Melting and viscous dissipation effect on upper‐convected Maxwell and Williamson nanofluid

This article mainly addresses the influence of the viscous dissipation, melting, and chemical reaction on Williamson and Maxwell nanofluids over a stretching sheet embedded in porous media. The system of partial differential equations which is obtained by conservation principles, is transformed by means of an appropriate similarity transformation into a system of ordinary differential equations. The numerical results are obtained by employing the Keller box method. The impacts of different germane parameters on velocity profiles, thermal and concentration fields, Nusselt number, skin friction coefficient, and Sherwood number are selected by means of graphical and tabular representations. Our numerical solution detects that the dimensionless melting parameter highly affects the velocity boundary layer of a Williamson nanofluid when compared with an upper‐convected Maxwell nanofluid. Moreover, the velocity, temperature, and concentration distributions decrease for both fluids when the permeability parameter increases. Furthermore, the temperature distribution increases with an increase of the Eckert number.

of Eyring-Powel nanofluid past isothermal sphere were deliberated by Swarnalathamma. 7 The thermal performance of single-walled carbon nanotube nanofluid under turbulent flow conditions was investigated by Fadodun et al. 8 Their results indicates that convective heat transfer (average Nusselt number) increases by 7.48% while the pressure drop and pumping power increase by 119% and 199%, respectively. Mathematical modeling of non-Newtonian fluid with chemical aspects via numerical technique was investigated by Hayat et al. 9 IjazKhan et al 10 examined new thermodynamics of entropy generation minimization in the presence of nonlinear thermal radiation and nanomaterial. Raju and Ojjela 11 examined the effects of the induced magnetic field motion on mixed convective Jeffrey nanofluid flow through a porous channel due to thermophoresis and Brownian. They found that when the values of viscoplastic Eyring-Powell fluid parameter increase the flow, thermal, and concentration boundary layer thickness decelerates. The Brownian motion and thermophoresis aspects in nonlinear flow of micropolar nanoliquid overstretching surface have been investigated by Hayat et al. 12 A substantial amount of experimental and theoretical work has been voted for to determine the role of natural convection in the kinetics of heat transfer escorted with melting or solidification effect. The study of heat transfer escorted by melting has received much attention because of its important applications in areas such as liquid polymer extrusion, frozen ground thawing, permafrost melting, casting, and welding processes as well as phase change material, hot extrusion, polymers, ceramics, geothermal energy recovery, silicon wafer process, thermal insulation, and so on. The dynamics of melting processes under different circumstances were described by Roberts, 13 Tien and Yen, 14 Epstein and Cho, 15 and Kairi and Murthy. 16 The result reveals that melting enhances heat and mass transfer. Moreover, different researchers such as, Krishnamurthy et al, 17 Kumar and Gireesha, 18 Khan et al, 19 Kairi and RamReddy, 20 and Babu and Narayana 21 have discussed the influences of melting on non-Newtonian fluids like Williamson and Burgers fluid exposed to different physical conditions like porosity, radiation, and so on. Still further, simultaneous effects of melting heat transfer and inclined magnetic field flow of tangent hyperbolic fluid over a nonlinear stretching surface with homogeneous-heterogeneous reactions was studied by Qayyum et al. 22 As indicated in different literatures viscous dissipation is the unidirectional processes by which the work done by a fluid on adjacent layers due to the action of shear forces are transformed into heat. It has significant impact on heat transfer; especially for high-velocity flows, fluids with a moderate Prandtl number, highly viscous flows at moderate velocities and moderate velocities with small wall-to-fluid temperature difference. Accordingly, Pop 23 and Yasin et al 24 remarked that energy dissipation and Joule heating in fluid dynamic has a great significance for the design of energy-conversion systems and energy-efficient circuits. Furthermore, the impact of Joule heating, viscous dissipation, and heat generation of fluid like Oldroyd B and Casson nanofluid has been discussed by Kumar et al, 25 Reddy et al, 26 Ajayi et al, 27 and Yohannes and Shankar. 28 A modified homogeneous-heterogeneous reaction for MHD stagnation flow with viscous dissipation and Joule heating was investigated by Khan et al. 29 Still further, Hayat et al 30 examined by entropy generation in magnetohydrodynamic radiative flow due to rotating disk in presence of viscous dissipation and Joule heating.
