Numerical Solution And Global Error Estimation of Peristaltic Motion Of A Jhonson-Segalman Fluid With Heat and Mass Transfer In A Planer Channel


Mokhtar A. Abd El Naby, Nabil T. Mohammed El Dabe



Runge-kutta-Marson Method and Newton Iteration in shooting and matching technique ware used to obtain the solutions of the system of the non-linear ordinary differential equations, which describe the two-dimensional flow of a Johnson-segalman fluid with heat and mass Transfer in a planer channel having walls that are transversely displaced by an infinite, harmonic traveling wave of large wavelength. Accordingly, we obtained the solutions of the momentum, the energy and the concentration distributions of the problem were illustrated graphically. Effect of some parameter of this problem such as, Weissenberg number W, total flux number F, Eckeret number, Prandtle number P, Soret number S, Schmidt S, Reaction number Rc, Reaction Parameter R, and reaction order m on these formula were were discussed. Also we estimate the global error for the numberical values of Solution by using Zadunaisky technique.


Johnson-Segalman fluid ,heat transfer ,mass transfer ,global error , peristaltic,


I. Hayat T., Wang Y., Hutter K., Asghar S, and Siddiqui A.M., “Peristaltic Transport of an Oldroyd_B Fluid in a planer channel”. Mathematical problems in Engineering, Vol.4 pp.347-376, (2004).

II. Ayukawa, K., Kawa T., and Kimura M, “Streamlines and path lines in peristaltic flow at high Reynolds number”. Bull. Japan Soc. Mech. Engrs. Vol.24, pp.948-955.(1981).

III. Hamin M., “The flow through a channel due to transversely Oscillating walls”, Israel  J. Tec., Vol.6, pp. 67-71, (1968).

IV. Hayat T.,  Wang Y., Siddiqui A.M. and Butter K.,  “Peristaltic motion of a Johnson-Segalman Fluid in a planer channel.”  Mathematical problems in Engineering, vol.1, pp. 1-23.(2003).

V.  Takabatake S. and Ayukawa K., “Numerical study of Two-dimensional peristaltic Flows”,  J.Fluid Mech., Vol.122, pp.439-465,(1982).

VI. Halfen LN., and Castenholz RW., “Gliding in the blue-green alga: a possible mechanism” Nature, Vol.225, pp.1163-1165, (1970).

VII. Kolkka RW., Malkus DS., Hansen MG., Lerly GR. and Worthing RA, “Spurt Phenomenon of the Johnson-Segalman Fluid and related models”, Journal of Non-Newtonian Fluid Mechanics, Vol.29, pp.303-335, (1988).

VIII. Mcleish TCB, and Ball RC., “A molecular approach to the spurt in polymer melt flow”, Journal of polymer Science (B) , Vol.24, pp.1735-1745, (1986).

IX. Malkus D.S., Nohel JA., and Ploher BJ., “Dynamics of Shear flow of a non-Newtonian fluid”, Journal of computational physics, Vol.87, pp.464-497, (1990).

X. Kalika DS., and Denn MM., “Well slip in and extrudate distortion in liner low-density Polyethylene”, Journal of Rheology, Vol.31, pp.815-834, (1987).

XI. Ramamurthy Av., “Wall slip in Viscous Fluids and influence of material of Construcation”, Journal of Rheology,Vol.30 pp.337-357, (1986).

XII. Kraynik AM., and Schowater WR., “Slip at the wall and extrudate roughness with aqueous solutions polyvinyl alcohol and sodium borate”. Journal of Rheology, Vol.25, pp.95-114, (1981).

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Three Dimensional Couette Flow With Transpiration Cooling Between Two Horizontal Parallel Porous Plates


R.C. Chaudhary, M.C. Ghoyal, Umesh Gupta



The couette flow between two horizontal parallel porous flat plates with transverse sinusoidal injection of the fluid at the lower plate and it's corresponding removal by constant suction through the upper plate has been analyzed when both the plates are in motion. Due to this type of injection, the flow becomes Three-dimensional. For small perturbation approximate, the analytical method is applied to obtain the expressions for the velocity and temperature fields. The effect of injection/ suction velocity on the flow field, skin Friction and heat transfer are reported and discussed with the help of graphs and tables.


