Maxwell Nanofluid Flow over an Infinite Vertical Plate with Ramped and Isothermal Wall Temperature and Concentration

Department of Mathematics, City University of Science and Information Technology, Peshawar 25000, Khyber Pakhtunkhwa, Pakistan Computational Analysis Research Group, Ton Duc (ang University, Ho Chi Minh City 70000, Vietnam Faculty of Mathematics and Statistics, Ton Duc (ang University, Ho Chi Minh City 70000, Vietnam Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia


Introduction
ere are several types of fluids that cannot be described by simple Navier-Stokes equations because of their complex rheology.
ese fluids are called non-Newtonian fluids which have many applications in our daily life such as biotechnologies, geophysics, astrophysics, industries, and engineering developments. erefore, researchers are attracted to study the non-Newtonian fluids. Non-Newtonian fluids may be further classified as differential, rate, and integral type fluids. Integral type fluids are those in which the actual Cauchy stress is calculated by reading an integral over the background of the general deformation gradient. Differential type fluids are those of which the stress is determined by the velocity inclination and its various higher time derivatives. As movement ceases in different rates compressible fluids, the stress reduces to an uncertain circular stress. In comparison to differential type fluids, where the stress is distributed unequally as a function of the velocity gradient and its higher time derivatives, rate type liquid models have an implied relationship between the stress and its higher time derivatives. e Maxwell fluid is a viscoelastic fluid of rate type that relates to the non-Newtonian fluid class. In terms of experimentation, rate type models are more realistic since they account for both memory and elastic effects [1]. As a consequence, the Maxwell model, a subclass of rate type models, has been included in the study. e concept of Maxwell fluid was given by Maxwell [2]. In the physical situation, many fluids such as engine oil (EO), glycerin, and some other plyometric materials behave like Maxwell fluid. Many researchers have considered Maxwell fluid in their studies like Maxwell nanofluid flow over a vertical plate was analyzed by Aman et al. [3] using the Laplace transform technique to find accurate solution to partial differential equations for initial and boundary conditions. Arif et al. [4] studied Maxwell nanofluid flow and discussed some applications of their results in EO with the influence of ramped wall conditions. Khalid et al. [5] calculated the results of ferrofluid in a permeable channel along a vibrating plate with ramped and isothermal wall temperature (RWT and IWT). Zhao et al. [6] reported the results of Maxwell nanofluid flow with the influence of heat and mass transfer with Dufour influences using the Caputo fractional model. e characteristics of wave motion occurring in the Maxwell nanofluid rheometer was studied by Huilgol [7]. Jamil [8] calculated the Maxwell nanofluid flow along with a vibrating plate, the effect of shear stress with no-slip condition using discreet Laplace transform to find the closed form of solutions. Convective flow of a nonsteady Maxwell nanofluid over an infinite vertical plate with ramped conditions was examined by Anwar et al. [9]. Na et al. [10] analyzed free convection Maxwell nanofluid flow between vertical plates with the influence of thermal radiation. Raza and Asad [11] derived the solutions of Maxwell fluid flow considering a vertical flat plate. ey observed that increasing the fluid velocity with respect to Grashof number (Gr) and Maxwell parameter λ. e numerical solutions were obtained by Sui et al. [12] calculating the Maxwell nanofluid over stretching sheet with the influence of heat and mass transfer using the homotopy analysis process. Wang and Tan [13] discussed the Maxwell nanofluid flow passing through a permeable channel with the Soret effect.
Heat is a form of energy that can be moved from higher concentration to a lower concentration. Heat can be transfered through radiation, conduction, and convection. Transfer of heat has many applications in industrial and manufacturing processes like cooling chambers, heating phenomena, engines, and drying processes. Similarly, the net movement from one region to another is known as mass transfer. Distillation, filtration, crystallization, precipitation, evaporation, and absorption are only some of the processes that include mass transfer. It has wide applications in modern sciences and technologies like purification of crude oil, petrochemical refining, fractional distillation, highpressure pumps, and refrigeration. Steeman et al. [14] analyzed mass and heat measurement for water vaporization. e effects of heat and mass transfer study of water and gas for waste heat recovery were investigated by Men et al. [15]. Chakkinga et al. [16] studied the heat and mass transfer in adiabatic cylindrical obstacles. Yan et al. [17] examined planar layer for fuel cell considering mass and heat transfer. Chu et al. [18] studied the physical characteristics of ammonia regeneration with mass and heat transfer. Hazarika et al. [19] calculated that fork's usages influenced the fin preparation project for instantaneous mass and heat transfer in a space-controlled position. Khan et