#### Abstract

In this work, fundamental ﬂow problems, namely, Couette flow, fully developed plane Poiseuille flow, and plane Couette–Poiseuille flow of a third-grade non-Newtonian ﬂuid between two horizontal parallel plates separated by a finite distance in a fuzzy environment are considered. The governing nonlinear differential equations (DEs) are converted into fuzzy differential equations (FDEs) and explain our approach with the help of the membership function (MF) of triangular fuzzy numbers (TFNs). Adomian decomposition method (ADM) is used to solve fundamental ﬂow problems based on FDEs. In a crisp environment, the current findings are in good accord with their previous numerical and analytical results. Finally, the effect of the -cut and other engineering constants on fuzzy velocity proﬁle are invested in graphically and tabular forms. Also, the variability of the uncertainty is studied through the triangular MF.

#### 1. Introduction

The non-Newtonian fluids have gained considerable attention from 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, yoghurt, syrups, and glues, etc. Due to the intricate nature of non-Newtonian ﬂuids, it is very hard to establish a single model that can describe the characteristics of all non-Newtonian ﬂuids. So, the ﬂuids of differential type [1] have received superior consideration by researchers. Here, we will consider the third-grade ﬂuids (differential type by a subclass), which have been studied effectively in numerous types of ﬂow mechanisms [2–9]. In ﬂuid dynamics, the study of three fundamental ﬂows specifically (Couette, Poiseuille, and generalized Couette ﬂow) attracts the researchers by various non-Newtonian ﬂuids, due to their uses in science, engineering, and industry. The unidirectional ﬂows are used in polymer engineering, for instance, die ﬂow, injection moulding, extrusion, plastic forming, continuous casting, and asthenosphere ﬂows [10–13]. ADM was introduced by Adomian [14–16]. ADM is a reliable, effective, and powerful technique to calculate linear and nonlinear DEs. It gives analytical solutions in the form of an inﬁnite convergent series The ADM has various imperative points of interest over other scientific techniques just as mathematical strategies, no linearization, discretization, perturbation, and spatial transformation. Siddiqui et al., [17] deliberated parallel plate ﬂow of a third-grade ﬂuid using ADM and compare the results with numerical technique.” Pirzada and Vakaskar [18] calculated the solution of the fuzzy heat equation with the help of fuzzy ADM. Paripour et al. [19] 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, Siddiqui et al., [20] provided a comparison of ADM and Homotopy perturbation method (HPM) in terms of squeezing ﬂow between two circular plates. Their analysis shows that ADM is better than HPM. In the review of literature, third-grade fluid problems are studied only for crisp cases.”

Fluid flow plays the main role in the field of science and engineering. The rise is 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 defects, experimental and measurement errors, etc. So in this situation, fuzzy sets theory (FST) is an effective tool for a better understanding of the considered phenomena and it is more accurate than assuming the crisp physical problems. More precisely, the FDEs play a major role to reduce the uncertainty and proper way to describe the physical problem which arises in uncertain parameters, initial and boundary conditions.

