Chapter III

Consequence OF MHD FLOWS DUE TO NON-COAXIAL ROTATIONS OF A POROUS DISK AND A FLUID AT INFINITY

3.1 Introduction

Revolving disc flows of electrically carry oning fluids have practical applications in many countries, such as revolving machinery lubrication, oceanology, computing machine storage devices, viscosimetry and crystal growing procedure etc. Disk shaped organic structures are frequently encountered in many technology applications. It has ever been interesting to transport out the flows which are revolving. Examples of such flows are upwind forms, atmospheric foreparts and ocean currents. The geophysical jobs are strongly influenced by diurnal rotary motion of the Earth, which is manifested in the equations of gesture as the Carioles force. The flow due to an infinite rotating disc is one of the classical jobs which were foremost introduced by von Karman [ 1921 ] . The flow of a conducting fluid above a revolving disc in the presence of an external unvarying magnetic field was studied by Mistikawy and Attia [ 1990, 1991 ] .

The consequence of unvarying suction or injection through a revolving porous disc on the steady hydrodynamic or hydromagnetic flow induced by the disc was investigated by Stuart [ 1954 ] , Kuiken [ 1971 ] , Attia [ 1998, 2001, and 2000 ] . Since the pioneering work of Von Karman [ 1921 ] , the flow due to a revolving disc has attracted increasing attending of research workers. In his work, he considered the instance of an infinite disc and gave a preparation of the hydrokineticss job. Next he introduced his celebrated transmutations which reduced the regulating partial derived function to ordinary differential equations. Cochran [ 1934 ] used the von Karman transmutations and obtained asymptotic solutions for the steady hydrodynamic job. Benton [ 1966 ] improved Cochran’s solutions and extended the job to a flow which starts impetuously from remainder. The ability of electromagnetic Fieldss to act upon fluid flow has long been known and used with changing grades of success.

In peculiar, the MHD fluid flow job of a revolving disc discoveries particular topographic points in several scientific discipline and technology applications, for case, in turbo machinery, in cosmical fluid kineticss, in gaseous and atomic reactors, in MHD power generators, flow metres, pumps and so on.

The incompressible flow between bizarre revolving discs has been studied by a Numberss of research workers. The flow between two discs revolving with same angular speed has considered by Beker [ 1963 ] . Three dimensional flows between analogue home bases which are revolving about a common axis or about distinguishable axes has been studied by Lai et Al. [ 1984 ] . The inertia consequence of the non-Newtonian flow between bizarre discs revolving at different velocities has been studied by Knight [ 1980 ] . The flow of a 2nd order fluid between two analogue home bases has been considered by Rajagopal [ 1981 ] .

Hydromantic flow between bizarre revolving discs with the same angular speed has been studied by Mohanty [ 1971 ] . Rao and Kasiviswanathan [ 1981 ] have considered the flow of an incompressible syrupy fluid between two bizarre revolving discs. Erdogan [ 1995 ] has studied the unsteady syrupy flow between bizarre revolving discs. Unsteady flow due to homocentric rotary motion of bizarre revolving discs has been studied by Ersoy [ 2003 ] . Ersoy and Baris [ 2002b ] studied the flow induced by the rotary motion at the same angular speed in the instance of a porous disc for a 2nd class / order fluid. Pop [ 1979 ] was the first to see the unsteady flow produced by a disc and a fluid at eternity. He considered the job for a Newtonian fluid and studied the unsteady flow induced when the disc and the fluid at infinity start impetuously to revolve with the same angular speed about non-coincident axes. Hayat et Al. [ 2001 ] examined Erdogan’s work [ 1997 ] for a porous disc in the presence of a magnetic field.

An effort has been made out to analyze the Magneto hydro dynamic syrupy fluid flow features on a revolving porous disc Hayat et Al. [ 2003 ] . In this chapter, the work is extended for the flows which are due to non – coaxal rotary motions of a porous disc and a fluid at eternity. The influence of an externally applied magnetic field on the speed distribution is discussed. The general category of solutions of the clip dependent Navier-Stokes equations due to arbitrary periodic oscillations of the disc is discussed. Both the disc and the fluid are in a province of non-coaxial rotary motions. A general periodic oscillation H ( T ) with periodis considered. The response of oscillations in the flow field can be built up utilizing Fourier series representation and the Laplace transform. The exact analytical solution ( due to arbitrary periodic oscillation ) depicting the flow at big and little times after the start is obtained. Thus the MHD consequence and arbitrary nature of oscillation, the graphical representation of the flow features is the particular characteristic of this chapter.

