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Abstract:

1. The problem of unsteady magneto hydrodynamic flow with heat transfer of a non-Newtonian fluid obeying power low fluid in a porous medium between two coaxial cylinders is investigated when the inner cylinder is at rest and the outer cylinder rotates with constant velocity, taking into account pulsation the pressure gradient and Darcy dissipation term. A Rung-Kutta-Merson method and a Newtown Iteration in a shooting and matching technique are used to obtain the solution of the system Equations of the problem. The velocity and temperature distributions are obtained as a perturbation technique. During this work we calculate an estimation of the global error by using Zadunaisky technique. The effects of behaviour index, Reynolds number, steady state part of the pressure gradient, the amplitude of the oscillatory part, the magnetic parameter, the permeability parameter, Forschheimer number, Prandtl number, Eckert number on the velocity and temperature distributions of Newtown and non- Newtown fluid are evaluated and depicted graphically.

Keywords:

Non-Darcain effect,Fluid flow ,Heat transfer,Rotating calender.,

Refference:

I. Long Q., Xua X.Y., Ramnarine K.V. and Hoskins P. Journal of Biomechanics 34(2001), 1229.

II. Phillips E.M. And Chiang S.H. Int. J.Engng Sci. 11(1973), 579.

III. McDonald D.A. “Blood flow in arteries”, Edward Arnold, London. 1974.

IV. Bitoun J.P. and Bellet D.: Biorheology, 23(1986), 51.

V. Ahmed S.A. and Giddens D.P.:  J. Biomech, 17(1984),695.

VI. Lieber B. pH. D. Thesis, Georgia Institute of Technology, Atlanta (1985).

VII. Ojha M., Cobbold R.S.C., Johnston K.W. and Hummel R.J. Fluid. Mech. 203 (1989), 173.

VIII. Hong J.T., Tien C.L. and Kaviany M.Int. J. Heat Mass Transfer. 28 No. 11(1985), 2149.

IX. Andresson H.I.,  Bech K.H. and Dandapat B.S.  Int. J. Nonlinear Mechanics. 27No.6(1992),929.

X. Nakayama A. Transactions of the ASME. 114(1992), 642.

XI. Nakayama A. Int. J. Heat and Fluid. 14 no.3(1993). 278.

XII. Agrawal A.K. and Sengupta S. Int. J. Heat and Fluid Flow. 11. No.1(1990), 54.

XIII. Mokhtar A. Abdelnaby, Nabil T.M Eldabe and Mohammed Y. Abo zeid: J.Ind.Theo.Phys. (2006), in press. 4164.

XIV. Nabil T.M. Eldabe, Sadeek G. and Asma F. El-Sayed. Phys. Soc. Japan. 64(1995), 4165.

XV.  Batra R.L. and Biggani Das: Fluid Dynamic Research, 9 (1992), 133.

XVI. Ibrahim F.N. phys. D: APPL. Phys. 24(1991), 1293.

XVII. Zadunaisky, P.E: Numer. Math. 27(1976),21.

 

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Abstract:

1. The object of this paper is to study the torsional vibration of an in-homogeneous elastic cone. For in-homogeneous of the material considered it is assumed that the elastic constants and the density of the material very exponentially as the radial distance. Two broad cases of end condition have been taken into account. Displacements and stresses for a particular case have been obtained and are shown in tabular from and graphically for different values of radial distance.

Keywords:

Torrtion vibration,Elastic cone,In-homogeneous,Stress,

Refference:

I. Abramovitz, M.: Handbook of Mathematical Functions, Dover and Publications, New York, 1970 stegun,A.

II. Banerjee, A.: ‘Torsional vibration in a circular cylinder’, Bull. Calcutta. Math. Soc. 72, 309-314, 1989.

III. Bullen, K.E. An Introduction to the Theory of Seismology 2nd. Edn. Cambridge University Press, 1976.

IV. Bhanja, N.: ‘ Torsional vibration problem for a cone of spherically anisotropic material, Ind. Jour. Mech. Math. Vol-8,2,160-167,1970.