porous plate, Couette flow,transpiration colling,


I. H. Schlichting: Boundary Layer Theory, McGraw Hill, New York (1960).

II. K.Gerstan and J.F. Gross: J. Appl. Maths. Phys., ZAMP, 25(1974) 399.

III. P.Singh, V.R.Sharma and U.N. Mishra: Appl. Sci. Res., 345(1978), 105.

IV. P.Singh, V.R.Sharma and U.N. Mishra: Int. J. Heat Mass transfer, 1,(1978) 1117.

V. K.D. Singh: ZAMM, 73 (1993) 58.

VI. R.C. Chaudhary; Pawan Kumar Sharma: Jour. of Zhejiang Univ. Sc., 4(2003), 181.

VII. E.R.G. Eckert: Heat and Mass transfer, McGraw Hill, New York(1958)

VIII. K.D. Singh; Rakesh Sharma: Z. Naturforsch., 56a(2001) 596.

IX. K.D. Singh: ZAMP, 50(1999) 661.

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Nonlinear Vibrations Of Axisymmetric Thin Circular Elastic Plates Under Thermal Loadi


Utpal Kumar Mandal, Paritosh Biswas



This paper studies Nonlinear free vibration of thin circular plates with clamped immovable boundary under thermal loading. A steady-state temperature, field, characterized by constant surface temperatures measured from stress free temperature, is considered. The basic governing differential equations have been derived in the von Karman sense in terms of displacement components and solved with the help of Galerkin Procedure. Parametric studies have been presented to understand the Nonlinear free vibrations of thin isotropic elastic circular plates under thermal loading. This study reveals some interesting Nonlinear dynamic features of such structures which may prove useful to the designers.


elastic plate ,vibration ,thermal loading ,surface temperature ,stress free temperature,


I. Pal, M.C., 1969, “Large Deflections of Heated Circular Plates,” ACTA Mechanica, Vol.8,pp.82-103.

II. Pal, M.C.,1970a, “Large Amplitude Free vibration of Circular plates subjected to Aerodynamic Heting,” International Journal of Solid and Structures, vol.6, pp.301-313.

III. Jones R. and Mazumdar J., 1974, “Transverse Vibrations of Shallow Shells by the Method of constant Deflection Contours”, Journalof Acoustical Society of American, vol.56, No.5 pp.1487-1492.

IV. Biswas, P. and Kapoor, P., 1984a, “Nonlinear free vibrations and Thermal Buckling of Circular Plate at Elevated Temperature”. Indian Journal of pure and Applied Mathematical, vol.15,no.7, pp.809-812.

V. Bswas, P. and Kapoor, P, 1984b, “Nonlinear free vibrations of orthotropic circular plates at Elevated Temperature”. Journal of the indian isstitute of Science, Bangalore, vol.65(B), pp.87-93.

VI. Sachdeva, R.C., (1988), “Fundamentals of Engineering Heat and Mass Transfer”, New Age International (P) Limited, Publishers (ISBN: 81-224-0076-0).

VII. Chia, C.Y., 1980, “Nonlinear Analysis of Plates,”  McGraw Hill International Book Company.

VIII. Nash,W. and Modeer, J., 1959, “Certain Approximate Analysis of the Nonliner Behavior of Plates and Shallow Shells,”  Proceedings of Symposium on the Theory of Thin Elastic Shells, Delft, The Netherlands,pp. 331.


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On The Generalised Order And Generalised Type Of Differential Monomials And Differential Polynomials


Sanjib Kumar Dutta



In the paper we study he relation between the generalised order (generalised type) of a transcendental meromorphic function and that of a differental monomial by it. We also establish some theorems on the relationship between the generalised order ( generalised type) of a meromorphic function and that of a differental polynomial generated by it under  different conditions.


differential monomials,meromorphic function,differential polynomial,


I. N. Bhattacharjee and I Lahiri: Growth and value distribution of polynomials, Bull. Math. Soc. Sc. Math. Roumanie Tome, Vol.39(87), np.1-4(1996), pp.85-104.

