Effects of activation energy and chemical reaction on unsteady MHD dissipative Darcy

Scientific Reports volume 13, Article number: 2666 (2023) Cite this article

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The impact of chemical reaction and activation energy plays a vital role in the analysis of fluid dynamics and its thermal properties. The application of the flow of fluid is significantly considered in nuclear reactors, automobiles, manufacturing setups, electronic appliances etc. This study explores the impacts of activation energy and chemical reaction on the magnetohydrodynamic Darcy–Forchheimer squeezed Casson fluid flow through a porous material across the horizontal channel where the two parallel plates are assumed to be in motion. By using similarity variables, partial differential equations are converted to ordinary differential equations. Numerical method is applied using MATLAB to solve the problems and acquire the data for velocity field, thermal distribution, and concentration distribution. The graphs indicate that fluid velocity and temperature increases as the plates are brought closer. In addition, there was a correlation between a rise in the Hartmann number and a decrease in the fluid's velocity because of the existence of strong Lorentz forces. The temperature and the concentration of the liquid will increase due to the Brownian motion. When the Darcy–Forchheimer and activation energy parameters are both increased, the velocity and concentration decreases.

Squeezing flow between two parallel discs has garnered much interest recently due to the wide range of applications in technical and industrial setups. The notion of flow between two squeezing surfaces is used in devices like, hydraulic brakes, the moving piston of an engine, chocolate fillers, and many more. Syringes and nasogastric tubes both include the process of squeezing flow while a moving disc is influencing it. A more profound understanding of these fluxes results in creating more effective and efficient machines that may be used for various mechanical and industrial applications. The manufacturing of hydrodynamical devices, accelerators, compression and injection moulding, lubrication equipment, and polymer processing are some of the places where squeezing flow can be observed. Stefan1 studied squeezing flow using the lubrication approximation; several scholars subsequently examined squeezing flow issues for various geometrical configurations using multiple approaches. Moore2 indicated that the influences such as surface finish, viscoelastic liquids, elastomeric surfaces, and molecular effects plays a vital role and hence it must be considered either partially or entirely depending on the degree of complexity of the problems. Gupta et al.3 noticed that the unsteady squeezing channel flow problem could be significantly simplified via similarity variables. The distance between the paralleled plates varies as the square root of a linear function of time. In this scenario, the similarity variables allow the problem to be significantly simplified. Duwairi et al.4 studied the effects of heat transfer on unstable squeezing channel flow, they assumed that the paralleled walls were heated evenly at a constant temperature. This allowed them to examine the impact of heat transfer on the flow. Furthermore, various scholars have looked into the heat transfer properties of nanofluid flowing between parallel plates5,6,7 by considering various physical conditions.

Squeezing flow between parallel plates finds its significance in the area of fluid dynamics as it finds applications in hydraulic machinery and tools, electric motors, food industry, bioengineering, and automobile engines. Other simpler but equally important examples are flow patterns occurring in syringes and compressible tubes. In these applications, flow patterns can be classified into laminar, turbulent, and transitional flows on the basis of the well-known Reynold's number. From an industrial perspective, it is necessary to study the effect of these different behaviors for non-Newtonian fluids and in this regards many scholars have studied the flow of Casson fluid8,9 as it is able to capture complex rheological properties of a fluid. It was observed that the movement of microorganisms within the Casson nanofluid will help in preventing the agglomeration of nanoparticles and provides a smoother flow10,11. Concentrated fluids like sauces, honey, juices, blood, and printing inks can be well described using this model. Casson fluid can be defined as a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear. Hussain et al.12 performed a non-similar analysis to study the EMHD flow of Casson nanofluid considering the shape of the suspended nanoparticle as a factor. Jamshed et al.13 implemented the Tiwari-Das model to examine the thermal properties of Casson nanofluid and found an increase in the temperature absorbed when the nanoparticle volume fraction was raised. Furthermore these studies were extended to anaylse the motion of Casson nanofluid over a Riga plate by Upreti et al.14.

