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

 
 
 
  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract


    The purpose of this paper is to indicate a class of exact solutions of the system of partial differential equations governing the steady, plane motion of incompressible fluid of variable viscosity with body force term to the right-hand side of Navier-Stokes equations. The class consists of the stream function characterized by the equation  in polar coordinates  and  where  and  are continuously differentiable functions and the function  is such that  where a non-zero constant is  and overhead prime represents derivative with respect to . When  or  we show exact solutions for given one component of the body force for both the cases when the function  is arbitrary and when it is not. For the arbitrary function case,  appears in the coefficient of a linear second order ordinary differential equation showing a large numbers of solutions of this equation. This in turn establishes an infinite set of exact solutions to the problem concerned however; we show three examples of such exact solutions. The alternate case fixes  and provides viscosity as derivative of temperature function for  and . Anyhow, we find an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution.


  • Keywords


    Some Exact Solutions in the Presence of Body Force; Exact Solutions to the Flow Equations of Incompressible Fluids; Exact Solutions of Var-iable Viscosity Fluids; Navier-Stokes Equations with Body Force.

  • References


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

      [2] 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.

      [3] 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.

      [4] 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.

      [5] 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.

      [6] 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.

      [7] 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.

      [8] 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.

      [9] 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.

      [10] 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.

      [11] 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.

      [12] 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.

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

      [14] 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.

      [15] 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.

      [16] 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.

      [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.

      [18] Daniel Zwillinger; Handbook of differential equations; Academic Press, Inc. (1989)].


 

View

Download

Article ID: 9998
 
DOI: 10.14419/ijamr.v7i2.9998




Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.