Impact of Activation Energy and Temperature-Dependent Heat Source/Sink on Maxwell–Sutterby Fluid
The present article aims to investigate the behaviour of Maxwell–Sutterby fluid past an inclined stretching sheet accompanied with variable thermal conductivity, exponential heat source/sink, magneto-hydrodynamics (MHD), and activation energy. By utilizing the compatible similarity transformations, the nondimensionless PDEs are converted into dimensionless ODEs and further these ODEs are tackled with the help of the bvp4c numerical technique. To check the legitimacy of upcoming results and reliability of the applied bvp4c numerical scheme, a comparison with existing literature and nonlinear shooting method is made. The numerical outcomes delivered here show that the temperature profile escalates due to an augmentation in the heat sink parameter and moreover mass fraction field escalates on account of an improvement in the activation energy parameter.
The fluids whose viscosity changes due to applied stress are termed as non-Newtonian fluids. Daily life examples of these kind of fluids are ketchup, blood, honey, glue, jellies, etc. Unlike Newtonian fluids, it is complicated to mathematically model the non-Newtonian fluids due to variation in viscosity and elasticity. Many researchers are working to explore the properties like viscosity and elasticity hidden in various non-Newtonian fluids. Prashu and Nandkeolyar  utilized finite difference scheme to achieve the numerical solution of three-dimensional Casson fluid under thermal radiation and Hall effect. Saidulu et al.  explored the coduct of MHD on radiative tangent hyperbolic nanofluid past an inclined stretchable surface and noticed that the velocity of the fluid diminishes owing to an increment in the magnetic number. Sajid et al.  contemplated the impact of heat source/sink and species diffusivity on radiative Reiner–Philippoff fluid past a stretchable surface. Williamson fluid accompanied with MHD, heat source, and nonlinear thermal radiation was deliberated by Parmar  who concluded that temperature gradient augments on account of an enhancement in the thermal radiation effect. Wang  studied the impact of free convection on vertical stretching surface and noticed that the Nusselt number upsurges because of an augmentation in the Prandtl number. Tlili  explored the marvels of MHD, mixed convection, and heat source on Jeffrey fluid flowing across a stretchable surface. They perceived that the temperature field escalates by escalating the magnetic parameter. Khan al.  studied the behaviour of non-Newtonian Carreau fluid flow over an inclined surface and found that the fluid velocity diminishes because of an augmentation in the Weissenberg number. Koriko et al.  analyzed the conduct of thermal stratification and nonlinear thermal radiation on micropolar fluid moving over a vertical surface. The phenomenon of heat transfer has been investigated by many researchers [9–14].
In recent years, the study of the phenomena like heat generation/absorption and temperature-dependent thermal conductivity has gained interest of the researchers because of its enormous applications in the field of computer technology and mechanical engineering. The phenomena such as heat source/sink alter the heat distribution which later on make a difference to the particle deposition rate in the system. Heat generation/absorption has immense applications in nuclear reactor engineering, chillers, and heat pumps. The ability of a material to conduct heat is called thermal conductivity. Thermal conductivity’s dependence on temperature is termed as variable thermal conductivity. In fluids, thermal conductivity of fluid changes due to intermolecular collision which gives rise to a gradual increase in temperature inside the fluid. Variable thermal conductivity is important in electrolytes which have been important for the preparation of batteries. Various researchers have studied the importance of heat generation/absorption and variable thermal conductivity for the purpose of heat transfer analysis. Patil et al.  contemplated the fluid above an exponentially stretching surface along with MHD and nonuniform heat source/sink. They found that temperature profile diminishes by escalating the thermal relaxation effect. Carreau fluid flow under the effect of joule heating and heat source/sink was elucidated by Reddy et al.  who determined that an augmentation in the Weissenberg number leads to an enrichment in the temperature distribution. Mahanthesh et al.  found the solution of the Casson liquid embedded with exponential temperature-dependent heat source/sink and cross diffusion effects and revealed that an embellishment in the velocity profile occurred by ascending the fluid relaxation effect. Konda et al.  studied the Williamson fluid past a vertical sheet with the inclusion of convective heat transfer at the boundary surface along with nonuniform heat source/sink and perceived that the temperature of the fluid abates because of an enrichment in heat sink parameter. Tsai et al.  adopted the Chebyshev finite difference scheme to achieve the numerical solution of fluid flowing across a stretchable surface accompanied with nonuniform heat source/sink. Yousif et al.  studied radiative Carreau nanofluid flow over an exponentially stretchable embedded with MHD, thermal radiation, and internal heat source/sink and found that the temperature distribution upsurges by mounting the thermal radiation parameter. Kumar and Varma  utilized the Runge–Kutta method to achieve the numerical solution of the fluid flow above a porous stretching surface along with suction and internal heat generation/absorption. Abel et al.  investigated the viscoelastic fluid flow over a permeable stretching surface together with MHD and heat source/sink. Khan et al.  contemplated the impact of heat generation/absorption and MHD on Sisko fluid past a porous stretching sheet. Khan et al.  examined the effect of heat source/sink on Maxwell nanofluid flow over a stretchable surface. Irfan et al.  scrutinized the behaviour of Maxwell nanofluid influenced by a stretched cylinder embedded with heat source/sink and MHD. Haq et al.  scrutinized the impact of heat source/sink and chemical reaction on Cross nanofluid moving over a wedge.
