Commenced in January 2007
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Some Rotational Flows of an Incompressible Fluid of Variable Viscosity

Authors: Rana Khalid Naeem, Waseem Ahmed Khan, Muhammad Akhtar, Asif Mansoor

Abstract:

The Navier Stokes Equations (NSE) for an incompressible fluid of variable viscosity in the presence of an unknown external force in Von-Mises system x,\ are transformed, and some new exact solutions for a class of flows characterized by equation y f x a\b for an arbitrary state equation are determined, where f x is a function, \ the stream function, a z 0 and b are the arbitrary constants. In three, out of four cases, the function f x is arbitrary, and the solutions are the solutions of the flow equations for all the flows characterized by the equationy f x a\b. Streamline patterns for some forms of f x in unbounded and bounded regions are given.

Keywords: Bounded and unbounded region, Exact solution, Navier Stokes equations, Streamline pattern, Variable viscosity, Von- Mises system

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1078003

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References:


[1] E. W. Hardiman and A. H. Nissan, "A rational basis for the viscosity index system." I. J. Inst. Petrol. 31, pp. 255-270, 1945.
[2] E. Z. Mohr, "Laminar and turbulent flow with special consideration of the effect of rigid walls". Phys. 121, (5-8), pp. 301-350, 1943.
[3] A. F. H. Ward, S. M. Neale. and Bilton. N. F., "Viscosity of liquids at high rates of shear." Letter in Nature. Lond. 166, pp. 905, 1950.
[4] A. V. Brancker, "Viscosity-temperature dependence." Letter in Nature. Lond. 166, pp. 905-906, 1950.
[5] J. Belval, "The principle of least constraint applied to the dynamics of incompressible fluids. Bull. Acad. Roy. Belge. Cl. Sci. 36, 8-9, pp. 639- 648, 1950.
[6] W. R. Dean, "Slow motion of viscous liquid in a semi infinite channel." Proc.Camb. Phil. Soc 47, pp. 127-141, 1951.
[7] E. Lahaye, "Contribution to the solution of the equations of motion of a fluid and of the equations of general atmospheric circulation." Inst. Roy. Met. Bleg. Mem. 38, pp. 78, 1950.
[8] D. G. Christopherson, "Hydrodynamic lubrication: general survey." Brit. J. Appl. Phys. 1. (Physics of lubrication), pp. 1-7, 1951.
[9] G. W. Morgan, "A study of motion in a rotating liquid." Proc. Roy. Soc. A. 206, pp. 108-130, 1950.
[10] V. I. Yatseev, "One class of the exact solution of the equation of motion of a viscous liquid." J. Exp. Theor. Phys. (USSR), 20, pp. 1031-1034, 1950.
[11] Y. Takaisi, "Note on the drag in a circular cylinder moving with low speeds in a viscous liquid between two parallel walls." J. Phys. Soc. Japan. 11, No. 9, pp. 1009-1013, 1956.
[12] A. Mukgerjee, "Flow of an incompressible jet with a distribution of tempreture on the axis." Indian J. Pure Appl. Phys. 5, No. 4, pp. 151-152. 1967.
[13] A. Davey and P. G. Drazin, "The stability of poisseuille flow in a pipe." J. Fluid Mech. (GB), 36, pt. 2, pp. 209-218. 1969.
[14] A. Yoshizawa, "Viscous flow past a semi infinite flat plate to the second Ocean-type approximation." J. Phys. Soc. Jap. 30 No. 6, 1757 et seq., 1971.
[15] J. Zhu, "The boundary integral equation method for incompressible viscous flows with slip boundary condition." Z. Angew. Math. Mech. 70, No. 6, pp. 717-719, 1990.
[16] M. N. Zakharenkov, "Spetial features of difference schemes for solving two-dimensional Navier-Stokes equations connected with the formulation of boundary conditions at a solid surface." Zh. Vychisl. Mat. Mat. Fiz. (USSR) 30, No. 8, pp. 1224-1236, 1990.
[17] K. Onishi and K. Yamasaki, "Thermal circulation in deep water reservoir," in Proc. First International-Conf. Advance Computational Methods in Heat Transfer. Southampotn UK, 1990, pp. 183-194.
[18] R. K. Naeem and S. S. Ali, "Exact solutions of the equations of motion of an incompressible fluid of variable viscosity." Kar. Univ. J. Sc. 22 (1&2), pp. 97-100, 1994.
[19] R. K. Naeem, "Exact solutions of flow equations of an incompressible fluid of variable viscosity via one-parameter group." JSE. 19 No. 1, pp. 111-114, 1994.
[20] R. K. Naeem, "Steady plane flows of an incompressible fluid of variable viscosity via Hodograph transformation method." Kar. Univ. J. Sc. 31 No.1, pp. 73-89, 1995.
[21] M. H. Martin, "The flow of viscous fluid." Arch. Rat. Mech. Anal. 41, pp. 266-286, 1971.
[22] R. K. Naeem and S. M. Ali, "Steady aligned flows of an incompressible second-grade fluid of finite electrical conductivity." Journal of Scientific Research, vol. XXVI, No. 1 & 2, pp. 1-11, 1997.
[23] R. K Naeem and M. Imtiaz, "On exact solutions of the equations of motion of an incompressible fluid of variable viscosity in the presence of unknown external force." Sindh Univ. Res. J. , vol.39, no.2, pp. 1-18, 2007.