Our key objective in the present work is to elaborate the concept of activation energy in chemically reactive flow with the help of modeling and computation. The model investigated is fluid flow over a vertical cylinder in the porous medium with chemical reaction and radiation effect. The similarity transform converted the resulting constitutive equations and partial differential equations (PDEs) into ordinary differential equations (ODEs). The resulting non-linear momentum, heat transfer, and mass transfer coupled equations are computed with the Range–Kutta–Fehlberg method. Both assisting and non-assisting buoyant flow conditions are considered, and observed numeric solutions vary with the transport properties. Characteristics of momentum, heat, and concentration under the applied boundary conditions are analyzed. In addition, the increment in activation energy parameters boosts the Lorentz force and mass transfer rate.
magnetic field (Tesla)
curvature parameter (m−1)
- C f
skin friction coefficient (Pascal)
solute diffusivity (m2 s)
activation energy (Joules)
radiation parameter (–)
- L, S
velocity and temperature slip length (m)
magnetic parameter (W m2)
constant number (–)
Nusselt number (dimensionless number)
Prandtl number (dimensionless number)
- q m
surface mass flux (kg m−2 s−1)
- q r
radiative heat flux (kg s−3)
- q w
surface heat flux (W m−2)
Reynold's number (–)
Schmidt number (dimensionless number)
- T w
cylinder temperature (Kelvin)
- T ∞
ambient temperature (Kelvin)
- u, v
x, r – directions velocity (m s−1)
kinematic viscosity (m2 s−1)
concentration (volume fraction) (mol m3)
dimensionless temperature (–)
density of fluid (kg m−3)
dimensionless similarity variable (–)
- σ e
mixed convective parameter (W m−2 K−1)
- c p
specific heat (J kg−1 K−1)
stream function (–)
- τ w
cylinder temperature (Kelvin)
thermal slip parameter (–)
temperature difference (Kelvin)
The theory of Newtonian fluid is named after Isaac Newton and has received much attention in the last several years because the non-Newtonian fluids contain fewer complex derivations and singularities. It can describe the characteristics of fluid flow precisely. Isaac was the first who had used the differential equation to postulate the relation between the shear strain rate (s−1) and shear stress (mPa) for such fluids. This relationship is now known as Newton's law of viscosity (i.e., shear stress (τ) = viscosity (η) × shear rate (γ)). Some η examples of Newtonian fluids are water, organic solvents, and honey. These fluid's viscosity is only dependent on temperature. Due to ideal behavior, these fluids have applications only in the theoretical studies to understand the non-Newtonian fluids.
A boundary-layer flow (BLF) is a thin-layer of viscous-fluid close to the solid surface in contact with a moving stream in which the thickness of the flow velocity varies from zero at the wall and increases up to free stream velocity at the boundary. The BLF of fluids attracted numerous researchers due to its vast applications over stretching/shrinking surfaces in different sciences and engineering fields. In the study of aerodynamical forces and their types, one can understand that the phenomena is dependent on the BLF. Few other applications of BLF are elastic sheets, tinning and strengthening of copper wires, hot rolling, fiber turning, non-stop cooling, wire drawing, paper, food processing, and expulsion of polymers. Sakiadis et al. was the first who examined the BLF due to its strong influence on the latest developments [1,2,3]. Later on, many researchers considered boundary-layer flow of Newtonian and non-Newtonian fluids. Crane et al. had given further directions to investigate the Sakiadis problem by utilizing the stretching boundary and evaluated the exact solutions . It should be noted that many works have been done on BLF. However, we considered recently published works of BLF over a stretching surface. Nazar et al. had studied the unsteady flow in a rotating fluid due to suddenly stretched surface . They found a smooth transition from the small-time solution to the large-time or steady-state solution. Van Gorder et al. had presented similar solutions for the nano BLF with the Navier boundary condition . They found expected results for fluid flows at nano-scale, i.e., the shear stress along the wall decreases with the increase in slip effect. Khan et al. had the investigated laminar fluid flow due to stretching of flat surface in a nanofluid . They found that the reduced Nusselt number is a decreasing function of each dimensionless number. Ullah et al. had evaluated non-linear stretching of cylinder place in the porous medium . They had noticed that wall shear stress magnitude is higher due to an increase in porosity. Fang et al. had studied the BLF over a stretching sheet with variable thickness . They had observed that the stretching surface's curvature has a direct effect on the formation of the boundary layer along the wall, the velocity profiles, and the shear stress distribution in the fluid. Gireesha et al. had investigated numerically the effect of Copper as a nanoparticle and water with suspended particles as its base fluid on the BLF . They found that momentum and thermal boundary layer are thinner due to the influence of suspended particles. Other researchers had also studied BLF [11,12,13,14] with different physical effects.
