MHD Flow caused by NonCoaxial Rotations of a Porous Disk and an Oscillating Third Grade Fluid at Infinity
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Chapter VI
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MHD FLOW CAUSED BY NONCOAXIAL ROTATIONS OF A POROUS DISK AND AN OSCILLATING THIRD GRADE FLUID AT INFINITY
6.1 Introduction
Non – Newtonian fluid flow play an of import functions in several industrial fabrication procedures. An illustration of such non Newtonian fluids includes boring clay, polymer solutions or thaws, certain oils and lubricating oils and many other emulsions. Some of the typical applications of non –Newtonian fluid flow are noticed in the boring of oil and gas Wellss, polymer sheet bulge from a dye, glass storyteller and paper production, pulling of plastic movies, waste fluids etc. Because of the complexness of nonNewtonian fluids, it is really hard to analyze a individual theoretical account that exhibits all its belongingss. The normal emphasis differences is describe in 2nd class in second of non – Newtonian fluid, but it can non foretell shear cutting or inspissating belongingss due to its changeless evident viscousness. The 3rd class fluid theoretical account effort includes such features of viscoelastic fluids. Despite assorted complexnesss in the constituent equations, several research workers have investigated the flows of 3rd – class fluids taking into history assorted facets. Several research workers discussed the faux pas consequence on fluid flow. For case, Asghar et Al. ( 2006 ) examines the effects of partial faux pas on the revolving flow of incompressible 3rd class fluid pat a uniformly porous home base. Sajid and Hayat ( 2007 ) discussed the two – dimensional boundary bed flow of a 3rd class fluid over stretching sheet. Sajit et Al.(2007 ) considered heat transportation features in an electrically carry oning 3rd grade fluid. Miccal and James ( 2008 ) discusses the consequence of replacing the standard no faux pas boundary status of fluid mechanics using for the so called FalknerSkan solutions, with a boundary status that allows some grade of digressive fluid faux pas. Ellahi ( 2009 ) discourse the faux pas status of an Oldroyd 8 – changeless fluid. Sahoo ( 2010 ) ) computed the numerical solutions for heat transportation in Heimenz flow of 3rd –grade fluid. Hayat et Al. ( 2010 ) examined the coincident effects of heat and mass transportation on an unsteady flow of thirdgrade fluid.
Flows over disc or between discs are popular among research workers in position of their applications in Engineering. Discoid organic structures are frequently encountered in Engineering. Examples include, revolving heat money changers, revolving disc reactors for biofluids production, gas or Marine turbine and chemical and car industries. In position of the above mentioned applications of flow over a disk/between discs, several research workers have analyzed such flows. In this pioneering work, Von Karman ( 1921 ) considered hydrodynamic flow over an infinite disc. Cochran ( 1934 ) derived asymptotic solutions for steady hydrodynamic flow over a rotating infinite disc. Benton ( 1966 ) extended Cochran’s work to the initial value job regulating flow induced by an impetuously started disc. Takar et Al. ( 2003 ) investigated the consequence of a magnetic field on unsteady assorted convection flow from a revolving perpendicular cone. Stuart ( 1954 ) examined the consequence of unvarying suction on the steady flow due to a revolving disc. Rashidi and Pour ( 2010 ) examined the three dimensional job of a condensation movie inclined on a revolving disc by a differential transform method. Recently Hayat and Nawaz ( 2011 ) discussed unsteady stagnancy point flow over a rotating disc.
In this chapter, the work is extended for Hussain, Zahid ( 2006 ) .The consequence of unsteady MHD flows induced by noncoaxial rotary motions of a porous disc and a 3rd class fluid is exhaustively studied. Here the job is solved numerically utilizing modified Crank –Nicolson implicit strategy which is so solved by MATLAB. The survey is performed for assorted interesting parametric quantities and is discussed. It is noted that the boundary bed thickness additions by increasing the 3rd class parametric quantities. An effort is made for the hydrodynamic job associated with the instance of blowing and the resonating frequence has been resolved by hydromagnetic state of affairs.