The widespread use of mass transport with activation energy in essential fields such as geothermal engineering, chemical engineering, oil emulsions, and food processing has drawn the attention of researchers. Arrhenius proposed the concept of activation energy in the year 1889. It is the minimal amount of energy that particles must obtain to undergo a chemical reaction. This energy might exist in kinetic or potential energy, and without it, reactants cannot make products. Activation energy has a comprehensive range of applications, including geothermal engineering, chemical engineering, oil emulsions, and food processing. In the first part of his study, Bestman15 looked at the convective flow of binary amalgam through a porous medium. Makinde et al.16 investigated the activation energy and the impacts of the nth order chemical process on a time-dependent radiated flat porous panel. Alsaadi et al.17 studied the nonlinear mixed convective flow of non-Newtonian nanoliquids across an absorbent stretched sheet. In contrast, the flow was subjected to the influence of nonlinear radiation and activation energy. Additionally, the researchers explored the rate at which entropy is generated. The researchers in this study concluded that an increase in the activation energy parameter led to a rise in concentration. Irfan et al.18 constructed an unsteady flow of Carreau nanofluid to gain the impacts of binary chemical reaction and activation energy. They reported the alterations in shear-thinning fluids and shear thickening fluids with the influence of the reaction rate parameter, revealing that the concentration decreased.

The amount of energy that must be present in a chemical system containing potential reactants to produce a chemical reaction is referred to as the activation energy. The Arrhenius equation, which explains the shift in rate constants as a function of temperature, is the formula used to calculate activation energy. In geothermal engineering, chemical engineering, mechano-chemistry, oil and water emulsions, and the degradation of materials, a mass transfer phenomenon with a chemical reaction is employed. Chemical reactions and mass transfer have a complicated relationship with one another. This connection may be investigated for both fluid flow and mass transfer by fabricating and digesting reactant species at varying speeds, which both involve the flow of fluids and the transfer of mass. Hsiao19 gave a numerical analysis of the manufacturing efficiency of a thermal extrusion system by applying an improved method of controlling parameters. This was accomplished via the use of an enhanced technique of controlling parameters. Majeed et al.20 investigated the accumulative impacts of a binary chemical reaction and activation energy in a fluid flow under a scenario of second-order momentum slip. Along with the topic of activation energy, Khan et al.21 examined the influence of nonlinear thermal radiation. They found a correlation between the more significant activation energy parameter and a rise in species concentration. Dhlamini et al.22 broadened the scope of the study by taking into account mixed convection. They discovered that using a heated plate increased the concentration of the chemical species. For the flow of Carreau fluids, Irfan et al.18 used nonlinear mixed convection as their transport mechanism. The nonlinear fluctuation of density, as opposed to the linear variation of density, may sometimes lead to a more significant increase in species concentration.

Concerning the petroleum industry, convective fluid flow phenomena in a porous space are essential when considering high flow rate regions, likely to occur near wellbores for gas and condensate reservoirs. This is because these regions will likely be where high flow rates arise. High velocities may be found in several contemporary applications that use porous areas. A modified version of the traditional Darcy's law, known as the non-Darcian porous space, considers the effects of both porous space and inertia. The majority of the research that has been done on this topic has modelled and analysed flow issues across porous areas using the traditional Darcy's equation. However, Darcy's theory in its classical form breaks down under the conditions of greater velocities and bigger porosities. Therefore, to take into account inertia's effects, Forchheimer23 included a square velocity factor in the equation for momentum. This component was referred to as "Forchheimer's word" in Muskat's24 analysis. Seddeek's25 used the Darcy–Forchheimer relation to investigate the mixed convective flow of nanofluid. Jha and Kaurangini26 discovered approximate solutions for the nonlinear Brinkman– Forchheimer-extended Darcy flow model they were working on. Pal and Mondal27 investigated the hydromagnetic Darcy–Forchheimer flow of a liquid with varying viscosity. Darcy–Sadiq and Hayat28 examined the Forchheimer flow of a magneto-Maxwell liquid limited by a convectively heated sheet. Shehzad et al.29 investigated the impact of the Cattaneo–Christov heat flux model on the Darcy–Forchheimer flow of an Oldroyd–B fluid with variable conductivity and nonlinear convection. Bakar et al.30 investigated the forced convection boundary layer stagnation-point flow in a Darcy–Forchheimer porous space towards a diminishing sheet. Hayat et al.31 studied a Maxwell material's Darcy–Forchheimer flow when it was subjected to a heat flux. Their research was based on the Cattaneo–Christov theory and included variable thermal conductivity. Using the Darcy–Forchheimer–Brinkman model, Umavathi et al.32 performed a computational analysis of the natural convective flow of nanofluids and the heat transfer that occurred inside a vertical rectangular duct. The Darcy–Forchheimer flows of viscoelastic nanofluids were the subject of comparative research carried out by Hayat et al.33. A recently improved model for the Darcy–Forchheimer flow of a Maxwell nanofluid with a convective surface condition was published by Muhammad et al.34.

