NUMERICAL SOLUTION OF RADIATIVE BOUNDARY LAYER FLOW IN POROUS MEDIUM DUE TO EXPONENTIALLY SHRINKING PERMEABLE SHEET UNDER FUZZY ENVIRONMENT

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INTRODUCTION
Flow and heat Transfer in boundary layer flow of viscous fluid due to deforming surface is pivotal in many industrial processes cutting across different realms.Specially radiative thermal regime in porous medium has drawn much attention recently due to large application in gasification of oil shale waste heat storage in aquifer and many more.
Vast application of radioactive thermal in porous medium we need to study this class of problems in different ways.Due to involvement of nonlinear differential equation, there is no direct process available to solve exactly.Here we consider such a mechanical problem for our discussion in which the governing equations of motion can have two nonlinear differential equations of motion (One for velocity profile and another one is for temperature profile) and four parameters in the governing equation of motion and one parameter in the boundary conditions.The specific problem is Radiative Boundary Layer Flow in Porous Medium due to Exponentially Shrinking permeable Sheet.
A few relevant research has been presented in recent years (2010 to cont.).Radiative flow of Jeffery fluid with variable thermal conductivity in a porous medium was discussed by Elbashbeshy and Emam (2011), Hayat et al. (2012) about the effects of radiation and heat transfer over an unsteady stretching surface embedded in a porous medium.Paresh Vyas and Nupur Srivastava studied (2016) about the flow past and exponentially shrinking placed at the bottom of fluid saturated porous medium taking variable thermal conductivity and radiation using fourth order Runge-kutta scheme together with shooting method.
Here we introduce a new approach of solving of the said problem using fuzzy set theory.In this chapter our objective is to find is there any kind of uncertainty involved in the specific problem i.e.Radiative Boundary Layer Flow in Porous Medium due to Exponentially Shrinking permeable Sheet using fuzzy environment.For the graphical interpretation we developed computer codes for the said problem and represent the parameter's effect on the uncertainty involved in the flow of motion.On the basic concept of fuzzy differential equations Chakraverty et al., (2016) proposed some numerical methods for fuzzy fractional differential equations.Hazarika and Bora (2017, 2018) studied about the fuzzification of some numerical problems.J. Bora et al (2020) discussed some fluids problems using fuzzy set theory.

FORMULATION OF THE PROBLEM 2.1. Derivation of The Basic Equation
Let us consider the steady 2D boundary layer flow of optical thick viscous Newtonian fluid and associated heat transfer over a permeable sheet placed at bottom of the fluid saturated porous medium having permeability of specific form.A Cartesian coordinate system is chosen where the x-axis is taken along the sheet and y-axis is normal to it.The flow is caused by the sheet shrinking in an exponential fashion.A suction is applied normal to sheet to contain the vorticity.The fluid considered here is without phase change, optically dense, absorbing-emitting radiation but a nonscattering medium.The thermal conductivity of the fluid is assumed to vary linearly with temperature.The radiation flux in the energy equation is presumed to follow Rosseland approximation.The boundary layer equations for the considered setup are With the boundary condition where u, v are the velocity components along x and y directions, respectively, k is the permeability,  is the specific heat at constant pressure, υ is the kinematic viscosity, ρ is the density, and T, μ, and κ are the temperature, viscosity and thermal conductivity of the fluid, respectively.Further, L is the characteristic length,  is the variable temperature at the sheet,  is the constant reference temperature, and  is the constant free stream temperature. and  are the shrinking velocity of the sheet and mass transfer velocity, respectively, where c > 0 is the shrinking constant and  is a constant (where  < 0 corresponds to mass suction).
Let us introduce the stream function  ,  as Thus equation (5.1) is identically satisfied and the similarity transformation can be written as On using (5.5) and (6.5) we obtain the expression for velocity component in non-dimensional form as  =  ()  and  = −  () + () In order to obtain the similarity solutions, it is assumed that the permeability k of the porous medium takes the following form Where  is the reference permeability.As in our setup the thermal conductivity of the fluid is assumed to vary with temperature in a linear function as Where ∈ is the thermal conductivity variation parameter.In general, ∈ > 0 for fluids such as water and air, while ∈<0 for fluids such as lubrication oils.The radiative heat flux in the energy equation is presumed to follow Rosseland approximation and is given by Where  is the Stephan-Boltzmann constant and  is the mean absorption constant.It is further assumed that the temperature difference within the fluid is sufficiently small sothat  may be expressed as a linear function of temperature T. This is done by expanding  in a Taylor series about  and omitting higher-order terms to yield Thus, the equation of momentum (5.2) and energy (5.3) reduces to the following non dimensional form With the boundary conditions Where  = ,  = ,  = Denote the permeability parameter, Prandtl number, and radiation parameter respectively.