6.3 Mathematical preparation
In a Cartesian coordinate system with z axis normal to the porous disc which lies in the plane omega = 0 is considered. The part z & A ; gt ; 0 is occupied by an incompressible 3rd grad fluid. The axis of rotary motion of both the disc and the fluid, are assumed to be in the plane x = 0, with the distance between the axes being l. The disc and the fluid at eternity are ab initio revolving about theaxis with the same angular speed a„¦ , and at clip T = 0, the disc starts to hover all of a sudden along the x – axis and to revolve impetuously about the omega axis with the same angular speed a„¦ , and the fluid at eternity continues to revolve about the– axis with the same angular speed. The fluid is electrically conducting and assumed to be permeated by a magnetic field, and holding no constituents in the ten, y waies. The speed field is of the signifier
( 6.1 )
where U, V, w are the constituents of the speed vector, in the waies x, y, z severally. Obviouslyis the suction speed andis the blowing speed. The speed field satisfieswhich is nil but the incompressibility status. For the job under consideration, the boundary and initial conditions can be written as
( or )
,at
asfor all T,
atfor( 6.2 )
where K is the frequence of nontorsional oscillations and U is the speed.
The hydromagnetic equations of gesture for an electrically carry oning incompressible fluid is
( 6.3 )
( 6.4 )
( 6.6 )
and Ohm’s jurisprudence for a moving music director
( 6.7 )
In the above equations,is the denseness,is the material clip derivative,is the electric current denseness,is the magnetic permeableness,is the electric field, B is the entire magnetic field and so thatB is the induced magnetic field,is the electric conduction of the fluid. In equations ( 6.4 ) to ( 6.7 ) the magnetic Reynolds figure[ ] is assumed to be little as is the instance with the most of the conducting fluids and therefore the induced magnetic field is little in comparing with the applied magnetic field and is hence non taken into history. The magnetic organic structure forcesnow becomes
The constituent equation of the 3rd – class fluid is
++( 6.8 )
Whereis the emphasis tensor,is the individuality,andare the Rivlin – Ericsen tensors of the first, 2nd and 3rd orders, severally.is the inactive fluid force per unit area (=is the dynamic viscousness, coefficients,are the material invariables. The first, 2nd and 3rd – order Rivlin – Ericksen tensors are severally as follows
The thermo kineticss of fluid modelled by equation ( 6.8 ) has been object of a elaborate survey by Fosdick and Rajagopal ( 1980 ) and Dunn and Rajagopal ( 1995 ) . They show that the equation ( 6.8 ) to be compatible with thermodynamics and the free energy to be minimal when the fluid is at remainder, the stuff changeless shoud satisfy the relation
( 6.9 )
And specific Helmholts free energyhas the signifier
( 6.10 )
where( 6.11 )
Here, the fluid is thermodynamically compatible ; hence the emphasis constituent relation ( 6.8 ) reduces to
++( 6.12 )
Utility equation ( 6.12 ) andinto equation ( 6.3 )
The flow is governed through the undermentioned equations.
= 0 ( 6.13 )
( 6.14 )
( 6.15 )
Extinguishingfrom above and so combination of ensuing equation outputs
( 6.16 )
( 6.17 )
where
( 6.18 )
On presenting nondimensional parametric quantity
, N=
Equation ( 6.16 ) and ( 6.17 ) becomes
( 6.19 )
( 6.20 )
It is noted that equation ( 6.19 ) is a 3rd order and non – additive partial differential equation. As a consequence, it seems to be impossible to obtain the general solution in closed signifier for arbitrary values of all parametric quantities originating in this nonlinear equation. Further equation ( 6.19 ) is parabolic with regard to clip which allows a clip processing solution to the equations. The nonlinearity of above equation must be suppressed in using the Von – Newmann stableness analysis by taking solution – dependent coefficients multiplying derived functions, temporarily frozen. The modified equation attack to analyzing nonlinear computational algorithm is applicable but the visual aspect of merchandises of higherorder derived functions makes the buildings of more accurate strategies less precise than the instance of additive equations. As this job is clip dependent and has mixed derivative with regard to clip and infinite coordinates so it is required to utilize an inexplicit strategy. Using inexplicit strategy to nonlinear equation ( 6.19 ) a figure of picks are available. Numeric methods for a parabolic partial differential equation include both ( I ) a boundary value job and ( two ) an initial value job.
Uniting these two jobs may ensue in really complicated or, at least inefficient methods ( e.g ) higher order Rungekutta methods or forecaster – corrector methods. This restriction leads to the consideration of the simplest group of numerical methods for initial value jobs. A modified CrankNicolson implicit preparation with forward clip and cardinal finite difference infinite estimate is used, so that the equation ( 6.19 ) is transformed to algebraic equation.
