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

Authors: Naveed Khan; Farhad Ali; Muhammad Arif; Zubair Ahmad; Aamina Aamina; Ilyas Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The aim of this study is to investigate how heat and mass transfer impacts the unsteady incompressible flow of Maxwell fluid. An infinite vertical plate with ramped and isothermal wall temperature and concentration boundary conditions is considered with the Maxwell fluid. Furthermore, in this study, engine oil has been taken as a base fluid due to its enormous applications in modern science and technologies. To see the importance of nanofluids, we have suspended molybdenum disulfide in engine oil base fluid to enhance its heat transfer rate. To investigate the flow regime, the system of equations was derived in the form of partial differential equations. The exact solutions to the complex system are obtained using the Laplace transform technique. Graphically, the impact of different embedded parameters on velocity, temperature, and concentration distributions has been shown. Through using the graphical analysis, we were interested in comparing the velocity, temperature, and concentration profiles for ramped and isothermal wall temperature and concentration. The magnitude of velocity, temperature, and concentration distributions is greater for an isothermal wall and less for a ramped wall, according to our observations. We observed that adding molybdenum disulfide nanoparticles to the engine oil increased the heat transfer up to 12.899%. Finally, the corresponding skin friction, Nusselt number, and Sherwood number have been calculated and presented in a tabular form.


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## Body

## 1. Introduction

There are several types of fluids that cannot be described by simple Navier–Stokes equations because of their complex rheology. These fluids are called non-Newtonian fluids which have many applications in our daily life such as biotechnologies, geophysics, astrophysics, industries, and engineering developments. Therefore, 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. The 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. The 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. The 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. They observed that increasing the fluid velocity with respect to Grashof number (Gr) and Maxwell parameter λ. The 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, high-pressure pumps, and refrigeration. Steeman et al. [14] analyzed mass and heat measurement for water vaporization. The 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.The ramped temperature on the fluid is a temperature variation during a given time period. The 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. The 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, 29–34].Heat and mass transfer have many physical applications in today’s sciences and technology. The 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. They 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 (MoS2) nanoparticles was investigated Jan et al. [38]. Some applications of nanoparticles in different regular fluids are discussed in [39–42]. Ali et al. [43] examined the applications of EO in modern sciences, taking molybdenum disulfide MoS2 nanoparticles in it. The 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. They 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. The 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. They observed that the dust phase has reduced velocity and temperature as compared to those of the fluid phase. The 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. They 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. They 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 shapedMoS2 nanoparticles have been spread in EO.