In most industries nowadays the importance of non-Newtonian fluids dominates the Newtonian fluids. The rheological possessions of non-Newtonian fluids cannot be elucidated by the classical Naiver-Stokes equations. Also no single model is sufficient to describe non-Newtonian fluids characteristics. To overcome this difficulty several models have come into being. The rheological models that were proposed were Williamson, Cross, Ellis, power law, Carreau fluid model, and so on. Typical of a non-Newtonian fluid model with shave retreating property is Williamson fluid model and was first projected by Williamson. 31 The examination of Williamson fluid has been carried out by researchers such as, Dapra and Scarpi, 32 Reddy et al, 33 Sreelakshmi et al, 34 Talha et al, 35 Immaculate et al, 36 Eswaramoorthi et al, 37 and AL-Qaissy and Abdulhadi 38 past different physical geometry such as stretching surface, vertical porous plate, and asymmetric channels. Furthermore, magnetohydrodynamic bio convective flow of Williamson nanofluid containing gyro tactic microorganisms subjected to thermal radiation and Newtonian conditions was studied by Zaman and Gul. 39 Still further, Hayata et al 40 studied mathematical modeling of non-Newtonian fluid with chemical aspects. From their simulation it can be seen that increasing values of Sc enhance the concentration field but opposite trend is seen for higher values of K 1 and K 2 .
A Maxwell fluids model can foresee the stress lessening and become the most popular model. This model excludes the complicated effects of shear-dependent viscosity and thus enables us to focus merely on the effects of fluid's elasticity on the characteristics of its boundary layer. Typically, many researchers have been studied non-Newtonian upper-convected Maxwell fluid flows. Accordingly, the slip effect on non-Newtonian upper-convected Maxwell and micropolar fluid flow over a stretching sheet were studied by Vijayalakshmi et al. 41 Furthermore, the study of the influence of cross diffusion on Casson and Maxwell fluid flows past a stretching surface was examined by Kumaran et al. 42 In a porous medium mass and heat transfer of a non-Newtonian Maxwell nanofluid above a stretching surface with variable thickness was investigated by Elbashbeshy et al. 43 With variable thermophysical possessions mass and heat transfer of upper-convected Maxwell fluid flow properties over a horizontal melting surface were studied by Adegbie et al. 44 The result indicated when thermal conductivity parameter increases the temperature of the fluid increases. Hayat et al 45 47 From their result it is observed that thermal, concentration, and motile density stratification parameters result in the reduction of temperature, concentration, and motile microorganisms' density distributions, respectively.
In view of all the above mentioned studies, it is decided that impacts of melting and viscous dissipation on boundary flow of upper-convected Maxwell and Williamson nanofluids in porous medium with chemical reaction is not inspected yet. Thus, to fill this gap, the main object of this article is to explore the effect of melting heat transfer and viscous dissipation on boundary flow of upper-convected Maxwell and Williamson nanofluids in porous medium with chemical reaction. For the information our work is innovative in terms of the proposed fluids and incorporated parameters in the boundary layer flow. By employing appropriate similarity transformations the nonlinear coupled dimensionless equations from the governing equations are achieved. Finally, the resulting dimensionless equations are solved computationally by instigating the Keller Box method via MATLAB software. We scrutinize the influence of dimensionless melting, magnetic field, thermal radiation, chemical reaction, Brownian motion, thermophoresis, permeability parameter, Prandtl number, Eckert number, and Lewis number on Williamson and Maxwell nanofliuds that are derived from the governing equations. From the result we observed that the dimensionless melting parameter affects highly the velocity boundary layer of Williamson nanofluid when compared with upper-convected Maxwell nanofliud.
