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Abstract:

• 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.

Keywords:

microstretch layer ,wave propagation ,micropolarity,wave velocity,

Refference:

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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Abstract:

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.

Keywords:

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

Refference:

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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Abstract:

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.

Keywords:

Regular space,Pairwise regular space,Paraconpactness,

Refference:

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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Abstract:

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.

Keywords:

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

Refference:

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).

III. Soundalgekar, V.M.; Vighnesam, N.V. and Takhar, H.S. IEEE Trans. Plasama Sci.PS-7, p.178(1979).

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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Abstract:

The paper is concerned with the radial vibration of a spherial shell whose young's modulus is a function of the radial distance from the sphere and the poisson's ratio is taken to be constant. The frequency equations for the period of vibration has been determined by perturbation method considering the vibration parameters to be small.

Keywords:

spherical shell,radial vibration,period of vibration,

Refference:

I. Love, A.E.H. (1944), A Treatise on the Mathematical theory of Elasticity, Dover Publication (1944) p-142.

II. Paris, G. (1963): Jour. Of Sci. and Engng. Res., 7,1-4.

III. Charaborty, J.G., Radial and rotatory vibration of a spherical shell of aeolotropic elastic material, Bull. Cal. Math. Soc. Vol.147,no.4 (1965)

IV. Sur, S.P., Radial and rotatory vibration of a Spherical shell of aeolotropic elastic material, Bull. Cal. Math. Soc. Vol..47, no.4 (1965).

V. De, P.K. (1968): Bull. Cal. Math. Soc. 60, no.4.

VI. Pan, M.(1975): pH.D.(Se.) Thesis, C.U.

VII. Sengupta, P.R. and Roy, S.K., Radial vibration of a sphere of general visco elastic solid, Garlands Beitr. Geophysik, Leipzig, 92, 5, s. 435-442(1983).

VIII. Ezzat, M.A., Fundamental Solution in Generalized megneto-thermo-elasticity with two relaxation times for perfect conductor cylindrical region, Int. J. Eng. Sci., 42, pp 1503-1519 (2004).

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

X. Sengupta, S., Roy, I., Chakraborty, H.S., Redial vibration of a non-homogeneous anisotropic elastic spherical shall with inclusion, J. Mech. Cont. and Math. Sci. Vol .2, no.1, pp1-9, (2007).

 

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Abstract:

The aim of the present paper is to investigate the propagation of waves in a magneto­visco-elastic initially stressed electrically conducting plate of finite thickness involving time rate of strain and stress of higher order. The initial stress is assumed to be of the nature of hydrostatic tension or compression. The normal mode analysis is used to obtain the wave velocity equations for the waves propagated in the plate bounded by stress free plane boundaries. The wave velocity equations in different cases, obtained in this paper may be considered as more general in the sense that the results presented by other authors may be obtained as special cases in the absence of additional fields. Numerical computations are carried out and the effects of higher order viscoelasticity, magnetic field and initial stress on the phase velocity ratio are exhibited graphically.

Keywords:

visco-elastic plate,initial stress,magnetic fluid,wave propagation,

Refference:

1) X. Wang and H. L. Dai: Magnetothermodynamic stress and perturbation
of magnetic field vector in an orthotropic thermoelastic cylinder, Int. J. Engng. Sci, 42 (2004), 539-556.
M. A. Ezzat: Fundamental solution in generalized magneto thermoelasticity
with two relaxation times for perfect conductor cylindrical region, Int. J. Engng. Sci, 42 (2004), 1503-1519.
3) Rakshit M. and Mukhopadhyay B. : An electro-magneto-thermo-visco
elastic problem in an infinite medium with a cylindrical hole, Int. J. Engng. Sei, 43 (2005), 925-936.
4) Bakshi A., Bera R. K. and Debnath L.: A study of magneto-thermo elastic
problems with thermal relaxation and heat sources in a three-dimensional infinite rotating elastic medium, Int. J. Engng. Sci, 43 (2005), 1419-1434.
5) Roy Choudhuri S. K. : Magneto-thermo-elastic waves in an infinite
perfectly conducting solid without energy dissipation, J. Tech. Phys., 47 (2006), 63-72.
6) M. I. A. Othman and Y. Song: The effect of rotation on the reflection of
magneto-thermoelastic waves under thermoelasticity without energy dissipation, Acta Mech., 184 (2006), 189-204.
Higuchi M., Kawamura R. and Tanigawa Y.: Magneto-thermo-elastic
stresses induced by a transient magnetic field in a conducting solid circular cylinder, Int. J. Solids Struct., 44 (2007) 5316-5335.Acharya D. P. and Sengupta P. R.: Magneto-thermo-elastic waves in an
initially stressed conducting Layer, Gerlands Beitr. Geophys., 87 (1978), 229-239.
9) Das S. C., Acharya D. P. and Sengupta P. R.: Magneto visco-elastic surface
waves in stressed conducting media, Sildhani , 19 (1994), 337-346.
10) Wang J. and Slattery S. P.: Thermoelasticity without energy dissipation for initially stressed bodies, Int. J. Math. Math. Sci, 31 (2002), 329-337.
11) Othman M. I. A. and Song Y.: Reflection of plane waves from an elastic
solid half-space under hydrostatic initial stress without energy dissipation, Int. J. Solids Struct., 44 (2007), 5651-5664.
12) Sharma M. D.: Effect of initial stress on reflection at the free surface of anisotropic elastic medium, J. Earth Syst. Sci., 116 (2007), 537-551.
13) Selim M. M.: Torsional waves propagation in an initially stressed dissipative
cylinder, AppL Math. Sci. 1 (2007), 1419-1427.
14) Gupta S., Chattopadhyay A. and Kumari P. : Propagation of shear wave in anisotropic medium, Appi. Math. Sci. 1 (2007), 2699-2706.
15) Yu C. P. and Tang S. : Magneto-elastic waves in initially stress’ conductors, Z. Angew. Math. Phys., 17 (1966), 766-775.
16) De S. N. and Sengupta P. R. : Magneto-elastic waves and disturbances in initially stressed conducting media, Pure Appi. Geophys. 93 (1972), 41-54.
17) Roy Choudhuri S. K. and Banerjee M.: Magneto-viscoelastic plane waves
in rotating media in the generalized thermoelasticity II, Int. J. Math. Math. Sci, 11 (2005), 1819-1834.
18) Addy S. K. and Chakraborty N. R.: Rayleigh waves in a viscoelastic half
space under initial hydrostatic stress in presence of the temperature field, Int. J. Math. Math. Sci, 24 (2005), 3883-3894.
19) Song Y. Q., Zhang Y. C., Xu H. Y. and B. H. Lu: Magneto
thermoviscoelastic wave propagation at the interface between two micropolar viscoelastic media, Appl. Math. Compu., 176 (2006), 785-802.
20) Sharma J. N. and Othman M. I. A.: Effect of rotation on generalized
thermo viscoelastic Rayleigh-Lamb waves, Int. J. Solids Struct., 44 (2007), 4243-4255.
21) Yin-feng Z. and Zhong-min W. : Transverse vibration characteristics of axially moving viscoelastic plate, Appl. Math. Mech., 28 (2007), 209-218.
22) Rakshit M. and Mukhopadhyay B. : Visco-elastic plane waves in two
dimensions using generalized theory of thermo-elasticity, Bull. Cal. Math. Soc, 99 (2007), 279-292.
23) Voigt W. : Theortische student uberdie elasticitats verhalinisse krystalle, Abh. Ges. Wiss Goetting, 34 (1887).
24) Sengupta P. R., De N., Kar M. and Debnath L.: Rotatory vibration of
sphere of higher order viscoelastic solid, Int. J. Math. Math. Sci, 17 (1994), 799-806.
25) Othman M. I. A. : Effect of rotation on plane waves in generalized thermo
elasticity with two relaxation times, Int. J. Solids Struct., 41(2004), 2939¬2956.
26) Ezzat M. A., Othman M. I., El-Karamany A. S. : Electromagneto-
thermoelastic plane waves with thermal relaxation in a medium of perfect conductivity, J. Thermal Stresses, 24 (2001), 411-432.
27) Rayleigh F. W. : On the free vibrations of an infinite plate of homogeneous isotropic elastic material, Proc, Math. Soc.20 (1989), 225-234.
28) Lamb H. : On waves in an elastic plate, Proc. Roy. Soc. (London), 93 (1916), 114-128.
29) Eringen A. C. : On Rayleigh surface waves with small wave lengths, Applied and Engineering Sciences, 1 (1973), 11-17.
30) Acharya D. P. and Mandal Asit : Effect of rotation on Rayleigh surface
waves under the linear theory of non-local elasticity, Ind. J. Theor. Phys., 52(1) (2004).