## 2. Mathematical Formulation

The unsteady incompressible Maxwell fluid flow over an infinite vertical plate with ramped and isothermal wall temperature and concentration was investigated in this research.MoS2 nanoparticles have been suspended uniformly in the base fluid EO to enhance the rate of heat transfer. The 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. T1∞ and constant concentration C1∞. After some time t=0+, the temperature and concentration of the plate rise slowly to T1w and C1w, respectively. In other words, at time t=0+, temperature of the plate and concentration become higher or lower to T1+T1w−T1∞t1/t0 and C1∞+C1w−C1∞t1/t0 when t1≤t0, respectively. When t1>t0, 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(1)V⟶=uy,t,0,0.Continuity equation is given as(2)∇·V⟶=0.Using equations (1) and (2) satisfies. Using equation (1), the governing equation for the above flow regime is given as(3)ρnf1+λ1∂∂t1∂u1y1,t1∂t1=μnf∂2u1y1,t1∂y12+1+λ1∂∂t1ρβTnfgT1y1,t1−T1∞+1+λ1∂∂t1ρβCnfgC1y1,t1−C1∞,and temperature equation is(4)ρCpnf∂T1∂t1=knf∂2T1∂y12,∂C1∂t1=Dnf∂2C1∂y12,subject to the following imposed physical condition.(5)u1y,0=0,∂u1y,t∂t|t=0=0,T1y,0=T1w,C1y,0=C1w,fort=0andy≥0,u10,t=0,T10,t=T1+T1w−T1∞t1t0,0<t1≤t0,T1w,t1≥t0,C10,t=C1∞+C1w−C1∞t1t0,0<t1≤t0,C1w,t1≥t0,fort>0andy=0,u1∞,t=0,T1∞,t=T1∞,C1∞,t=C1∞,fort>0andy⟶∞.In the above system of equations, the velocity component of Maxwell nanofluid along the axis isu1, T1 represents the temperature distribution, T1w is the wall temperature which is fixed, and g is the gravitational acceleration. The thermophysical properties of the EO and MoS2 nanoparticles are given in Table 1.Table 1
Thermophysical properties of EO and MoS2 [43]:
Materialρkm−3cpJkg−1K−1kwm−1K−1β×10−5K−1Engine oil86820480.14040.00007MoS2506×103397.21904.42.8424(6)ρnf=ρf1−ϕ+ϕρs,ρβTnf=ρβTf1−ϕ+ϕρβTs,ρβCnf=ϕρβCs+ρβCf1−ϕ,ρcpnf=1−ϕρcpf+ϕρcps,Dnf=Df1−ϕ,knf=kfks+2kf−2ϕkf−ksks+2kf+ϕkf−ks,μnf=μf1−ϕ−2.5.μnfandμf represent the dynamic viscosity of the nanofluid and based fluid, respectively. Here, ϕ used to represent the volume fraction [43]. The following dimensionless variables were added to make the above system of equations dimensionless:(7)w=u1U0,ζ=y1υτ,τ=t1τ0,τ0=υ1/2gβT1w−T1∞2/3,Θ=T1−T1∞T1w−T1∞,Φ=C1−C1∞C1w−C1∞.Using equation (7), the momentum, temperature, and concentration equations nondimensionalized to the following form:(8)d1+λ∂∂τ∂wζ,τ∂τ=d1∂2wζ,τ∂ζ2+d21+λ∂∂τGrΘ+d31+λ∂∂τGmΦ,(9)Prm3∂Θζ,τ∂τ=∂2Θζ,τ∂ζ2,(10)Sc∂Φζ,τ∂τ=1−ϕ∂2Φζ,τ∂ζ2.Equations (8), (9), and (10) in more simplified form can be written as(11)1+λ∂∂τ∂wζ,τ∂τ=m∂2wζ,τ∂ζ2+m21+λ∂∂τGrΘζ,τ+m41+λ∂∂τGmΦζ,τ,(12)∂2Θζ,τ∂ζ2−b∂Θζ,τ∂τ=0,(13)∂2Φζ,τ∂ζ2−b1∂Φζ,τ∂τ=0.With the dimensionless physical IC’s and BCs are(14)wζ,0=0,∂wζ,τ∂τ|τ=0=0,Θζ,0=0,Φζ,0=0,w0,τ=0,Θ0,τ=τ,0<τ≤1,1,τ>1,=tHτ−τ−1Hτ−1,Φ0,τ=τ,0<τ≤1,1,τ>1,=tHτ−τ−1Hτ−1,w∞,τ=0,Θ∞,τ=0,Φ∞,τ=0,where Hτ is the Heaviside step function, and some dimensionless numbers/constants which appear during calculations are given as follows.(15)λ=λ1τ0,Gr=gxβTfT1w−T1∞τ0U0,Pr=μCpfkf,m=dd1,m1=d2d1,m2=m1Gr,m3=d3d4,b=m3Pr,Gm=gxβCfC1w−C1∞τ0U0,b1=Sc1−ϕ,m4=m1Gm,d=1−ϕ+ϕρsρf,d1=11−ϕ2.5,d2=1−ϕ+ϕρβTsρβTf,d3=1−ϕ+ϕρcpsρcpf,d4=ks+2kf−2ϕkf−ksks−2kf+kf−ks,Sc=νfDf,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.


## 3. Solution of the Problem

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


### 3.1. Exact Solution of Energy Equation

We use the Laplace transform technique to solve equation (12), and incorporating the ICs and BCs, we get(16)Θ¯ζ,p=1p2e−ζbp−e−pp2e−ζbp,inverting the Laplace transform, equation (16) for ramped wall temperature becomes(17)Θζ,τramp=Θ1ζ,τ−Θ1ζ,τ−1Hτ−1.