II. W. Doeringer: Exceptional values of differential polynomials, Pacific J.Math, vol.98, no.1(1982), pp.55-62.

III. W.K.Hayman: Meromorphic function, The Clarendon Press, oxford (1964).

IV. I.Lahiri: Generalised order of the derivative of a Meromorphic function, Soochow J.Math. Vol.14, no.1(1988), pp.85-92.

V. I.Lahiri: Deficiencies of differental polynomials, Indian J.Pure Appl. Math,, pp.435-447.

VI. I.Lahiri and S.K.Dutta : Growth and Value distribution of differential monomials, Indian J. Pure Appl. Math., vol.32, no.12(2001), pp.1831-1841.

VII. D. Sato : On the rate of growth of entire functions of fast growth, Bull. Amer. Math.Soc., vol.69(1963), pp.411-414.

VIII. L.R.Sons : Deficiencies of monomials, Math.Z, vol.111(1969), pp.53-68.

IX. L.Yang : Value distribution theory and new research on it, Science press, Beijing.(1982).

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Expansion OF A Spherical Cavity At The Center Of A Non-Homogeneous Sphere Of Ductile Metal Under Internal And External Pressures


L.K. Roy



The aim of this paper is to investigate the distribution of stresses due to expansion of a spherical cavity at the center of a non-homogeneous metallic sphere of finite radius for an elasto-plastic solid under an increasing internal pressure, the external pressure remaining constant. The non-homogeneity of the elastic material is characterised by supposing that the lame constrants very exponentially as the function of radial distance. The case of ideal plastic solid has been deducted from this general case.


non-homogeneous sphere, ,ductile metal, ,internal and external pressure, spherical cavity, ,


I.  R. Hill (1950) : Theory of Plasticity, Oxford University Press, p-317

II.  A.E.H. Love(1952) : The Mathematical Theory of Elasticity, Dover Publication, p-164, London.

III.  Saint-Venant(1865) : Jour. De-Math, Primes at appl.(lonvilla) t-10.

IV.  S.G. Lekhnitskii (1963) : Theory of elasticity of an Anisotropic Elastic Body, Holden-Dey, Inc, p-390.

V.  H.C.Hopkins (1960) : Progress in solid Mechanics, Vol.1, p-80, Edited by I.N.Eneddon and R.Hill, North Holland Publishing Company, Amsterdem.

VI.  P.R.Sengupta (1969) : Ind.Jour. Mech and Math., Special issue, aprt-II, p-80, prof. B.sen, D.Se., F.N.I., 70th Birth Anniversary volume.

VII.  L.K. Roy (1992) :  Proc. Nat. Acad. Sci. India, 62(A), III p-445.

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Krylov-Bogoliubov-Mitropolskii (KBM) Method For Fourth Order More Critically Damped Nonlinear System


M. Ali Akber, Md. Sharif Uddin, Mo. Rokibul Islam, Afroza Ali Soma



Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended for sotaining of forth order more Critically Damped Nonlinear Systems. The results obtained by the presented KBM method show good coincidence with numerical results obtained by Runge-Kutta method. The method is illustrated by an example.


critically damped,non-linear system,KBM method,Runge-Kutta method,


I. Ali Akber, M., M.A. Sattar and A.C. Paul, An Asymptotic Method of Krylov-Bogoliubov for Forth Order-Damped Nonliner Systems, Ganit. J. Bangladesh Math. Soc., vol.22, pp.83-96, 2002.

II. Ali Akber, M., M.Shyamsul Alam and M.A.Satter, Asymptotic Method for Forth Order-Damped Nonliner Systems, Ganit. J. Bangladesh Math. Soc., vol.23, pp.41-49, 2003.

III. Ali Akber, M., M.Shyamsul Alam and M.A.Satter, A Simple Technique for Obtaining Certain Over-damped Solutions of n-th order Nonlinear Differential equation, Soochow Journal of Mathematics vol.31(2), pp.291-299, 2005.

IV. Bogoliubov, N.N. and Yu. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York. 1961.

V. Krylov, N.N and N.N. Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947.

VI. Mendelson, K.S., Perturbation Theory for Damped Nonlinear Oscillations, J. Math. Physics, Vol.2, pp.3413-3415,1970.

VII. Murty, I.S.N., B.L.Deekshatulu and G. Krishna, on an Asymptotic Method of krylov-Bogoliubov for Over-damped Nonlinear Systems, J. Frank. Inst. Vol.288, pp.49-65. 1969.