The ability of a material to conduct heat is called thermal conductivity. Thermal conductivity solely depends upon the nature of the material and changes with temperature. Metals have high thermal conductivity in comparison to solids and liquids. In fluids, thermal conductivity depends upon intermolecular collisions of the atoms inside the molecules. As a result of intermolecular collision, the molecules exchange energy with each other and fluid temperature increases which consequently amplifies the thermal conductivity. Thermal conductivity has enormous utilization in the field and technology like electrolytes, steam generators, concrete heating, laminating, catalysis, and molding blow. Gbadeyan et al.  analyzed the conduct of variable thermal conductivity and diffusivity on two-dimensional Casson nanofluid past a stretching sheet and found that the temperature field magnifies by amplifying the thermal conductivity parameter. Hamid et al.  adopted the Galerkin approach to achieve the numerical solution of nanofluid past a stretchable surface accompanied with variable thermal conductivity. It is noted that the temperature profile escalates as a result of an augmentation in the thermal conductivity parameter. Si et al.  contemplated the pseudoplastic fluid moving along a vertical stretching plate embedded with variable thermal conductivity and noted that the mass fraction field augments owing to an embellishment in the power law index. Kumar et al.  deliberated the performance hyperbolic tangent fluid with the inclusion of variable thermal conductivity past a sensor stretching surface and found that the velocity field by enhancing the Weissenberg number. Viscous fluid flow between two parallel plates in the presence of variable thermal conductivity, variable viscosity and first order chemical reaction is observed by Umavathi and Shekar  and deduced that the temperature profile embellish by embellishing the thermal conductivity variation effect. Salawu and Dada  pondered the conduct of inclined magnetic field on incompressible fluid flow over a stretching medium with variable thermal conductivity and found that an escalation in the temperature profile occurred on account of an enrichment in thermal conductivity effect. Lin et al.  studied the impact of variable thermal conductivity and thermal radiation on pseudoplastic non-Newtonian nanofluid and found that the temperature distribution increases because of an enlargement in the value of thermal radiation parameter. Aziz et al.  scrutinized the conduct of temperature-dependent thermal conductivity and heat generation on inclined radiating plate and observed that the velocity distribution decreases as a result of an improvement in the Prandtl number.
Recently, many researchers studied the influences of electrical conductivity fluid in the presence of magnetic field. These studies have important applications in generators, pumps, bearings, magneto-hydrodynamic (MHD) generators, etc. One of the basic and important problems in this area is the unsteady magnetic fluid behaviour of boundary layers along fixed or moving stretching surfaces. Khan et al.  studied the effect of MHD on Carreau nanofluid moving along a bidirectional stretching surface. They noted that a positive improvement in magnetic parameter increases the temperature field. Sharma and Mishra  studied the impact of MHD and internal heat generation/absorption on micropolar fluid moving along a stretchable sheet. Prasad et al.  developed a mathematical model of electrical conductivity fluid moving along a slender elastic sheet under the effect of temperature-dependent thermal conductivity, and it is remarkable that a positive amplification in magnetic parameter brings about a decrement in velocity field. Dessie and Kissan  investigated heat transfer characteristic of magnetic hydrodynamic fluid past a stretching sheet under the effect of viscous dissipation and heat source/sink utilizing the shooting method. From their numerical study, it is noted that a positive variation in heat sink parameter leads to a reduction in temperature field. Awati  scrutinized the behaviour of electrically conducting fluid flow over a stretching sheet accompanied with suction/blowing effects. The effect of variable internal heat generation/absorption and variable thermal conductivity on Carreau fluid moving along a stretchable surface has been debated in detail by Irfan et al. . Mishra et al.  analyzed the magneto power law fluid past a porous stretching sheet embedded with nonuniform heat source/sink and found that the velocity of the fluid depreciates owing to an improvement in porosity parameter. Ganga et al.  studied the behaviour of MHD on nanofluid past a vertical stretching plate. It is observed that velocity field abates owing to an enhancement in magnetic parameter. Khan et al.  contemplated the impact of MHD on Burger’s nanofluid past a stretchable surface accompanied with nonlinear thermal radiation and motile gyrotactic microorganisms. Khan et al.  studied the impact of MHD and thermal radiation on 3D Sisko fluid moving along a stretchable surface. The impact of MHD on Sisko fluid moving over a stretchable surface embedded with homogenous/heterogenous reaction and thermal radiation has been debated in detail by Khan et al. .
