Journal Vol – 2 No -1, July 2007

Radial Vibration Of a Non-Homogeneous Anisotropic Elastic Spherical Shell With Inclusion


Sudipta Sengupta, Indrajit Roy, H.S. Chakraborty



The aim of this paper is to study the radial vibration of a non-homogeneous sherically anisotroplc elastic spherical shell with an isotropic elastic inclusion as the core. The non-homogeneous of the material is characteised by taking linear vibration with radial distance of elastic parameters and mass density. This property of non-homogeneity is assumed to be satisfied by the entire shell of the sphere, while the core of the spherical shell behaves like an inclusion of isotropic homogeneouselastic mass. Satisfying the appropriate boundary conditions, the frequency of vibration of the composite solid sphere has been determined. results obtained by other authors may be deduced from our more general result as special cases.


spherical shell,anisotropic elasti ,radial vibration ,elastic inclusion,


I. A.E.H. Love , A treatise on the Mathematical theory of Elasticity, Dover publication.(1952).

II. A.E.H. Love, Some problem of Geodynamics, Cambridge University pPress, London.(1911).

III. R.F.S. Hearmon, An Introduction to Applied Anisotropic Elasticity, Oxford University Press, London.(1961).

IV. S.G. Lekhnitski, Theory of Elasticity of an Anisotropic elastic body. Holden Day Inc.(1963).

V. W. Olszak, non-homogeneous in Elasticity and Plasticity-Proceeding of the Internal Union of Theoretical and Applied Mechanics Symposium, pergamon press.

VI. P.R. Sengupta, Problem of twised elastic sphere with a concentric inhomogeneous spherical inclusion, Jour. Sci. Engg. Res. Vol-8, No-2, pp.193-203 (1964).

VII.P.R. Sengupta, Inclusion in elastic solids of work-hardening materil, Ind. Jour. Mech. & Math. Part-II, Special Issue. pp.80-89 (70th birth anniversasry volume of Prof. B.Sen, F.N.A.) (1969).

VIII.P.R. Sengupta, & A.N. Basumallick, Radial deformation of a non-homogeneous spherically anisotropic elastic sphere with a concentric spherical inclusion, Ind. Jour. Mech. & Math, Vol-8, No-2, pp.1-9(1970).

IX. J.G. Chakraborty, radial and rotatory vibration of a spherical shell of aeolotropic elastic material, Bull. Soc. Vol-47, No.4 (1965).

X. S.P. Sur, Radial and rotatory vibration of a sphere of non-homogeneous spherically aeolotropic material of unifrom density. Ind. Jour. Mech. Math. Vol-II, No.1 (1964).

XI. P.R. Sengupta & S.K. Roy, Radial vibration of a sphere of general viscoelastic solid, Gerlands Beitr. Geophysik, Leipzig, 92, 5, s.435-442 (1983).

XII. P.R. Sengupta & S.K. Roy, Rotatory vibration of a sphere of general viscoelastic solid, Gerlands Beitr. Geophysik, Leipzig, 92, 1, s.70-76 (1983).

XIII. H.M. Youssef, dependence of modulus of elasticity and thermal conductivity on reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity, Appl. Math. Mech., 26, 4,pp.470-475. (2005).

XIV. S. Banerjee & S.K. Roy Chowdhury, Spherically symmetric thermo-visco-elastic waves in a visco-elastic medium with a spherical cavity, Computers Math. Applic., 30,1, pp.91-98. (1995).

XV. M.Rakshit & B. Mukhopadhyay: An electro-magneto-thermo-visco-elastic problem in an infinite medium with a cylindrical hole, Int. J. Eng. Sci., 43, pp.925-936, (2005).

XVI. M.A.Ezzat.Fundamental solution in generalized magneto-thermoelasticity with two relaxation times for parfact conductor cylindrical region, Int. J. Eng. Sci., 42, pp.1503-1519, (2004).

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Refined Measure Spaces


Manoj Kumar Bose, Arup Roy Choudhury, Rupesh Tiwari



We introduce and study a refined real valued measure on a  -algebra


refined real number ,refined measure space,non-archimedean,


I. M.K. Bose and R. Tiwari, ‘The refined real number system and the refined measure’ , submitted for publication.

II. K.Hrbacek and T. Jech, introduction to set theory, Marcel Dekker, Inc., 1999.

III. A.Robinson, Nonstandard Analysis, Princeton University Press, 1996.

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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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