Conversion of The Basic Equation into
Fuzzified Form Now we Applying Zadeh fuzzy Extension theorem in (5.11-5.12)and (5.13-5.14) And the boundary condition became as (Fuzzy Environment) Considering the Fuzzified (5.15) equations as triangular fuzzy number then the Fuzzified equation became the following: Using fuzzy arithmetic we have, Where  = ( ) is the local Reynolds number.

Result and Discussion
The system of equations (26-28), the fuzzified equations of motion with fuzzified boundary conditions are solved numerically by using finite difference scheme.The discretized fuzzified equations are solved using an iterative method based on Gauss Seidel iterative method by developing suitable codes in python.
The numerical computations carried out for different sets of values of the parameters entering into the problem have been depicted through graphs and tables.Result is obtained for different values for the parameter  = 1,  = .25,∈= .1, = 0.7 and for different  −  of the fuzzified system of equations (26-28) In each of the following graphs the blue curve is the solution for the right values of the of the fuzzified velocity profile, green curve is the solution for the mid values of the fuzzified velocity profile which is same as the crisp velocity profile and blue curve is the solution for the right values of the of the Fuzzified velocity profile.
The Figure (1-3) exhibits the Fuzzified temperature profile for  −  ℎ  = 0.3,0.6,0.9, and  = 0.25,  = 0.5 ,  = 0.7 ,∈= 0.1,  = 1.It is observed from the graph that there is a deflection on the curve in the right solution of the temperature profile as compare to the left solution of the temperature profile from the mid value solution (i.e.crisp solution).Which is the indication of the uncertainty involved in the temperature profile.7 that with the increasing values of suction parameter  the velocity decrease.Whereas velocity decreases with increase of  in Figure 8.It is found that the pattern of the flows is almost similar in the temperature profile for the changes of the parameter.Also, we see that () decay with the increase of .Whereas () increases with increasing value of ∈ in Figure 9.
As the parameter changes are not affect in the uncertainty of the solution of the temperature profile so we are discussed the effect of the parameter in Crisp Solution i.e.,  −  = 1.It is observed from the table that with the increasing values of permeability parameter the values of the Skin friction coefficient increases.Similarly with the increasing values of permeability parameter the values of Nusselt number also increases.The results are well agreed with those of crisp values.The effect of fuzzification is also observed from the above Table.

Conclusion
In this chapter, the Radiative boundary layer flow in Porous medium due to exponentially shrinking steady MHD stagnation point flow due to shrinking permeable sheet has been theoretically considered under fuzzy environment.The effect of suction parameter, velocity ration parameter, Prandlt number on the flow and heat transfer have been studied under fuzzy environment.The numerical results have been obtained by developing computer codes on PYTHON.Thus, we conclude the followings from the above discussion: (1) The involvement of uncertainty in the equation of motion of this problem.

Figure ( 10 )
Figure(7)(8)(9) represent the crisp velocity profile for different values of , ∈  .It is observed in Figure7that with the increasing values of suction parameter  the velocity decrease.Whereas velocity decreases with increase of  in Figure8.It is found that the pattern of the flows is almost similar in the temperature profile for the changes of the parameter.Also, we see that () decay with the increase of .Whereas () increases with increasing value of ∈ in Figure9.As the parameter changes are not affect in the uncertainty of the solution of the temperature profile so we are discussed the effect of the parameter in Crisp Solution i.e.,  −  = 1. Figure (10) is the Fuzzified velocity profile for  −  ℎ  = 0.5, and  = 0.25,  = 0.5 ,  = 0.7, ∈= 0.1,  = 1.It is observed from the graph that there is a deflection on the curve in the right solution of the velocity profile (Green curve) as compare to the left solution of the velocity profile (Light yellow curve) from the mid value solution i.e. crisp solution (Violet curve).Which is the indication of the uncertainty involved in the solution of the velocity profile.

( 3 )
The crisp solution of velocity profile as well as temperature profile and the fuzzified velocity profile as well as temperature profile are in good agreements.The flow pattern for both the case velocity profile as well as Temperature profile are almost similar for different values of parameters.(4) With the increasing values of permeability parameter, the values of both the numbers Skin friction coefficient as well as Nusselt number are increases.(5) The effect of fuzzification is observed in the values of the physical quantities of the Skin friction coefficient  and local Nusselt number  .

Funding:
Not Applicable Consent Statement: Not Applicable Data Applicability: Not Applicable Conflict of Interested: Here with I declare there is no conflict of interested.