al. [20] examined intermittent microwave convective drying modelling of the combined influence of heat and mass. Luikov [21] analyzed the fluid flow in a capillary absorbent organ with the mutual influence of mass and heat. Riaz et al. [22] described Maxwell nanofluid flow by taking the effect of heat and mass transfer. Chen et al. [23] analyzed the experimental study of the mass and heat properties of water and air in an elliptical plane tube. e ramped temperature on the fluid is a temperature variation during a given time period. e ramped or nonuniform wall temperature has many applications in modern sciences and technologies such as engineering, industrial, and several other physical applications such as automobile, electric circuit, heat transfer in the building, solar collector, nuclear energy, geothermal, air condition, and photovoltaic mechanism. Similarly, isothermal temperature means constant temperature. Generally, it describes a thermodynamic process where the temperature is constant. It has many modern science and technologies applications such as the factory automation system, robot's system, water handling system, automobiles, airplanes and boats, vehicles, mechanical structure, electrical power, plants, animals, transpiration, and computers. Chandran et al. [24] calculated the natural convection fluid flow by taking RWT near the plate. Khalid et al. [25] discussed the free convection nanofluids with RWT. e transient free convection fluid flow with RWT is analyzed by Narahari [26]. RWT conditions as given by Ghara et al. [27] were used to investigate the impact of magnetohydrodynamic (MHD) free convection flow over a fluctuating layer. RWT and slip conditions investigated by Haq et al. [28] were used to the unsteady viscous fluid flow. Some other real-world and industrial applications of ramped and isothermal wall concentration and temperature like generator systems, biofluids, blood flow, and nanofluids can be found in [9,[29][30][31][32][33][34].
Heat and mass transfer have many physical applications in today's sciences and technology. e thermal conductivity of traditional fluids such as ethylene glycol, water, polyethylene glycol, and EO is poor. To improve the thermal conductivity of their materials, Maxwell [35] developed the concept of microparticle dispersion in regular fluids. Still, there were few restrictions in the Maxwell model because these particles are very heavy and have very quick settled down, clogging of narrow channels, and cause destruction of these channels. To overcome this issue, Choi [36] developed a new concept for interrupting nanosized particles in regular fluids to improve thermal conductivity. In a rotating frame, the Jeffery nanofluid flow was examined by Ali et al. [37] with the impact of heat transfer analysis. ey observed that the heat will enhance up to 12.37% by adding silver nanoparticles from 0.00 to 0.04 in the base fluid EO. Using EO as a base fluid, the Brinkman-type fluid flow with spherical form molybdenum disulfide (MoS 2 ) nanoparticles was investigated Jan et al. [38]. Some applications of nanoparticles in different regular fluids are discussed in [39][40][41][42]. Ali et al. [43] examined the applications of EO in modern sciences, taking molybdenum disulfide MoS 2 nanoparticles in it. e applications of different nanoparticles to improve the heat transfer rate are discussed by many researchers [44,45]. Turkyilmazoglu [46] studied the flow past a vertical infinite isothermal plate in a viscous electrically conducting natural convective incompressible nanofluid. ey take into account the effects of heat absorption, heat generation, and radiation. Akinshilo et al. [47] investigated the steady incompressible flow of a non-Newtonian sodium alginate (SA) fluid conveying copper nanoparticles (Cu) which flow within two vertical parallel plates.
e obtained results can be used off the sodium alginate in processes such as manufacturing and biomedical applications. Turkyilmazoglu [48] analyzed the phenomenon of interaction of suspended particles within the fluid over a stretchable rotating disk. ey observed that the dust phase has reduced velocity and temperature as compared to those of the fluid phase. e heat and mass transfer of a radiating non-Newtonian sodium alginate transported through parallel squeezing plates was examined by Akinshilo et al. [49] Turkyilmazoglu [50] analyzed the flow and heat transfer of the nanofluid film flow over a moving inclined substrate. ey observed that the motion of nanoparticles is induced by the action of both the gravitational force as well as the substrate movement. Akinshilo and Sobamowo [51] investigated the flow of blood with gold nanoparticles equally suspended through a porous channel. ey developed models to show the effects of the nanoparticles on the concentration, temperature, and velocity of the fluid as it flows through the porous medium.
Based on the abovementioned literature, no work has been performed to obtain Maxwell nanofluid close form solutions with the combined effect of heat and mass transfer. We also focused at how ramped and isothermal wall temperatures and concentrations influence the results. Furthermore, contributing to its wide physical and mechanical engineering applications, EO is used as a base fluid. To increase thermal conductivity, spherical shaped MoS 2 nanoparticles have been spread in EO.