The minimum energy provided to the system to start a chemical reaction is called activation energy. Two important energies called kinetic and potential energies are responsible for breaking the bonds during chemical reactions. Sometimes the reaction between molecules are not complete due to the loss of kinetic energy or improper collision. At this stage, minimum amount of energy is needed to start the chemical reaction. Activation energy has distinguished applications in hydrodynamics, geothermal, and oil storage industry. Various researchers studied the marvels of activation energy. Kumar et al.  examined the conduct of chemical reaction and activation energy on Carreau fluid and observed that the concentration field increases because of an increment in the activation energy parameter. Gireesha et al.  studied the behaviour of activation energy and exponential temperature-dependent heat source on nano-Casson fluid and observed that the concentration distribution depreciates owing to an enhancement in Lewis parameter. Zaib et al.  determined the entropy of stagnated nano-Williamson fluid past a moving plate along with activation energy and found that the velocity distribution reduces as a result of an improvement in the Williamson parameter. Alshomrani et al.  examined the unsteady Eyring–Powell fluid embedded with MHD and activation energy and observed that the concentration distribution reduces by raising the fitted rate constant parameter. Nanofluid past a stretching sheet along with activation energy and convective boundary conditions was analyzed by Dhlamini et al.  who found that the concentration distribution showed a decreasing trend in the case of improvement in the reaction rate constant parameter. Chetteti and Chukka  manipulated the consequences of activation energy and convective boundary conditions on nanofluid flow over a cone. They analyzed that the temperature distribution increases by rising the activation energy parameter. Majeed et al.  pondered on the impact of activation energy and momentum slip condition on non-Newtonian fluid. They perceived that a positive variation in the Schmidt number abates the fluid concentration. Babu and Sathian  treated the water flow through carbon nanotubes along with activation energy. Ahmed and Khan  studied the impact of activation energy on Sisko nanofluid moving over a curved surface. The impact of activation energy and binary chemical reaction on 3D Cross nanofluid has been scrutinized in detail by Khan et al. . The impact of activation energy along with nanoparticles on Cross fluid has been discussed by Sultan et al. . Muhammad et al.  studied the heat and mass transfer analysis of cross magneto liquid accompanied with thermal conductivity and activation energy. The activation energy was used by many authors to observe its impact on the fluid flow [58–61].
The objective of the present study in the light of above mentioned literature is to explore the importance of various effects on Maxwell–Sutterby fluid flow over an inclined stretching surface. Heat transfer analysis has been carried out in the presence of variable thermal conductivity along with temperature-dependent heat source sink, and both parameters have important applications in energy sector like electrolytes, steam generators, cooling reactors, and nuclear reactors, whereas mass transfer analysis has been debated in the presence of activation energy and variable molecular diffusivity. This paper is important in its sense that no literature has been reported yet to study the mixture of Sutterby and Maxwell fluids past an inclined stretching surface in terms of heat and mass analysis with the aforementioned effects most importantly with variable molecular diffusivity and activation energy.
2. Mathematical Formulation
Two-dimensional incompressible electrically conducting Maxwell–Sutterby fluid flows over a stretching sheet inclined at an angle under the effect of magnetic field acting perpendicular to the sheet as shown in Figure 1. Magnetic field inclination is actually the angle made with the horizontal by the magnetic field lines. Positive values of inclination indicate that the field is pointing downward, into the sheet surface. In this article, the angle of inclination is . It is due to the fact that with an increase in angle of inclination , the effect of magnetic field on fluid particles increases which enhances the Lorentz force and furthermore depreciates the fluid flow. It is quite notable that maximum resistance is offered for the fluid particles when . The is taken along the leading edge of the inclined stretching sheet with stretching velocity and is normal to the surface. The magnetic Reynolds number is considered very small so that the effect of electric current and induced magnetic field can be neglected as compared to the applied magnetic field. Temperature and concentration at the surface of the sheet are denoted by and , whereas ambient temperature and concentration are indicated by and . The sheet temperature is ; moreover, fluid concentration is . Furthermore, the effects like activation energy, exponential temperature-dependent heat source/sink, and variable thermal and molecular diffusivity are also considered during mathematical formulation of the problem.
