A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force

Authors

  • Mushtaq Ahmed

    University of Karachi

  • Waseem Ahmed Khan

    University of Karachi

  • S M. Shad Ahsen

    University of Blochistan

Received date: May 1, 2018

Accepted date: July 17, 2018

Published date: August 27, 2018

DOI:

https://doi.org/10.14419/ijamr.v7i3.12326

Keywords:

Exact Solutions in the Presence of Body Force, Exact Solutions for Incompressible Fluids, Variable Viscosity Fluids, Na-Vier-Stokes Equations with Body Force, Martin’s Coordinates.

Abstract

This paper determines a class of exact solutions for plane steady motion of incompressible fluids of variable viscosity with body force term in the Navier-Stokes equations. The class consists of stream function  characterized by equation , in polar coordinates ,  where ,  and  are continuously differentiable functions, derivative of  is non-zero but double derivative of  is zero. We find exact solutions, for a suitable component of body force, considering two cases based on velocity profile. The first case fixes both the functions ,  and provides viscosity as function of temperature. Where as the second case fixes the function , leaves  arbitrary and provides viscosity and temperature for the arbitrary function . In both the cases, we can create infinite set of expressions for streamlines, viscosity function, generalized energy function and temperature distribution in the presence of body force.

 

 

References

  1. [1] Giga, Y.; Inui, K.; Mahalov; Matasui S.; Uniform local solvability for the Navier-Stokes equations with the Coriolis force: Method and application of Analysis, 2005, 12 , 381-384.

    [2] Gerbeau, J. -F. Le Bris, C., A basic Remark on Some Navier-Stokes Equations with Body Forces: Applied Mathematics Letters, 2000, 13(1), 107-112. https://doi.org/10.1016/S0893-9659(99)00194-9.

    [3] Landau L. D. and Lifshitz E. M.; Fluid Mechanics, Pergmaon Press, vol 6.

    [4] Naeem, R. K.; Mushtaq A.; A class of exact solutions to the fundamental equations for plane steady incompressible and variable viscosity fluid in the absence of body force: International Journal of Basic and Applied Sciences, 2015, 4(4), 429-465. http:/www.sciencepubco.com/index.php/IJBAS. https://doi.org/10.14419/ijbas.v4i4.5064.

    [5] Mushtaq A., On Some Thermally Conducting Fluids: Ph. D Thesis, Department of Mathematics, University of Karachi, Pakistan, 2016.

    [6] Mushtaq A.; Naeem R.K.; S. Anwer Ali; A class of new exact solutions of Navier-Stokes equations with body force for viscous incompressible fluid: International Journal of Applied Mathematical Research, 2018, 7(1), 22-26. http:/www.sciencepubco.com/index.php/IJAMR. https://doi.org/10.14419/ijamr.v7i1.8836.

    [7] Mushtaq Ahmed, Waseem Ahmed Khan ,: A Class of New Exact Solutions of the System of PDE for the plane motion of viscous incompressible fluids in the presence of body force,: International Journal of Applied Mathematical Research, 2018, 7 (2) , 42-48. http:/www.sciencepubco.com /index.php /IJAMR.

    [8] Wang, C. Y. on a class of exact solutions of the Navier-Stocks equations: Journal of Applied Mechanics, 33 (1966) 696-698. https://doi.org/10.1115/1.3625151.

    [9] Kapitanskiy, L.V.; Group analysis of the Navier-Stokes equations in the presence of rotational symmetry and some new exact solutions: Zapiski nauchnogo sem, LOMI, 84 (1) (1979) 89-107.

    [10] Dorrepaal, J. M. an exact solution of the Navier-Stokes equations which describes non-orthogonal stagnation –point flow in two dimensions: Journal of Fluid Mechanics, 163 (1) (1986) 141-147. https://doi.org/10.1017/S0022112086002240.

    [11] Chandna, O. P., Oku-Ukpong E. O.; Flows for chosen vorticity functions-Exact solutions of the Navier-Stokes Equations: International Journal of Applied Mathematics and Mathematical Sciences, 17 (1) (1994) 155-164. https://doi.org/10.1155/S0161171294000219.

    [12] Naeem, R. K.; Exact solutions of flow equations of an incompressible fluid of variable viscosity via one – parameter group: The Arabian Journal for Science and Engineering, 1994, 19 (1), 111-114.

    [13] Naeem, R. K.; Srfaraz, A. N.; Study of steady plane flows of an incompressible fluid of variable viscosity using Martin’s System: Journal of Applied Mechanics and Engineering, 1996, 1(1), 397-433.

    [14] Naeem, R. K.; Steady plane flows of an incompressible fluid of variable viscosity via Hodograph transformation method: Karachi University Journal of Sciences, 2003, 3(1), 73-89.

    [15] Naeem, R. K. on plane flows of an incompressible fluid of variable viscosity: Quarterly Science Vision, 2007, 12(1), 125-131.

    [16] Naeem, R. K. and Sobia, Y. Exact solutions of the Navier-Stokes equations for incompressible fluid of variable viscosity for prescribed vorticity distributions: International Journal of Applied Mathematics and Mechanics, 2010, 6(5), 18-38.

    [17] Martin, M. H. The flow of a viscous fluid I: Archive for Rational Mechanics and Analysis, 1971, 41(4), 266-286. https://doi.org/10.1007/BF00250530.

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How to Cite

Ahmed, M., Ahmed Khan, W., & M. Shad Ahsen, S. (2018). A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force. International Journal of Applied Mathematical Research, 7(3), 77-81. https://doi.org/10.14419/ijamr.v7i3.12326

Received date: May 1, 2018

Accepted date: July 17, 2018

Published date: August 27, 2018