Zadeh [21] established the FST in 1965. FST is an extremely useful technique for defining situations in which information is ambiguous, hazy, or uncertain. FST is entirely defined by its MF or sense of belonging. In FST, the MF allocates a number between 0 and 1 to each element in the discourse universe. On the other hand, the degree of nonbelongingness is a complement to “one” of the membership degree. The fuzzy number (FN) is a generalization of the classical real number (which uses absolute 0 and 1 only, and nothing in between) with additional properties. FN can be expected as a function whose domain is speciﬁed zero to one. This domain is called an MF. Every numerical value in the domain is allocated a deﬁnite grade of MF where 0 signiﬁes the minimum possible grade and 1 is the maximum possible grade. FNs are competent in modelling epistemic uncertainty and its circulation. FNs are a very useful tool for FDEs, fuzzy analysis and other applications in fuzzy logic. Arithmetic operations on FNs were developed by Dubois and Prade [22]. The impreciseness or vagueness can be deﬁned mathematically using FNs. The information of dynamical systems modelled by ODEs or PEDs is commonly incomplete, vague, or uncertain, while FDEs represent a proper way to model the dynamical systems under vagueness or uncertainty. Seikala [23] introduced the fuzzy differentiability concept. Later on, Kaleva [24] presented fuzzy differentiation and integration. Kandel and Byatt [25] introduced the FDEs in 1987. Buckley et al. [26] used two methods extension principle and FNs for the solution of FDEs. Nieto [27] studied the Cauchy problem for continuous FDEs. Lakshmikantham and Mohapatra studied the initial value problems for FDEs that have been commenced in [28]. Park and Han [29] used successive approximation methods for the existence and uniqueness solution of FDE. Hashemi et al. [30] used HAM (Homotopyianalysis method) to calculate the analytical solutions for the system of fuzzy differential equations (SFDEs). Mosleh [31] used universal approximation and fuzzy neural network methods to solve the SFDEs. Gasilov et al. [32] developed the geometric method to solve SFDEs. Khastan and Nieto [33] used a generalized differentiability concept to solve the second-order FDE. A few decades ago, there have been many studies revolving around the concept of FDEs. [34–47] Biswal et al. [48] studied the Natural convection of nanofluid flow between two parallel plates using HPM in a fuzzy environment. The volume fraction of nanoparticles is considered as TFN and also shows the fuzzy result is better than a crisp result. Borah et al. [49] discussed the MHD flow of second-grade fluids in a fuzzy environment using fractional derivatives Atangana-Baleanu and Caputo-Fabrizio. The nondimensional governing equations convert into fuzzified governing equations with the help of the Zadeh extension principle and triangular fuzzy number. MHD and ohmic heating on the third-grade fluid in an inclined channel in a fuzzy environment was investigated by Nadeem et al. [50]. To discuss the uncertainty, the triangle membership function was used, as shown in Figure 1. Furthermore, FST has been employed by several researchers to accomplish well-known scientific and engineering conclusions [51–59].

The above-mentioned works motivated us to develop a model to describe the fuzzy\uncertain analysis for fundamental ﬂow problems, namely, plane Couette, fully developed plane Poiseuille, and plane Couette–Poiseuille ﬂow of the third-grade ﬂuid between two parallel plates. The basic purpose of this article is to show the uncertain ﬂow mechanism through FDEs and the generalization to the work of Siddiqui et al. [17] in the circumstance of fuzzy environment.

The article is structured as follows. Basic preliminaries are given in Section 2. In Section 3, the governing equations of the proposed study and changed governing equations in the fuzzy form for solving by a fuzzy ADM are developed. Results and discussion in graphical and tabular forms are presented in Section 4. In Section 5, some conclusions are given.

#### 2. Preliminaries about FST

*Definition 1 (see [21, 50]). *Fuzzy set is deﬁned as a set of ordered pairs such that here is the universal set, is MF of and mapping deﬁned as

*Definition 2 (see [22, 50]). *- cut or - level of a fuzzy set deﬁned by 0 where is crisp set and

*Definition 3 (see [22, 50]). *Let with MF called a TFNs ifThe TFNs with peak (center) right width left width and representation of ordered pair functions through -cut approach is written as where TFNs satisfy the following conditions: (i) is nondecreasing on [0, 1]. (ii) is nonincreasing on [0, 1]. (iii) on [0, 1]. (iv) and are bounded on left continuous and right continuous at [0, 1], respectively. (v) If then it is called a crisp solution.

*Definition 4 (see [23, 50]). *Let be an interval such that A mapping is called a fuzzy process, deﬁned as and The derivative of a fuzzy process is deﬁned as

*Definition 5 (see [23, 50]). *Let and be a fuzzy valued function deﬁne on Let for all -cut. Assume that and have continuous derivatives or differentiable, for all and then Similarly, the higher-order derivatives can be deﬁned in the same way. Then satisfy the following conditions: (i) is nondecreasing on [0, 1]. (ii) is nonincreasing on [0, 1]. (iii) on [0, 1].

#### 3. Basic Equations

The basic equations which govern the flow of an incompressible fluid ignoring the thermal effects are as follows [17]:where *f* is the body force, is the pressure, is the density of ﬂuid, V is velocity vector, is the material derivative, *I* is the unit tensor, is the stress tensor, and is the extra stress deﬁned as follows:Here, represents the coefficient of shear viscosity, and are material constants. The tensors are, respectively, given by the following:

“For the problem under consideration, we assume a velocity profile for one-dimensional flow and stress tensor of the form.”

