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As shown in Scheme problem, the fluid flow in vertical channel, are from bottom to upward. Fluid flow in vertical channel is from bottom upwards. Velocity input profile to the channels for the Non-Newtonian fluid Pavrla is like parabolic and for Newtonian fluid appears as a uniform are considered. Temperature profile from entering to channel is considered uniform Non-Newtonian Pavrla fluid model belongs to only those Non-Newtonian non-elastic fluid which are usually generalized as extended Newtonian fluid
(laminar type of fluid flow, incompressibility, permanent, two-dimensional and considered developed has all of the properties of the fluid are independent of the temperature Default) .stress tensor is written like this:
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In which is the fluid viscosity is the function of tensor (tensor change shape), which is defined as below:
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For Non-Newtonian fluid model Pavrla (power) stress tensor is:
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As can be seen here, in this function just only variable number2 tenssor deformation rate, is used that showed application of shear flow model (in simple shear flow variables 1 and 3 both are zero) based on these two-dimensional flow model shear viscosity of the fluid is:

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Power model (Pavrla) n are among the fluid properties, can also be easily checked that in the model above the Newtonian fluid has a special case(n=1). Flow equations, including continuity equations, Navyr - Stokes two-dimensional, energy and entropy equation which followed in the below:
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Range of computational work is half the channel by the length of 30 non-dimensional and width of 5/0 non-dimensional. Although the effect of natural heat transfer is applied in the conditions of problem and Mvmntvm equations, continuity and energy are affected by the temperature, the governing equations except entropy equation should be solved simultaneously. The end results of solving the above equations will be considered as the input of entropy production equation. According to the two-dimensional of flow and non-possible density of fluid, the formulation of function of flow - Vorticity used for the formulation of the problem. Considering to Vorticity which is represented the period and pressure and gravity only act through the mass center of particles and are not able to create the period, so, They do not make any changes in the Vorticity , the good example of this is the absence of pressure in equations. Now consider the non-dimensional following parameters under different boundary conditions will apply. [5]

Using non-dimensional parameters and considering the flow function formulation of Vorticity continuity equations, and Mvmntvm and energy we will reach to following three equations:
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Ending of equations are solved by finite different method.Whereas the method with the expression, the reduction steps are limited, Upwind methods can be used for eliminating this restriction which is stable for small steps. In the second derivative of the sentences we use central difference that ultimately Vorticity and temperature equations are obtained as follows:
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In the equation above shows or . A, B, C are three diameter matris.In order to achieve more convergence of Relaxation Under or Over Relaxation parameter used.
Problem solving, begins with the initial guess for the current function - Vorticity and temperature begins. In the next step, boundary conditions applied and solving algorithm to determine the function flow - Vorticity are used. The results of this stage are considered as the results of the next stage and repeated to reach more carefully of encouraged.