V. Campbell, J.D.:  ‘Torsional vibration in a circular cylinder and Q.J.M.A.

VI. M.25, 74-84, 1972. Tsao, M.C.C

VII. Davies, R.M. “Torsional vibration in a circular cylinder “, Surveys of Mechanics, Cambridge University Press, 1959.

VIII. Koslky, H. “Torsional vibration in stress waves in solids.

IX. Love, A.E.H.: “The Mathematics Theory of Elasticity” 4th Edn. Cambridge University Press, 1962.

X. Mitra, A.K.: “Torsional vibration in a circular cylinder” J.Sci. Engng. Res. No.2, 251-28, 1961.

XI. Mondal, N.C: “Free Torsional vibration of a non-homogeneous semi-solid circular cylinder” Proc. Indian Bath. Sci. Acad. 52A, 512-20, 1986.

XII. Mukherjee, S. : ‘ Torsional vibration of an isotropic material.’ Ind. Jour, Mech. Math. Vol. V, 1967.

XIII. Rao, Y.B. : ‘ Torsional vibration in an isotropic homogeneous infinitely long solid circular cylinder with rigid boundary.’  Proc. Indian Natn. Sci. Acad. 52A, 497-501, 1986.

XIV. Watson,  G.N.: ” A treatise on Theory of Bessel Functions”.

XV. Cambridge University Press, NY 10022, USA, 1922.

XVI. Watson, G.N.: ” The Theory of Bessel Functions” 2nd Ed.,  Cambridge University Press,1948.

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Abstract:

In this paper, development of secondary flow and unsteady by using the spectral method. Numberical calculations are carried out for the Grashof number Gr=1000 over a wide range of the Dean number,0≤Dn≤1000, and the curvature,0<ẟ≤0.5, where the outer well is heated and the inner wall is cooled. First steady solutions are obtained by the Newton-Raphson Iteration method. As a result, we obtain five branches of asymmetric steady solutions with one -two-four-six and eight-vortex Solution at the same Dean number. Then,time evolution calculations of the unsteady solutions are performed, and it is found that the steady flow turns into chaotic flow through periodic flows, no matter with the curvature is Finally, the complete unsteady Solution, covering the wide range of dn and ẟ are shown by a phase diagram.

Keywords:

Secondary flow,Curved duct,Chaotic flow,Vortex Solution,

Refference:

I. Berger, S.A. Talbot, L. And Tai, L.S., Flow in curved pipes, Annual Review of Fluid Mechanics, Vol 35, pp. 461-512,1983.

II. Dean, W.R.., Note on the motion of Fluid in a curved pipe, Philosophical.

III. Ito, H., Flow in curved pipes, JSME International Journal, vol.30, pp.543-552,1987.

IV. Mondal, R.N. 2006. Isothermal and non-isothermal flows through curved  ducts with square and rectangular cross sections, PH.D. Thesis, Department of Mechanical Engineering, Okayama University, Japan.

V. Mondal, R.N., Kaga, Y., Hyakutake, T. and Yanase, S., Effect of curvature and convective heat transfer in curved square duct flows, Trans. ASME, Journal of Fluids Engineering, vol.128(9), pp. 1013-1023,2006.

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Abstract:

The aim of the present paper is to investigate interface waves (surface waves) of Earthquakes in second order thermo-viscoelastic solid media under the influence of gravity. The displacement components are expressed in terms of displacement potentials. The problem of surface waves, particularly, Rayleigh waves, Love waves and stoneley waves have been determined. All final results and Equations are in fair agreement with the corresponding classical results when the effect of temperature, viscosity and gravity are ignored.  

Keywords:

Thermo-viscoelastic solid,Surface wave,Rayleigh wave,Wave velocity,,Gravity,

Refference:

I. Love AEH(1911) Some problems of Geodynamics. Dover, New York.

II. Stoneley R (1924) Elastic waves at the surface of separation of two solids. Proc. Roy. Soc. London A-106, pp. 416-428.

