A Circular Cylinder. 1 Introduction. RITA CHOUDHURY 1 and BIBHASH DEKA 2

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1 MHD V -E F F H T A Circular Cylinder RITA CHOUDHURY 1 and BIBHASH DEKA 2 1,2 Department of Mathematics, Gauhati University, Guwahati-78114, Assam, INDIA. 1 rchoudhury66@yahoo.in 2 bibhashdeka66@gmail.com Abstract:-An investigation is made on the temperature distribution within the thermal boundary layer region due to the flow of a second-grade fluid around a heated circular cylinder, maintained at a constant temperature higher than that of the fluid at infinity in presence of magnetic field applied transversely to the direction of the main flow. The problem has been solved by the application of steepest descent method used by Meksyn. The impact of various pertinent parameters on flow characteristics have been discussed through graphical illustrations. Newtonian results are found to emerge as limiting cases of the present analysis. Key-words:-Visco-elastic, Boundary layer, MHD, Circular Cylinder, Heat transfer. 1 Introduction The boundary layer concept has tremendous achievement in the interdisciplinary activities concerning engineering and technology developments. The mechanism of thermal boundary layer flow of magnetohydrodynamic visco-elastic fluids are used in different manufacturing processes such as extrusion of plastic sheets, fabrication of adhesive tapes, coating layers into rigid surfaces etc. The influence of magnetic field on an electrically conducting viscous incompressible fluid past a cylinder in presence of heat transfer has practical significance in many engineering applications viz. solar power collectors, compact heat exchanges and nuclear reactors. Dennis and Chang [1] have presented the numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 1. An experimental investigation of the steady separated flow past a c ircular cylinder has been studied by Grove et al. [2]. Fornberg [] has analysed the numerical study of steady viscous flow past a circular cylinder. Steady two-dimensional viscous flow of an incompressible fluid past a Circular cylinder has been investigated by Takami and Keller [4]. Also, the authors viz. Thoman and Szewczyk [5], Kawaguti [6], Hamielec and Raal [7], Gschwendter [8], Sen et al. [9] etc. have remarkable contribution in this field. The study of visco-elastic fluid flow has been the objective of immense research due to its applications in industries of chemical processes such as food processing and polymer production etc. The differential models of visco-elastic fluid encountering the effects of shear thinning/thickening and normal stress differences are known as secondgrade fluids which follow from generalized Rivlin- Ericksen s fluid. The viscous feature of secondgrade fluid is caused by the transport phenomenon of the molecules of fluids whereas the elastic characteristic is assignable to the chemical structure and configuration of polymer molecules. Many scientists have contribution on this field but a few of them are mentioned here. The second-order thermal boundary layer equation for the flow of a secondorder fluid past heated body has been investigated by Srivastava [1] using the order of magnitude approach. The temperature distribution for the flow near a t wo-dimensional stagnation point occurring on a flat plate maintained at a temperature higher than that of the fluid at infinity has been analysed. The thermal boundary layer on a steadily rotating sphere in infinitely extending second-order fluid has been discussed by Bhatnagar and Palekar [11]. Srivastava and Maiti [12] have analysed the flow of a second-order fluid past a cylinder by expanding the flow functions in series and obtaining the first four terms by Karman-Pohlhausen method. The heat transfer in a sec ond-grade fluid for flow around a E-ISSN: X 98 Volume 12, 217