Under the aforementioned assumptions and after utilizing the necessary boundary layer approximations the Cartesian form of governing equations regarding continuity, momentum, energy, and concentration are enumerated underneath [10, 13]:
The boundary conditions are
By utilizing transformation  given below, we convert dimensionless PDEs to nondimensional ODEs.
The boundary conditions arewhere
The skin friction coefficient, rate of heat transfer, and mass transfer on the wall are denoted bywhereas the expressions regarding , , and are given by 
The dimensionless form of heat transfer and mass transfer is given by
3. Numerical Scheme
The nonlinear nondimensional transformed problem equations (9)–(11) along with boundary conditions (12) have been solved with the help of the MATLAB built-in function [62–64] and nonlinear shooting scheme. In the shooting method, first-order ODEs along with initial conditions are integrated with the utilization of RK4 method and modified missing initial conditions with the utilization of Newton’s scheme until solution meets the specified accuracy. The asymptotic convergence is observed to be achieved for . bvp4c is one of the boundary value problem solvers in MATLAB package. We use MATLAB software where we performed a finite difference method which is a collocation method of order four. All the numerical results achieved in this problem are subjected to an error tolerance . The system of partial differential equations (PDEs) is converted into first-order ordinary differential equations (ODEs) by utilizing the variables enumerated underneath:
To the reader’s convenience, a detailed procedure of bvp4c numerical technique is prescribed in Figure 2.
4. Results and Discussion
In this section, the behaviour of various physical parameters emerging during numerical simulation of the problem on the velocity, temperature, and concentration profiles has been debated in the form of graphs, and likewise their impact on skin friction coefficient and heat transfer and mass transfer rate is also discussed in the form of tables. The comparison analysis of numerical results is available in the literature. In order to check the authenticity of the numerical scheme and results, a problem can be tackled with nonlinear shooting scheme as well. A comparison analysis of the present scheme with shooting scheme reveals that the computed numerical results are quite reliable and authentic.
Table 2 explores the conduct of distinguished parameters on the surface drag coefficient. From the table, it is quite clear that an improvement in parameters such as Maxwell fluid Deborah number , magnetic parameter , Reynolds number , Sutterby fluid Deborah number , angle of inclination , and power law index leads to an enhancement in the surface drag coefficient but a variation in the thermal Grashof number and solutal Grashof number brings about an abatement in the surface drag coefficient. Table 3 shows the effect of various parameters on the heat transfer and mass transfer rates. From the table, it is observed that an augmentation in the values of Prandtl number , heat source , heat sink , reaction rate constant , Schmidt number , exponential index , temperature difference parameter , and thermal conductivity produces a decrement in the heat transfer rate, but an improvement in the heat transfer rate is observed on account of an improvement in the fitted rate constant , activation energy , and species diffusivity . The mass transfer rate enlarges because of an augmentation in the values of various parameters such as , , , , , , and but mass transfer rate decreases in the case of , , , and .
Figure 3 explores the consequences of solutal Grashof number against the velocity profile. Grashof number is actually the ratio of buoyancy forces to the viscous forces. It is quite interesting to note that increasing the value of lessens the viscous forces and strengthens the shear forces which eventually improves the velocity field. Figure 4 is sketched to show the impact of Maxwell fluid Deborah number on the velocity field. The fluids behave like liquids in the case of smaller Deborah number and become more viscous in the case of large values of Deborah number which eventually slows down the fluid velocity and eventually leads to a decrement in the velocity field. Figure 5 is sketched to witness the features of thermal Grashof number on the velocity field. A positive variation in the thermal Grashof number leads to an abatement in the viscous forces which thickens the boundary layer and moreover lessens the velocity profile. The variation of magnetic parameter on the velocity field is shown in Figure 6. Fluid moving through an electric field in the presence of magnetic field experiences a force called Lorentz force which reduces the movement of the fluid flow. The impact of Sutterby fluid Deborah number on the velocity profile is shown in Figure 7. Deborah number is defined as the ratio of the characteristic time to the time scale of deformation. The Deborah number is used to represent the viscoelastic nature of the material. It is observed that the greater the Deborah number, the more solid the material; the smaller the Deborah number is, the more fluid it is. From the figure, it is quite clear that an augmentation in the Deborah number brings about an abatement in the fluid movement. As a result, velocity field reduces. Impression of Reynolds number on the velocity field is shown in Figure 8. Reynolds number is defined as the ratio of inertial forces to the viscous forces. The fluid becomes more viscous in the case of the higher values of