Fluid Mechanics General Perspective and Application

Department of MathematicsCOMSATS 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

ρ

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

(1)

If the density ( ) is constant (incompressible flow), Eq. (1) reduce to the simple equation:

d iv V = 0 (2)

( ) 0 ( ) 0D div V or div VDt tρ ρρ ρ∂+ = + =

∂

2 2

2 21u u u p u uu v

t x y x x yν

ρ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

+ + = − + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠2 2

2 21v v v p v vu v

t x y x x yν

ρ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

+ + = − + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ (4)

The well known Navier-Stokes equations (momentum Equation) for unsteady, incompressible viscous fluid in rectangular coordinate system are given by

(3)

DD twhere 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.

where is the dissipation function and is thermaldiffusivity of the fluid.

If are velocity components and the temperature of the fluid respectively, then the energy equation is given by:

2D T TD t

α= ∇ + Φ

( , , )andu v w T

α

(5)

Φ' ii j

j

ux

τ ∂Φ =

∂

' ( ) div where 3 +2 =0i iij ij

j j

u ux x

τ μ δ λ λ μ∂ ∂= + +

∂ ∂v

(6)

(7)

• 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 asemi-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

From the continuity equation we have( ) u u y=

And the momentum equation takes the following form2

2

10 ,d p d ud x d y

νρ

⎛ ⎞= − + ⎜ ⎟

⎝ ⎠

The boundary conditions are: 0 0u a t y a n d y h= = =

(9)

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

Geometry

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.

x-axis

y-axis a

Porous wall

Porous wall y=a

y=-a

Under the above assumptions the continuity and momentum equations reduce to the following form

0,u vx y∂ ∂

+ =∂ ∂

2 2

2 2

1 ,u u p u uu vx y x x y

νρ

⎛ ⎞∂ ∂ ∂ ∂ ∂+ =− + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

(10)

(11)

(12)2 2

2 2

1 ,v v p v vu vx y y x y

νρ

⎛ ⎞∂ ∂ ∂ ∂ ∂+ = − + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

The appropriate boundary conditions are( , ) 0, ( , ) at ,wu x y v x y v 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 v+ =0,x y∂ ∂∂ ∂

2 2

2 21 ,u u u p u uu v

t x y x x yν

ρ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

+ + =− + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

(16)

(17)

(18)2 2

2 21 ,v v v p v vu v

t x y y x yν

ρ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

+ + = − + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠The appropriate boundary conditions are

wu=0, v=-v at y=a(t), (19)

Geometry of the bulk fluid motion

0, 0 at 0,u v yy∂

= = =∂

0 at 0.u x= = (20)

X-axis

y-axis a(t)

Porous wall

Porous wall y=a(t)

y=a(t)

The governing equations are u v+ =0,x y∂ ∂∂ ∂

2 2

2 21 ,u u u p u u uu v

t x y x x y kνεν

ρ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

+ + = − + + −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

(21)

(22)

(23)2 2

2 21 ,v v v p v v vu v

t x y y x y kνεν

ρ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂

+ + = − + + −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

The appropriate boundary conditions arewu=0, v=-v at y=a(t),

(24)0, 0 at 0,u v y

y∂

= = =∂

Where ɛ is the porosity and k is the permeability.

1. Exact solutions using symmetry methods and conservation laws for the viscous flow through expanding–contracting channels.S. Asghar, M. Mushtaq, A.H. KaraApplied Mathematical Modelling,Volume 32, Issue 12, 2008.

2. Application of Homotopy perturbation method to deformable channelwith wall suction and injection in a porous medium.M. Mahmood, M. A. Hussain, S. Asgar, T. HayatInternational 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. HayatInternational Journal of Nonlinear Sciences and Numerical Simulation.

References

• 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

360 Degree Peristaltic PumpBolus move into the esophagus

2( , ) sin ( )h x t a b x ctπλ

= + −The wall motion is described by:

Coordinate system and the channel under consideration

The equations that govern the flow are the continuity equation and Navier Stokes equations

λ

b

(25)

2( , ) ( ) sin ( )h x t a t b x ctπλ

= + −The wall motion is described by:

The governing equations are the continuity equation and Navier Stokes equations

Coordinate system and the channel under consideration

λ

b

where the distance between the walls is changing in (26)

time

References1. 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. HayatInternational 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. HayatInternational 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

y

xg

v

u

δ

δT

u=0

u=U(x) v =0

T w

O

T∞

Mixed convection flowalong a heated continuously moving surface subject to non-uniform surface temperature assuming that the surface velocity is u0x and the wall temperature is T0x2.

Boundary-layer equationsMaking the usual boundary-layer approximations

2

2

2 2 3

2 2 3

2

2

0

( )

ux y

u u uu g T Tx y y

u u uK ux yy y y

T T Tux y y

ν β

α

∞

∂ ∂+ =

∂ ∂

∂ ∂ ∂+ = + −

∂ ∂ ∂

⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂ ∂+ + +⎢ ⎥⎜ ⎟∂ ∂∂ ∂ ∂⎝ ⎠⎣ ⎦

∂ ∂ ∂+ =

∂ ∂ ∂

v

v

v v

v

(27)

(29)

(28)

Boundary conditionsWe have assumed that the flow is caused by thestretching of the wall and the buoyancy effectdue to variable surface temperature

( ), 0, ( ) at 00, as

u U x T T T x yu T T y

∞

∞

= = = +Δ =→ = →∞

v

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

20 0( ) , ( )T x T x U x u xΔ = = (31)

(30)

References1. Mixed convection flow of second grade fluid along a vertical stretching flat

surface with variable surface temperature. M. Mushtaq, S. Asghar and M. A. HossainHeat 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.AsgarHeat 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