2 circular cylinder by Karman-Pohlhausen method has been studied by Srivastava and Saroa [1].The heat transfer in the boundary layer region of a secondorder fluid past a plate by presenting a uniform constant suction and temperature at the plate has been studied by Agarwal and Bhatia [14]. Srivastava and Saroa [15] have investigated the heat transfer in a second-order fluid for flow around a circular cylinder. Numerical simulation of visco-elastic flow past a cylinder has been analysed by Hu and Joseph [16]. Chhabra et al. [17] has presented the steady non Newtonian flow past a ci rcular cylinder: a numerical study. In this paper, we have studied the problem of flow and heat transfer of a second-order fluid around a circular cylinder in presence of magnetic field by series expansion used by Meksyn [18] and found that the point of separation for the Newtonian case comes out to be 19.9 whereas the exact value is 19.6 [Schlichting,[19]]. Srivastava and Saroa [14] have obtained the separation point at 11.8 by taking the boundary layer thickness to be variable along the cylinder. By using Karman-Pohlhausen method, Srivastava and Maiti obtained the corresponding point of separation at The constitutive equation for the second-order incompressible fluid is taken in the form 2 σσ = PPPP + μμ (1) AA (1) + μμ (2) AA (2) + μμ () AA (1) (1) where σσ is the stress tensor, AA (nn) (nn = 1,2) are the kinematic Rivlin-Ericksen tensors, μμ (nn) (nn = 1,2,) are the material co-efficients describing the viscosity, elasticity and cross viscosity respectively. The case μμ (2) = μμ () = corresponds to an incompressible Newtonian fluid. On thermodynamic considerations μμ (2) is found to be negative whereas μμ (1) and μμ () are positive. Coleman and N oll [2] derived the equation from the simple fluid by assuming that stress is more sensitive to the recent deformation than to the deformation that occurred in the distant past. Markovitz and Brown determined experimentally the material constants for solutions of poly-isobutylene in cetane of various concentrations. 2 Basic Equations Consider an incompressible MHD second-grade fluid moving with a uniform velocity UU at infinity in presence of a fixed circular cylinder of radius aa maintained at a constant temperature TT ww. Let the temperature of the fluid at infinity be TT where TT ww > TT. Let (rr, θθ, zz) be the cylindrical polar coordinates with zz-axis coincides with the axis of the cylinder. The fluid flow is two-dimensional in rr and θθ directions. The two-dimensional velocity boundary layer equations are uu + vv = νν 2 uu 1 + νν 2 2 uu uu + vv uu 2 + σσbb 2 yy ρρ 2 uu + 2 vv (UU uu) + UU (2) + = () where yy = rr aa, xx = aaaa and uu, vv are velocity components along xx and yy-axis respectively within the viscous boundary layer region, UU is the main stream velocity, ρρ is the fluid density, BB yy is the strength of the magnetic field and νν ii = μμ ii, (ii = 1,2). ρρ Also, equations (2) and () are independent of the curvature of the wall and thus are applicable to the case of a flat wall. Srivastava [1967] has derived the thermal boundary layer equation for the second-grade incompressible fluid against a heated wall as: ρρcc pp uu uu + vv = μμ μμ 2 vv 2 uu uu + kk 2 TT (4) 2 where CC pp is the specific heat, kk is the thermal conductivity and TT is the