In this paper, we analyse the effects of activation energy and chemical reaction on magnetohydrodynamic (MHD) Darcy–Forchheimer squeezing Casson fluid flow across porous media along a horizontal channel. Using similarity variables, one may successfully convert partial differential equations (PDEs) to ordinary differential equations (ODEs). Numerical method is applied using MATLAB to solve the problems and acquire the data for velocity field, thermal distribution, and concentration distribution. Results have been plotted graphically. In addition, the correctness of the solution has been checked by comparing them to the results published in peer-reviewed papers.

The symmetric and time-dependent Magnetohydrodynamic Darcy–Forchheimer squeezed Casson nanofluid flow through a porous material with activation energy and chemical reaction is presented. The movement of water through the channel is caused by the compression of two of its surfaces. The separation between the upper and lower surface if y = ℎ(t) = (1 − αt)1/2. The magnetic field B(t) is assumed to be perpendicular to the bottom plate35. The geometrical representation of the Casson fluid flow is shown in Fig. 1. In addition, the influence of a homogeneous chemical reaction of the first order is considered in the concentration equation. Given these assumptions, the continuity, momentum, energy, and concentration equations that regulate the current physical issue with required circumstances are as follows36:

Physical model of problem.

The correlated boundary conditions (BCs) are

The Rosseland approximation can be used for the radiative heat flux vector \({q}_{r}\) because there is also self-absorption in addition to emission for an optically thick fluid. Since the absorption coefficient is typically wavelength dependent and significant, we can use the Rosseland approximation. Therefore, the definition of \({q}_{r}\) is37,49.

In this equation, k1 denotes the Rosseland mean absorption co-efficient and σ1 stands for the Stefan–Boltzmann constant.

We are working under the assumption that the temperature changes inside the flow are not very significant, allowing us to describe T4 as a linear function. We extend T4 about the free stream temperature T using Taylor's series, ignoring higher order variables in the process. The following is an approximation that may be derived from this:

The equation for energy may be obtained by combining Eqs. (8) and (9), as shown in the following:

The following similarity transformation is employed to obtain the ODEs from the PDEs:

where η is the local similarity variable, f(η), θ(η) and ϕ(η) are the dimensionless velocity, temperature and concentration of the fluid in the boundary layer region, respectively.

In order to produce the following non-dimensional equations, substitute Eq. (11) into Eqs. (2), (3), and (10) respectively, and then gives.

The associated dimensionless boundary conditions are as follows:

In the equations that do not include dimensions, the important parameters are defined as

The local skin friction coefficient Cfx, the local Nusselt number Nux, and the local Sherwood number Shx are the physical quantities of relevance that influence the flow. These numbers have the following definitions:

where τw, qw and qs are the wall skin friction, wall heat flux and wall mass flux respectively given by

The coefficient of skin friction, the Nusselt number, and the Sherwood number are all expressed in their non-dimensional versions in terms of the similarity variable as follows:

where Rex = lvw/νf is the local Reynolds number based on the squeezing velocity vw.

In computational analysis, the ND-Solve (Shooting) procedure is a procedure is a technique for tacking a boundary value problem (BVP) by reducing it to first order differential equation (Initial Value Problem IVP). It comprises finding solutions to the IVP for various initial conditions until one finds the solution that also fulfils the boundary conditions of the BVP. For the consider flow problem, the system of Eqs. (12–14) with boundary conditions (15, 16) are solved numerically with the help of ND-Solve (Shooting) technique. For this purpose, the higher order differential equations are first altered into first order with the help of new transformations. The new transformations procedure are listed as:

The Eqs. (12, 13, 14) with boundary conditions (15) takes the form

with

The system whose equations are Eqs. (12)–(14) has been numerically solved. In this example, a ND-Solve (Shooting) technique is utilized. An algorithm is designed in the program known as MATLAB to create both numerical and graphical answers. The findings are compared with those obtained by Noor et al.36 and Naduvinamani and Shankar38 to verify the precision of the current numerical scheme as shown in Table 1. In addition, it can be seen from Table 1 that the absolute values of wall shear stress go up as the values of the squeezing number go up, whilst the Nusselt and Sherwood numbers go down. This is the opposite of what happens when the squeezing number goes down. Additionally, it should be no surprise that the passage of heat from the surface of parallel plates to the fluid in between the plates is indicated by the Nusselt number having negative values.