#### 3.1.1. Error Function

The error function equals twice the integral of a normalized Gaussian function between0andx/σ2, where σ is the standard distribution.(18)y=erfx=2π∫0xe−t2dt,forx≥0,y0,1.


#### 3.1.2. Complementary Error Function

In terms of error function, the complementary error function can be expressed as follows:(19)1−y=erfcx=1−erfx=2π∫x∞e−t2dt,forx≥0,y0,1,and solution for isothermal wall temperature becomes(20)Θ1ζ,τiso=erfcζb2τ,where(21)Θ1ζ,τ−1=erfcζ2bττ+bζ22−ζbτπe−bζ/4τ.


### 3.2. 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:(22)Φ¯ζ,p=1p2e−ζb1p−e−pp2e−ζb1p,inverting the Laplace transform, equation (22) for ramped wall concentration becomes(23)Φζ,τramp=Φ1ζ,τ−Φ1ζ,τ−1Hτ−1,and solution for isothermal wall concentration becomes(24)Φ1ζ,τiso=erfcζb12τ,where(25)Φ1ζ,τ−1=erfcζ2b1ττ+b1ζ22−ζb1τπe−b1ζ/4τ.


### 3.3. Solution of Momentum Equation

We can get the following result by applying the Laplace transform technique to equation (10) and using equation (13):(26)w¯ζ,p=−1pm21+λp1−exp−pp2ψλp+ψ−bexp−ζψp1+λp−1pm31+λp1−exp−pp2`ψλp+ψ−b1exp−ζψp1+λp+1pm21+λp1−exp−pp2ψλp+ψ−bexp−ζbp+1pm31+λp1−exp−pp2ψλp+ψ−b1exp−ξb1p.Equation (26) can be written in more simplified form as(27)w¯ζ,p=1−exp−ppΨ¯ζ,p+Ψ¯1ζ,p+Ψ¯2ζ,p+Ψ¯3ζ,p,where Ψ¯ζ,p,Ψ¯1ζ,p,Ψ¯2ζ,p,andΨ¯3ζ,p are given as(28)Ψ¯ζ,p=−m21+λpp2ψλp+ψ−bexp−ζψp1+λp,(29)Ψ¯1ζ,p=−m31+λpp2ψλp+ψ−b1exp−ζψp1+λp,(30)Ψ¯2ζ,p=m21+λpp2ψλp+ψ−bexp−ζbp,(31)Ψ¯3ζ,p=m31+λpp2ψλp+ψ−b1exp−ζb1p.By taking the invers Laplace transform equations (27)–(31), we get the following form:(32)w¯ζ,τ=∫0ττHτ−τ−1Hτ−1∗Ψζ,τ−1+Ψ1ζ,τ−1+Ψ2ζ,τ−1+Ψ3ζ,τ−1dτ,where ∗ is the convolution product:(33)Ψζ,τ=−m2ψλ∫0τHτ−ξφλ−1φ2+τ−ξφ+exp−φτ−ξ1−φλφ2f1ζ,τ−ξdτ,Ψ1ζ,τ=−m3ψλ∫0τHτ−ξφλ−1φ2+τ−ξφ+exp−φτ−ξ1−φλφ2f1ζ,τ−ξdτ,f1ζ,τ−ξ=erfcξψ2λψττ+ψλ22−ζψλτψλe−ψλζ/4τ,Ψ2ζ,τ=m2ψλ∫0τHτ−ξφλ−1φ2+τ−ξ1φ+exp−φτ−ξ1−φλφ2ζbexp−ζ2b/4τ2τπτdτ,Ψ3ζ,τ=m3ψλ∫0τHτ−ξφλ−1φ2+τ−ξφ+exp−φτ−ξ1−φλφ2ζb1exp−ζ2b1/4τ2τπτdτ.


## 3.1. Exact Solution of Energy Equation

We use the Laplace transform technique to solve equation (12), and incorporating the ICs and BCs, we get(16)Θ¯ζ,p=1p2e−ζbp−e−pp2e−ζbp,inverting the Laplace transform, equation (16) for ramped wall temperature becomes(17)Θζ,τramp=Θ1ζ,τ−Θ1ζ,τ−1Hτ−1.


