Some Rotational Flows of an Incompressible Fluid of Variable Viscosity
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:60858", author = "Rana Khalid Naeem and Waseem Ahmed Khan and Muhammad Akhtar and Asif Mansoor", title = "Some Rotational Flows of an Incompressible Fluid of Variable Viscosity", 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", volume = "3", number = "9", pages = "721-6", }