the Reynolds number. The viscous forces dominate the inertial forces which bring about an abatement in the fluid flow. Figure 9 is portrayed to explore the marvels of inclination angle against the velocity profile. It is revealed that an augmentation in the inclination angle parameter depreciates the buoyancy forces which furthermore reduces the fluid velocity. Figure 10 shows the contribution of power law index on the velocity field. A shear thinning attitude is observed in liquid in the case of an enhancement in the power law index which brings about a reduction in the velocity profile. Figure 11 demonstrates the impact of heat sink parameter on the temperature profile. From the figure, it is quite evident that a positive variation in the heat sink parameter generates more heat into the liquid which leads to an improvement in the thermal boundary layer thickness and temperature profile. Figure 12 is sketched to interpret the features of on the temperature field. Prandtl number is defined as the ratio of momentum diffusivity to the thermal diffusivity. It is quite evident that amplification in thermal diffusivity leads to a reduction in Prandtl number which depreciates the temperature field. Figure 13 demonstrates the conduct of exponential index on the temperature profile. It is found that a positive variation in depreciates the heat source/sink term and thermal boundary layer thickness. It is found that a positive variation in the exponential index brings about a decrement in the fluid temperature. Figure 14 shows the influence of thermal conductivity on temperature field. It is noteworthy that when the molecules collide with each other, they shift energy which consequently improves the temperature. As a result, an improvement in the thermal conductivity parameter causes amelioration in the fluid temperature and furthermore results in an embellishment in the temperature profile. The impression of reaction rate constant on the mass fraction field is shown in Figure 15. It is found that an augmentation in reaction rate constant causes an improvement in the factor . As a result, destructive chemical reaction occurs and moreover an abatement in the mass fraction field takes place. Figure 16 is designed to depict the effect of Schmidt number on the mass fraction field. Schmidt number is actually the ratio of momentum diffusivity to the Brownian diffusivity. It is observed when molecules collide randomly, the Brownian diffusivity parameter increases which depreciates the Schmidt number. The concentration boundary layer thickness increases due to an improvement in which leads to a reduction in the mass fraction field. Figure 17 explores the conduct of temperature difference parameter on the mass fraction field. When the difference between surface temperature and ambient temperature rises, the concentration boundary layer thickness increases which eventually makes a decrement in the mass fraction field. Figure 18 demonstrates the consequences of activation energy on the mass fraction field. According to the definition, minimum energy required to start a reaction is called activation energy. It is revealed that at lower temperature and high activation energy brings about a decrement in the reaction rate constant which eventually slows down the chemical reaction and furthermore an enhancement in the mass fraction field takes place. Figure 19 shows the behaviour of variable molecular diffusivity on the mass fraction field. It is monitored that an augmentation in the species diffusivity gives rise to an elevation in the concentration boundary layer thickness. It is also observed that the species diffusivity is directly proportional to the concentration. That is why an embellishment in the mass fraction field takes place as a result of an improvement in species diffusivity parameter. Figure 20 is portrayed to analyze the conduct of fitted rate constant on the velocity profile. It is found that a boost in brings about an enhancement in factor which favours the destructive chemical reaction and leads to an increment in the mass fraction field.
5. Final Remarks
The forthright aim of this correspondence is to investigate the influence of exponential temperature dependent-heat source/sink, variable thermal and molecular diffusivity, MHD, and activation energy on Maxwell–Sutterby fluid. The main findings of the present study are enumerated underneath:(i)The velocity profile decreases on account of an improvement in magnetic parameter .(ii)Shear thickening behaviour is observed on account of an improvement in power law index .(iii)The temperature profile increases on account of an improvement in the thermal conductivity , porosity parameter , and Deborah number .(iv)Both thermal conductivity parameter and the heat sink parameter boost the temperature field.(v)A positive variation in heat sink depreciates the velocity field.(vi)An improvement in fitted rate constant amplifies the concentration field.(vii)Mass fraction field augments owing to an improvement in and .
|:||Solutal Grashof number|
|:||Dimensionless thermal conductivity|
|:||Maxwell fluid Deborah number|
|:||Temperature-dependent thermal conductivity|
|:||Power law index|
|:||Surface heat flux|
|:||Dimensionless reaction rate constant|
|:||Temperature difference parameter|
|:||Fitted rate constant|
|:||Radiative heat flux|
|:||Magnetic field strength|
|:||Temperature ratio parameter|
|:||Reaction rate constant|
|:||Dimensionless heat source|
|:||Dimensionless heat sink|
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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