3.3 Mathematical preparation

The flow of an incompressible electrically carry oning fluid is considered. The fluid is electrically carry oning in the presence of a magnetic field. The disc ( z = 0 ) is assumed to be a porous disc. The fluid fills the infinite z = 0 and is in contact with the disc. The axes of rotary motion of both the disc and the fluid are assumed to be in the plane x = 0. The distances between axes are being considered as l. Initially the disc and the fluid are revolving about omega¹– axis with changeless angular speed a„¦ . At clip T = 0, the disc and the unstable start rotate at omega and omega¹axes severally with changeless angular speed a„¦ . The disc besides oscillates its ain plane with frequence N, at clip T & gt ; 0.

Under the above premises, the equations regulating the unsteady gesture of the carry oning syrupy incompressible fluid are those refering to the preservation of impulse and of mass which are

( 3.1 )

( 3.2 )

The equations regulating the flow consists of the Maxwell equations and a generalized Ohm’s jurisprudence which after pretermiting the supplanting currents are

( 3.3 )

( 3.4 )

( 3.5 )

( 3.6 )

whereis the unstable speed with u, V, and tungsten as the speed constituents in the ten, Y and omega – waies severally,is the unstable denseness,is the scalar force per unit area,is the material derived function,is the kinematic viscousness,is the current denseness,is the entire magnetic field which is the amount of applied magnetic fieldand induced magnetic field B,is the magnetic permeableness and E is the electric field andis the electrical conduction of the fluid

For the derivation of Lorentz force in equation ( 3.1 ) , it is assumed that the magnetic field is normal to the speed field, the electric field is negligible and the induced magnetic field is little compared with the applied magnetic field. The last premise is valid when the magnetic Reynolds figure is really little and there is no displacement current.

In position of the above premises the electromagnetic organic structure force involved in equation ( 3.1 ) takes the signifier

=

( 3.7 )

The relevant boundary and initial conditions are taken in the signifier

,

at z = 0 for T & gt ; 0,

as omegafor all T,

at t = 0 for omega & gt ; 0, ( 3.8 )

whereis the speed and H ( T ) is the general periodic oscillation of a disc.

The Fourier series representation of H ( T ) is given by

( 3.9 )

where=( 3.10 )

whereis the non nothing hovering frequence.

The coefficients {are Fourier series coefficients or the spectral coefficients of H ( T ) . The boundary and initial conditions show that the gesture is a summing up of a coiling and translatory gesture with the speed profile being

( 3.11 )

Using equation ( 3.2 ) , the unvarying porous disc is of the signifier

( 3.12 )

where (& gt ; 0 is the suction speed and& lt ; 0 is the matching blowing speed )

From the above equation ( 3.11 ) ,

& A ;

Using equations ( 3.1 ) , ( 3.7 ) , ( 3.11 ) and ( 3. 12 ) , an equation can be written as

( 3.13 )

in which( 3.14 )

utilizing the above equation, the boundary and initial conditions are

=

=

+()

for all T & gt ; 0 ( 3.15 )

( 3.16 )+

( 3.17 )

In order to happen the solution of equation ( 3.13 ) topic to equations ( 3.15 ) and ( 3.16 ) the Laplace transform brace can be defined as

( 3.18 )

( 3.19 )

Taking( 3.20 )

utilizing the Laplace parametric quantity s, the Equations ( 3.13 ) becomes

and allow

( 3.21 )

Using equations ( 3.15 ) , ( 3.9 ) and equation ( 3.17 )

Therefore,( 3.22 )

Therefore,as( 3.23 )

The subsidiary equation of ( 3.19 ) is

= 0

=

Therefore the roots are

=

Therefore the general solution of the ordinary differential equation ( 3.19 ) is

++( 3.24 )

Whereandarbitrary invariables

Using equation ( 3.20 ) and ( 3.21 ) and taking omega = 0 in ( 3.23 )

=

Substitutingin ( 3.23 )

=

Since

=

=

=

and hence

,=( 3.25 )

Now replacingandin ( 3.23 )

=( 3.26 )

Taking the Laplace transform for the above consequence, the speed field is of the signifier

( 3.27 )

whereindicates the reverse Laplace transform and it is known that

=

=( 3.28 )

Using whirl theorem of Laplace transform,

=( 3.29 )

Herein the above equation is indicated for the whirl.

