Abstract
The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow due to a porous disk rotating with a constant angular speed. The three-dimensional hydromagnetic equations of motion are treated analytically to obtained exact solutions with the inclusion of suction and injection. The well-known thinning/thickening flow field effect of the suction/injection is better understood from the constructed closed form velocity equations. Making use of this solution, analytical formulas for the angular velocity components as well as for the permeable wall shear stresses are derived. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation and the Joule heating. As a result, exact formulas are obtained for the temperature field which take different forms corresponding to the condition of suction or injection imposed on the wall.
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References
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mech. 63, 193–248 (1934)
Engquist, B., Schmid, W.: Mathematics Unlimited-2001 and Beyond. Springer-Verlag, New York (2001)
Batchelor, G. K.: An Introduction to Fluid Dynamics. Cambridge University Press, America (1967)
Polyanin, A. D.: Exact solutions to the Navier-Stokes equations with generalized separation of variables. Dokl. Phys. 46, 726–731 (2001)
Asghar, S., Hanif, K., Hayat, T.: The effect of the slip condition on unsteady flow due to non-coaxial rotations of disk and a fluid at infinity. Mechanica 42, 141–148 (2007)
Sai, K. S., Rao, B. N.: Magnetohydrodynamic flow in a rectangular duct with suction and injection. Acta Mech. 140, 57–64 (2000)
Tsangaris, S., Kondaxakis, D., Vlachakis, N. W.: Exact solution for flow in a porous pipe with unsteady wall suction and/or injection. Commun. Nonlinear Sci. Numer. Simul. 12, 1181–1189 (2007)
Wang, C. Y.: Flow due to a stretching boundary with partial slip an exact solution of the Navier-Stokes equations. Chem. Eng. Sci. 57, 3745–3747 (2002)
Rosenhead, L.: Laminar Boundary Layers. Oxford University Press, England (1963)
Sherman, R. S.: Viscous Flow. McGraw-Hill (1990)
Mehta, K. N., Jain, R. K.: Laminar hydrodynamic flow in a rectangular channel with porous walls. Proc. Nat. Inst. Sci. India 28, 846–856 (1962)
Berman, A. S.: Laminar flow in channels with porous walls. J. Appl. Phys. 24, 1232–1235 (1953)
Terril, R. M.: Laminar flow through a porous tube. J. Fluids Eng. 105, 303–307 (1983)
Tsangaris, S., Kondaxakis, D., Vlachakis, N. W.: Exact solution of the Navier-Stokes equations for the pulsating Dean flow in a channel with porous walls. International Journal of Engineering Science 44, 1498–1509 (2006)
Kármán, T. V.: Uber laminare und turbulente reibung. Zeitschnift fur Angewante Mathematik und Mechanik 1, 233–252 (1921)
Benton, E. R.: On the flow due to a rotating disk. J. FluidMech. 24, 781–800 (1966)
Boedewadt, U. T.: Die drehstroemung uber festem grund. Zeitschnift fur AngewanteMathematik und Mechanik 20, 241–253 (1940)
Cochran, W. G.: The flow due to a rotating disk. Proc. Camb. Phil. Soc. 30, 365–375 (1934)
Federov, B. I., Plavnik, G. Z., Prokhorov, I. V., et al.: Transitional flow conditions on a rotating-disk. J. Eng. Phys. 31, 1448–1453 (1976)
Gregory, N., Stuart, J. T., Walker, W. S.: On the stability of three dimensional boundary layers with applications to the flow due to a rotating-disk. Philos. Trans. R. Soc. London Ser. A 248, 155–199 (1955)
Hall, P.: An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating-disk. Proc. Roy. Soc. London Ser. A 406, 93–106 (1986)
Jarre, S. L. G., Chauve, M. P.: Experimental study of rotatingdisk instability. II. Forced flow. Phys. Fluids 8, 2985–2994 (1996)
Kohama, Y.: Study on boundary layer transition of a rotatingdisk. Acta Mech. 50, 193–199 (1984)
