropriate boundary conditions are<br />
⎞<br />
⎟,<br />
⎠<br />
(16)<br />
(17)<br />
(18)<br />
u=0, v=-v<br />
w<br />
at y=a(t), (19)
∂ u = 0, v = 0 at y = 0,<br />
∂y<br />
u<br />
= 0 at x = 0. (20)<br />
y-axis<br />
Porous wall<br />
a(t)<br />
X-axis<br />
Porous wall<br />
y=a(t)<br />
y=a(t)<br />
Geometry of the bulk fluid motion
The govern<strong>in</strong>g equations are<br />
∂u<br />
∂x<br />
∂v<br />
+ =0,<br />
∂y<br />
∂ ∂ ∂ ∂ ⎛∂ ∂<br />
∂ ∂ ∂ ∂ ⎝∂ ∂<br />
2 2<br />
u u u 1 p u u νεu<br />
+ u + v =− + ν ⎜ +<br />
2 2⎟−<br />
t x y ρ x x y k<br />
∂ ∂ ∂ ∂ ⎛∂ ∂<br />
∂ ∂ ∂ ∂ ⎝∂ ∂<br />
2 2<br />
v v v 1 p v v νεv<br />
+ u + v =− + ν ⎜ +<br />
2 2 ⎟−<br />
t x y ρ y x y k<br />
Where ɛ is the porosity and k is the permeability.<br />
The appropriate boundary conditions are<br />
u=0,<br />
v=-v at y=a(t),<br />
w<br />
∂ u = 0, v = 0 at y = 0,<br />
∂y<br />
⎞<br />
⎠<br />
⎞<br />
⎠<br />
,<br />
,<br />
(21)<br />
(22)<br />
(23)<br />
(24)
References<br />
1. Exact solutions us<strong>in</strong>g symmetry methods and conservation laws for the<br />
viscous flow through expand<strong>in</strong>g–contract<strong>in</strong>g channels.<br />
S. Asghar, M. Mushtaq, A.H. Kara<br />
Applied Mathematical Modell<strong>in</strong>g,Volume 32, Issue 12, 2008.<br />
2. Application of Homotopy perturbation method to deformable channel<br />
with wall suction and <strong>in</strong>jection <strong>in</strong> a porous medium.<br />
M. Mahmood, M. A. Hussa<strong>in</strong>, S. Asgar, T. Hayat<br />
International Journal of Nonl<strong>in</strong>ear Sciences and Numerical Simulation.<br />
3. Application of Homotopy perturbation method to deformable channel with<br />
wall suction and <strong>in</strong>jection <strong>in</strong> a porous medium.<br />
M. Mahmood, M. A. Hussa<strong>in</strong>, S. Asgar, T. Hayat<br />
International Journal of Nonl<strong>in</strong>ear Sciences and Numerical Simulation.
• The flow of ur<strong>in</strong>e from the kidneys <strong>in</strong>to the<br />
bladder through tubular organs<br />
• Bile from the gallbladder <strong>in</strong>to the duodenum<br />
• Peristalsis pushes <strong>in</strong>gested food through the<br />
digestive tract towards its release at the<br />
anus<br />
• Worms propel themselves through peristaltic<br />
movement<br />
• Spermatic flow is also due to the peristalsis<br />
motion<br />
• Peristaltic pump
Bolus move <strong>in</strong>to the<br />
esophagus<br />
360 Degree Peristaltic Pump
The equations that govern the flow are the<br />
cont<strong>in</strong>uity equation and Navier Stokes equations<br />
The wall motion is described by:<br />
2<br />
hxt ( , ) = a+ bs<strong>in</strong> π ( x−ct)<br />
λ<br />
(25)<br />
b<br />
λ<br />
Coord<strong>in</strong>ate system and the channel under consideration
The govern<strong>in</strong>g equations are the cont<strong>in</strong>uity equation<br />
and Navier Stokes equations<br />
The wall motion is described by:<br />
2<br />
hxt ( , ) = at ( ) + bs<strong>in</strong> π ( x−ct)<br />
λ<br />
where the distance between the walls is chang<strong>in</strong>g <strong>in</strong> (26)<br />
time<br />
b<br />
λ<br />
Coord<strong>in</strong>ate system and the channel under consideration
References<br />
1. Peristaltic flow <strong>in</strong> a deformable channel.<br />
D. N. Khan Marwat, S. Asghar<br />
2. Application of Homotopy perturbation method to deformable<br />
channel with wall suction and <strong>in</strong>jection <strong>in</strong> a porous medium.<br />
M. Mahmood, M. A. Hussa<strong>in</strong>, S. Asgar, T. Hayat<br />
International Journal of Nonl<strong>in</strong>ear Sciences and Numerical<br />
Simulation.<br />
3. Application of Homotopy perturbation method to deformable<br />
channel with wall suction and <strong>in</strong>jection <strong>in</strong> a porous medium.<br />
M. Mahmood, M. A. Hussa<strong>in</strong>, S. Asgar, T. Hayat<br />
International Journal of Nonl<strong>in</strong>ear Sciences and Numerical<br />
Simulation.