III. Bullen KE(1965) An introduction to the Theory of Seismology. Cambridge University Press,. London. pp.85-99.

IV. Jeffreys. Sir H (1970) The Earth, Cambridge University Press.

V. Bromwich TJIA(1898) On the influence of gravity on elastic waves and in particular on the vibrations of an elastic globe. Proc. London Math. Soc.30, pp.98-120.

VI. De SN and Sengupta PR (1975) Therm-elastic Rayleigh waves under the influence of gravity. Gerlands Betir. Geophysik. 84(6), pp.509-514.

VII. De SN and Sengupta PR (1976) Surface waves under the influence of gravity. Gerlands Betir. Geophysik.85(4), pp.311-318.

VIII. Das TK and Sengupta PR(1992) Effect of gravity on visco-elastic surface waves in solids involving time rate of strain and stress of first order. Sadhana. 17(2), pp.315-323.

IX. Das TK, Sengupta PR, and Debnath L (1995). Effect of gravity on visco-elastic surface waves in solids involving time rate of strain and stress of higher order. Int. J. Math. Sci. 18(1), pp. 71-76.

X. Ghosh NC, Roy I and Biswas PK. (2000)  Effect of gravity and couple-stress on thermo-visco-elastic Rayleigh waves involving stress rate and strain rate. J.Bih. Math. Soc. 20, pp.89-99.

XI. Boot MA (1965)  Mechanics of incremental deformations, Theory of elasticity and viscoelasticity of initially stressed stressed solids and fluids, including thermodynamic foundation and application to finite strain, John Wiley & Sons, New York. 44-45, 273-281.

XII. Mukherjee A and Sengupta PR(1991) Surface waves in thermo visco-elastic media of higher order. Ind. J. Pure. APPL. Math.22(2), pp.159-167.

 

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Abstract:

In this paper some properties of anti fuzzy subgroups have been introduced.

Keywords:

Fuzzy set,Fuzzy subgroup,Anti-fuzzy subgroup,

Refference:

I. Kandasamy w.b. Vasantha, Samarandache Fuzzy Algebre- American Research Press, Rehoboth (2003), 22-26.

II. Majumder. S.K and Sardar. S.K; On fuzzy Magnified Translation (Communicated).

III. Rosenfeld. A, Fuzzy groups, J. Math. Anal. APPL. 35(1971), 512-517.

IV. Zadeh. L.A, Fuzzy Sets, Inform And Control 8(1965) 338-365.

 

 

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Abstract:

In this paper dynamic stresses and displacements are calculated in a large thick composite plate due to Torsional vibration under sharing forces applied on the free plane boundary being in contact with a rigid foundation. The applied shearing force on the free plane boundary is expressed in terms of Four lier -- Bessel integrals; in particular, case of Gaussian load has been treated in details to find distributions of stresses and displacements.

Keywords:

Torrtional vibration,Composite plate,Shearing force,Plane surface.,

Refference:

I. Chatterjee, P.P (1957), Proc. Th. And Appl. Mech P. 129.

II. Love, A.E.H. (1994), A Treatise on the Mathematical theory of Elasticity.

III. Timoshenko and Goofier (1951), theory of Elasticity.

IV. Acharya, D P. and Mondal, A Effect of rotation on Rayleigh surface waves under the linear theory of non-local elasticity, Ind. J. Theory. Phys. 52(1) (2004).

V. Yu, Yi-Yuan(1954), Quar. Journal of Mech. And App. Math., Vol.7 p. 287.

VI. Sengupta, P.R. (1964), Bull. Cal. Math. Soc.vol.56 , No.1, march, 1967. Pp. 33-40.

VII. Sengupta, P.R. (1965), Bull. Cal. Math. Soc.vol.57, No.2 & 3, June and Sept. (1965), pp.69-70.

VIII. Bhattachryya, P.C. and Sengupta,  P.R., propagation of waves in composite elastic layer, Jour. Sci. Res. Vol.3. (1981) pp.211-214.