Most researchers kept up their work with the addition of assumption for different physical aspects – all this about getting close to the occurrence of real scenarios in the presence of a magnetic field, existence of magneto-hydrodynamics (MHD) flow. The projects related to engineering and technology, we mostly used MHD factor to control mass and heat transfer. In chemical engineering, experiments and numerical simulations are commonly conducted in the presence of MHD factor, see e.g. [15,16,17,18] for better understanding. The magnetic field and heat source/sink with different conditions were considered meticulously in refs. [19,20,21]. The presence of a magnetic field to control the different mechanisms in the material industry is important, and many researchers contributed their efforts in this direction [22,23,24,25].
The drag reduction phenomena in fluid dynamics with the surface walls are considered the slip effect. It is noticed that slip has altogether variable effects on the heat and thermal boundary layers. Slip flow due to a stretching cylinder was studied in ref. . They discovered that slip significantly decreases the magnitudes of the velocities and shear stress. Hatte et al. had studied the analytical model for the evaluation of effective slip . They discovered that liquid-infused engineered non-wetting surfaces have fluid flow with alternating no-slip and partial slip boundary conditions, resulting in decreased friction at the interface. Mukhopadhyay et al. explored the slip effects in the presence of MHD flow over a stretchable cylinder . Numerous researchers discussed the flow problems by considering the slip conditions on the boundary [29,30,31].
Thermal-radiation has an important role in the surface heat transfer when the convection heat transfer coefficient is small. Its few significant applications are heating the earth by the sun, room by an open-hearth of the fireplace, circulating blood to the body, and light/heat sources in control heating systems. Due to this phenomenon, energy transfers from the source object in the form of rays. Sheikholeslami et al. had examined the thermal radiation effect on the MHD nanofluid flow between two horizontal rotating plates . Li et al. had investigated the combined radiation and MHD effect on momentum and heat transfer in a vertical cylindrical annulus . They observed a substantial change in temperature profile when scattering albedo reaches one. Alsagri et al. had founded the enhancement in the magnetic-radiations yields a decrease in the velocity field for nanoparticles . Sinha et al. had presented the heat transfer characteristics by taking into account a steady heat supply at the wall and completely formed blood-flow through the capillary . Their results revealed that the blood temperature can be controlled by regulating Joule-heating parameter. Few important studies in this physical aspect are conducted by refs. [36,37,38,39].
The heat and mass transfer in chemical reaction has important role in fluid mechanics. Due to its nature and engineering applications, it received stellar recognition. Some important applications include human transpiration, chemical catalytic reactors, nuclear reactors, electronic-equipment, gas turbines and propulsion devices, aerodynamic-extrusion of plastic sheets, filtration, refrigeration, and medicine diffusion in blood veins. The effect of the Arrhenius activation energy under the different physical conditions was discussed in [40,41,42]. They found that the activation energy of a chemical reaction is closely related to its rate, because molecules can only complete the reaction once they have reached the top of the activation energy barrier. Seddeek et al. had investigated the effect of chemical reaction on free convective flow and mass transfer over a stretching surface . They noticed that an increase in the chemical parameters decreases the velocity, temperature and concentration in the boundary layer flow. Olanrewaju et al. analyzed the effects of thermal diffusion, including magnetic field intervention and suction/injection, on the chemical-reacting boundary-layer heat distribution and mass transfer onto a moving vertical plate . Shahzed et al. had investigated the impacts on porous media of Casson fluid via chemical reaction, transfer of mass, and MHD flow .