### 3.1.1. Error Function

The error function equals twice the integral of a normalized Gaussian function between0andx/σ2, where σ is the standard distribution.(18)y=erfx=2π∫0xe−t2dt,forx≥0,y0,1.


### 3.1.2. Complementary Error Function

In terms of error function, the complementary error function can be expressed as follows:(19)1−y=erfcx=1−erfx=2π∫x∞e−t2dt,forx≥0,y0,1,and solution for isothermal wall temperature becomes(20)Θ1ζ,τiso=erfcζb2τ,where(21)Θ1ζ,τ−1=erfcζ2bττ+bζ22−ζbτπe−bζ/4τ.


## 3.1.1. Error Function

The error function equals twice the integral of a normalized Gaussian function between0andx/σ2, where σ is the standard distribution.(18)y=erfx=2π∫0xe−t2dt,forx≥0,y0,1.


## 3.1.2. Complementary Error Function

In terms of error function, the complementary error function can be expressed as follows:(19)1−y=erfcx=1−erfx=2π∫x∞e−t2dt,forx≥0,y0,1,and solution for isothermal wall temperature becomes(20)Θ1ζ,τiso=erfcζb2τ,where(21)Θ1ζ,τ−1=erfcζ2bττ+bζ22−ζbτπe−bζ/4τ.


## 3.2. 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:(22)Φ¯ζ,p=1p2e−ζb1p−e−pp2e−ζb1p,inverting the Laplace transform, equation (22) for ramped wall concentration becomes(23)Φζ,τramp=Φ1ζ,τ−Φ1ζ,τ−1Hτ−1,and solution for isothermal wall concentration becomes(24)Φ1ζ,τiso=erfcζb12τ,where(25)Φ1ζ,τ−1=erfcζ2b1ττ+b1ζ22−ζb1τπe−b1ζ/4τ.


## 3.3. Solution of Momentum Equation

We can get the following result by applying the Laplace transform technique to equation (10) and using equation (13):(26)w¯ζ,p=−1pm21+λp1−exp−pp2ψλp+ψ−bexp−ζψp1+λp−1pm31+λp1−exp−pp2`ψλp+ψ−b1exp−ζψp1+λp+1pm21+λp1−exp−pp2ψλp+ψ−bexp−ζbp+1pm31+λp1−exp−pp2ψλp+ψ−b1exp−ξb1p.Equation (26) can be written in more simplified form as(27)w¯ζ,p=1−exp−ppΨ¯ζ,p+Ψ¯1ζ,p+Ψ¯2ζ,p+Ψ¯3ζ,p,where Ψ¯ζ,p,Ψ¯1ζ,p,Ψ¯2ζ,p,andΨ¯3ζ,p are given as(28)Ψ¯ζ,p=−m21+λpp2ψλp+ψ−bexp−ζψp1+λp,(29)Ψ¯1ζ,p=−m31+λpp2ψλp+ψ−b1exp−ζψp1+λp,(30)Ψ¯2ζ,p=m21+λpp2ψλp+ψ−bexp−ζbp,(31)Ψ¯3ζ,p=m31+λpp2ψλp+ψ−b1exp−ζb1p.By taking the invers Laplace transform equations (27)–(31), we get the following form:(32)w¯ζ,τ=∫0ττHτ−τ−1Hτ−1∗Ψζ,τ−1+Ψ1ζ,τ−1+Ψ2ζ,τ−1+Ψ3ζ,τ−1dτ,where ∗ is the convolution product:(33)Ψζ,τ=−m2ψλ∫0τHτ−ξφλ−1φ2+τ−ξφ+exp−φτ−ξ1−φλφ2f1ζ,τ−ξdτ,Ψ1ζ,τ=−m3ψλ∫0τHτ−ξφλ−1φ2+τ−ξφ+exp−φτ−ξ1−φλφ2f1ζ,τ−ξdτ,f1ζ,τ−ξ=erfcξψ2λψττ+ψλ22−ζψλτψλe−ψλζ/4τ,Ψ2ζ,τ=m2ψλ∫0τHτ−ξφλ−1φ2+τ−ξ1φ+exp−φτ−ξ1−φλφ2ζbexp−ζ2b/4τ2τπτdτ,Ψ3ζ,τ=m3ψλ∫0τHτ−ξφλ−1φ2+τ−ξφ+exp−φτ−ξ1−φλφ2ζb1exp−ζ2b1/4τ2τπτdτ.