The influence of the Forchheimer parameter (Fr) on the velocity and the temperature profile is seen in Figs. 2 and 3, respectively. It should be noted that the decrease in fluid flow velocity occurs when the value of Fr is increased. Figure 3 demonstrates that an increase in the value of the Frochheimer parameter increases both the thermal gradients and its corresponding boundary layer thickness. The rise in the Forchheimer parameter signifies that the resistive forces are generated within the system. This leads to a decrease in the fluid's motion; consequently, the velocity decreases. Additionally, the temperature of the fluid increases due to the drag force, which inclines the thermal gradients. An increase in the Hartmann number results in a reduction in the standard component of the velocity profile in the flow zone is shown clearly in Fig. 4. It is reasonable to anticipate that a negligible increase in the Hartmann number leads to the increased Lorentz forces associated with magnetic fields. Because of this strong Lorentz forces, a resistance is offered to the fluid passage through the channel. As a consequence, the velocity field will become less intense as the value of the Hartmann number rises. The effect of the porosity parameter on the velocity is seen in Fig. 5. It has been discovered that a more considerable porosity value results in a fluid with a lower velocity and a higher temperature and concentration.

Effect of Fr on f(n).

Effect of Fr on temperature.

Effect of Ha on velocity.

Effect of Da on velocity.

Figures 6 and 7 illustrate the influence that the thermophoresis parameter (Nt) on the temperature and the concentration, respectively. Temperature and concentration both increase when there is an increase in Nt. Physically, the thermophoresis force is assumed otto be stronger as the value of Nt is raised. This is because the nanoparticles are being pushed away from the hot zone and towards the cold area resulting in an increase in the temperature. A rough draught of Fig. 8 is shown here to explain the change in (ϕ(η)) that results from the Brownian motion parameter (Nb). It was found that the correlation coefficient tends to decrease with increasing Nb values. This phenomenon occurred due to the fluid's nanoparticles moving in an erratic pattern.

Effect of Nt on temperature.

Effect of Nt on concentration.

Effect of Nb on temperature.

The relationship between an increase in the thermal radiation parameter and a subsequent decrease in the temperature profile in the flow zone is seen quite clearly in Fig. 9. It is possible to draw the following conclusion from Fig. 9. An increase in the thermal radiation parameter results in a higher temperature value, which might benefit various thermodynamic sectors. Figure 10 show how the temperature profile is affected by the effect of the heat production or absorption parameter (Q). It demonstrates an increase in the temperature field when the parameter Q is increased. In addition, the thickness of the thermal boundary layer grows with an increase in the value of Q. It is to be anticipated that, in the process that generates heat, a higher temperature will typically be discharged into the working fluid. As a consequence of this factor, the heat production parameter increases the temperature profile. Additionally, a temperature rise may be attributed to exothermic chemical processes.

Effect of Rd on temperature.

Impact of Q on temperature.

Figure 11 is an illustration of the influence that the chemical reaction parameter on the concentration profile. In general, it has been shown that in many instances, a lower concentration field is found for destructive chemical reactions. Figure 12 demonstrates that an increase in the Schmidt number results in a reduction in the concentration field in the flow zone. A slight increase in the Schmidt number reduces the coefficient of mass diffusion, bringing about drops in the concentration field inside the flow zone. In addition to this, it has been shown that the concentration field is a function that decreases as Sc increases. Additionally, when the concentration of the boundary layer increases, the thickness of the coating decreases. Figure 13 discusses the impact that activation energy, denoted by E, has on concentration, represented by (ϕ(η)). The increasing the values of E lowers the Arrhenius energy function, as a result, increasing the rate of the generative chemical reaction that enhances the concentration.

Impact of R on concentration.

Impact of Sc on concentration.

Impact of E on concentration.

In the present study, the radiative Darcy–Forchheimer squeezed flow of unstable magneto-hydrodynamic non-Newtonian Casson fluid between two parallel plates with Joule heating and heat production or absorption in the presence of Activation energy and homogenous chemical reaction effects is reported. In the context of the current issue, flow is brought about by the movement of parallel plates. After obtaining the findings of the non-Newtonian Casson fluid flow model, the authors solve the highly nonlinear coupled two-dimensional unsteady partial differential equations by using the classic Runge–Kutta fourth order integration approach in conjunction with the shot technique. Refs.39,40,41,42,43,44,45,46,47,48 indicates some latest and modern studies on fluid flow, materials analysis and different computational and numerical techniques. The numerical simulations are carried out for each of the many control parameters that have been selected. The following significant inferences may be made based on the previous numerical simulations.