The rest of this article is structured as follows. The second section describes statement of the problem and mathematical formulation. Solution of methodology and scheme of methodology are provided under Section 3. Section 4 tells us that result and discussion of the article. Conclusion is provided under Section 5.

MATHEMATICAL FORMULATION
The main theme in this article is based on a time-independent flow of an upper-convected Maxwell and Williamson nanofluid with chemical reaction and viscous dissipation effect past a stretching surface embedded in porous medium. The study considered uniform stretching velocity u w and the uniform free stream velocity u e in the similar direction as revealed in Figure 1. Above the stretching sheet the concentration and the ambient concentration, respectively, represented by C w and C ∞ . The melting surface of wall subjected to fixed temperature T m and the free stream condition T ∞ with T ∞ > T m is considered. In addition, thermal radiation influence is taken into account. The Williamson fluid model essential equations are specified as follows: where p is pressure, I is identity vector, S is the Cauchy stress tensor, is extra stress tensor, 0 and ∞ are the limiting viscosities at zero and at infinite shear rate, Γ > 0 is the time constant, A 1 is the first Rivlin-Ericson tensor anḋis defined as followṡ= √ where is the second invariant strain tensor. Here, we have only considered the case for which ∞ = 0 and Γ̇< 1. Then we obtain. Under these assumptions the governing equations can be described in Cartesian system as listed below.
The appropriate boundary conditions are: where u and v are the velocity components along the x and y directions, respectively, f the density of the base fluid, the thermal diffusivity, is the relaxation time parameter of the fluid, B 0 is the strength of the magnetic field, is the kinematic viscosity of the fluid, k is the thermal conductivity of the fluid, D B is the Brownian diffusion coefficient, is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid, c is the volumetric volume expansion coefficient and is the density of the particles, k ′ is the permeability of porous medium, is the latent heat of the fluid and c s is the heat capacity of the solid surface, c f is the heat capacity of the fluid, is the electrical conductivity. We can write for the radiation using Rosseland approximation where * is the Stefan-Boltzman constant, k * is the absorption constant. Assuming the temperature difference within the flow such that T 4 may be expanded in a Taylor series about T ∞ and neglecting higher orders we get Introducing similarity transformations We choose the stream function (x, y) such that By applying the similarity transformation in Equation (13) and Equations (6) to (8) are transformed into the nondimensional ordinary differential equation form as follows: With corresponding boundary conditions.
where f ′ is dimensionless velocity, is dimensionless temperature, is dimensionless concentration, and is the similarity variable. The prime denotes differentiation with respect to . The overall governing parameters are defined The skin friction C f , local Nusselt number Nu x , and the Sherwood number Sh x are the important physical quantities of interest in this problem which are defined as Here, w = − (1 + ) is the surface mass flux.
where 1 2 x = xu w (x) is local Reynolds number.

The finite difference scheme
We write the governing third-order momentum Equation (15) and second-order energy and concentration Equations (16) and (17) in terms of a first-order equations. For this purpose we introduce new dependent variable u, v, t, = s(x, ), Thus, Equations (15) to (17) can be written as The boundary conditions are where prime denotes the differentiation with respect to . We know consider the net rectangle in the x − plane shown in Figure 2 and the net points defined as below.
x 0 = 0, x i = x i−1 + k i , i = 1, 2, 3, … , I, where k i is the x-spacing and h j is the -spacing. Here i and j are the sequence of numbers that indicate the coordinate location, not tensor indices, or exponents.

