Study of Fuzzified Boundary Value Problems for MHD Couette and Poiseuille Flow of a Differential Type Fluid


 In this work, discuss the magneto-hydro-dynamics (MHD) flow in three fundamental flaws of the third-grade fluid between two parallel plates in a fuzzy environment by the fuzzy Adomian decomposition method (ADM). We extend the work of Kamran and Siddique [19], using fuzzy differential equations (FDEs) and explain our approach with the help of membership function of triangular fuzzy numbers (TFNs). In the end, the effect of the fuzzy parameter (\(\alpha \in [0,\,1]\)), and other engineering parameters on fuzzy velocity profiles are investigating in graphically and tabular representation.


INTRODUCTION
The study of non-Newtonian fluids has gained considerable attention to scientists because of extensive applications in engineering, science and industry. Various industrial ingredients fall into this bunch, such as biological solutions, soap, paints, cosmetics, tars, shampoos, mayonnaise, blood, yogurt, syrups, and glues, etc. Due to the intricate nature of non-Newtonian fluid, it's very hard to establish a single model that can describe characteristics of all non-Newtonian fluid. So, the fluids of differential type [1] have received superior consideration by researches. Here, we will consider the third-grade fluids (differential type by a subclass), which have been studied effectively in numerous types of flow mechanisms [2][3][4][5][6][7][8][9]. In fluid dynamics, the study of three fundamental flows (namely, Couette, Poiseuille, and generalized Couette flow) attracts the researchers by various non-Newtonian fluids, because of their uses in engineering and industry. The unidirectional flows are used in polymer engineering for instance die flow, injection molding, extrusion, plastic forming, continuous casting, and asthenosphere flows [10][11][12][13]. Magneto-hydrodynamics (MHD) deals with the study of the motion of electrically conducting fluids in the presence of the magnetic field. MHD flow has significant importance applications between infinite parallel plates in various areas such as geophysical, astrophysical, metallurgical processing, MHD generators, pumps, geothermal reservoirs, polymer technology, and mineral industries, etc. MHD fluids use as a lubricant in industrial and other applications, for stops the unexpected variation of lubricant viscosity with temperature under the certain norms. In this way there is large number of literature as Khan et al. [14], Hayat et al. [15, 16, and 17], and Islam et al. [18]. Kamran and Siddique [19] calculated the analytical solutions for MHD flow between infinite parallel plates.
ADM is reliable and effective technique to compute the linear and non-linear differential equations. So, it gives analytical solutions in the form of an infinite convergent series. The ADM has some vital advantages over other analytical methods as well as numerical methods, needs no linearization, discretization, perturbation, and spatial transformation. Siddiqui et al., [23] deliberated parallel plate flow of a third grade fluid using ADM and compare the results with numerical technique. Pirzada and Vakaskar [24] calculated the solution of the fuzzy heat equation with the help of fuzzy ADM. Paripour et al., [25] studied the analytical solution of hybrid FDEs by using the fuzzy ADM and predictor-corrector method, which shows ADM is better than the predictor-corrector method. Also, Siddiquie et al., [26] provided a comparison of ADM and homotopy perturbation method (HPM) in terms of squeezing flow between two circular plates. Their analysis shows the ADM is better than HPM. Biswal et al. [27] studied Natural convection of nanofluid flow between two parallel plates using HPM in a fuzzy environment. The volume fraction of nanoparticle considered as TFN and also shows the fuzzy result is better than a crisp result.
Fluid flow plays a main role in the field of science and engineering. The rise in an extensive range of problems like chemical diffusion, magnetic effect and heat transfer etc. After governing these physical problems convert into linear or nonlinear DEs. In general, the physical problems with involved geometry, coefficients, parameters, initial, and boundary conditions greatly affect the solution of DEs. Then the coefficients, parameters, initial, and boundary conditions are not crisp due to the mechanical defect, experimental error, and measurement error etc. So in this situation, fuzzy sets theory is an effective tool for a better understanding of the considered phenomena and it is more accurate than assuming the crisp or classical physical problems. For more precisely the FDEs play a major role to reduce the uncertainty and proper way to describe the physical problem which arise in uncertain parameters, initial and boundary conditions.
In 1965, Zadeh [28] presented the fuzzy set theory (FST). FST is very valuable tool to define the situation in which information is imprecise, vague or uncertain. FST is completely defined by its membership function or belongingness. In FST, the membership function describes each element of the universe of discourse by a number from [0, 1] interval. On the other hand, the degree of nonbelongingness is a complement to "one" of the membership degree or belongingness. Fuzzy number (FN) can be expected as a function whose range is specified zero to one. Every numerical value in the range is allocated a definite grade of membership function where, 0 signifies the minimum possible grade and 1 is the maximum possible grade. As FST is the generalization of crisp (classical) set theory and similarly, FNs are the generalization of real intervals. Arithmetic operations on FNs are developed by Dubois and Prade [29]. Different types of FNs can be categories in triangular, trapezoidal and Gaussian fuzzy numbers. Here we consider TFNs for the sake of completeness.
The modeling of physical problems in terms of ordinary differential equations (ODEs) includes uncertain parameters or variables. This impreciseness or vagueness can be defined mathematically using FNs. Seikala [30] introduced the fuzzy differentiability concept. Later on, Kaleva [31] presented the fuzzy differentiation and integration. Kandel and Byatt [32] introduced the FDEs in 1987. Buckleyet et al., [33] used two methods extension principle and FNs for the solution of FDEs. Nieto [34] studied the Cauchy problem for continuous FDEs. Lakshmikantham and Mohapatra studied the initial value problems for FDEs has been commenced in [35].
Salahsour et al. [36] studied the fuzzy logistic equation and alley effect using FDE with the help of TFNs. Hashemi et al. [37] used HAM to calculate the analytical solutions for system of fuzzy differential equations (SFDEs). Tzimopoulos et al. [38] studied the solution of vertical infiltration equation using FDE and get well-known results in a fuzzy environment. Gasilov et al., [39] developed the geometric method to solve SFDEs. Khastan and Nieto [40] used generalized differentiability concept to solve the second order FDE. In few decades ago, there have been many studies revolving around the concept of "fuzzy differential equations". Many scholars have applied FST to obtained well-known results in the field of commerce and science.
In the review of literature, third-grade problems studied for only crisp or classical cases. So, the above-mentioned works motivated us to develop a model to describe the fuzzy analysis for unidirectional MHD flow in three fundamental flow problems namely, plane Couette, fully developed plane Poiseuille and plane Couette-Poiseuille flow of a third-grade fluid between two parallel plates. The basic purpose of this article is to show the uncertain flow mechanism through FDEs and generalize the work of Kamran and Siddique [19] in the circumstance of fuzzy environment.
The article is structured as follows; basic preliminaries are given in section 2. In section 3, developed the governing equations of the proposed study and changed governing equations in the fuzzy form for solving by a fuzzy ADM. Results and discussion in graphical with tabular forms are presented in section 4. In section 5, some conclusions are given.