IX. Sharma M.D. Effect of initial stress on reflection at the free surface of anisotropic elastic medium, J. Earth Syst. Sci. 116 (2007), pp.537-557.

X. Guptav, S. Chattopadhyaya, A and Kumari, P., Propagation of share waves in anisotropic medium ., Appl. Math. Sci., 1(2007) pp.2699-2706.

XI. Selim. M.M., Torsional Waves Propagation in an initially stresses dissipative cylender., Appl. Math. Sci. 1 (2007) pp.1419-1427

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Abstract:

An asymptotic method has been found to obtain approximate solution of a second of a second order Nonlinear Differential system based on the extension of Krylov-Bogoliubov-mitropolskii method, whose coefficients change slowly and periodically with time. Moreover a non-autonomous case also investigated in which an external periodic force acts in the system. The solutions for different initial conditions show a good agreement with those obtained by numerical method. The method is illustrated by examples.  

Keywords:

Non-linear system, ,Varying coefficent,Periodic force,Asymptotic Method,

Refference:

I. Poincare H. Les Methods Nouvelles de ls Mecanique Celeste, Paris, 1892.

II. Wetzel, G.Z. Physio 38 (1926) 518.

III. Kramer’s H.A., G.Z. physik 39 (1926) 828.

IV. Beillouin L., Compt. Tend. 183(1926) 24.

V. Frshcnko S.F., Shkil N.I. and Nikolenko L.D.,  Asymptotic Method in the theory of linear differential Equation, (in Russia), Naukova Dumka, Kiev 1966 ( English translation, Amer Elsevier publishing co. INC. New York 1967.

VI. Nayfeh A.H., Perturbation Methods, J. Wiley, New York, 1973.

VII. Krylov N.N. and N.N., Bogoliubov, Introduction to Nonlinear Mechanics. Princeton University Presses, New Jersey, 1947.

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

IX. Mitropolskii Yu., Problemson Asymptotic Method of non-stationary Oscillations (in Russian), Izdat, Nauka, Moscow,(1964).

X. Bojadziev G.  And Edwards j, On some asymptotic methods for non-oscillatory and processes, Nonlinear vibration problem 20(1981) 69-79.

XI.  Murty I.S.N., A Unified Krylov-Bogoliubov method for second order Nonlinear Systems, Int. J. Nonlinear Mech. 6 (1971) 45-53.

XII. Shamsul Alam M., Unified Krylov-Bogoliubov- Mitropolskii method for Solving n-th order Nonlinear Systems with slowly varying coefficients, Journal of sound and vibration 256 (2003) 987-1002.

XIII. Hung Cheng and Tai Tsum Wu. An aging spring, Studies in Applied Mathematics 49(1970) 183-185.

XIV. K.C. Roy, Shamsul Alam M, Effect of higher approximation of Krylov-Bogoliubov-Mitropolskii solutions and matched asymptotic Solution of a differential system with slowly varying coefficients and damping near to a turning point, Vietnam Journal of Machanics, VAST, 26 (2004) 182-192.

XV. Shamsul Alam M, Perturbation theory for damped Nonlinear Systems with large damping, India J. Pure and APPL. Math. 32(2001) 1453-1461.

XVI. Shamsul Alam M, Bellal Hossain M. Shanta S.S., Approximate Solution if non-oscillatory systems with slowly varying coefficients, Ganit (Bangladesh J. of Math. Soc.) 21 (2001) 55-59.

XVII. Popov I.P., A Generalization of the Bogoliubov Asymptotic Method in the Theory of Nonlinear Oscillations (in Russia), Doll. Akad. Nauk. SSSR 111(1956) 308-310.

XVIII. Minorski N., Nonlinear Oscillations, Princeton, Von Nostrand Co. 1962.

XIX. G.N. Bojadziev and C.K. Hung Damped oscillation modeled by a 3-dimesionaltime dependent differential system, Acta Mech, 53(1984) 101-114.

XX. Shamsul Alam, Damped oscillation modeled by an n-th order time dependent quasi-linear differential system, Acta Mech. 169(2004) 111-122.

 

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