#### Abstract

The heat and mass transfer characteristics of unsteady flow of incompressible chemically reactive upper-convected Maxwell fluid along a stretching surface in the presence of temperature-dependent viscosity has been studied. The theoretical analysis on heat and mass transfer over a stretching sheet is investigated for numerical analysis. With the use of stream function formulation, the governing boundary layer equations of momentum, energy, and concentration are reduced to a set of linked ordinary differential equations. The nonlinear ordinary differential equations are then solved numerically by using the Keller Box method. The physical behavior of governing parameters on velocity, temperature, and concentration profiles and the local skin friction coefficient and heat and mass transfer rates are graphed and tabulated. The physical impact of Maxwell parameter *β*, unsteadiness parameter M, Schmidt number Sc, Prandtl number Pr, reaction rate parameter *γ*, and variable viscosity parameter on the heat and mass transfer has been examined along the stretching surface numerically. The novelty of the present work is to examine the importance of destructive reaction and contractive reaction on the dynamics of upper-convected Maxwell fluid flow in the presence of temperature-dependent viscosity effects. It is observed an interesting behavior of temperature distribution and concentration profile is noted for lower value of viscosity parameter in the presence of chemical reaction. It is also found that skin friction and the rate of heat transfer are decreased by increasing the variable viscosity parameter .

#### 1. Introduction

Heat and mass transfer together with the momentum of laminar boundary flow on a stretching sheet is important both theoretically and practically due to its wide application in polymer technology. The flow characteristics of non-Newtonian fluids are quite different in comparison to Newtonian fluids. In order to obtain a clear idea of non-Newtonian fluids and their various applications, it is necessary to study their flow behavior. Heat is a sort of energy that is transferred between different temperature systems or objects. The mass transfer phenomenon is used in various scientific disciplines for different systems and mechanisms that involve molecular and convective transport of atoms and molecules. The dynamics of a material having the properties of elasticity and viscosity when undergoing deformation is a fundamental topic in fluid dynamics. This kind of material referred to as Maxwell fluid has attracted the attention of many researchers due to its wide industrial and technical applications. The upper-convected Maxwell model can be described as the generalization of the Maxwell material for the case of large deformation using the upper-convected time derivative in the coordinate system stretching with the fluid. The study of boundary layer flow over a stretching surface is always essential from a technological point due to its multiple applications in engineering for non-Newtonian fluid flow mechanisms such as printer’s ink, melts, slurries, and groceries. Siddheshwar et al. [1] worked on forced flow mechanism for Newtonian liquids along the various stretchable geometries numerically. Aman et al. [2] developed a fractional mathematical model of Maxwell fluid along the porous geometry with second-order slip effects numerically. A similarity analysis was presented by Abd El-Aziz [3] to investigate a special distribution of the stretching velocity and surface heat flux for unsteady boundary layers over a stretched sheet. Abel et al. [4] performed a nonisothermal analysis on the stretching shape with heat flux and variable viscosity effects numerically.

In all of the mentioned studies, fluid viscosity has been assumed to be a constant function of temperature within the boundary layer. However, it is known that the physical properties of the fluid may change significantly when exposed to internal generated temperature. Viscosity depends strongly on temperature. In liquids, it usually decreases with increasing temperature, whereas in gases, viscosity increases with increasing temperature. Hassanein [5] developed a mathematical analysis for continuous shapes to examine heat transfer with variable viscosity. Khan et al. [6] analyzed the viscosity change in the Maxwell fluid mechanism under the influence of solutal stratification effects. Khan et al. [7] discussed the technical note for thin film flow behavior over shrinking geometry with thermal properties. Mukhopadhyay and Bhattacharyya [8] discussed the effects of chemical reactions on unsteady mixed convection flow and heat transfer over a porous stretching sheet. Some authors [9, 10] studied the effects of ohmic dissipation on viscoelastic flow and heat transfer under different conditions. Cortell [11] examined the numerical phenomena on second-grade viscoelastic flow to predict the mass and momentum transfer behavior across the porous mechanism theoretically. He obtained valid solutions compared to the previous mentioned results. An analytical solution of chemically reacting species in viscous flow across the plane elastic geometry under homogeneous reactions has been illustrated by Andersson et al. [12]. The technical problem about thermocapillary over a stretchable shape for thin film flow has been studied in [13, 14].

Elbashbeshy and Bazid [15] examined the velocity behavior of flow for unsteadiness along the stretching geometry numerically. Tsai et al. [16] studied the quiescent fluid medium problem with heat source effects across the unsteady stretching shape. The flow behavior of Maxwell fluid in a channel has been obtained theoretically in [17]. Chamkha et al. [18] produced similar results in porous shape for mass and heat behavior with chemical reaction. The mass transfer characteristics along the stretching geometry under slip and suction/injection effects by using shooting technique have been obtained by Bhattacharyya et al. [19, 20]. Gupta and Gupta [21] illustrated the mass transfer phenomena with blowing or suction impacts along the stretching surface mathematically. Ishak et al. [22] investigated the heat mechanism on permeable geometry with stretching and wall temperature effects numerically. The combined effects of transpiration and suction/blowing on unsteady mixed convection flow via a stretching sheet were studied by Mukhopadhyay and Vajravelu [23].

