# A class of new exact solutions of the equations governing the steady plane flows of incompressible fluid of variable viscosity

## DOI:

https://doi.org/10.14419/ijbas.v4i4.5064## Published:

2015-11-07## Abstract

The objective of this paper is to indicate a class of new exact solutions of the equations governing the steady plane flows of incompressible fluid of variable viscosity. The class consists of the stream function characterized by equation (2). Exact solutions are determined for and When is arbitrary we can construct an infinite set of streamlines and the velocity components, viscosity function, generalized energy function and temperature distribution . Therefore, an infinite set of solutions to flow equations. When is not arbitrary, there are two values of and therefore, two exact solutions to flow equations. The streamlines are presented through Fig.(1â€“56) for some chosen from of f(r).

## References

[1] Berker R (1982), an exact solution of the Navier-Stokes equations, *International Journal of Engineering and Science*, 22, pp.217-230. http://dx.doi.org/10.1016/0020-7225(82)90017-9.

[2] Chandna OP and Oku-Ukpong EO (1994). Flows for chosen vorticity functions-Exact Solutions of the Navier-Stokes Equations, *International Journal of Applied Mathematics and Mathematical Sciences*, 17, pp.155-164. http://dx.doi.org/10.1155/S0161171294000219.

[3] Dorrepaal JM (1986), An exact solution of the Navier-Stokes equations which describes non-orthogonal stagnation â€“point flow in two dimensions, *Journal of Fluid Mechanics*, 163, pp. 141-147. http://dx.doi.org/10.1017/S0022112086002240.

[4] Hui WH (1987), Exact solutions of the unsteady two-dimensional Navier-Stokes equations, *Journal of Applied Mathematics and Physics, *38, pp. 689-702 http://dx.doi.org/10.1007/BF00948290.

[5] Kovasznay LIG (1948), Laminar flow behind a two-dimensional grid, *Proceedings of the Cambridge Philosophical Society*, 44, pp. 58-62. http://dx.doi.org/10.1017/S0305004100023999.

[6] Kapitanskiy LV (1979), Group analysis of the Navier-Stokes equations in the presence of rotational symmetry and some new exact solutions, *Zapiski nauchnogo sem, LOMI*. 84, pp. 89-107.

[7] Kambe T (1986), A class of exact solutions of the Navier-Stokes equations, *Fluid Dynamics Research*, 1, pp.21-31http://dx.doi.org/10.1016/0169-5983(86)90004-3.

[8] Martin M H (1971), the flow of a viscous fluid I, *Archive for Rational Mechanics and Analysis*, 41(4), pp.266-286. http://dx.doi.org/10.1007/BF00250530.

[9] Rana Khalid Naeem (1994), Exact solutions of flow equations of an incompressible fluid of variable viscosity via one â€“ parameter group, *The Arabian Journal for Science and Engineering*, 19, pp.111-114.

[10] Rana Khalid Naeem and Syed Anwer Ali (1996), Exact solutions of the equations of motion of an incompressible fluid of variable viscosity, *Karachi University Journal of Science*, 24, pp. 35-40.

[11] Rana Khalid Naeem and Sarfaraz Ahmed Nadeem (1996), Study of steady plane flows of an incompressible fluid of variable viscosity using Martinâ€™s System, *Journal of Applied Mechanics and Engineering*, 1, pp.397-433.

[12] Rana Khalid Naeem and Syed Anwer Ali (2001), 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*, 6, pp.395-436.

[13] Rana Khalid Naeem (2003), Steady plane flows of an incompressible fluid of variable viscosity via Hodograph transformation method, *Karachi University Journal of Sciences*, 3, pp. 73-89.

[14] Rana Khalid Naeem and Muhammad Jamil (2005), 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*, 6(1, 2), pp. 11-18.

[15] Rana Khalid Naeem and Muhammad Jamil (2006), on plane steady flows of an incompressible fluid with variable viscosity, *International Journal of Applied Mathematics and Mechanics*, 2(3), pp. 32-51.

[16] Rana Khalid Naeem (2007), on plane flows of an incompressible fluid of variable viscosity, *Quarterly Science Vision*, 12, pp.125-131.

[17] Rana Khalid Naeem and Sobia Younus (2010), exact solutions of the Navier-Stokes equations for incompressible fluid of variable viscosity for prescribed vorticity distributions, *International Journal of Applied Mathematics and Mechanics*, 6(5), pp. 18-38.

[18] Rana Khalid Naeem and Sobia Younus (2014), A class of exact solutions of the Navier-Stokes equations for incompressible fluid of variable viscosity, *International Journal of Applied Mathematics and Mechanics*, 10(2), pp. 93-121

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

[20] Wang C Y (1989), Exact solutions of the unsteady Navier-Stokes Equations, *Applied Mechanics Review*, 42, pp.269-282. http://dx.doi.org/10.1115/1.3152400.

[21] Wang C Y (1990). Exact solutions of the Navier- Stokes equations-the generalized Beltrami flows, review and extension, *Acta Mechanica*, 81, pp.69-74. http://dx.doi.org/10.1007/BF01174556.

[22] Wang C Y (1991), Exact solutions of the steady-state Navier-Stocks equations, Annual Review, *Fluid Mechanics*, 23, pp.159-177. http://dx.doi.org/10.1146/annurev.fl.23.010191.001111.

[23] Wang C Y (2003), Stagnation flows with slip: exact solutions of the Navier-Stokes equations, *Zeitschrift fur Angewandte Mathematik und Physik*, 54(1), pp.184-189.http://dx.doi.org/10.1007/PL00012632.