VIII. Murty, I.S.N., A Unified Krylov-Bogoliubov method for Solving Order Nonlinear Systems, Int. J. Nonlinear Mech. Vol.6, pp.45-53, 1971.

IX. Popov, I.P., A Generalization of the Bogoliubov Asymptotic Method in the Theory of Nonlinear Oscillations (in Russia), Doll. Akad. USSR vol.3, pp.308-310, 1956.

X. Rokibul Islam M., M.Ali Akber, M.Samsuzzoha and Afroza Ali Soma, New Technique for Third order Critically Damped Nonlinear Systems, Acta Mathematics Vietnamica.

XI. Rokibul Islam M.,  Md. Sharif Uddin,  M. Ali Akber, M. Azmol Huda and S.M.S Hossain, New Technique for Fourth Order Critically Damped Nonlinear Systems, Calcutta Math. Soc.

XII. Sattar, M.A., An Asymptotic Method for second order Critically Damped Nonlinear Equations,  J.Frank. Inst. Vol.321, pp.109-113,1986.

XIII. Sattar, M.A., An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems, Ganit, J. Bangladesh Math. Soc., Vol.13, pp.1-8, 1993.

XIV. Shamsul Alam, M. and M.A. Satter, an Asymptotic Method for third order Critically Damped Nonlinear Equations, J. Mathematical and physical Sciences, vol.30, pp.291-298,1996.

XV.  Shamsul Alam M. Asymptotic Methods for Second order Over-damped and Critically Damped Nonlinear Systems, Soochow Journal of Math. Vol.27, pp.187-200, 2001.

XVI. Shamsul Alam M., Bogoliubov’s method for third Order Critically Damped Nonlinear Systems, Soochow J. Math. vol.28, pp.65-80,2002.

XVII. Shamsul Alam M., On some Special Conditions of Third order Over-damped Nonlinear Systems, Indian J. Pure appl. Math. vol.33, pp.727-742, 2002.

XVIII. Shamsul Alam M., A Unified Krylov-Bogoliubov-Mitropolskii  Method for Solving n-th order Nonlinear Systems, J. Frank. Inst. vol.339, pp.239-248, 2002.

XIX. Shamsul Alam M.,  Asymptotic Method for non-oscillatory Nonlinear Systems, Far East J. Appl. Math., vol.7, pp.119-128, 2002.



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Propagation Of Waves In A Microstretch Elastic Solid Layer


D.P.Acharya, Chaitali Maji



  • Starting from the fundamental equations of motion for liner homogeneous isotropic microstretch elastic solid media, two dimensional wave propagation in a microstretch layer has been investigated in this paper. Under suitable boundary conditions concerned frequency equations involving a eighth order determinant has been obtained. Expressing the determinant as a product of two fourth order determinants, several possibilities and the corresponding wave velocities have been found out in closed forms. Two interesting particular cases when the large of the wave is very small or large relative to the thickness of the layer have been discussed. Graphs have been drawn to highlight the effect of microstretch and micropolarty in the propagation of waves. It is found that the wave velocity increases with the increase of the microelastic parameter while the stretch character of the medium causes diminution of the wave velocity.


microstretch layer ,wave propagation ,micropolarity,wave velocity,


I.  Acharya, D.P. and Sengupta, P.R., 1976, surface waves in the micropolar thermo-elasticity, Acta Geophysica, Vol XXV, No.4.

II.  Acharya, D.P. and Sengupta, P.R., 1979, Two dimensional wave propagationin a micropolar thermo-wlastic layer with stretch, Int.J.Engng, sci. vol-17, pp-1109-1116.

III.  De, S.N. and Sengupta, P.R., 1974, Surface waves in micropolar elastic media, Bull., Del’Acad Polon Sci., Ser.Sci techn, vol XXII, no.3 pp-137-146.

IV. Eringen, A.C., 1990, Theory of thermo- microstretch elastic solids, Int. J. Engng. Sci. 28 1291-1301.

V. Eringen, A.C., 1994, Mechanics of micromorphic materials, in: H. Gortler (Ed.), Proc. 11th Int. Congress of Appl. Mech., Springer Verlag, New York.