Qiu, X. M., Huang, L., Jian, G. D.: Finite larmor radius magnetohydrodynamic analysis of the Rayleigh-Taylor instability in Z pinches with sheared axial flow. Phys. Plasmas 14, 032.111 (2007)
Rogers, M. H., Lance, G. N.: The rotationaly symmetric flow of a viscous fulid in the presence of an infinite rotating disk. J. Fluid Mech. 7, 617–631 (1960)
Turkyilmazoglu, M.: Linear absolute and convective instabilities of some two and three dimensional flows. [Ph.D. Thesis]. University of Manchester (1998)
Berker, R.: An exact solution of the Navier-Stokes equation the vortex with curvilinear axis. International Journal of Engineering Science 20, 217–230 (1982)
Rajagopal, K. R.: A class of exact solutions to the Navier-Stokes equations. International Journal of Engineering Science 22, 451–455 (1984)
Rajagopal, K. R.: Swirling flows of viscoelastic fluids. International Journal of Engineering Science 30, 143–149 (1988)
Rao, A. R., Kasiviswanathan, S. R.: On exact solutions of the unsteady Navier-Stokes equation the vortex with instantaneous curvilinear axis. International Journal of Engineering Science 25, 337–349 (1987)
Erdogan, M. E.: Flow due to parallel disks rotating about noncoincident axis with one of them oscillating in its plane. Int. J. Non-Linear Mech. 34, 1019–1030 (1999)
Erdogan, M. E.: Flow induced by non-coaxial rotation of a disk executing nontorsional oscillations and a fluid rotating at infinity. International Journal of Engineering Science 38, 175 (1999)
Millsaps, K., Pohlhausen, K.: Heat transfer by laminar flow from a rotating-plate. J. Aero. Sci. 19, 120–126 (1952)
Riley, N.: The heat transfer from a rotating-disk. Q. J. Mech. Appl. Math. 17, 331–349 (1964)
Sparrow, E. M., Gregg, J. L.: Heat transfer from a rotating disk to fluids of any Prandtl number. J. Heat Transfer. 81, 249–251 (1959)
Ackroyd, J. A. D.: On the steady flow produced by a rotating disc with either surface suction of injection. J. Eng. Phys. 12, 207–220 (1978)
Ariel, P. D.: On computation ofMHD flow near a rotating-disk. Z. Angew. Math. Mech. 82, 235–246 (2001)
Hayat, T., Asghar, S., Siddiqui, A. M., et al.: Unsteady MHD flow due to non-coaxial rotations of a porous disk and a fluid at infinity. Acta Mech. 151, 127–134 (2001)
Hossain, M. A., Hossain, A., Wilson, M.: Unsteady flow of viscous incompressible fluid with temperature-dependent viscosity due to a rotating disc in the presence of transverse magnetic field and heat transfer. Int. J. Therm. Sci. 40, 11–20 (2001)
Kaloni, P. N., Venkatasubramanian, S.: Physical mechanisms of laminar-boundary layer transition. Journal of Magnetism and Magnetic Materials 320, 142–149 (2008)
Kumar, S. K., Thacker, W. I., Watson, L. T.: Magnetohydrodynamic flow and heat transfer about a rotating disk with suction and injection at the disk surface. Comput. Fluids 16, 183–193 (1988)
Sharma, P. K., Khan, S.: MHDflow in porous medium induced by torsionally oscillating disk. Comput. Fluids 39, 1255–1260 (2010)
Sparrow, E. M., Cess, R. D.: Magnetohydrodynamic flow and heat transfer about a rotating disk. J. Appl. Mech. 29, 181–187 (1962)
Huang, L., Qiu, X. M., Jian, G. D., et al.: Effects of compressibility on the finite larmor radius stabilized Rayleigh-Taylor instability in Z-pinch implosions. Phys. Plasmas 15, 022–103 (2008)
Schlichting, H.: Boundary-Layer Theory. McGraw-Hill (1979)
Stuart, J. T.: On the effects of uniform suction on the steady flow due to a rotating disk. Q. J. Mech. Appl. Math. 7, 446–457 (1954)
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Turkyilmazoglu, M. A class of exact solutions for the incompressible viscous magnetohydrodynamic flow over a porous rotating disk. Acta Mech Sin 28, 335–347 (2012). https://doi.org/10.1007/s10409-012-0042-6
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DOI: https://doi.org/10.1007/s10409-012-0042-6