By utilizing equation (7), the continuity equation (2) is indistinguishably fulfilled and the equation of motion (3), without gravitational impact, becomes as follows:

On introducing the modiﬁed pressure

Using equation (10) in (9), we find that

From equation (11) and as a result, equation (8) becomes as follows:where for simplicity we have introduced

“Equation (12) is a second-order nonlinear ordinary differential equation. This equation governs the unidirectional flow of a non-Newtonian third-grade fluid between two horizontal infinite parallel plates.”

##### 3.1. The Adomian Decomposition Method (ADM)

In this section, we briefly outline the decomposition method [14]. To clarify the basic idea, we write the underlying nonlinear differential equation as follows:where and are linear and nonlinear operators, respectively, and is the source term.

In general, the operator can be written in the formwhere is the highest order derivative in and is assumed to be easily invertible, is the remaining operator in whose order is less than the order of

Using equation (14) in (13), we have the following:

Applying on the above equation, we have the following:where signiﬁes the terms arising after integration of and calculate the constants of integration with the help of initial/boundary conditions. According to Adomian [14–16], and can be uttered individually in the formwhere are called Adomian polynomials [14, 15].

The algorithm of the general ADM can be communicated as follows:

Thus, by calculating all components of the solution can be written as follows:

Many researchers have established the convergence of this method [16]. In this continuation, we apply the ADM in the fuzzy sense to three flow problems *d* problems.

##### 3.2. Plane Couette Flow

Let us consider the steady laminar ﬂow of an incompressible third-grade ﬂuid between two infinite horizontal parallel plates. The lower plate is ﬁxed while the upper plate at a distance is moving with unvarying speed Also assume that is normal to the ﬂow while is taken in the direction of ﬂow as shown in Figure 2. In the absence of pressure gradient, the resultant differential equation (12) for such flow with boundary conditions reduces to the following:

Introduce the following dimensionless parameters

The boundary value problem (22) and (24) after dropping “” becomes

###### 3.2.1. Solution of the Problem in Fuzzy Environment

To handle these problems, we have taken TFNs and discretization in the form of and for the fuzzy parameters. This discretization is used in the boundary of the parallel plates for certain flow behavior because the boundary is taken as fuzzified. In above, the governing equation (26) is taken as FDE

Subject to fuzzy boundary conditions by using the - cut approach, the fuzzy boundary conditions can be decomposed into an interval form regarding the - cut. Therefore, under the - cut, the interval boundary conditions can be written as follows:where operator defines the multiplication of fuzzy numbers with a real number and is a fuzzy valued function [23]. Also , say, is the fuzzy velocity proﬁle, and represent the fuzzy first and second-order derivatives. Then are the lower and upper bounds of fuzzy velocity proﬁles, respectively, while (30) are the fuzzy boundary conditions. So, equation (29) with fuzzy boundary conditions becomes

For lower bound of velocity proﬁle, we apply the ADM to equation (31) with the fuzzified boundary conditions (32) as follows:where and inverse operator is Applying on both sides of equation (35), we have the following:where the constants of integration are and .

To solve equation (36) by the ADM, we letwhere

In view of equations (37)–(39), equation (36) provides the following:

We identify the zeroth-order component as follows:

And the remaining components as the recurrence relation,where are the Adomian polynomials that represent the nonlinear term in (39). The first few Adomian polynomials are as follows:

The corresponding fuzzy boundary conditions, after using (37) in (32), become as follows:

And

Solving equations (41) and (44), we obtain the zeroth-order solution as follows:

Using polynomial (43) into (42), with the boundary conditions (45), we obtain the following:

Inserting equations (46) and (47) in (37), the solution of the differential equation (31) takes the form

Similarly, the upper bound of velocity proﬁle is as follows:

Equations (48) and (49) represent the solutions of the fuzzy velocity proﬁle for the flow of a fuzzified third-grade ﬂuid between two parallel plates. In these solutions, the non-Newtonian parameter does not appear. Hence, solutions of fuzzy velocity proﬁle give the same solution as for a Newtonian ﬂuid.

