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

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

2 Archimedes( B.C.) observed floating objects on water and reasoned out the principle of bouncy. Da Vinci( ) built the first chamber canal lock. Castelli( ) stated the continuity principle for the river flow. Torricelli( ) perfected the barometer. Pascal( ) discover the scalar nature of pressure Newton( ) developed the resistance law. Bernoulli( )developed developed energy equation for inviscid fluids

3 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.

4 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.

5 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

7 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)

8 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 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)

9 ( 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)

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

11 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.

12 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

13 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)

14 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.

15 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 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)

16 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 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)

17 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

18 The governing equations are u x v + =0, y 2 2 u u u 1 p u u νεu + u + v = + ν t x y ρ x x y k 2 2 v v v 1 p v v νεv + u + v = + ν 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)

19 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, 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.

20 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

21 Bolus move into the esophagus 360 Degree Peristaltic Pump

22 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

23 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

24 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.

25 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

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

27 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 δ T v δ g y T u=0

28 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 u u v u + K u v 2 3 x y y y y 2 T T T + v = α x y y 2 (27) (28) (29)

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 (30) Δ Tx () = Tx, Ux () = ux (31)

30 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, 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.

Contents. I Introduction 1. Preface. xiii

Contents. I Introduction 1. Preface. xiii Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................