=

( 3.30 ) Applying ( 3.30 ) in ( 3.28 ) , and utilizing ( 3.14 ) so

( 3.31 ) whereis the complementary mistake map and is defined by

( 3.32 )

Clearly the existent and fanciful parts of equation ( 3.30 ) isandseverally.

Substituting

( 3.33 )

( 3.34 )

and equation ( 3.31 ) becomes

=

( 3.35 )

In which

( 3.36 )

Substituting

( 3.37 )

( 3.38 )( 3.39 )

( 3.40 )

The solution obtained in the equation ( 3.35 ) is the complete analytic solution for the speed field due to an arbitrary periodic oscillation in its ain plane. As a particular instance of this oscillation, the flow field for different oscillations is obtained by an appropriate pick of the Fourier coefficients which give rise to different disc oscillations. The periodic oscillation and their corresponding Fourier coefficients are given in Table 1

Table 1: The periodic oscillations and their corresponding Fourier coefficients

Sl.No	Oscillations h ( T )	Fourier coefficients
1		=1 and( K
2	Cos National Trust	=andotherwise
3	Sin National Trust	=andotherwise
4	1, 0	==for all K
5		=for all K

The flow Fieldss in the above instances can easy be obtained by utilizing in turn the appropriate Fourier coefficients in equation ( 3.35 ) . The ensuing flow Fieldss for these instances are as follows

=

( 3.41 )

=

( 3.42 )

=

( 3.43 )

( 3.44 )

( 3.45 )

Takingin equations ( 3.35 ) and ( 3.41 ) to ( 3.45 ) and utilizing asymptotic expression for the complementary mistake map i.e

( 3.46 )

Therefore the equation ( 3.34 ) and ( 3.40 ) to ( 3.44 ) takes the signifier

( 3.47 )

( 3.48 )

( 3.49 )

( 3.50 )

,( 3.51 )

( 3.52 )

where( 3.53 )

and

is the steady province solution.

In the instance of blowing the steady province solutions are obtained from equations ( 3.35 ) and ( 3.41 ) toby replacingbyprovided

( 3.54 )

( 3.55 )

( 3.56 )

( 3.57 )

,( 3.58 )

( 3.59 )

whereandare obtained by replacingbyin the equations ( 3.37 ) to ( 3.40 ) and ( 3.53 ) severally.

Consequences and Discussions

Graph is depicted for the equation ( 3.47 ) which is given in figure 3.2 to 3.5. These figure shows that the solution represents cross moving ridges occurs both the suction and blowing instances forand. Figure 3.6 to 3.9 show that the boundary bed thickness decreases with an addition of the suction parametric quantity for bothandIt is farther noted from these figures for hydrodynamic fluidand the solution ( 3.54 ) to ( 3.59 ) do non fulfill the boundary status in the resonating instance n = a„¦ . In hydromagnetic state of affairs, figures 3.10 to 3.15. are depicted and it is noted thatandlessenings with the addition of magnetic field and the boundary conditions are satisfied.

The consequence of magnetic field onandin the presence of suction at

;a„¦=1

Figure 3.2

Figure 3.3

The consequence of magnetic field onandin the presence of blowing at

;a„¦=1

Figure 3.4

Figure 3.5

The consequence of hydrodynamic field onandin the presence of increasing suction at;a„¦=1

Figure 3.6

Figure 3.7

The consequence of hydrodynamic field onandin the presence of increasing blowing at;a„¦=1

Figure 3.8

Figure3.9

The consequence of magnetic field onandat;a„¦=1

Figure 3.10

Figure 3.11

The consequence of magnetic field onandat;a„¦=1

Figure 3.12

Figure 3.13

The consequence of magnetic field onandat;a„¦=1

Figure 3.14

Figure 3.15

The consequence of magnetic field whenw0 = [ 1 2 3 4 5 ] ;= 1 ;= 1 ;= 2.5 ; T = 3 ; V = 1 ; n = 2 ;

Figure 3.15

Figure 3.16

3.5 Decision

• The solutions for suction and blowing instances are derived for all values of frequences including at resonating frequence.
• The effects of the magnetic parametric quantity and suction/blowing parametric quantities on the speed are seen, from where it is observed that an addition in the magnetic parametric quantity leads to a lessening in the boundary bed thickness.
• The consequence of suction parametric quantity on the speed is similar to that of magnetic parametric quantity.
• It is further noted that diffusing moving ridges occur in the hydromagnetic system.
• It is confirmed that for big times the get downing solution tends to the steady province solution.