##### 3.3. Fully Developed Plane Poiseuille Flow

We consider the steady laminar ﬂow of a third-grade ﬂuid between two stationary infinite parallel plates under a constant pressure gradient. Assume that distance between two plates is 2*d* and origin of the rectangular coordinates in between the plates as shown in Figure 3.

Thus, the resulting differential equation for the problem under consideration takes the form of equation (12) with the following boundary conditions

Introducing the dimensionless parameters

Equations (12) and (51) after dropping “” take the following form:

Now, convert equations (53) and (55) into the form of FDE as follows:

Lower and upper bounds of velocity profile are as follows:

For lower bound of velocity proﬁle, we apply the ADM to equation (58) as we have applied in Section 3.2.1, with the fuzzified boundary conditions (59)where Applying the inverse operator on both sides of equation (62) yields the following:where the constants of integration are and .

Given equations (37) and (38), equation (63) is provided as follows:

Consequently, the decomposition method yields the recurrence relation,where the first few terms of the Adomian polynomial that represent the nonlinear term are defined in (43). Insight of expressions (66), we know that

The corresponding fuzzy boundary conditions become as follows:

And so on

By solving equations (65) and (66) with the fuzzified boundary conditions (68) and (69), using expression of adomian polynomials (43), equation (63) gives the solution of lower bound of velocity profile as follows:where and

Equation (70) is the approximate solution of the fully developed plan Poiseuille ﬂow and is a non-Newtonian parameter. By setting we have the solution for a viscous fluid.

Similarly, we can find the solution of the upper bound of velocity profile as follows:where and

##### 3.4. Plane Couette–Poiseuille ﬂow

Again consider the steady laminar flow of a third-grade fluid between two infinite horizontal parallel plates at a distance *d* apart. The upper plate is moving with constant speed while the lower plate is stationary. We choose along with the lower plate and perpendicular to it as shown in Figure 4. The resulting differential equation in dimensionless form is (53) and the corresponding dimensionless form boundary conditions for this flow are given as follows.

Now we convert equations (53) and (72) into the form of FDE as follows:

Lower and upper bounds of velocity proﬁles with fuzzy boundary conditions are as follows:

Following the same process as in previous sections and applying ADM to equation (75) with the fuzzified boundary conditions (76), we find the solution of lower bound of velocity profile as follows:where Similarly, the solution of an upper bound of the velocity profile is as follows:where and

#### 4. Results and Discussion

In this section, we present a numerical solution of Plane Couette flow, fully developed plane Poiseuille flow, and plane Couette–Poiseuille flow for the third-grade non-Newtonian fluid with fuzzified boundary conditions. Firstly, convert the governing equations of these problems into FDEs, then find the solutions for fuzzy velocity proﬁles by using ADM. Achieved fuzzy velocity profiles are plotted in Figures 5–17 for different values of -cut It can be observed that as increases from 0 to 1, we have a narrow width of fuzzy velocity profiles and uncertainty decreases drastically, which finally provide crisp results for

Tables 1–3 show the comparison of the crisp velocity profile with Siddiqui et al. [4] and Yürüsoy [9]. The validation of the present study findings was determined to be in excellent agreement.

##### 4.1. For Plane Couette Flow

The non-Newtonian parameter does not exist in this solution. As a result, solutions for fuzzy velocity profiles are the same as for a Newtonian fluid.

##### 4.2. For Fully Developed Plane Poiseuille Flow

Figures 5–8 show the effect of non-Newtonian parameter on the fuzzy velocity proﬁles with constant pressure gradient at different values of fuzzy parameter It is observed that the lower and upper bounds of velocity proﬁles decrease with increasing non-Newtonian parameter as well fuzzy parameter Figure 8 shows the good agreement of crisp solution or classical solution that lower and upper bounds of velocity proﬁles are the same at Figure 9 describes the lower and upper bounds of fuzzy velocity proﬁles at the different values of So, for the fuzzy velocity profile falls into classical velocity profile, which shows the present problem is a generalization of Siddiqui et al. [17]. Figure 10 shows the uncertain behavior in terms of the triangular fuzzy plot by ﬁxing the values of and The horizontal axis display the velocity profile and the vertical axis expresses the which range from 0 to 1. In this figure, uncertain width gradually decreases with increasing -cut. We observed that increases and decreases with increasing of -cut, so the solution is strong. When increases the width between lower and upper bounds of fuzzy velocity proﬁles decreases and when they concur with one another. Also, the width between and for different values of beta is the same. This means that the uncertainty of fuzzy velocity is minimum. Table 4 shows the analysis of lower, mid and upper bounds of velocity proﬁles at different values of *x* with constant pressure gradient *p* = −0.4. The mid-value of a TFN concurs with the crisp or classical value of the original problem.