• A cont<strong>in</strong>uously mov<strong>in</strong>g surface through a<br />
quiescent medium:<br />
• Hot roll<strong>in</strong>g, wire draw<strong>in</strong>g, sp<strong>in</strong>n<strong>in</strong>g of laments,<br />
metal extrusion, crystal grow<strong>in</strong>g, cont<strong>in</strong>uous<br />
cast<strong>in</strong>g, glass fiber production, and paper<br />
production.<br />
• The flow over a cont<strong>in</strong>uous material mov<strong>in</strong>g<br />
through a quiescent fluid is <strong>in</strong>duced by the<br />
movement of the solid material and by thermal<br />
buoyancy.<br />
• Cool<strong>in</strong>g of electronic devices
Cool<strong>in</strong>g of electronic devices<br />
CPU heat s<strong>in</strong>k with fan<br />
attached<br />
Radial isotherm and swirl<strong>in</strong>g<br />
forced convection flow<br />
trajectories
Mixed convection flow along a vertical<br />
stretch<strong>in</strong>g plate with variable plate<br />
temperature<br />
x<br />
Mixed convection flow<br />
along a heated<br />
cont<strong>in</strong>uously mov<strong>in</strong>g<br />
surface subject to nonuniform<br />
surface<br />
temperature assum<strong>in</strong>g<br />
that the surface velocity<br />
is u 0 x and the wall<br />
temperature is T 0 x 2 .<br />
u=U(x)<br />
v =0<br />
T w<br />
O<br />
u<br />
g<br />
v<br />
δ T<br />
T ∞<br />
δ<br />
u=0<br />
y
Boundary-layer equations<br />
Mak<strong>in</strong>g the usual boundary-layer approximations<br />
∂<br />
∂<br />
u<br />
x<br />
∂v<br />
+ =<br />
∂y<br />
0<br />
2<br />
u u u<br />
∂ ∂ ∂<br />
u + v = ν + g β ( T − T )<br />
2<br />
∞<br />
∂x ∂y ∂ y<br />
u<br />
2 2 3<br />
⎡ ∂ ⎛ ∂ u ⎞ ∂u ∂ v ∂ u<br />
+ K ⎢ ⎜ u<br />
2 ⎟ + + v<br />
2 3<br />
⎣ ∂x<br />
⎝ ∂y ⎠ ∂y<br />
∂y ∂y<br />
2<br />
T T T<br />
∂ ∂ ∂<br />
+ v = α<br />
∂x ∂y ∂ y<br />
2<br />
⎤<br />
⎥<br />
⎦<br />
(27)<br />
(28)<br />
(29)
Boundary conditions<br />
We have assumed that the flow is caused by the<br />
stretch<strong>in</strong>g of the wall and the buoyancy effect<br />
due to variable surface temperature<br />
u= U(), x v = 0, T= T +Δ T() x at y=<br />
0<br />
u→ 0, T= T as y→∞<br />
∞<br />
∞<br />
where T ∞ is the temperature of the ambient fluid.<br />
Here we consider the follow<strong>in</strong>g form of the<br />
surface temperature and the stretch<strong>in</strong>g velocity<br />
of the surface<br />
2<br />
0 0<br />
(30)<br />
Δ Tx () = Tx, Ux () = ux<br />
(31)
References<br />
1. Mixed convection flow of second grade fluid along a vertical stretch<strong>in</strong>g flat<br />
surface with variable surface temperature.<br />
M. Mushtaq, S. Asghar and M. A. Hossa<strong>in</strong><br />
Heat and Mass Transfer, Volume 43, Number 10 / August, 2007.<br />
2. Squeezed flow and heat transfer over a porous surface for viscous fluid.<br />
M. Mahmood, M.A. Hussa<strong>in</strong>, S.Asgar<br />
Heat and mass transfer .<br />
3. Hydro-magnetic squeezed flow of a viscous <strong>in</strong>compressible fluid past a<br />
wedge with permeable surface.<br />
M. Mahmood, M.A. Hussa<strong>in</strong>, S. Asgar,<br />
ZAMM.<br />
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