The flow of fluids may be categorized into Newtonian fluid flow, viscoelastic fluid flow, a flow of non-Newtonian fluid with time-dependent viscosity, and the flow of non-Newtonian fluids with time-independent viscosity. However, there exists no fluid flow which accurately fits the definition of a Newtonian fluid flow (i.e., shear stress is proportional to the rate of deformation). By taking an idea from Mukhopadhyay and Bhattacharyya [8], the unsteady two-dimensional flow of Maxwell fluid in the presence of variable viscosity and chemical reaction has been studied. The governing partial differential equations are transformed into ordinary ones using appropriate transformations, and the reduced ordinary differential equations are solved numerically using the Keller Box scheme. With the use of graphical representations, the impacts of controlling factors on velocity, temperature, and concentration fields with their slopes are explored and analyzed.

#### 2. Research Methodology

Over an unstable stretched sheet, we study laminar boundary-layer two-dimensional flow and mass transfer of an incompressible non-Newtonian Maxwell fluid. Let represent the concentration on the sheet surface and represent the concentration away from the sheet. Furthermore, the species’ reaction is a first-order homogeneous chemical reaction with a rate that varies with time. We assume that the fluid and mass fluxes are constant at time . At , the unsteady fluid and mass fluxes begin. The sheet emerges from a slit at the origin and moves with a nonuniform velocity , where are positive constants with dimensions (time)-1, is the initial stretching rate, and is the effective stretching rate that increases with time. The material properties of the extruded sheet may change over time in the case of polymer extrusion. The governing equations are given as follows [8]:

In Figure 1, and are the components of velocity in and directions, are the fluid’s dynamic and kinematic viscosity, is the fluid’s concentration (of the species), and is the diffusing species’ diffusion coefficient. Furthermore, is the time-dependent reaction rate, where denotes destructive reaction and denotes constructive reaction, is a constant, is the period’s relaxation time, is a constant, denotes temperature, and denotes free-stream temperature.

The suitable boundary conditions for the problem are given as follows:

Here, is concentration at the sheet surface equal to , where is the concentration far away from the surface, is variable temperature, is constant, is the thermal conductivity of fluid, and is the specific heat.

##### 2.1. Stream Function Formulation

The following physical dimensionless similarity transformations along with the stream function formulation were used by Mukhopadhyay [8, 23] to reduce the governing partial differential equations into the system of ordinary differential equations:where is the stream function and is the similarity variable.

With the help of the above relations, the governing partial differential equations are finally reduced to nonlinear ordinary differential equations given as follows:where Pr is the Prandtl number, is the similarity variable, is the dimensionless temperature, is the kinematic fluid viscosity, is the variable viscosity parameter, is the Schmidt number, is the unsteadiness parameter, is the Maxwell parameter, and is the reaction rate parameter. Moreover, shows the destructive reaction and shows reaction is constructive, but corresponds to no reaction.

The boundary conditions in (5) then becomes

The mathematical expression for skin friction is , the Nusselt number is , and the value of Schmidt number is defined as the ratio of momentum diffusivity and mass diffusivity. It is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection process. The mathematical expression for the Sherwood number is . The values of , and are .

Thus, the values of skin friction, Nusselt number, and Sherwood number are given by .

##### 2.2. Computational Technique

The coupled partial differential equations for the above model are reduced into a set of ordinary differential equations with the help of suitable stream function formulation. The reduced nonlinear ordinary differential equations (8)–(10) along with the boundary conditions in (11) are solved numerically by using the Keller Box scheme with an iterative method. We start by introducing new independent variables , and by using the following equation:

To overcome the difficulty of solving the equations, equations (8)–(11) become

The reduced boundary conditions are as follows:

We now consider the segment and with as the midpoint given as follows:

The central difference form and average form are given as follows:

By applying equations (18)–(22), governing equations (13)–(16) becomealong with boundary conditions

Now, by applying the iterative Newton–Raphson method for smooth algorithm, we get

As in the standard Newton–Raphson technique, we neglect all appearances of powers of greater than the first power and hence equations become

Again, by using equations (27)–(31) in equations (23)–(26), the reduced form of equations is given as follows:

We recall the boundary conditions that can be satisfied exactly with no iteration. Therefore, in order to keep up these correct values in all iterates, we take

Now, this is a very important step to arrange the above difference equations in matrix form. If it is carried out incorrectly, then either the method becomes very inefficient due to the absence of any discernible structure in the matrix or the matrix solution method breaks down due to having a singular matrix (determinant = 0) or submatrix. The matrix form is given as