VI. Eringen, A.C., Suhubi, E.S., 1964, Nonlinear theory of simple microelastic solids- I, Int. J. Engng. Sci. 2, 189-203.

VII. Eringen, A.C., 1998, Mechanics of micromorphic continua, in: Kroner (Ed.), IUTAM Symposium Mechanics of Generalized continua, Springer-Verlag, New-York, pp. 18-35.

VIII.Eringen, A.C., 1999, Microcontinuum Field Theories: Foundations and Solids, Springer-verlag, New York, Inc.

IX.Eringen, A.C., 2004, Electromagnetic theory of microstretch elasticity and bone modeling, Int. J. Engng. Sci. 42, 109-122.

X. Iesan, D., Scalia, A., 2003on complex potentials in the theory of microstretch elastic bodies, Int. J. Engng. Sci. 41, 1989-2003.

XI. Iesan, D., Quintanilla, R.1964, Existence and continuous dependence results in the theory of microstretch elastic bodies, Int. J. Engng. Sci.32, 991-1002.

XII. Iesan, D., Nappa,  L., 1994, Saint venant’s problem for microstretch elastic solids, Int. J. Engng. Sci. 32, 229-236.

XIII. Kumar, R., Deswal, S., 2001, Disturbance due to mechanical and tharmal sources in a generalized thermo-microstretch elastic half-space, Sadhana 26(6) 529-547.

XIV. Lamb, H., 1916, On waves in an elastic plane, Proc. Roy Soc. S. 93, 114-128.

XV. Mondal, A.K. and Acharya, D.P., 2006, Surface waves in a micropolar elastic solid containing voids, Acta Ceophysica, Vol.54, No.4, pp 430-452.

XVI. Nowacki, W.,1970, Theory of Micropolar elasticity, International centre for Mechanical Sciences, udine couses and iecture No.25, Springer-verlag, Berlin.

XVII. Parfitt, V.R., Eringen, A.C., 1969, Reflection of plane waves from a flat boundary of a micropolar elastic half-space, J. Acoust. Soc. Am. 45, 1258-1272.

XVIII. Rayleigh, F.W., 1889, on the free vibration of an infinite plate of homogeneous isotropic elastic material, Proc, Math, Soc. 20, 225-234.

XIX. Singh , B., 2002, Reflection of nplane waves from free surface of a microstretch elastic solid, Proc. Indian Acad. Sci. (Earth Planet. Sci.) 111, 29-37.

XX. Singh, B., Kumar, R., 1998, Wave propagation in a generalized thermo_microstretch elastic soild, Int. J. Engng. Sci. 36, 819-912.

XXI. Suhubi, E.S. Eringen, A.C., 1964, Nonlinear theory of microelastic solids II, Int. J. Engng. Sci.36, 891-912.

XXII. Tomar, S.K., Gogna, M.L., 1992, Reflection and refraction of a longitudinal microrotational wave at an Interface between two micropolar elastic solids in welded contack, Int. J. Engng.Sci. 30, 1637-1646.

XXIII.Tomar, S.K., Gogna, M.L., 1995, Reflection and refraction of coupled transverse and microrotational waves at an interface between two different micropolar elastic media in welded contact, Int. J. Engng. Sci.33, 485-496.

XIV. Tomar, S.K., Kumar, R., 1999, Wave propagation at liquid/micropolar  elastic soloid interface, J. Sound. Vibr. 222(5), 858-869.

XV. Tomar, S.K., Garg, M; 2005, Reflection and transmission of waves from a plane interface between two microstretch solid half-spaces, International Journal of Engng. Sci. 43, 139-169.



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Iterative Solution For Pulsatile Flow Of Blood Through An Artery


A.K. Maity



The effect of magnetic field on pulsatile flow of blood through an artery is considered treating blood be a suspension of small uniformly sizes spherical particles. Following an iterative scheme, the solution with three significant correction terms over the classical solution is obtained. The numerical computation of velocities (of the suspension and the particles) for varying radial coordinates and the wall shear strees for varying time are carrid out, graphed and discussed.


pulsalite flow of blood,spherical particle ,shear stress, artery ,


I. Poiseuille, J.L.M., Memoris Present par Divers Savants a L`Academic Royal des Sciences- de I’Institut de France, 9(1946) 433.

