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Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model. (English) Zbl 1185.76523
Summary: This paper is concerned with the exact analytic solutions for the velocity field and the associated tangential stress corresponding to a potential vortex for a fractional Maxwell fluid. The fractional calculus approach is taken into account in the constitutive relationship of a non-Newtonian fluid model. Exact analytic solutions are obtained by using the Hankel transform and the discrete Laplace transform of sequential fractional derivatives. The solutions for a Maxwell fluid appear as the limiting cases of our general solutions by setting α=1. The influence of fractional coefficient on the decay of vortex velocity is also analyzed by graphical illustrations.
76D17Viscous vortex flows
26A33Fractional derivatives and integrals (real functions)
[1]Rajagopal, K. R.: Mechanics of non-Newtonian fluids, Pitman research notes in mathematics 291, 129-162 (1993) · Zbl 0818.76003
[2]Rajagopal, K. R.; Bhatnagar, R. K.: Exact solutions for simple flows of an Oldroyd-B fluid, Acta mech. 113, 233-239 (1995) · Zbl 0858.76010 · doi:10.1007/BF01212645
[3]Hayat, T.; Siddiqui, A. M.; Asghar, S.: Some simple flows of an Oldroyd-B fluid, Int. J. Eng. sci. 39, 135-147 (2001)
[4]Hayat, T.; Khan, M.; Ayub, M.: Exact solutions of flow problems of an Oldroyd-B fluid, Appl. math. Comput. 151, 105-119 (2004) · Zbl 1039.76001 · doi:10.1016/S0096-3003(03)00326-6
[5]Tan, W. C.; Masuoka, T.: Stokes first problem for an Oldroyd-B fluid in a porous half space, Phys. fluids 17, 023101 (2005) · Zbl 1187.76517 · doi:10.1063/1.1850409
[6]Fetecau, C.; Fetecau, Corina: Decay of a potential vortex in a Maxwell fluid, Int. J. Nonlinear mech. 38, 985-990 (2003)
[7]Fetecau, C.; Fetecau, Corina; Vieru, D.: On some helical flows of Oldroyd-B fluids, Acta mech. 189, 53-63 (2007) · Zbl 1108.76008 · doi:10.1007/s00707-006-0407-7
[8]Fetecau, C.; Fetecau, Corina: The first problem of Stokes for an Oldroyd-B fluid, Int. J. Nonlinear mech. 38, 1539-1544 (2003)
[9]Chen, C. I.; Chen, C. K.; Yang, Y. T.: Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different volume flow rate conditions, Heat mass trans. 42, 203-209 (2004)
[10]Hayat, T.; Khan, M.; Ayub, M.: On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int. J. Eng. sci. 42, 123-135 (2004) · Zbl 1211.76009 · doi:10.1016/S0020-7225(03)00281-7
[11]Hayat, T.; Khan, M.; Asghar, S.: Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta mech. 168, 213-232 (2004) · Zbl 1063.76108 · doi:10.1007/s00707-004-0085-2
[12]Zhang, Z.; Fu, C.; Tan, W. C.: Onset of oscillatory convection in a porous cylinder saturated with a viscoelastic fluid, Phys. fluids 19, 098104 (2007) · Zbl 1182.76871 · doi:10.1063/1.2773739
[13]Fu, C.; Zhang, Z.; Tan, W. C.: Numerical simulation of thermal convection of a viscoelastic fluid in a porous square box heated from below, Phys. fluids 19, 104107 (2007) · Zbl 1182.76257 · doi:10.1063/1.2800358
[14]Tan, W. T.; Masuoka, T.: Stability analysis of a Maxwell fluid in a porous medium heated from below, Phys. lett. A 360, 454-460 (2006)