## 4. Limiting Cases

For validation of the present work, some limiting cases are described as follows.Case 1.
In the absence ofGm⟶0, our solutions were reduced to the following form:(34)Θ¯ζ,p=1p2e−ζbp−e−pp2e−ζbp,w¯ζ,p=−m21+λp1−exp−pp2ψλp+ψ−bexp−ζψp1+λp+m31+λp1−exp−pp2ψλp+ψ−bexp−ζbp.
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].(35)Θζ,τramp=Θ1ζ,τ−Θ1ζ,τ−1Hτ−ξ.
Solution of isothermal temperature is(36)Θisoζ,τ=erfcζb2τ,where(37)Θ1ζ,τ=erfcζ2bττ+bζ22−ζbτπe−bζ/4τ,Ψζ,τ=−m2ψλ∫0tHτ−ξφλ−1φ2+1φτ−ξ+exp−φτ−ξ1−φλφ2f1ζ,τ−ξdt,Ψ2ζ,τ=−m3ψλ∫0τHτ−ξφλ−1φ2+1φτ−ξ+exp−φτ−ξ1−φλφ2ζbexpζ2b/4τ2τπτdτ.Case 2.
In the absence of RWT and IWT,(38)Θ¯ζ,p=1pe−ζbp,w¯ζ,p=−m21+λp1−exp−pp2ψλp+ψ−bexpζψp1+λp+m31+λp1−exp−pp2ψλp+ψ−bexpζbp.
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].(39)Θζ,τ=erfcζb2τ,Ψζ,τ=−m2ψλ∫0τHτ−ξφλ−1φ2+τ−ξ1φ+exp−φτ−ξ1−φλφ2f1ζ,τ−ξdτ,Ψ2ζ,τ=−m3ψλ∫0τHτ−ξφλ−1φ2+1φτ−ξ+exp−φτ−ξ1−φλφ2ζbexpζ2b/4τ2τπτdτ.Case 3.
In the absence ofGr⟶0, and λ⟶0, our obtained exact results were reduced to the following form [25].(40)Θ¯ζ,p=1p2e−ζbp−e−pp2e−ζbp,(41)Ψζ,τ=−m2ψ−b∫0ττ+ψζ22erfcζ2ψτ−ζψτπe−ζψ/4τdτ+m3ψ−b∫0ττ+ψζ22erfcζ2bτ−ζbτπe−ζ2ψ4τdτ.


## 5. Nusselt Number, Sherwood Number, and Skin Friction

### 5.1. Nusselt Number

The dimensional form of Nusselt number for Maxwell nanofluid is given by(42)Nu=−knfkfυτT1w−T1∞∂T1∂y1|y=0.By using equations (6) and (7), the dimensionless form of equation (40) becomes(43)Nu=−knfkf∂Θ∂ζ|ζ=0.


### 5.2. Sherwood Number

The dimensional form of Sherwood number for the Maxwell nanofluid is given by(44)Sh=−DnfντC1w−C1∞∂C1∂y1|y1=0.By using equations (6) and (7), the dimensionless form of equation (42) becomes(45)Sh=−Dnf∂Φ∂ζ|ζ=0.


### 5.3. Skin Friction

The dimensional form of nonzero shear stress for Maxwell fluid is given as(46)τxy∗=μf1+λ1∂/∂t1∂u1∂y1|y1=0.For Maxwell nanofluid, equation (46) takes the following form:(47)τxy∗=μnf1+λ1∂/∂t1∂u1∂y1|y1=0.By using equations (6) and (7), the dimensionless form of equation (47) becomes(48)τxy=μf1−ϕ2.511+λ∂/∂τ∂w∂ζ|ζ=0,where τxy=τxy∗νt1/τ0/μfU0 which is the dimensionless form of nonzero is shear stress and λ=λ1/τ0 is the dimensionless Maxwell parameter.The mathematical representation for the skin frictionSf for Maxwell nanofluid is given as(49)Sf=11−ϕ2.511+λ∂/∂τ∂w∂ζ|ζ=0.


