A Class of New Exact Solutions of Navier-Stokes Equations with Body Force For Viscous Incompressible Fluid

  • Authors

    • Mushtaq Ahmed University of Karachi
    • Rana Khalid Naeem University of Karachi
    • Syed Anwer Ali University of Karachi
    2018-02-14
    https://doi.org/10.14419/ijamr.v7i1.8836
  • Abstract

    This paper is to indicate a class of new exact solutions of the equations governing the two-dimensional steady motion of incompressible fluid of variable viscosity in the presence of body force. The class consists of the stream function $\psi$ characterized by equation $\theta=f(r)+ a \psi + b $ in polar coordinates $r$, $\theta$ , where a continuously differentiable function is $f(r)$ and $a\neq 0 , b $ are constants. The exact solutions are determined for given one component of the body force, for both the cases when $f(r)$ is arbitrary and when it is not. When $f(r)$ is arbitrary, we find $a=1$ and we can construct an infinite set of streamlines and the velocity components, viscosity function, generalized energy function and temperature distribution for the cases when $R_{e}P_{r}=1$ and when $R_{e}P_{r}\neq 1$ where $R_{e}$ represents Reynolds number and $P_{r}$Prandtl number. For the case when $f(r)$ is not arbitrary we can find solutions for the cases $R_{e}P_{r}\neq a$ and $R_{e}P_{r}=a$ where $"a"$ remains arbitrary. 
  • References

    1. Wang, C. Y.; On a class of exact solutions of the Navier-Stocks equations: Journal of Applied Mechanics, 33 (1966) 696-698.

      [2] 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.

      [3] 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.

      [4] 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.

      [5] 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.

      [6] Naeem, R. K.; Anwer Ali, S.; Exact solutions of the equations of motion of an incompressible fluid of variable viscosity: Karachi University Journal of Science, 1996, 24(1), 35-40.

      [7] 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.

      [8] Naeem, R. K.; Anwer Ali, S.; A class of exact solutions to equations governing the steady plane flows of an incompressible fluid of variable viscosity via von-Mises variables: International Journal of Applied Mechanics and Engineering, 2001, 6(1), 395-436.

      [9] 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.

      [10] Naeem, R. K.; Jamil, M.; A class of exact solutions to flow equations of an incompressible fluid of variable viscosity: Quaid-e-Awam University Research Journal of Engineering Science and Technology, 2005, 6(1,2), 11-18.

      [11] Naeem, R. K.; Jamil, M.; On plane steady flows of an incompressible fluid with variable viscosity: International Journal of Applied athematics and Mechanics, 2006, 2(3), 32-51.

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

      [13] Naeem, R. K.; 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.

      [14] 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.

      [15] 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.

      [16] Naeem, R. K.; Aurnangzeb, et. al.; Exact solution to the steady Plane Flows of an Incompressible Fluid of Variable Viscosity Using OR Coordinates: World Academy of Science, Engineering and Technology, (2009), 3(1), 839-846.

      [17] Naeem, R. K.; Razia S.;, Some exact solution to the unsteady Navier-Stokes equations Viscous Incompressible fluid in the presence of Unknown External Force: International Journal of Appl. Math. And Mechanics, 2011, 7(11), 83-97.

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

      [19] 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. doi:10.14419/ijbas.v4i4.5064

      [20] Martin, M. H.; The flow of a viscous fluid I: Archive for Rational Mechanics and

      Analysis, 1971, 41(4), 266-286.

      [21] Weatherburn C.E., Differential geometry of three dimensions, Cambridge University Press, 1964

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  • How to Cite

    Ahmed, M., Naeem, R. K., & Ali, S. A. (2018). A Class of New Exact Solutions of Navier-Stokes Equations with Body Force For Viscous Incompressible Fluid. International Journal of Applied Mathematical Research, 7(1), 20-26. https://doi.org/10.14419/ijamr.v7i1.8836

    Received date: 2017-12-11

    Accepted date: 2018-01-22

    Published date: 2018-02-14