temperature. This equation is valid within the boundary layer over both a curved wall and a flat wall when xx-axis is taken in the tangential direction and yy-axis along the normal to the surface. The relevant boundary conditions are, uu =, vv =, TT = TT ww at yy = (5) uu UU, TT TT in yy (6) The velocity distribution UU outside the velocity boundary layer region created by the cylinder is given by, UU(θθ) = 2UU sin θθ and the stream-function ψψ is given by, ψψ = νν 1aa 2UU 2UU θθff 1 (ηη) 8! UU θθ ff (ηη) ! UU θθ 5 ff 5 (ηη) 16 7! UU θθ 7 ff 7 (ηη) + (7) E-ISSN: X 99 Volume 12, 217

3 where ηη = yy 2UU 1 2 νν 1 aa Now, the velocity components uu and vv within the velocity boundary layer region are given by, uu = = 2UU θθ ff 1 (ηη) 2 θθ2 ff (ηη) θθ4 ff 5 (ηη) 1 6 θθ6 ff 7 (ηη) + (8) and vv = = 2νν 1UU ff aa 1 (ηη) 2θθ 2 ff (ηη) θθ4 ff 5 (ηη) 1 9 θθ6 ff 7 (ηη) + (9) Here prime denotes differentiations w.r.t ηη. The temperature TT within the thermal boundary layer region should be taken in the form TT = TT TT TT ww TT = 4EE{TT 1 (ηη) θθ 2 TT (ηη) + θθ 4 TT 5 (ηη) θθ 6 TT 7 (ηη) + } (1) where EE is the Eckert number and it is given by UU2 EE =. For simplicity, we shall confine CC pp (TT ww TT ) ourselves to terms up to ff 7 (ηη) and TT 7 (ηη) only. Now, the boundary conditions of ff ii (ηη) and TT ii (ηη), (ii = 1,,5,7) are, At ηη = ; ff ii (ηη) =, ff ii (ηη) =, TT 1 = 1 4EE, TT = TT 5 = TT 7 = (11) In ηη ; ff 1 1, ff 1, ff 4 5 1, ff 6 7 1, TT 8 ii (12) Now, substituting the values of uu, vv and TT from equations (8) to (1) into (2) and (4) and equating the co-efficient of like powers of θθ with the neglect of higher order terms on both sides of the equations we have, ff 1 + ff 1 ff 1 = 1 + MM(ff 1 1) + ff αα 1 (ff ff 1 ff 1 2ff 1 ff 1 ) (1) ff + ff 1 ff = 1 + MM ff 1 + 4ff 4 1 ff ff ff 1 + αα 1 [ 4(ff 1 ff + ff ff 1 ) + 4ff 1 ff + (ff 1 ff + ff ff 1 )] (14) ff 5 + ff 1 ff 5 = 8 + MM ff ff 1 ff 5 5ff 5 ff (ff 2 ff ff ) + αα 1 6(ff 1 ff 5 + ff 5 ff 1 ) + 6ff 1 ff 5 + (ff 1 ff 5 + 5ff 5 ff 1 ) + 8 ( 2ff ff + ff ff + ff 2 ) (15) ff 7 + ff 1 ff 7 = 8 + MM ff ff 8 1 ff 7 7ff 7 ff ff ff 5 6ff ff 5 15ff 5 ff + αα 1 [ 8(ff 1 ff 7 + ff 7 ff 1 ) + (ff 1 ff 7 + 7ff 7 ff 1 ) 168(ff ff 5 + ff ff 5 ) + (15ff 5 ff + 6ff ff 5 ) + 168ff ff 5 + 8ff 1 ff 7 ] (16) TT 1 + PPPPff 1 TT 1 = (17) TT + PPPPff 1 TT = Pr[2TT ff 1 2ff TT 1 + ff αα 1 {ff 1 (ff 1 ff 1 ff 1 ff 1 )}] (18) TT 5 + PPPPff 1 TT 5 = Pr 4TT 5 ff 1 2ff TT + 4 TT ff 1 ff 4 5TT ff 1 ff + 2 αα 1{ff 1 (ff 1 ff + 4ff 1 ff ) ff 1 (ff 1 ff + ff ff 1 ) ff 1 ff 1 ff } (19) TT 7 + PPPPff 1 TT 7 = Pr 6TT 7 ff 1 2ff TT TT 5ff 1 ff 4 5TT + 1 TT 1 ff 5 1 ff 9 7TT ff 1 1 ff ff αα 1 1 ff 2 1 ( ff 1 ff 5 + 6ff 1 ff 5 + ff 1 ff 5 5ff 5 ff 1 ) 4 ff 9 (ff 1 ff + ff ff 1 4ff 1 ff ) + 1 (8ff 6 1 ff 2 8ff 1 ff ff ff 5 ff 1 ff 1 ) (2) where αα 1 = 2UU νν 2 is the visco-elastic parameter, aaνν 1 PPPP = ρρνν 1CC pp is the Prandtl number and MM = σσbb2 aa is kk 2UU ρρ the magnetic parameter. Solution of the Problem The equations (1) to (2) subject to the boundary conditions (11) and (12) have been solved by the application of steepest decent method used by Meksyn followed by the method of Laplace. In this method we express the functions ff ii (ηη) and TT ii (ηη) in power series of ηη as ff ii (ηη) = AA ii 2! ηη2 + BB ii! ηη + CC ii 4! ηη4 + DD ii 5! ηη5 + EE ii 6! ηη6 + (21) TT 1 (ηη) = 1 + aa 4EE 1ηη + bb 1 2! ηη2 + cc 1! ηη + dd 1 (22) 4! ηη4 + ee 1 5! ηη5 + E-ISSN: X 1 Volume 12, 217