## 5.1. Nusselt Number

The dimensional form of Nusselt number for Maxwell nanofluid is given by(42)Nu=−knfkfυτT1w−T1∞∂T1∂y1|y=0.By using equations (6) and (7), the dimensionless form of equation (40) becomes(43)Nu=−knfkf∂Θ∂ζ|ζ=0.


## 5.2. Sherwood Number

The dimensional form of Sherwood number for the Maxwell nanofluid is given by(44)Sh=−DnfντC1w−C1∞∂C1∂y1|y1=0.By using equations (6) and (7), the dimensionless form of equation (42) becomes(45)Sh=−Dnf∂Φ∂ζ|ζ=0.


## 5.3. Skin Friction

The dimensional form of nonzero shear stress for Maxwell fluid is given as(46)τxy∗=μf1+λ1∂/∂t1∂u1∂y1|y1=0.For Maxwell nanofluid, equation (46) takes the following form:(47)τxy∗=μnf1+λ1∂/∂t1∂u1∂y1|y1=0.By using equations (6) and (7), the dimensionless form of equation (47) becomes(48)τxy=μf1−ϕ2.511+λ∂/∂τ∂w∂ζ|ζ=0,where τxy=τxy∗νt1/τ0/μfU0 which is the dimensionless form of nonzero is shear stress and λ=λ1/τ0 is the dimensionless Maxwell parameter.The mathematical representation for the skin frictionSf for Maxwell nanofluid is given as(49)Sf=11−ϕ2.511+λ∂/∂τ∂w∂ζ|ζ=0.