#### 3. Analysis and Discussion of Results

The numerical computations of Maxwell fluid are executed for several values of dimensionless parameters involved in the equations such as variable viscosity parameter , the Prandtl number (Pr), the Maxwell parameter , and the unsteadiness parameter M. To illustrate the computed results, some figures are plotted and physical explanations are given for velocity , concentration , and temperature with their slopes. Table 1 presents the validated numerical data for skin friction for accuracy of results by comparing them with existing results available in the study by Chamkha et al. [18] and Mukhopadhyay and Bhattacharyya [8]. The validated results with good agreement are found by keeping unsteady flow of incompressible Newtonian flow fluid . The most favorable numerical results are obtained by adding the energy equation to the present phenomena.

Figures 2(a)–2(c)) present the physical behavior of , and profiles for numerous values of , and with some fixed parameters. From Figure 2(a), it is noted that the fluid velocity becomes maximum at a higher value of while the minimum value of velocity profile is examined at a lower value of . In Figure 2(b), an interesting behavior of the temperature profile is reported for various values of with chemical reaction effects. The concentration profile increases with lower value of but decreases as increases with prominent variations in Figure 2(c) in the presence of chemical reaction. The maximum value of concentration is noted at lower . The temperature profile is increased with suitable variations at but decreases as the parameter decreases significantly. In the presence of variable viscosity, the prominent behavior in concentration is illustrated for each value and temperature distribution is also found to be in good agreement.

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Figures 3(a)–3(c) demonstrate fluid velocity profile, fluid temperature profile, and fluid concentration profile for the various values of Prandtl number Pr , and by keeping other parameters , and constant along the stretchable surface. The velocity profile increases with a lower value of Pr and gradually decreases with increasing Pr in Figure 3(a). The maximum increasing behavior in temperature is depicted at a lower value of Pr , but this decreases as the value of Pr increases with prominent variations in Figure 3(b) significantly. The similar trend in concentration is examined for each value of Pr graphically, but the concentration increases with decreasing Pr values by satisfying the given boundary conditions in Figure 3(c). In the presence of viscosity effects, a suitable change is examined in velocity and temperature but a similar trend is examined in concentration for . The suitable response in concentration is illustrated in the presence of . The fluid flow caused solely, and the concentration is maximum than free-stream concentration along the stretching sheet.

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From Figures 4(a)–4(c), it is concluded that the fluid velocity is maximum along the surface of a given shape for a lower value of and the minimum value is obtained at the largest value of with good response. The velocity profile showed good variations in the presence of in Figure 4(a). In Figure 4(b), the temperature profile is represented graphically for different values of , and by keeping other parameters constant. In Figure 4(c), it is noted that the concentration distribution has a similar behavior at lower values of but increasing change is noted as increases. It is observed that temperature shows similar behavior for each value of and gradually increases with increasing . The boundary layer thickness is reduced by increasing the effect of Maxwell parameter .

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In Figures 5(a)–5(c), the velocity, temperature, and concentration profiles are drafted against various values of unsteadiness parameter ,and with some fixed parameters , and Pr . The most favorable change in each profile is noted against the unsteadiness parameter in the presence of viscosity and chemical reaction effects. The velocity, temperature, and concentration increase at a minimum value of but decrease gradually with increasing with prominent variations in the presence of . The increasing effect of unsteadiness parameter exhibits the reduction in thickness of thermal boundary layer near the wall of stretching surface. In Figures 6(a)–6(c), the effects of viscosity parameter on skin friction , heat transfer , and mass transfer are clearly shown. The maximum value of and is noted at a lower value of viscosity parameter , and this gradually decreases with increasing in the presence of chemical reaction. The decreasing behavior in mass transfer is drafted by increasing but increases as decreases significantly.

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#### 4. Conclusion

The graphical and numerical outcomes of Maxwell fluid with variable viscosity and chemical reaction effects are executed for several values of dimensionless parameters involved in the equations such as the variable viscosity parameter , the Prandtl number (Pr), the Maxwell parameter , and the unsteadiness parameter M. To illustrate the computed results, some figures are plotted and physical explanations are given for velocity , concentration , and temperature with their slopes skin friction , mass transfer , and heat transfer , respectively. The main outcomes are given as follows:(i)The fluid velocity is depicted maximum at a higher value of , while the minimum value of velocity profile is examined at a lower value of (ii)The similar trend in concentration is examined for each value of Pr graphically but it increases with decreasing Pr by satisfying the given boundary conditions(iii)The increasing effect of the unsteadiness parameter exhibits the reduction in thickness of the thermal boundary layer near the wall of stretching surface(iv)The maximum increasing behavior in temperature is depicted at a lower value of Pr , but it decreases as the value of Pr increases with prominent variations.(v)The maximum value of and is noted at a lower value of viscosity parameter , and it gradually decreases with increasing in the presence of chemical reaction.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.