4 TT jj (ηη) = aa jj ηη + bb jj 2! ηη2 + cc jj! ηη + dd jj 4! ηη4 + ee jj 5! ηη5 + (2) where ii = 1,,5,7 and jj =,5,7. We have taken the above forms of ff ii, TT 1 and TT jj for sufficiently small values of ηη and all of them satisfy the boundary conditions (11) at ηη =. Now substituting the expressions of ff ii, TT 1 and TT jj from (21), (22) and (2) into (1) to (2) and equating the co-efficient of different powers of ηη to zero, we obtain the constants BB ii, CC ii, DD ii, EE ii ; bb jj, cc jj, dd jj, ee jj as functions of AA ii s and aa ii s only. So, if AA ii and aa ii are known, the velocity and the temperature distributions are completely determined. Now we write equations (1) to (16) and (17) to (2) in the following forms: ff ii and TT ii + ff 1 ff ii = HH ii (ηη), ii = 1,,5,7 (24) + PPPPff 1 TT ii = MM ii (ηη), ii = 1,,5,7 (25) Here we have denoted the right-hand sides of the equations (1) to (16) and (17) to (2) by HH ii (ηη) and MM ii (ηη) respectively. Now integrating twice the equations (24) and (25) w.r.t ηη from to ηη, we get and ee GG(ηη) ψψ 1 (ηη) dddd = 1 4E ee GG(ηη) ψψ (ηη) dddd = ee GG(ηη) ψψ 5 (ηη) dddd = ee GG(ηη) ψψ 7 (ηη) dddd = (29) The above integrals can be evaluated asymptotically by Laplace s method. Putting FF(ηη) = GG(ηη) = δδ, transforming the equations (28) and (29) to the variable δδ and integrating in the gamma functions, we have, mm 1 Γ1 + mm 11 Γ2 + mm 12 Γ 1 + mm 1 Γ4 + + = 1 mm 14 Γ5 mm Γ1 + mm 1 Γ2 mm 5 Γ1 + mm 51 Γ2 mm 7 Γ1 + mm 71 Γ2 + mm 2 Γ 1 + mm Γ4 mm 4 Γ5 + = mm 52 Γ 1 + mm 5 Γ4 mm 54 Γ5 + = mm 72 Γ 1 + mm 7 Γ mm 74 Γ5 + = 1 8 () ff ηη ii (ηη) = ee FF(ηη) φφ ii (ηη) dddd (26) ηη TT ii (ηη) = aa + ee GG(ηη) ψψ ii (ηη) dddd (27) where ηη FF(ηη) = ff 1 (ηη) dddd ηη GG(ηη) = PPPP ff 1 (ηη) dddd ηη φφ ii (ηη) = AA ii + ee FF(ηη) HH ii (ηη) dddd ηη ψψ ii (ηη) = aa ii + ee GG(ηη) MM ii (ηη) dddd Now, taking ηη in the equations (26) and (27) we get, nn 1 Γ1 + nn 11 Γ2 + nn 12 Γ 1 + nn 1 Γ4 +nn 14 Γ5 + = 1 4EE nn Γ1 + nn 1 Γ2 +nn 4 Γ5 nn 5 Γ1 + nn 51 Γ2 +nn 54 Γ5 nn 7 Γ1 + nn 71 Γ2 +nn 74 Γ5 + nn 2 Γ 1 + nn Γ4 + = + nn 52 Γ 1 + nn 5 Γ4 + = + nn 72 Γ 1 + nn 7 Γ4 + = (1) ee FF(ηη) φφ 1 (ηη) dddd = 1 ee FF(ηη) φφ (ηη) dddd = 1 4 ee FF(ηη) φφ 5 (ηη) dddd = 1 6 ee FF(ηη) φφ 7 (ηη) dddd = 1 8 (28) where the constants ar e not presented here due to sake of brevity. Now solving the equations () and (1) using MATLAB, we get the values of AA ii s and aa ii s (ii = 1,,5,7) for different flow parameters involved in the equations (Tables 1 to 5). Throughout the computations, we use different values of magnetic parameter MM, Prandtl number PPPP E-ISSN: X 11 Volume 12, 217

5 and visco-elastic parameter αα 1 with fixed value of Eckert number EE =.1. In the study, the Newtonian fluid flow phenomenon is illustrated by αα 1 = and αα 1 characterizes the visco-elastic fluid. Now using the values of AA ii s and aa ii s in equations (21) to (2), we get the expressions for ff ii, TT 1 and TT jj. Finally putting the values of ff ii, TT 1 and TT jj in equations (8) to (1) we get the analytical expressions for velocity components uu, vv within the velocity boundary region and temperature TTP* within the thermal boundary layer region. 4 Results and Discussion Knowing the velocity and temperature fields, we obtain some important flow characteristics of the problem viz. wall shear stress and local heat flux. The non-dimensional shearing stress ττ at the wall ηη = is given by, σσ ττ = xxxx 1 = θθθθ 1 (ηη) 4! θθ ff (ηη) + 2 2ρρUU 2νν 1UU ll 6 5! θθ5 ff 5 (ηη) 8 7! θθ7 ff 7 (ηη) + ηη= (2) We get the location of the point of separation if the shearing stress at the wall vanishes. The conditions for which the shearing stress ττ vanishes at the surface of the wall for αα 1 =,.1, and. with magnetic