II. Lambossy, P., Helv. Physical Acta, 25 (1952) 371.

III. Womersley, J.R., J. Physiology, 127 (1955) 553.

IV. Womersley, J.R., Phys. Med. Boil. 2 (1957) 178.

V. Lieber Stein, H.M., Mathematical physiology, Elsevier, N.Y. (1973).

VI. Sankarasubramanian, K. and Naidu, K.B., Ind. J. Pure & Appl. Math., 18(1987) 557.

VII. Skalak, R., Machanics of Microcirculation in ‘Biomechanics’, It’s Foundation and objectives, Edited by Y.C. FUNG, Prentice-  Hall Englewood Cliffs, New Jersey. (1966).

VIII. Sobin, S.S., Tremer, H.M. and FUNG, H.C., Circulation Res. 26 (1970) 397.

IX. Dasgupta, S.and Chaudhary, S. J. Ind. Acad. Math. 16(1994) 56.

X. Liu, J.T.C., Astronaut. Acta 13(1067) 369.

XI. Healy, J.V. and YANG, H.T., Astronaut. Acta 17 (1972) 851.

XII. Gupta, R.K. and Gupta, S.C., ZAMP 27(1976) 119.


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On Bitopological Spaces


Ajoy Mukherjee, Arup Roy Choudhury, M.K. Bose



In this paper, we introduce weakly pairwise regular spaces and considering a weakly pairwise regular spaces, we prove a theorems on pairwise paracompactness as analogue of Michael's characterized of paracompactness of regular spaces.


Regular space,Pairwise regular space,Paraconpactness,


I. P. Fletcher, H.B. Hoyle III, and Patty C.W., ‘The comparison of topologies,’ Duke Math. J. 36(1969), 325-331.

II. Kelly J.C., ‘Bitopological space’, Proc. London Math. Soc. (3)13(1963), 71-89.

III. E.Michael, ‘A note on paracompact spaces’ Proc. Amer. Math. Soc. 4 (1954), 831-838.

IV. T.G. Raghavan and I.L. Reilly, A new bitopological paracompactness’ J. Austral. Math. Soc. (Series A) 41 (1986), 268-274.

V. S. Willard, General Topology, Addison-Wesley, Reading, 1970.

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MHD Flow And Heat Transfer Of Micropolar Visco-Elastic Fluid Between Two Parallel Porous Plates With Time Varying Suction


N.T.M. Eldabe, Mona A.A. Mohamed , Mohamed A. Hagag



Magnetohydrodynamic (MHD) flow and transfer of an incompressible electrically conducting micropolar visco-elastic fluid between two infinite parallel horizontal non conducting plates is studied taking into consideration the action of a transverse magnetic flied that is perpendicular to the plates. The two plates are kept at different but constant temperatures. The solutions of equations which governing the flow are obtained by using perturbation technique equations and finite difference approximation. The effects of various physical parameters acting on the problem are discussed and graphical representation for the velocity, angular velocity, the induced magnetic field and temperature are also given.


MHD flow ,heat transfer ,micropolar Visco-elastic fluid ,plates,


I. Cramer, K.R. and Pai, S -1 Magentofluid Dynamics for Engineers and Applied Physicits, McGraw- Hill NY, USA (1973).

II. Tani, I.J. of Aerospace Sci. Vol.29 p, 287(1962).

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IV. Soundalgekar,  Soundalgekar, .. and  Uplekar, A.G. IEEE Trans. Plasma Sci. PS-14, p.579 (1986).

V. Attia, H.A. Can. J. Phys. vol.76, p.739 (1998).

VI. A.C. Eringen, Int. J. Eng. Sci.2(1964), 205.

VII. Eldabe, N.T. and Elmohandis, M.G. Fluid Dynamic Research,15(1995), 313-324.

VIII. Eldabe, N.T. and Hassan. A.A. Can. J. Phys. Vol. 69. 1991.

IX. K.Walters. Second-order effect in Elasticity, Plasticity and fluid dynamics. Pergamon Press Ltd. Oxford. 1964.P.507.

X. R. Kant. Indian J. Pure Appl. Math.11.(4), 468(1980).

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