Definition
(v) Z  must be closed interval for every 01   also  , is called level of credibility or presumption. Membership function or grade is also named as grade of possibility or grade of credibility for a given number. [28]: Let ( , , )

Definition
, for , The TFNs with peak (center) ,  right width 0,   left width 0,   and these TFNs are transformed into interval numbers through  -cut approach, is written as

Basic equations and formulation of the problem
Let us consider the steady flow of a third-grade fluid between two horizontal parallel plates at distance d apart. A uniform magnetic field is applied transversely to the plates. We choose coordinate system in which the x-axis is taken perpendicular and y -axis is parallel to the plates.
For incompressible fluid the basic equations are: "where  is constant density, V is the velocity vector, f represents the body force per unit mass, *  is the stress tensor, J is the electric current density,  is the viscosity, B is the total ,  B denotes the imposed magnetic field and b represents the induced magnetic field and    is the material time derivative." In the absence of displacement currents, the modified Ohm's law and Maxwell's equations [19] are wherein, E is the electric field, m  is the magnetic permeability and  is the electrical conductivity.
"From Ohm's law and Maxwell's equations an evolution for the magnetic flux B can be obtained easily. This is known as the magnetic induction equation and it suggests that the motion of an electrically conducting fluid in an applied magnetic field induces a magnetic field in the medium. We assume that the total magnetic field B is perpendicular to the velocity field V and the induced magnetic field b is negligible compared with the applied magnetic field  B , so that magnetic Reynolds number is small. Since no external electric field is applied and the effect of polarization of the ionized fluid is negligible, the fluid flow region is assumed as no electric field."Under these assumptions, the MHD force involved in Eq. (2) can be put into the form [19] For third-grade fluid, the stress tensor *  is given by [1][2][3] where  is the coefficient of viscosity, p is the pressure, I is the unit tensor, The flow is one dimensional, we define velocity field in component form as: By using the above assumptions and Eq. (7), continuity Eq. (1) is identically satisfied and the momentum Eq. (2) in the absence of body forces reduces to the forms On introducing the modified pressure  p   Eq. (12) is a second-order non-linear ordinary differential equation.