Since the Lorentz forces become more powerful for higher Hartmann number, the velocity decreases.

An increase in the Forchheimer parameter causes a decrease in the velocity profile and an inclination in the temperature gradients.

The higher activation energy decreases the concentration profile significantly.

A rise in the value of the Schmidt number brings about a fall in the concentration profile.

An increase in the porosity parameter causes a reduction in the velocity of the fluid, as well as an increase in the thermal concentration gradients and the thickness of the boundary layer associated with it.

The concentration field becomes more intense due to the destructive chemical reaction, while the concentration field becomes less intense due to the constructive chemical response.

All the data are clearly available in the research work.

Stefan, J. "Versucheuber die scheinbare adhäsion," ¨Äkadenie deer Wissenschaften in Wien. Math. Naturw. 69, 713–721 (1874).

Google Scholar

Moore, D. F. A review of squeeze films. Wear 8(4), 245–263 (1965).

Article Google Scholar

Gupta, P. S. & Gupta, A. S. Squeezing flow between parallel plates. Wear 45(2), 177–185 (1977).

Article CAS Google Scholar

Duwairi, H. M., Tashtoush, B. & Damseh, R. A. On heat transfer effects of a viscous fluid squeezed and extruded between two parallel plates. Heat Mass Transf. 41(2), 112–117 (2004).

ADS Google Scholar

Verma, R. L. A numerical solution for squeezing flow between parallel channels. Wear 72(1), 89–95 (1981).

Article CAS Google Scholar

Mustafa, M., Hayat, T. & Obaidat, S. On heat and mass transfer in the unsteady squeezing flow between parallel plates. Meccanica 47(7), 1581–1589 (2012).

Article MathSciNet MATH Google Scholar

Hayat, T., Qayyum, A., Alsaadi, F., Awais, M. & Dobaie, A. M. Thermal radiation effects in squeezing flow of a Jeffery fluid. Eur. Phys. J. Plus 128(8), 85–91 (2013).

Article Google Scholar

Anandika, R., Puneeth, V. & Manjunatha, S. The Darcy–Frochheimer multilayer model of Casson nanofluid squeezed by Newtonian nanofluid under asymmetric slip conditions. Eur. Phys. J. Plus 137(12), 1–14 (2022).

Article Google Scholar

Puneeth, V., Khan, M. I., Jameel, M., Geudri, K. & Galal, A. M. The convective heat transfer analysis the Casson nanofluid jet flow under the influence of the movement of gyrotactic microorganisms. J. Indian Chem. Soc. 99(9), 100612 (2022).

Article CAS Google Scholar

Puneeth, V., Khan, M. I., Narayan, S. S., El-Zahar, E. R. & Geudri, K. The impact of the movement of the gyrotactic microorganisms on the heat and mass transfer characteristics Casson nanofluid. Waves Random Complex Media 1–24 (2022)

Puneeth, V., Manjunatha, S., Gireesha, B. J. & Gorla, R. S. R. Magneto convective flow of Casson nanofluid due to Stefan blowing in the presence of bio-active mixers. Proc. Inst. Mech. Eng. Part N J. Nanomater. Nanoeng. Nanosyst. 235(3–4), 83–95 (2021).

CAS Google Scholar

Hussain, M., Farooq, U. & Sheremet, M. Nonsimilar convective thermal transport analysis of EMHD stagnation Casson nanofluid flow subjected to particle shape factor and thermal radiations". Int. Commun. Heat Mass Transf. 134, 106230 (2022).

Article Google Scholar

Jamshed, W., Kumar, V. & Kumar, V. Computational examination of Casson nanofluid due to a non-linear stretching sheet subjected to particle shape factor: Tiwari and Das model. Numer. Methods Partial Differ. Equ. 38(4), 848–875 (2022).

Article MathSciNet Google Scholar

Upreti, H., Pandey, A. K., Uddin, Z. & Kumar, M. Thermophoresis and Brownian motion effects on 3D flow of Casson nanofluid consisting microorganisms over a Riga plate using PSO: A numerical study. Chin. J. Phys. 78, 234–270 (2022).

Article MathSciNet Google Scholar

Bestman, A. R. Natural convection boundary layer with suction and mass transfer in a porous medium. Int. J. Energy Res. 14(4), 389–396. https://doi.org/10.1002/er.4440140403 (1990).

