Analytical solutions for Navier-Stokes equations in the cylindrical coordinates

We consider the problem of convective heat transport in the incompressible fluid flow and the motion of the fluid in the cylinder which is described by the Navier-Stokes equations with the heat equation.The exact solutions of the Navier-Stokes equations, the temperature field and the vorticity vector are obtained.


INTRODUCTION
The fluid mechanics studies the behavior at every point within the domain under various physical conditions. For describing the physical phenomena in fluid mechanics one uses the mathematical model of motion such as Navier-Stokes equations. It is a well known fact that a few exact solutions of the Navier-Stokes equations are known even now. This has been largely due to the complexity of the system of differential equations. In the absence of a general solution it is often convenient to experiment with models to obtain information on the flow phenomena e.g. the velocity distribution, flow pattern, pressure losses, etc. In [1] one applied the "background" method to the arbitrary Prandtl number problem to derive a scaling lower bound on the space-time averaged temperature of the layer along with an explicit prefactor and one used a multiple boundary layer asymptotic theory to sharpen the estimate, increasing the prefactor in the lower bound by a factor. More recently,the existence of the global regular solutions to Navier-Stokes equations is proven in [8]. In this paper we investigate the analytical solutions to Navier-Stokes equations in cylindrical coordinates since the problem of transport of mass,momentum and heat in the case of flow is of great importance for engineering applications. The rest of this paper is organized as follows :in the next section we present the details of the models we analyze.In the section 3 we present the solutions to the Navier-Stokes equations and to the heat equation .In the conclusion, we summarize our results.

Governing equations
The fluid layer is confined between two parallel plates of horizontal extent L x and L y separated by vertical (z) distance (d). The no-slip upper and lower plates are held at fixed temperatures T 0 and T 1 respectively A uniform volumetric heat flux H (with units power/volume) is pumped into the layer. The governing equations for the velocity field u, the pressure p and the temperature T in the standard Boussinesq approximation are [1]: where is the velocity of the fluid; the pressure; the temperature; ; v is the viscosity, g is the acceleration of gravity along the z axis (in the k -direction ),  is the thermal expansion coefficient, k is the thermal diffusion coefficient and , where  is the density and c is the specific heat capacity of the fluid. We impose periodic boundary conditions in the horizontal directions with periods L x and L y. Using as the unit of time, d as the unit of length and as the unit of temperature, the governing equations are put into the nondimensional form: where is the Prandtl number and is the heat Rayleigh number. By the dot we denote the scalar product in R 3 In order to describe the domain Ω and the motion of the fluid, we introduce the cylindrical coordinates g, , z which are associated with the cartesian coordinates x 1 , x 2 , x 3 by the relations: Where  is the stream function.
Our aim is to construct the exact solutions to these equations.

Analytical solutions
Now, we can formulate the main results of the paper.

Proposition 1.
The solution of the equation (14) satisfying the boundary conditions (8) has the following form: where the function is positive such that remains a finite quantity and a sign function.

Proof
As the fluid flows are axisymmetric and helical , we can write [3] : (17) Applying the operator rot to the relation (17) and using the explicit relations between the cylindrical components of the vorticity and velocity, we get : where  is the laplacian in cylindrical coordinates r,z. Using the divergence-free condition

Proof
Applying the operators to the relations (10), (12) respectively and suming the obtained results, we get which can be written through the stream function  as follows : The Bulletin of Society for Mathematical Services and Standards Vol. 2