Adomian decomposition method
In this section, we discuss the basic sketch of ADM and write the basic non-linear differential equation as where 1 , qL and 1 N are source term, linear and non-linear operators respectively. Also the operator 1 L can be written as 11 , here, L is the highest order derivative in 1 L and is assumed to be easily invertible, 1 R is the remaining operator in 1 L whose order is less than the order of ˆ. L From Eqs. (13) and (14) we Applying 1 L  on the above Eq. (14), we have where   gx signifies the terms arising after integration of   qx and calculate constants of integration with the help of boundary conditions. So, can be write as [20][21][22], where * , ns A are called Adomian polynomials [20][21].
The algorithm of the general ADM can be communicated as Many researchers have been established the convergence of this method [22]. In this continuation, we apply the ADM in fuzzy sense to three flow problems.

Plane Couette flow
"Let us consider the steady flow of an incompressible third grade fluid between two horizontal infinite parallel plates. The lower plate is fixed while, the upper plate at xd  is moving with uniform velocity U. The plates are non-conducting and a magnetic field is applied in the vertical upward direction. Let y-axis be taken normal direction to flow and x-axis in the direction of flow (See in Fig. 1)." In the absence of pressure gradient, the differential equation (12) for such a flow reduces to  (22) and the boundary conditions are Introducing the following non-dimensional parameters and dropping the ' m' notation (for simplicity), the boundary value problem (22)

Solution of the problems in fuzzy environment
To handle these problems we have taken TFNs and discretization in the form of ( , , )    and ( , , ) d e f for the fuzzy parameters. This discretization is used in the boundary of the parallel plates for certain flow behavior because of the boundary is taken as fuzzified. The governing equation (24) with boundary conditions (25) are converted into FDEs as given below  (27) where operator e defines the multiplication of fuzzy numbers with a real number and  are lower and upper fuzzy velocity profiles, while boundary conditions in (27) are fuzzy boundary conditions [29]. So, Eqs. (26) and (27) Now we use the ADM in fuzzy boundary value problems (28) where the constants of integration are 1 c and 2 c .
In view of Eqs. (16) and (17) We identify the zeroth component as and the remaining components as the recurrence relation, Eqs. (41) and (42) with the boundary conditions Presenting the following non-dimensional parameters   2  2  3  0  2  2  3  2  2  2  2  2 , , , into Eqs. (43) and (44) To solve the above boundary value problem, we adopt the same procedure as in sec 4.1. For this Eqs. (46) and (47) (55) where the constants of integration are 1 c and 2 c .

Generalized Couette Flow
"Again we consider the steady laminar flow of a third grade fluid between two infinite horizontal parallel plates at a distance d apart under a constant pressure gradient. The lower plate is fixed while the upper plate at xd  is moving with uniform velocity U. The plates are nonconducting and a magnetic field is applied in the vertical upward direction. Let y-axis be taken normal direction to flow and x-axis in the direction of flow (See in Fig. 3)." The resulting non dimensional differential equation (46) with boundary conditions (25), under the transversal magnetic field and constant pressure gradient is

Results and Discussion
In this section we discuss graphically (plotted in Figs  In Fig. 12 the uncertain width gradually decreases with increasing input the magnetic parameter m while Fig. 13 shows the uncertain width gradually increases with increasing .  Furthermore, it is seen that less width of fuzzy velocity profiles at the centre of the plates so the uncertainty is less than the boundary is less sensitive. While, width of fuzzy velocity profiles are greater so uncertainty is maximum then the boundary is more sensitive. Also it is observed that  Fig. 43 shows the uncertain width rapidly increases with increasing the value of . dp dy p  Furthermore, it is seen that less width of fuzzy velocity profiles at the centre of the plates so uncertainty is less than the boundary is less sensitive. While, width of fuzzy velocity profiles are greater so uncertainty is maximum then the boundary is more sensitive. Table 3

Conclusions
In this work, we have studied the three fundamental flow problems that frequently arise in the