The undermentioned algebraic equation is obtained from equation ( 6.19 )
( 6.21 )
Equation ( 6.19 ) together with equation can besides be written as
( 6.22 ) 

( 6.23 ) 
In which


( 6.24 ) 

( 6.25 )
Whenso equation ( 6.22 ) is
( 6.26 )
The value of F^{*}at the fabricated point ?_{1}is approximated by agencies of the Lagrange multinomial of 3rd grade
( 6.27 )
in which
From equations ( 6.26 ) and ( 6.27 )
Sinceis known, so equation ( 6.27 ) gives
=+,=+,
=+,=( 6.28 )
Forwe have
( 6.29 )
or
in which
( 6.30 )
Whenthe equations are
Forwe have
( 6.31 )
For knownequation ( 6.21 ) is
Where
( 6.32 )
For, one obtains
( 6.33 )
To happen the value of H_{1}at the clip degree J, we must hold the value of. Since Eq. ( 4.7 ) is of 3rd order and boundry conditions are two. Therefore we use the augmentation procedure and compose
0 ( 6.34 )
This boundary status is discredited to give
( 6.35 )
Therefore forequation ( 6.23 ) is
where
( 6.36 )
It is noted that there are N1 equations. In matrix signifier, it can be written as
( 6.37 )
Obviously the matrix involved in Equation ( 6.27 ) is penta diagonal.
6.4 Consequences and Discussion
This subdivision concerns with the influence of emerging parametric quantities on the speed profile. Attention has been focused to the parametric quantities c, ? , ? , i?Zand ? . The fluctuations of the indicated parametric quantities are shown with the aid of several Figures 6.4.1 to 6.4.24.
The behavior of speed profiles for assorted values of 3rd grade stuff parametric quantity i?? ( = 0.0, 1.0, 2.0 ) is shown in Figures 6.4.1 to 6.4.6. It is revealed thatlessenings andadditions for big value of i?? . However, the boundary bed thickness is found to increase.
Figures 6.4.7 to 6.4.12 are drawn in order to analyze the effects of porousness parametric quantity i?Z ( = 1.0, 0.0, 1.0 ) on the flow. It is found from these figures, addition in i?Z ,additions andlessenings near the disc. It is farther observed from these figures that suction reduces the boundary bed thickness. These figures depict that blowing enhances the bed thickness in comparing to the instance of suction.
Figures 6.4.13 to 6.4.18 have been made for the illustration of the effects of magnetic parametric quantity ? ( = 0.0, 1.0, 2.0 ) on the speed. Here it is noted thatadditions andlessenings for big values of ? . These figures besides indicate that the addition in magnetic parametric quantity plays a function of cut downing the boundary bed thickness.
Figures 6.4.19 to 6.4.24 are plotted merely to see the influence of clip i?? on the speed distribution. These figures elucidatelessenings andadditions by increasing i?? .
The fluctuation of the speed field with distance from the disc for assorted values of 3rd class parametric quantitywhen, c = 1.5,
The fluctuation of the speed field with distance from the disc for assorted values of 3rd class parametric quantitywhen, c = 1.5,
The fluctuation of the speed field with distance from the disc for assorted values of 3rd class parametric quantitywhen, c = 1,
The fluctuation of the speed field with distance from the disc for assorted values of 3rd class parametric quantitywhen, c = 1,
The fluctuation of the speed field with distance from the disc for assorted values of magnetic parametric quantity N when,,,and
The fluctuation of the speed field with distance from the disc for assorted values of magnetic parametric quantity N when,,and
The fluctuation of the speed field with distance from the disc for assorted Valuess of Magnetic parametric quantity N, when,,and
The fluctuation of the speed field with distance from the disc for assorted values of clipwhen degree Celsius =1.5,,, and n = 0
The fluctuation of the speed field with distance from the disc for assorted values of clipwhen degree Celsius =0.5,,,, and n=0
The fluctuation of the speed field with distance from the disc for assorted values of clipwhen degree Celsius =0.5,,,, and n=0
Decision
 The boundary bed thickness is increased for big values of the 2nd grade stuff parametric quantity
 The boundary bed thickness is decreased when suction parametric quantity is increased
 The boundary bed thickness is increased when suction parametric quantity is increased
 The boundary bed thickness is increased when magnetic parametric quantity is increased.
 is decreased andis increased when i?? is increased.
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