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Abstract:

Krylov-Bogolov-Mitropolskii (KBM) method has been extended and applied to certain third order non-oscillatory nonlinear systems characterizing critically damped .stems, For different· set oj tnisia! Conditions as well as for different eigenvalues the solutions obtained by the extended (KBM), ·method show good coiricidetlce with those obtained by the numerical method. The method is iIIustrated by all example

Keywords:

microstretch layer,wave propagation,micropolarity,wave velocity,

Refference:

1) Ali Akbar, M., M. Shamsul Alam and M. A. Sattar, A Simple Technique for Obtaining Certain Over-damped Solutions of an n-th Order Nonlinear Differential Equation, Soochow Journal of Mathematics Vol. 31(2), pp. 291-299, 2005.
2) Bogoliubov, N. N. and Yu. Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961.
3) Bojadziev, G. N., Damped Nonlinear Oscillations Modeled by a 3-dimensional Differential System, Acta Mechanica, Vol. 48, pp. 193-201, 1983.
4) Krylov, N. N. and N. N. Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947.
5) Mendelson, K. S., Perturbation Theory for Damped Nonlinear Oscillations, J. Math. Physics, Vol. 2, pp; 3413-3415, 1970.
6) Murty, I. S. N., Deekshatulu, B. L. and Krishna, G. On an Asymptotic Method of Krylov-Bogoliubov for Over-damped Nonlinear Systems, J. Frank. Inst., Vol. 288, pp. 49-65, 1969.
7) Murty, I. S. N., and Deekshatulu, B. L., Method of Variation of Parameters for Over-Damped Nonlinear Systems, J. Control, Vol. 9, no. 3, pp. 259¬266,1969.
8) Murty, I. S. N., A Unified Krylov-Bogoliubov Method for Solving Second Order Nonlinear Systems, Int. J. Nonlinear Mech. Vol. 6, pp. 45-53, 1971.
9) Popov, I. P., A Generalization of the Bogoliubov Asymptotic Method in the Theory of Nonlinear Oscillations (in Russian), Doki. Akad. USSR Vol. 3, pp. 308-310, 1956.
10) Rokibul Islam, M., M. Ali Akbar, M. Samsuzzoha and Afroza Ali Soma, A New Technique for Third Order • Critically Damped Non-linear Systems, Research Journal Applied Science (Accepted for Publication).
11) Sattar, M. A., An asymptotic Method for Second Order Critically Damped Nonlinear Equations, J. Frank. Inst., Vol. 321, pp. 109-113, 1986.
12) Sattar, M. A., An Asymptotic Method for Three-dimensional Over-damped Nonlinear Systems, Ganit, J. Bangladesh Math. Soc., Vol. 13, pp. 1-8, 1993.
13) Shamsul Alam, M. and M. A. Sattar, An Asymptotic Method for Third Order Critically Damped Nonlinear Equations, J. Mathematical and Physical Sciences, Vol. 30, pp. 291-298, 1996.
14) 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.
15) Shamsul Alam, M., Bogoliubov’s Method for Third Order Critically Damped Nonlinear Systems, Soochow J. Math. Vol. 28, pp. 65-80, 2002.
16) 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.
17) 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.
18) Shamsul Alam, M., On Some Special Conditions of Over-damped Nonlinear Systems, Soochow J. Math. Vol. 29, pp. 181-190, 2003.

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Abstract:

The paper is concerned with the investigation of the steady unidirectional flow of a Micropolar fluid of finite depth due to tangential stress applied at the surface. Numerically it is found that the velocity component (u) of the Micropolar fluid increases to a significant extent over the classical fluid.. The miocropolar effect increases the velocity of the classical fluid and rotation component a increases with the increase of the depth

Keywords:

micropolar fluid,unidirectional floe,tangential stress ,

Refference:

1. Eringen, A.C. Theory of Micropolar fluids. J. Math. Mech. 16.1 (1967)
2. Willson, A.J. Basic Flows of Micropolar liquid. Appl. Sci. REs. 20, 338 (1969)
3. Willson, The flow of a Micropolar liquid layer down on inclined plane. Proc. Camb. Phil. Soc. 64, 513 (1968)
4. Willson, A.J. Boundary layer in Micropolar liquid, Proc. Camb. Phil. Soc. 67, 469 (1970)
5. Goldstem, Proc. Land. Math. Soc. (2), 34 (1931).51.
6. Schlichting, H. Boundary layer theory, p. 230, New York. Mc-Graw¬Hill Co. 1955.
7. Whittake, A.G. and Robinson, C. The calculus of observation, p. 125, Blackie and Son, 1944.

8. Gupta, P.S. and Gupta, A.S. Steady flow of Micropolar Liquids, Acta Mechanica, 15, 141-149 (1972).
9. Ghosh, P.C. and Sengupta, P.R. N.B.U. review (Sc. and Tech.) Vol 4 (No.– 2); (1983).
10. Ghosh, P.C. and Sengupta, (1983), Journal of Technology, (Shibpur E. College), Vol — XXVII, No. 1
11. Lamb, H. (1975) Hydrodynamics, Cambridge University Press, Sixth
12. Batchelor, G. K. (1967) An Introduction of fluid Dynamics, Cambridge University Press, Cambridge.
13. Panja,S (2006), J. Mech. Cont. Sci. Vol 1, No. 1, July, 2006, pp 46-52

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