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Nonlinear fluid flows in pipe-like domain problem using variational-iteration method. (English) Zbl 1128.76019
Summary: We investigate the propagation of nonlinear waves in viscoelastic tube filled with incompressible viscous fluid. Various modified KdV equations and modified Burgers equations derived from the discussed problem are solved analytically by He’s variational-iteration method.

76D33 Waves for incompressible viscous fluids
76M30 Variational methods applied to problems in fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
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[1] Nakamura, M.; Swada, T., Numerical study on the unsteady flow of non-Newtonian fluid, J biomech eng trans ASME, 112, 100-103, (1990)
[2] Pak, B.; Young, Y.I.; Choi, S.U.S., Separation and re-attachment of non-Newtonian fluid flows in a sudden expansion pipe, J non-Newtonian fluid mech, 37, 175-199, (1990)
[3] Misra, J.C.; Patra, M.K.; Misra, S.C., A non-Newtonian fluid model for blood flow through arteries under stenotic conditions, J biomech, 26, 1129-1141, (1993)
[4] Tu, C.; Deville, M., Pulsatile flow of non-Newtonian fluid through arterial stenosis, J biomech, 29, 899-908, (1996)
[5] Das, B.; Johnson, P.C.; Popel, A.S., Effect of nonaxisymmetric hematocrit distribution on non-Newtonian blood flow in small tubes, Biorheology, 35, 69-87, (1998)
[6] Mandal, P.K., An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis, Int J non-linear mech, 40, 151-164, (2005) · Zbl 1349.76943
[7] Pedley, T.J., The fluid mechanics of large blood vessels, (1980), Cambridge University Press Cambridge · Zbl 0449.76100
[8] Fung, Y.C., Biodynamics: circulation, (1984), Springer New York
[9] Atabek, H.B.; Lew, H.S., Wave propagation through viscous incompressible fluid contained in an initially stressed elastic tube, Biophys J, 7, 480-503, (1966)
[10] Rachev, A.I., Effects of transmural pressure and muscular activity on pulse waves in arteries, J biomech eng, 102, 119-123, (1980)
[11] Demiray, H., Wave propagation through a viscous fluid contained in a prestressed thin elastic tube, Int J eng sci, 30, 1607-1620, (1992) · Zbl 0764.73065
[12] Rudinger, G., Shock waves in mathematical models of the aorta, J appl mech, 37, 34-37, (1970)
[13] Anliker, M.; Rockwell, R.L.; Ogden, E., Nonlinear analysis of flow pulses and shock waves in arteries, Z angew math phys, 22, 217-246, (1971)
[14] Tait, R.J.; Moodie, T.B., Waves in nonlinear fluid filled tubes, Wave motion, 6, 197-203, (1984) · Zbl 0545.76178
[15] Johnson, R.S., Nonlinear equations incorporating damping and dispersion, J fluid mech, 42, 49-60, (1970) · Zbl 0213.54904
[16] Hashizume, Y., Nonlinear pressure waves in a fluid filled elastic tube, J phys soc jpn, 54, 3305-3312, (1985)
[17] Yomosa, S., Solitary waves in large blood vessels, J phys soc jpn, 56, 506-520, (1987)
[18] Erbay, H.A.; Erbay, S.; Dost, S., Wave propagation in fluid filled nonlinear viscoelastic tubes, Acta mech, 95, 87-102, (1992) · Zbl 0756.76099
[19] Demiray, H., Solitary waves in prestressed elastic tubes, Bull math biol, 58, 939-955, (1996) · Zbl 0856.92003
[20] Demiray, H., Weakly nonlinear waves in elastic tubes filled with layered fluid, Int J nonlinear sci numer simul, 3, 2, 89-98, (2002) · Zbl 1079.74021
[21] Demiray, H., On the derivation of some non-linear evolution equations and their progressive wave solutions, Int J non-linear mech, 38, 63-70, (2003) · Zbl 1346.74034
[22] Demiray H. Interaction of nonlinear waves governed by Boussinesq equation. Chaos Solitons & Fractals, in press, doi:10.1016/j.chaos.2005.08.185, 2005. · Zbl 1142.35564
[23] He, J.H., A new approach to nonlinear partial differential equations, Comm nonlinear sci numer simul, 2, 4, 230-235, (1997)
[24] He, J.H., A variational iteration approach to nonlinear problems and its application, Mech appl, 20, 1, 30-31, (1998)
[25] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput methods appl mech eng, 167, 1&2, 57-68, (1998) · Zbl 0942.76077
[26] He, J.H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput methods appl mech eng, 167, 1&2, 69-73, (1998) · Zbl 0932.65143
[27] He, J.H., Variational iteration method – a kind of nonlinear analytical technique: some examples, Int J nonlinear mech, 34, 699-708, (1999) · Zbl 1342.34005
[28] He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl math comput, 114, 2&3, 115-123, (2000) · Zbl 1027.34009
[29] He, J.H.; Wan, Y.Q.; Guo, Q., An iteration formulation for normalized diode characteristics, Int J circuit theory appl, 32, 629-632, (2004) · Zbl 1169.94352
[30] Abdou, M.A.; Soliman, A.A., Variational-iteration method for solving burger’s and coupled burger’s equations, J comput appl math, 181, 245-251, (2005) · Zbl 1072.65127
[31] Momani, S.; Abuasad, S., Application of he’s variational-iteration method to Helmholtz equation, Chaos solitons fractals, 27, 5, 1119-1123, (2006) · Zbl 1086.65113
[32] Soliman AA. A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations, Chaos Solitons Fractals (In Press) doi:10.1016/j.choas.2005.08.054, 2005.
[33] Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss, Chaos Solitons Fractals (In Press) doi:10.1016/j.choas.2005.08.044, 2005.
[34] Finlayson, B.A., The method of weighted residuals and variational principles, (1972), Academic press New York · Zbl 0319.49020
[35] Prandtl, O.G.; Tietjens, Fundamentals of hydro- and aeromechanics, (1957), Dover New York · Zbl 0078.39603
[36] Demiray, H., Propagation of weakly nonlinear waves in fluid-filled thick viscoelastic tubes, Appl math model, 23, 779-798, (1999) · Zbl 0990.76099
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