parameter MM =.8 are given by,.229 XX XX XX.8664 = () XX.1489 XX XX.8692 = (4).86 XX XX XX.875 = (5) respectively with XX = θθ 2. The acceptable roots after solving these cubic equations (), (4) and (5) are XX =.6218, XX =.2282 and XX = respectively. Thus the points of separation for αα 1 =,.1, and. occur at θθ = 19.9P, θθ = 12.99P and θθ = 9.76P respectively. Thus, we observe that the separation point diminishes with the enhancement of absolute value of visco-elastic parameter. Again, if we increase the value of magnetic parameter, (MM =.15) then we get the separation point for αα 1 =,.1, and. at θθ = 111.9P, θθ = 14.2P and θθ = 94.42P respectively. So, it can be remarked that the enhancement of magnetic parameter increases the value of separation point in both Newtonian and non-newtonian cases. The heat generation flux gg from the cylinder to the fluid is given by, gg = kk = 4kkkk (TT yy= aa ww TT ){TT 1 (ηη) θθ 2 TT (ηη) + θθ 4 TT 5 (ηη) θθ 6 TT 7 (ηη)} ηη= (6) where RRRR = 2UU ρρ aaμμ 1 is the Reynolds number. Defining the Nusselt number NNNN = have, aaaa kk(tt ww TT ) NNNN = 4EE{TT 1 (ηη) θθ 2 TT (ηη) + θθ 4 TT 5 (ηη) θθ 6 TT 7 (ηη)} ηη= (7) we Figures 1 to 5 demonstrate the variation of shearing stress ττ and NNNN against θθ with various values of other flow parameters. The various angles (in degrees) are converted in radians (1 =.175 radian) while plotting the graphs in different cases. The graphs reveal that the shearing stress ττ and NNNN gradually diminish for Newtonian and non- Newtonian fluids. Also, it is observed that the growth of absolute value of visco-elastic parameter αα 1, decelerate the shearing stress ττ and NNNN in comparison with Newtonian fluid flow phenomenon. The rising values of magnetic parameter MM accelerate the shearing stress for both types of fluids (Figures 1 and 2). Again from the figures, 4 and 5, it can be revealed that the growth NNNN of magnetic parameter increases but an opposite trend is demonstrated during the growing behaviour of Prandtl number PPPP. It has been observed that the Meksyn method is useful and depicts better result when skin friction, heat flux etc. are calculated at the wall of the solid body. 5 Conclusion The flow and heat transfer of a v isco-elastic fluid around a circular cylinder in presence of a magnetic field has been analysed using the application of steepest descent method used by Meksyn. This study leads to the following conclusions: The velocity and temperature fields are significantly affected by visco-elasticity. E-ISSN: X 12 Volume 12, 217

6 The point of separation for Newtonian case comes to be 19.9 whereas the exact value is The point of separation has been found to shift towards the forward stagnation point due to the elasticity of the fluid. The enhancement of the value of separation point is noticed for both Newtonian and visco-elastic fluids with the increasing value of magnetic parameter. With the growth of the absolute value of the visco-elastic parameter, the shearing stress diminishes and the identical result is seen for Nusselt number. Both the shearing stress and Nusselt number show diminishing trends with the increase of the angle θθ in both types of fluids. Acknowledgement The financial support by UGC in the form MRP- Major General (NER), F.No /214 (SR) dated 1 st September 215 is gratefully acknowledged by the authors. References [1] Dennis, S. C. R. and Chang, G., Numerical Solutions for Steady Flow Past a Circular Cylinder at Reynolds Numbers up to 1, J. Fluid Mech., vol. 42, 197, pp [2] Grove, A. S., Shair, F.H., Petersen, E.E. and Acrivos, A., An