It is obvious from the aforementioned debate that these physical effects have limited research data in a Newtonian fluid. This study has much importance to get the ideal results for understanding the behavior of the non-Newtonian fluid. The investigators plan on the collective impact of binary chemical, magnetic, and thermal radiations due to significant discussions. The model addressed here has fundamental importance. We used the numerical procedure because of the non-linearity of the mathematical model and the additional complexity factors. The numeric procedure used for the transformed dimensionless governing equations [46,47] is the Runge–Kutta–Fehlberg (RKF) method with shooting technique [48,49,50,51]. Further, surface friction and rate of heat and mass transfers are tabulated and examined for various pertinent parameters. Also, a comparison with the previous studies is mentioned for the validity of the computational process.
2 Governing equations
We will discuss the laminar BLF generated due to the non-linear stretching of the vertical cylinder (radius R). The cylinder is placed in a porous medium on the fixed surface and stretched upward. The induced magnetic field has less influence as compared to the applied magnetic field, i.e., . The stretching of the cylinder sources the fluid which is placed around the cylinder. The below-given Scheme 1 shows its geometry.
where u x and u r are the velocity components, respectively, in x and r directions; u e is free stream velocity; kinematic viscosity; ρ fixed fluid density; c p specific heat; μ coefficient of viscosity; σ (e) electrical conductivity; B 0 constant magnetic field; α thermal diffusivity; and q r radiative heat flux which is calculated by Quinn Brewster . D is solute diffusivity, k is Boltzmann's constant, is response rate, m is fitted rate, and E a is the activation energy. It is assumed that temperature difference inside BLF is such that T 4 may be expanded in a Taylor's series about T ∞, and removing the higher order terms we will obtain
The appropriate Boundary Conditions (BCs) for the above problem are,
where stretching velocity; ; surface temperature; L is the length velocity slip; C w concentration of mass near the wall; and S is the length of the thermal slip, and U (0) is the reference velocity, T (0) is the reference temperature, T (∞) is the ambient temperature, C (∞) is the solute concentration.
Making use of the similarity transformations,
Plugging Eq. (8) in Eqs. (1–7), we will obtain the required governing equations with associated BCs. Stream function identically satisfies the continuity equation in cylindrical coordinates,
Abovementioned pertinent parameters are curvature , magnetic , porosity , mixed convection , velocity ratio , temperature ratio , Prandtl number , velocity slip , thermal slip , non-linear radiation , activation energy , temperature difference , reaction rate , and Schmidt number .
The industrial interest quantities like skin friction coefficient (C f ), local Nusselt number (Nu x ), and local Sherwood number (Sh x ) are defined as,
where the shear stress (τ w ), surface heat flux (q w ), and surface mass flux (q m ) are given by,
Using the non-dimensional variables, we obtain
where is the local Reynold's number.
3 Numerical technique
In order to study the flow model, the numerical scheme RKF method is used, which is reliable and efficient. Following are the mathematical steps for the RKF method:
where y is 4th and z is 5th order RKF. To calculate next step, we have
We use for calculation. The criteria used for convergence is the variation in the dimensional velocity, and the temperature should be less than between any two consecutive iterations. The asymptotic BCs in Eq. (12) is approximated by using a value of 10 for as follows:
4 Results and discussion
To analyze the impact of the various parameters with the fixed values of these physical parameters includes nonlinearity n = 2.0, curvature C = 1.0, magnetic M = 1, porosity P = 0.3, mixed convection λ = 1.0, temperature ratio N r = 0.3, Prandtl number Pr = 0.5, thermal radiation K = 2.0, velocity slip and thermal slip , Schmidt number S c = 0.1, activation energy E a = 0.1, reaction rate , fitted rate m = 0.1, and temperature difference . Related important results are presented both in graphical and tabulated forms.
Table 1 presents a comparison between the results obtained currently and the previous results for the heat transfer rate .
|Prandtl number||Magnetic parameter||Radiation parameter||||||||||Present|
Influence of governing parameters on velocity, temperature, and concentration profiles are depicted in Figures 1–14. Figure 1 exhibits the effect of variation in n on the fluid velocity. It is noticed that an increase in it decreases the momentum of the fluid. This behavior occurs due to the disturbance generated in the fluid particles/molecules. As a result, collisions between them enhances, which decreases the momentum.