##### 4.3. For Plane Couette–Poiseuille Flow

Figures 11–14 shows the effect of non-Newtonian parameter on the fuzzy velocity proﬁles with constant pressure gradient at different values of fuzzy parameter The fuzzy velocity proﬁles increases with increasing non-Newtonian parameter and fuzzy parameter Figure 14, shows the good agreement of crisp solution or classical solution for and of velocity proﬁles at Figure 15 describes the lower and upper bounds of fuzzy velocity proﬁles at the different values of So, for the fuzzy velocity profile fall into classical velocity profile, which shows the present problem is a generalization of Siddiqui et al. [17]. Figure 16 represents the fuzzy velocity profile for different ranges of the imposed pressure gradient. Figure 17 shows the uncertain behavior in terms of a triangular fuzzy plot by ﬁxing the values of and We observed that increases and decreases with increasing so the solution is strong. The crisp or classical solution lies among the fuzzy solutions when increases the width between lower and upper bounds of fuzzy velocity proﬁles decreases and at the coherent with one another. Since the boundary conditions are fuzzy, the uncertain width gradually decreases with increasing and non-Newtonian parameter Table 5 shows the analysis of lower, mid, and upper bounds of velocity proﬁles at different values of *x* with constant pressure gradient *p* = −0.6. The mid-value of a TFN concurs with the crisp or classical value of the original problem. Furthermore, fuzzy velocity profiles always change with a certain range for any fixed -cut and the range gradually decreases with increasing the values of - cut.

This whole discussion concludes that the fuzzy velocity profile of the fluid is a better option as compared to the crisp or classical velocity profile of the fluid. Crisp or classical velocity profile of fluid gives the single flow situation of the fluid while fuzzy velocity profile of fluid gives the interval flow situation like lower and upper bounds of the velocity profile.

###### 4.3.1. Fully Developed Plane Poiseuille Flow

Fuzzy velocity profiles are provided in Figures 5–10.

###### 4.3.2. Plane Couette–Poiseuille Flow

Fuzzy velocity profiles, inﬂuence of pressure gradient, and triangular MFs of fuzzy velocity profiles are shown in Figures 11–17, respectively.

#### 5. Conclusions

In this work, we have studied the three basic fundamental flow problems in a fuzzy environment. The dimensionless nonlinear governing equations are converted into FDEs with fuzzified boundary conditions and find their solutions using ADM. For the case of a plane Couette flow, we find the same solution as in the incident of viscous fluid. For plane Poiseuille and generalized Couette flows, triangular fuzzy numbers are used for uncertainty on the dynamic behavior of fuzzy velocity proﬁles. The most important findings are presented below:(i)Fuzzy velocity proﬁles increases with increasing the non-Newtonian ﬂuid parameter and fuzzy parameter .(ii)The results are indicated that the range of calculated lower and upper-velocity profiles depends upon a fuzzy parameter.(iii)The results are always an envelope of solutions with a crisp solution between the upper and lower bounds of the solutions. So fuzzy velocity proﬁles are the generalization of the crisp velocity proﬁle for third-grade ﬂuid between two parallel plates.(iv)Furthermore, it is observed that, in triangular MFs, if the width of fuzzy or uncertain velocity becomes more, then the boundary conditions are more sensitive, while for less width of fuzzy or uncertain velocity, the assumed boundary conditions are less sensitive.(v)The present crisp results obtained from ADM are found to be in excellent agreement as compared to existing results.(vi)In future work, for easier comprehension, the TFN is visualized. As a result, TFNs may be used to a variety of heat transfer challenges.

#### Data Availability

No data were used to perform this research.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.