[15]Song, D. Y.; Jiang, T. Q.: Study on the constitutive equation with fractional derivative for the viscoelastic fluids-modified Jeffreys model and its application, Rheol. acta 37, 512-517 (1998)
[16]Tan, W. C.; Xu, M. Y.: The impulsive motion of flat plate in a generalized second order fluids, Mech. res. Commun. 29, 3-9 (2002) · Zbl 1151.76368 · doi:10.1016/S0093-6413(02)00223-9
[17]Tan, W. C.; Pan, W. X.; Xu, M. Y.: A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, Int. J. Nonlinear mech. 38, 615-620 (2003)
[18]Qi, H. T.; Jin, H.: Unsteady rotating flows of viscoelastic fluid with the fractional Maxwell model between coaxial cylinders, Acta mech. Sinica 22, 301-305 (2006) · Zbl 1202.76016 · doi:10.1007/s10409-006-0013-x
[19]Qi, H. T.; Xu, M. Y.: Stokes first problem for a viscoelastic fluid with the generalized Oldroyd-B model, Acta mech. Sinica 23, 463-469 (2007) · Zbl 1202.76017 · doi:10.1007/s10409-007-0093-2
[20]Khan, M.; Nadeem, S.; Hayat, T.; Siddiqui, A. M.: Unsteady motions of a generalized second grade fluid, Math. comput. Modell. 41, 629-637 (2005) · Zbl 1080.76007 · doi:10.1016/j.mcm.2005.01.029
[21]Khan, M.; Maqbool, K.; Hayat, T.: Influence of Hall current on the flows of a generalized Oldroyd-B fluid in a porous space, Acta mech. 184, 1-13 (2006) · Zbl 1096.76061 · doi:10.1007/s00707-006-0326-7
[22]Khan, M.; Hayat, T.; Asghar, S.: Exact solution for MHD flow of a generalized Oldroyd-B fluid with modified Darcy’s law, Int. J. Eng. sci. 44, 333-339 (2006) · Zbl 1213.76024 · doi:10.1016/j.ijengsci.2005.12.004
[23]Tan, W. C.; Fu, C.; Xie, W.; Cheng, H.: An anomalous subdiffusion model for calcium spark in cardiac myocytes, Appl. phys. Lett. 91, 183901 (2007)
[24]El-Shahed, M.: MHD of a fractional viscoelastic fluid in a circular tube, Mech. res. Commun. 33, 216-268 (2006) · Zbl 1192.76067 · doi:10.1016/j.mechrescom.2005.02.017
[25]Shen, F.; Tan, W. C.; Zhao, Y. H.; Masuoka, T.: Decay of vortex velocity and diffusion of temperature in a generalized second grade fluid, Appl. math. Mech. 25, 1151-1159 (2004) · Zbl 1088.76004 · doi:10.1007/BF02439867
[26]Khan, M.: Partial slip effects on the oscillatory flows of a fractional Jeffrey fluid in a porous medium, J. porous media 10, 473-488 (2007)
[27]Hayat, T.; Khan, M.; Asghar, S.: On the MHD flow of a fractional generalized Burgers fluid with modified Darcy’s law, Acta mech. Sinica 23, 257-261 (2007) · Zbl 1202.76155 · doi:10.1007/s10409-007-0078-1
[28]Vieru, D.; Fetecau, Corina; Fetecau, C.: Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate, Appl. math. Comput. 200, 459-464 (2008) · Zbl 1158.76005 · doi:10.1016/j.amc.2007.11.017
[29]Hilfer, R.: Applications of fractional calculus in physics, (2000)
[30]Fetecau, C.; Fetecau, Corina: Decay of a potential vortex in an Oldroyd-B fluid, Int. J. Eng. sci. 43, 340-351 (2005) · Zbl 1211.76008 · doi:10.1016/j.ijengsci.2004.08.013
[31]Sneddon, I. N.: Fourier transforms, (1951)
[32]Miller, K. S.: An introduction to the fractional calculus and fractional differential equation, (1993)
[33]Debnath, L.; Bhatta, D.: Integral transforms and their applications, (2007)