Figure 2 depicts the influence of C on the velocity profile. It is observed that with the increase in the cylinder bending (or reduction in the area of cross-section), the momentum of fluid also enhances. It is actually due to less friction accounted for because of a decrease in the total surface area of the cylinder.
Figure 3 elucidates the effect of M on the velocity profile. It is evident that the increasing values of M reduces the fluid velocity. Also, the momentum boundary layer becomes thinner when n increase. This happens due to the resistance generated by the Lorentz force.
Figure 4 presents the effect of the variation in velocity slip on the velocity profile. As expected, the strength of the slip enhances the fluid flow, but in this case, it reduces. This change may occur due to the thermal/magnetic-radiations and chemical activity.
Figure 5 displays the effect of variation in A on the velocity profile, for A > 1 momentum and boundary layer thickness increase and vice versa for A < 1. However, for A = 1 no significant change is found.
Figure 6 depicts the impact of C on the heat profile. By increasing curvature parameter the heat transfer reduces.
Figure 7 portrays the effect of variation in N r on the heat profile. The thermal boundary layer becomes thicker with the increase in the temperature ratio parameter. This is because the fluid temperature is much higher than the ambient temperature for increasing values of N r , which increases the thermal state of the fluid.
Figure 8 demonstrates the effect of K on the heat profile. As radiation parameter is representing the thermal dissipation via electromagnetic radiation discharge, the fluid's main energy is transformed into electromagnetic energy that enhances the internal kinetic development and collisions between the fluid molecules. Therefore, larger values of K increases the rate of heat transfer and linked thermal boundary layer. As more and more heat shift to the fluid, the temperature increases.
Figure 9 demonstrates the effect of γ on the heat profile. By increasing thermal slip parameter we noticed that the heat transfer decreases.
Figure 10 illustrates the effect of variation in S c on the mass profile. For higher values of S c , mass transfer reduces because the Brownian diffusion coefficient rises in such a way that it retards the mass transfer and decreases the boundary layer thickness.
Figure 11 displays the influence of ε on mass transfer. The mass transfer and boundary thickness both are decreasing due to increment in a temperature gradient. Our results contradict the evaluation of ref. ; this may be due to the stretching and shrinking effect or the radiations, and a chemical reaction occurs.
Figure 12 reveals the effect variation in E on the mass profile. From the graph, it can be seen that the concentration enhanced due to the increment in the activation energy parameter. This phenomenon is not only chemical but also physical. Therefore, to interpret the right results, more simulation and experimental results are needed by considering different kinetic parameters.
Figures 13 and 14 examine the effect variation in σ and m on the mass profile. It is clear that due to the influence of reaction rate, fitted rate increases the factor . An increase in the values of the reaction rate parameter implies more interaction of species concentration with the momentum boundary layer. In contrast, an increase in the fitted rate parameter illustrates the marginal rise in the concentration.
In Table 2, the influence of variation in different parameters on the surface friction, heat transfer, and mass transfer rates are mentioned in detail.
|n||C||M||P||λ||δ||N r||Pr||K||γ||S c||ε||E||σ|
5 Summary and conclusion
Two-dimensional vertical cylindrical model considered with non-linear stretching in the porous medium. Few important physical aspects included are slip, radiation, and chemical reaction. The numeric data obtained from the model is compared with previously mentioned cases. It is found that the comparison of available open literature and obtained results are in good agreement:
An increment in the velocity slip parameter decreases the momentum of the fluid.
The influence of the thermal slip parameter increases the heat transfer rate.
An increase in the curvature of the cylinder makes the thermal boundary layer thinner.
The addition of radiation energy enhances the internal kinetic development and collisions between the fluid molecules.
Temperature gradient contains dual effects due to stretching/shrinking, radiation, and chemical reaction.
Enhancement of reaction rate parameter generates more interaction between the species.
6 Future perspectives
Much work needed for the future perspectives,
In the presence of radiation and chemical reaction models (specific reaction kinetics for different process).
The geometries used in industries with 3D modeling.
Comparison of the numerical methods and techniques for accuracy of these models are still required.
More relevant mathematical models related to engineering applications through simulation are needed.
Improvement needed in the numerical method algorithms.
Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
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