## 6. 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 MoS2 particles to improve the thermal conductivity of EO. Through the graphical analysis, the effect of λ, ϕ, Gm, and Sc on the velocity profile wζ,t, temperature distribution Θζ,t, and concentration distribution Φζ,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. The 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 Figures 10–13.Figure 1
Physical configuration of the problem.Figure 2
The effect of different values ofτ on velocity distribution of MoS2 EO-based Maxwell nanofluid when ϕ=0.02,Gr=0.05,Pr=6300,sc=15,Gm=0.5.Figure 3
The impact of sundry values ofλ on the velocity distribution of Maxwell nanofluid when ϕ=0.01,Gr=0.05,Pr=6300,sc=9,τ=0.7,τ=1.7,Gm=0.5.Figure 4
The impact of different values of Gm on velocity distribution of MoS2 EO-based Maxwell nanofluid when ϕ=0.02,Gr=0.05,Pr=6300,sc=9,τ=0.7,τ=1.7,λ=0.1.Figure 5
The impact of different values of Gr on velocity distribution of MoS2 EO-based Maxwell nanofluid when ϕ=0.01,Gm=3,Pr=6300,sc=15,τ=0.7,τ=1.7,λ=0.4.Figure 6
The impact of different values of Sc on velocity distribution of MoS2 EO-based Maxwell nanofluid when ϕ=0.02,Gr=0.05,Gm=0.9,τ=0.7,τ=1.7,λ=0.5,Pr=6300.Figure 7
The impact of different values ofϕ on the velocity distribution of MoS2 EO-based Maxwell nanofluid when Gr=0.05,Pr=6300,Gm=1,sc=9,τ=0.7,τ=1.7,λ=0.1.Figure 8
The impact of different values ofτ on temperature distribution of MoS2 EO-based Maxwell nanofluid when ϕ=0.05,Pr=6300.Figure 9
The impact of different values ofϕ on temperature distribution of MoS2 EO-based Maxwell nanofluid for when Pr=6300,τ=0.7,τ=1.7.Figure 10
The effect of different value ofτ on concentration distribution of MoS2 EO-based Maxwell nanofluid when ϕ=0.1,sc=5.Figure 11
The impact of different values ofϕ on concentration distribution of MoS2 EO-based Maxwell nanofluid when sc=5,τ=0.7,τ=1.7.Figure 12
The effect of different values of Sc on concentration distribution of MoS2 EO-based Maxwell nanofluid when ϕ=0.01,τ=0.7,τ=1.7.Figure 13
Velocity distribution of MoS2 EO-based Maxwell nanofluid when Gr = 0.5 ϕ=0.01,Gm=0,Pr=6300,sc=15,τ=1.5,λ=0.4.The influence of various values of timeτ on velocity wζ,t is sketched in Figure 2. The figure illustrates that the magnitude of velocity distribution increases when RWT, IWT, RWC, and IWC increase. The impact of different values of the Maxwell parameter λ on the velocity profile is demonstrated 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. The 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. The 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. The physics behind this is that by increasing ϕ, the flow becomes more vicious results the friction forces that retard the velocity of nanofluid.Figure8 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. The physics behind this behavior of rising temperature is the collision between the molecules of Maxwell nanofluid rises, and as a result, the temperature rises.Figure10 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. The concentration distribution decreases for significant values of ϕ and Sc, as can be seen in both of these figures. The 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.Figure13 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.Figure 14
Velocity distribution of MoS2 EO-based Maxwell nanofluid when ϕ=0.01,Gm=0,Gr=5,Pr=6300,sc=15,τ0=0.7,τ=1.7,λ=0.4.Figure 15
Velocity distribution of MoS2 EO-based Maxwell nanofluid for the different value of ϕ=0.01,Gm=0,Gr=5,Pr=6300,sc=15,andτ=1.7.Table2 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. The 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.Table 2
Skin friction of RWT and IWT EO-based MoS2.
λGmGrPrττ0ϕScCframpCfiso0.10.50.563001.70.70.02100.20.50.563001.70.70.021010.59822.7490.11.50.563001.70.70.021010.58125.6220.10.51.563001.70.70.021025.65961.7640.10.50.573001.70.70.02108.36120.1530.10.50.563001.90.70.02109.0623.4560.10.50.563001.71.90.021012.73921.8460.10.50.563001.70.70.031013.43432.3960.10.50.563001.70.70.02159.32322.476The values in bold show the variation in parameter values against skin friction.Table 3
Distinction in the rate of heat transfer RWT EO-based MoS2.
ϕττ0PrNurampHeat transfer enhancement (%)0.001.70.7630023.1320.011.70.7630024.2314.7500.021.70.7630024.9767.9710.031.70.7630025.3529.5970.041.70.7630026.11512.895Table 4
Variation in the rate of heat transfer IWT EO-based MoS2.
ϕττ0PrNuisoHeat transfer enhancement (%)0.001.70.7630013.4040.011.70.7630014.4045.4600.021.70.7630014.4727.9670.031.70.7630014.4918.1090.041.70.7630015.13312.899Table 5
Variation in the rate of mass transfer RWC EO-based MoS2.
ϕττ0ScShrampDecrease in mass distribution (%)0.001.70.751.0560.011.70.751.041.5150.021.70.751.0342.0830.031.70.751.0292.5560.041.70.751.0243.030Table 6
Variation in the rate of mass transfer IWC EO-based MoS2.
ϕττ0ScShisoDecrease in mass distribution (%)0.001.70.750.6120.011.70.750.6031.4700.021.70.750.5992.1240.031.70.750.5962.6140.041.70.750.5933.104


## 7. Concluding Remarks

The 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. The Laplace transform method is used to obtain precise solutions for velocity, temperature, and concentration distributions. Additionally, MoS2 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.The significant outcomes of our study are as follows:(i)
The velocity of Maxwell nanofluid slows down by increasing the amount of MoS2 nanoparticles(ii)
The 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 risingλ, Gm, Gr, and τ, the Maxwell nanofluid velocity increases(v)
By rising the value ofϕ for RWT enhanced heat transfer up to 12.895% while 12.899% for IWT of regular EO(vi)
The 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)
MoS2 is also used as dry lubricant. By making use of MoS2 in the engine oil, it increases the lubricity of EO which will help you to decrease the corrosion and friction of the engine parts.



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