Mathematical Formulation
e unsteady incompressible Maxwell fluid flow over an infinite vertical plate with ramped and isothermal wall temperature and concentration was investigated in this research. MoS 2 nanoparticles have been suspended uniformly in the base fluid EO to enhance the rate of heat transfer. e plate is taken along the x-axis, while the fluid occupies the space (y ≥ 0). Both the plate and the fluid are initially at rest with ambient temperature. T 1∞ and constant concentration C 1∞ . After some time t � 0 + , the temperature and concentration of the plate rise slowly to T 1w and C 1w , respectively. In other words, at time t � 0 + , temperature of the plate and concentration become higher or lower to respectively. When t 1 > t 0 , no change will occur in concentration and temperature and remains at constant temperature and concentration. Keeping in mind the above assumptions, the governing equations for the above flow regime are derived [3].
For the above flow regime, the velocity distribution for unidirectional flow is defined as Continuity equation is given as Using equations (1) and (2) satisfies. Using equation (1), the governing equation for the above flow regime is given as and temperature equation is Mathematical Problems in Engineering subject to the following imposed physical condition.
In the above system of equations, the velocity component of Maxwell nanofluid along the axis is u 1 , T 1 represents the temperature distribution, T 1w is the wall temperature which is fixed, and g is the gravitational acceleration. e thermophysical properties of the EO and MoS 2 nanoparticles are given in Table 1.
μ nf and μ f represent the dynamic viscosity of the nanofluid and based fluid, respectively. Here, ϕ used to represent the volume fraction [43]. e following dimensionless variables were added to make the above system of equations dimensionless: Using equation (7), the momentum, temperature, and concentration equations nondimensionalized to the following form: Sc Equations (8), (9), and (10) in more simplified form can be written as With the dimensionless physical IC's and BCs are  [43]: where H(τ) is the Heaviside step function, and some dimensionless numbers/constants which appear during calculations are given as follows.
where Pr denotes the Prandtl number, Gr and Gm denote the thermal and mass Grashof numbers, Sc represents the Schmidt number, and λ represents the dimensionless Maxwell nanofluid parameter.

Solution of the Problem
To find the solution of the dimensionless system, we first consider the energy equation.

Exact Solution of Energy Equation.
We use the Laplace transform technique to solve equation (12), and incorporating the ICs and BCs, we get inverting the Laplace transform, equation (16) for ramped wall temperature becomes 3.1.1. Error Function. e error function equals twice the integral of a normalized Gaussian function between 0 and x/σ , where σ is the standard distribution.

Complementary Error Function.
In terms of error function, the complementary error function can be expressed as follows: and solution for isothermal wall temperature becomes where

Exact Solution of Concentration Equation.
Using the Laplace transform technique to solve equation (13) and including ICs and BCs, we will find the solution to the concentration equation: inverting the Laplace transform, equation (22) for ramped wall concentration becomes and solution for isothermal wall concentration becomes where

Limiting Cases
For validation of the present work, some limiting cases are described as follows.

Case 1.
In the absence of Gm ⟶ 0, our solutions were reduced to the following form: By the inverse Laplace transform technique applied using the convolution theorem, we obtain the following results which are relatively same to results Arif et al. [4].
Solution of isothermal temperature is where Case 2. In the absence of RWT and IWT,

Mathematical Problems in Engineering
By the inverse Laplace transform technique applied using the convolution theorem, we obtained the following results which are relatively same to results Sidra et al. [3].
Case 3. In the absence of Gr ⟶ 0, and λ ⟶ 0, our obtained exact results were reduced to the following form [25].

Nusselt Number. e dimensional form of Nusselt number for Maxwell nanofluid is given by
By using equations (6) and (7), the dimensionless form of equation (40) becomes

Sherwood Number. e dimensional form of Sherwood number for the Maxwell nanofluid is given by
By using equations (6) and (7), the dimensionless form of equation (42) becomes

Skin Friction.
e dimensional form of nonzero shear stress for Maxwell fluid is given as For Maxwell nanofluid, equation (46) takes the following form: By using equations (6) and (7), the dimensionless form of equation (47) becomes where τ xy � τ * xy ������� ](t 1 /τ 0 )/μ f U 0 which is the dimensionless form of nonzero is shear stress and λ � λ 1 /τ 0 is the dimensionless Maxwell parameter. e mathematical representation for the skin friction (S f ) for Maxwell nanofluid is given as