Experimental Investigation of the Steady Separated Flow past a Circular Cylinder, J. Fluid Mech., vol. 19, 1964, pp [] Fornberg, B., A Numerical Study of Steady Viscous Flow past a Circular Cylinder, J. Fluid Mech., vol. 98, 198, pp [4] Takami, H. and Keller, H.B., Steady twodimensional viscous flow of an incompressible fluid past a circular cylinder, Phys. Fluids Suppl., vol.12,1969, pp [5] Thoman, D.C. and Szewczyk, A. A., Timedependent viscous flow over a ci rcular cylinder. Phys. Fluids Suppl., vol. 12, 1969, pp [6] Kawaguti, M., Numerical solution of the Navier Stokes equations for the flow around a circular cylinder at Reynolds number 4, J. Phys. Soc. Jpn, vol.8, 195, pp [7] Hamielec, A.E. and Raal, J.D., Numerical studies of viscous flow around circular cylinders, Phys. Fluids, vol. 12, 1969, pp [8] Gschwendter, M.A., The Eckert number phenomena: Experimental investigation on the rotating cylinder, Heat and Mass transfer, vol. 4, 24, pp [9] Sen, S., Mittal, S. and Biswas, G., Steady separated flow past a c ircular cylinder at low Reynolds numbers, J. Fluid Mech., vol.62, 29, pp [1] Srivastava, A.C., Thermal boundary layer in second-order fluids, Archwm. Mech. Stowow, vol.19, 1967, pp [11] Bhatnagar, R.K. and Palekar, M.G., The thermal boundary layer on a sp here rotating in an infinitely extending non-newtonian fluid, Rhcol. Acta.,vol. 11, 1972, pp [12] Srivastava, A.C. and Maiti, M.K., Flow of a second-order fluid past a symmetrical cylinder, Phys. Fluids, vol. 9, 1966, pp [1] Srivastava, A.C. and Saroa, M.S., Phenomenon of separation in second-order fluids, Int. J. nonlinear Mech., vol. 6, 1971, pp [14] Agarwal, R.S. and Bhatia, S.P.S., Heat transfer in second-order fluid with suction and constant heat sources, Int. J. Phys., vol.47, 197, pp [15] Srivastava, A.C. and Saroa, M.S., Heat transfer in a seco nd-order fluid for flow around a ci rcular cylinder, Int. J. non-linear Mech., vol. 1, 1978, pp [16] Hu, H.H. and Joseph, D.D., Numerical simulation of visco-elastic flow past a cylinder, J. non- Newtonian Fluid Mech., vol. 7, 199, pp [17] Chhabra, R.P., Soares, A.A. and Ferreira, J.M., Steady non Newtonian flow past a circular cylinder: a numerical study, Acta Mech., vol.172, 24, pp E-ISSN: X 1 Volume 12, 217

7 [18] Meksyn, D., New Method in Laminar Boundary layer theory, Pergamon Press, [19] Schlichting, H., Boundary layer theory, McGraw Hill, New York, [2] Coleman, B.D. and Noll, W., An application theorem for functional with applications in continuum mechanics, Archs Ration Mech. Analysis, vol.6,196,pp ττ -51,575 2,1 2,625,15,675-1 α₁ = -15 α₁ = α₁ = θθ Fig. 1 Variation of ττ against θθ for MM = ,575 2,1 2,625,15,675-5 ττ α₁ = α₁ = -.1 α₁ = θθ Fig. 2 Variation of ττ against θθ for MM =.15. E-ISSN: X 14 Volume 12, 217

8 -5 1,75 2,275 2,8,25,85 NNuu/ RRee θθ α₁= α₁= -.1 α₁= -. Fig. Variation of NNNN against θθ for MM =.8, PPPP =, EE = ,75 2,275 2,8,25,85 NNuu/ RRee θθ α₁ = α₁ = -.1 α₁ = -. Fig. 4 Variation of NNNN against θθ for MM =.15, PPPP =, EE =.1. NNuu/ RRee -51,75 2,275 2,8,25, α₁ = α₁ = -.1 α₁ = θθ Fig. 5 Variation of NNNN against θθ for MM =.8, PPPP = 5, EE =.1. E-ISSN: X 15 Volume 12, 217

9 αα AA AA AA AA Table 1. Values of AA ii ( ii = 1,,5,7) at MM =.8 αα AA AA AA AA Table 2. Values of AA ii ( ii = 1,,5,7) at MM =.15. αα aa aa aa aa Table. Values of aa ii ( ii = 1,,5,7) at MM =.8, PPPP = and EE =.1. E-ISSN: X 16 Volume 12, 217

10 αα aa aa aa aa Table 4. Values of aa ii ( ii = 1,,5,7) at MM =.15, PPPP = and EE =.1 αα aa aa aa aa Table 5. Values of aa ii ( ii = 1,,5,7) at MM =.8, PPPP = 5 and EE =.1. E-ISSN: X 17 Volume 12, 217