F I G U R E 2 Net rectangle for difference approximations
Since only first derivatives appear in the governing equations, centered differences, and two-point averages can be constructed involving only the four corner nodal values of the "box." For example, if p represents any of the dependent variables u, v, s, and t then Now write the finite difference approximations for first-order ordinary differential equation for the mid-point ( of the segment P 1 P 2 using centered difference derivatives. This process is called centering about . We get involve only known quantities if we assume that the solution is known on x = x i − 1 . In terms of the new dependent variables, the boundary conditions become Equations (27) are imposed for j = 1, 2, 3, … , J and the transformed boundary layer thickness J is sufficiently large so that it is beyond the edge of the boundary layer Cebeci and Bradshaw. 48 The boundary conditions yields at x = x i are

Newton's method
Equation (21) are nonlinear algebraic equations and therefore have to be linearized before the factorization scheme can be used. Let us write the Newton iterates as follows: For (k + 1)th iterates, we write Equation (27) can be written as By dropping the quadratic and higher order terms in j , t (i) j , and z (i) j a linear tridiagonal system of equations will be obtained, as follows: To complete the system (33) we recall the boundary conditions (30) which can be satisfied exactly with no iteration. Therefore, in order to maintain these correct values in all the iterates, we take In general case Equation (33) in vector matrix form; In Equation (34) the elements are defined by To solve Equation (34), we assume that A is nonsingular and can be factored in to where where [I] is the identity matrix of order 7 and [ i ] and [Γ i ] are 7 × 7 matrices which elements are determined by the following equations: Equation (36) can now be substituted into Equation (34), and we get If we define Equation (41) becomes where , and the [ W j ] are 7 × 1 column matrices. The elements W can be solved from Equation (43): The step in which Γ j , j , and W j are calculated is usually referred to as the forward sweep. Once the element of W are found, Equation (38) then gives the solution in the so-called backward sweep, in which the elements are obtained by the following relations: These calculations are repeated until some convergence criterion is satisfied and calculations are stopped when where 1 is small arranged value.

RESULTS AND DISCUSSION
Figures 3-5 portray the temperature, velocity, and concentration profiles for different values of dimensionless melting parameter. It is noticed that when dimensionless melting parameter heightens the flow and velocity boundary layer rises, whereas, the thermal and concentration profiles decreases. This is due to the fact that an increase in Q will upsurge the intensity of melting, which acts as blowing boundary condition at the stretching surface and hence tends to thicken the boundary layer. Melting parameter affects adversely the velocity boundary layer thickness of Williamson nanofluid when compared with upper-convected Maxwell nanofliud. The influence of thermal radiation parameter on temperature and concentration profiles is recounted through Figures 6 and 7. From the figures it is noticed that an increment in the thermal radiation parameter causes the reduction in temperature profiles, but opposite effect on concentration profile. This is because an increase in the radiation parameter R leads to a decrease in the boundary layer thickness and enhances the heat transfer rate on melting surface in the presence of chemical effect. The behavior of the concentration profiles in contradiction of chemical reaction parameter is offered in Figure 8. From the graph it can be seen that if the chemical reaction parameter increased the concentration boundary layer thickness and concentration profiles reduced. This is due to the fact that chemical reaction in this system results in ingesting of the chemical and hence results in reduction of concentration profile. The most important outcome is that the first-order chemical reaction has a tendency to reduce the overshoot in the profiles of the solute concentration in the solutal boundary layer. Prandtl number is a ratio of momentum diffusivity to thermal diffusivity. When Prandtl number is high fluid has low thermal conductivity, which decreases the thermal boundary layer thickness and conduction. Accordingly, form Figure 9 it is seen that with increasing Prandtl number the thermal boundary layer and temperature profiles upsurges. The impact of magnetic field parameter on velocity, temperature, and concentration profiles is plotted in Figures 10-12. From the figures it is noticed that as the values of magnetic parameter goes up, the velocity of the fluid, the temperature, and The