Results and Discussion
We studied the exact solutions of Maxwell fluid flow with ramped and isothermal wall boundary conditions in this study. We used EO as the base fluid and widely scattered MoS 2 particles to improve the thermal conductivity of EO. rough the graphical analysis, the effect of λ, ϕ, Gm, and Sc on the velocity profile w(ζ, t), temperature distribution Θ(ζ, t), and concentration distribution Φ(ζ, t) are shown. More specially, Figure 1 is the physical configuration of the problem. In Figures 2-7, the impact of various parameters on the velocity profile has been seen. e effect of sundry parameters on temperature is shown graphically in Figures 8  and 9. Finally, the impact of embedded parameters on concentration distribution is shown graphically in  Figure 3. From the figure, it is clear that the Maxwell fluid velocity increases with relaxation time λ. It is because, when we raise the value of λ, the gradual response obtains to shear stress and accelerates the Maxwell nanofluid velocity. Figures 4 and 5 indicate the impact of sundry values of Gm and Gr on velocity distribution, respectively. It is seen that Gm and Gr accelerate the fluid motion. e fact behind this is that increasing Gm and Gr emerges to improve the thermal and mass buoyancy forces that raise the fluid motion. Increasing the mass buoyancy forces, the viscosity of the Maxwell nanofluid decreases which accelerates the fluid motion. Figure 6 shows how Sc influences the Maxwell nanofluid velocity. e Maxwell nanofluid velocity is found to be reduced when Sc is increased. Since Sc is the ratio of mass diffusion to viscous forces, increasing Sc increases viscous forces, while decreasing mass diffusion, decreasing velocity. Figure 7 shows the influence of various values of ϕ on the Maxwell nanofluid velocity. By increasing the volume fraction, the Maxwell nanofluid velocity retards for both cases RWT, IWT, RWC, and IWC. e physics behind this is that by increasing ϕ, the flow becomes more vicious results the friction forces that retard the velocity of nanofluid. Figure 8 explains the impact of time on temperature distribution. By increasing τ, RWT rises, while IWT remains. Figure 9 illustrates the behavior of ϕ on temperature profile. From the figure, one can clearly see that increasing ϕ increases the temperature profile. e physics behind this behavior of rising temperature is the collision between the molecules of Maxwell nanofluid rises, and as a result, the temperature rises. Figure 10 shows how is the effect of τ on the concentration profile. We can see from the graph that increasing τ increases RWC while keeping IWC constant. Figures 11 and  12 show the effects of ϕ and Sc on the distribution of concentrations, respectively. e concentration distribution decreases for significant values of ϕ and Sc, as can be seen in both of these figures. e explanation for this behavior is that as the concentration distribution slows down, viscous forces rise. Sc, on either extreme, is the ratio of mass diffusion to viscous forces. As Sc rises, viscous forces increase when mass diffusion decreases, resulting in a decreasing of the concentration distribution. Figure 13 illustrates the difference between our results and those of the published studies of Sidra et al. [3]. Our findings strongly overlap with the reported results, as seen in the figure, when RWT and IWT are not available. From Figure 14, our findings are consistent with the findings of the published article of Arif et al. [4] by taking Gm ⟶ 0. Figure 15 compares our findings to the findings of the published article of Khalid et al. [25] to validate our obtained solutions. From this figure, our results matched the results of Khalid et al. [25] by taking λ ⟶ 0 and Gm ⟶ 0. Table 2 presents how various embedded parameters affect the skin friction of a Maxwell nanofluid flow. For both ramped and isothermal wall temperatures and concentrations, skin friction is determined. Skin friction for isothermal wall conditions is higher than skin friction for ramped wall conditions. e rate of heat transfer for ramped and isothermal wall temperatures is given in Tables 3 and 4. When considering ramped wall temperature, the heat transfer rate increases by 12.895%, and when considering isothermal wall temperature, the heat transfer rate increases by 12.899%. Sherwood numbers for ramped and isothermal wall concentrations are given in Tables 5 and 6. We found that 3.030% of mass transfer decreases for ramped wall concentration and 3.104% increases by considering isothermal wall concentration for ϕ � 0.04.
Velocity profile

Maxwell nanofluid
Temperature profile Concentration profile

Concluding Remarks
e aim of this study is to find exact solutions for mass and heat transfer in a Maxwell nanofluid flow with ramped and isothermal wall temperatures and concentration boundary conditions. e Laplace transform method is used to obtain precise solutions for velocity, temperature, and concentration distributions. Additionally, MoS 2 nanoparticles are dispersed in EO. Additionally, the results obtained for ramped and isothermal wall temperature and concentration are analyzed numerically using various graphs and tables. e significant outcomes of our study are as follows: (i) e velocity of Maxwell nanofluid slows down by increasing the amount of MoS 2 nanoparticles (ii) e velocity of a Newtonian fluid is greater than the velocity of a Maxwell nanofluid (iii) RWT and RWC have narrower velocity, temperature, and concentration distributions, while IWT and IWC have greater distributions. (iv) By rising λ, Gm, Gr, and τ, the Maxwell nanofluid velocity increases (v) By rising the value of ϕ for RWT enhanced heat transfer up to 12.895% while 12.899% for IWT of regular EO (vi) e rate of mass transfer for RWC decreases to 3.030% while 3.104% for IWC of regular EO (vii) Using nanoparticles in the engine oil increases the heat transfer rate which will off course increase life and efficiency of engines. (viii) MoS 2 is also used as dry lubricant. By making use of MoS 2 in the engine oil, it increases the lubricity of EO which will help you to decrease the corrosion and friction of the engine parts.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.