impact of thermophoresis and Brownian motion parameter on concentration and temperature profiles are portrayed in Figures 13-16. A phenomenon in which small particles are pulled away from the hot surface to cold one is called thermophoresis. So, when the surface is heated large number of nanoparticles is moved away which raises the temperature of fluid. Therefore, the temperature of fluid increases, whereas, the concentration of the fluid declines. Because of more heat is produced an increase in Nb enhances the random motion of fluid particles. This causes an increase of fluid temperature and concentration. Figures 17-19 are conspired to demonstrate the influence of permeability parameter on the flow of the fluid, temperature, and concentration profiles. It is obvious that the presence of porous medium causes higher restriction to the fluid flow, which in turn slows its motion. Therefore, an increment of permeability parameter causes the reduction in boundary layer thickness and the profiles of velocity, temperature, and concentration. In particular, the boundary layer of Williamson nanofluid more affected by permeability parameter when compared with the boundary layer of upper-convected Maxwell nanofluid. From Figure 20 it is noticed that as the value of Lewis number escalated the concentration boundary layer and concentration profiles rises. This is probably due to the fact that higher values of Lewis number create the smaller Brownian diffusion coefficient.   Figure 21 reveals that when the values of Eckert number upsurges the thermal boundary layer and the temperature profiles are enhanced. This is because of the point that heat energy is stowed in the liquid because of the frictional heating.
To check the validity of the numerical solution obtained a comparison of Nusselt and Sherwood number with Krishnamurthy et al 17 for different values of d and is exhibited in Table 1 and reasonable agreement with them has been found. The changes in physical quantities at various relevant parameters are displayed in Tables 2 and 3. From the tables it is noticed that with an increment in Eckert number Ec and thermophoresis parameter Nt a decrement in friction factor and mass transfer rate attained, whereas, the heat transfer rate is amplified. Moreover, when magnetic field parameter M upsurges skin friction coefficient upsurges but Nusselt number −Nu x (0) and Sherwood number −Sh x (0) are decreased. Furthermore, from the tables it can be seen that with an increase in Brownian motion parameter Nb there is a decrement in a friction factor, in contrast, there is an increment in both Sherwood and Nusselt number.
To check the validity of the numerical solution obtained a comparison of Nusselt and Sherwood number with Kairi and Murthy et al 16 for different values of d and is exhibited in Table 1 and reasonable agreement with them has been found. The changes in physical quantities at various relevant parameters are indicated in Tables 2 and 3. From the tables it is noticed that with an increment in Eckert number Ec and thermophoresis parameter Nt a decrement in friction factor and mass transfer rate attained, whereas, the heat transfer rate is amplified. Moreover, when  magnetic field parameter M upsurges skin friction coefficient upsurges but Nusselt number −Nu x (0) and Sherwood number −Sh x (0) are decreased. Furthermore, from the tables it can be seen that with an increase in Brownian motion parameter Nb there is a decrement in a friction factor, in contrast, there is an increment in both Sherwood and Nusselt number.

CONCLUSION
This article was aimed at exploring the effect of melting heat transfer, viscous dissipation, and chemical reaction on the flow of upper-convected Maxwell and Williamson nanofluids above a stretching surface. By employing well-known similarity transformations the nonlinear coupled dimensionless equations from the governing equations are achieved. Then, the resulting dimensionless equations are solved computationally by implementing Keller Box method with MATLAB software. The graphs and tables for velocity profiles, skin-friction coefficient, temperature field, local Nusselt number, concentration field, and local Sherwood number for different values of dimensionless melting, magnetic field, thermal radiation, chemical reaction, Brownian motion, thermophoresis, permeability parameter, Prandtl number, Eckert number, and Lewis number are revealed and pondered. The result shows that when the melting parameter enhance the fluid flow and boundary layer thickness increases but opposite trends happens to the temperature and concentration profiles. In addition upper-convected Maxwell fluids are less affected by melting parameter when compared with Williamson fluids. The interpretations of the conclusion are summarized as follows: ORCID Wubshet Ibrahim https://orcid.org/0000-0003-2281-8842