Article CAS Google Scholar

Makinde, O. D., Olanrewaju, P. O. & Charles, W. M. Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture. Afr. Mat. 22(1), 65–78. https://doi.org/10.1007/s13370-011-0008-z (2011).

Article MathSciNet MATH Google Scholar

Alsaadi, F. E., Ullah, I., Hayat, T. & Alsaadi, F. E. Entropy generation in nonlinear mixed convective flow of nanofluid in porous space influenced by Arrhenius activation energy and thermal radiation. J. Therm. Anal. Calorim. 140, 799–809. https://doi.org/10.1007/s10973-019-08648-0 (2019).

Article CAS Google Scholar

Irfan, M., Khan, W. A., Khan, M. & Gulzar, M. M. Influence of Arrhenius activation energy in chemically reactive radiative flow of 3D Carreau nanofluid with nonlinear mixed convection. J. Phys. Chem. Solids 125, 141–152. https://doi.org/10.1016/j.jpcs.2018.10.016 (2019).

Article ADS CAS Google Scholar

Hsiao, K. L. To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-nanofluid with parameters control method. Energy 130, 486–499 (2017).

Article Google Scholar

Majeed, A. et al. Analysis of activation energy in magnetohydrodynamic flow with chemical reaction and second order momentum slip model. Case Stud. Therm. Eng. 12, 765–773 (2018).

Article Google Scholar

Khan, M. I., Hayat, T., Khan, M. I. & Alsaedi, A. Activation energy impact in nonlinear radiative stagnation point flow of Cross nanofluid. Int. Commun. Heat Mass Transf. 91, 216–224 (2018).

Article CAS Google Scholar

Dhlamini, M., Kameswaran, P. K., Sibanda, P., Motsa, S. & Mondal, H. Activation energy and binary chemical reaction effects in mixed convective nanofluid flow with convective boundary conditions. J. Comput. Des. Eng. 6, 149–158 (2019).

Google Scholar

Forchheimer, P. Wasserbewegung durch boden. Zeitschrift Ver. D. Ing. 45, 1782–1788 (1901).

Google Scholar

Muskat M. The flow of homogeneous fluids through porous media (Edwards, 1946).

Seddeek, M. A. Influence of viscous dissipation and thermophoresis on DarcyForchheimer mixed convection in a fluid saturated porous media. J. Colloid Interface Sci. 293, 137–142 (2006).

Article ADS CAS PubMed Google Scholar

Jha, B. K. & Kaurangini, M. L. Approximate analytical solutions for the nonlinear Brinkman-Forchheimer-extended Darcy flow model. Appl. Math. 2, 1432–1436 (2011).

Article Google Scholar

Pal, D. & Mondal, H. Hydromagnetic convective diffusion of species in DarcyForchheimer porous medium with non-uniform heat source/sink and variable viscosity. Int. Commun. Heat Mass Transfer 39, 913–917 (2012).

Article Google Scholar

Sadiq, M. A. & Hayat, T. Darcy–Forchheimer flow of magneto Maxwell liquid bounded by convectively heated sheet. Results Phys. 6, 884–890 (2016).

Article ADS Google Scholar

Shehzad, S. A., Abbasi, F. M., Hayat, T. & Alsaedi, A. Cattaneo–Christov heat flux model for Darcy–Forchheimer flow of an Oldroyd-B fluid with variable conductivity and nonlinear convection. J. Mol. Liq. 224, 274–278 (2016).

Article CAS Google Scholar

Bakar, S. A., Arifin, N. M., Nazar, R., Ali, F. M. & Pop, I. Forced convection boundary layer stagnation-point flow in Darcy–Forchheimer porous medium past a shrinking sheet. Front. Heat Mass Transf. 7, 38 (2016).

Google Scholar

Hayat, T., Muhammad, T., Al-Mezal, S. & Liao, S. J. Darcy–Forchheimer flow with variable thermal conductivity and Cattaneo–Christov heat flux. Int. J. Numer. Methods Heat Fluid Flow 26, 2355–2369 (2016).

Article Google Scholar

Umavathi, J. C., Ojjela, O. & Vajravelu, K. Numerical analysis of natural convective flow and heat transfer of nanofluids in a vertical rectangular duct using Darcy-ForchheimerBrinkman model. Int. J. Thermal Sci. 111, 511–524 (2017).

Article CAS Google Scholar

Hayat, T., Haider, F., Muhammad, T. & Alsaedi, A. On Darcy–Forchheimer flow of viscoelastic nanofluids: A comparative study. J. Mol. Liq. 233, 278–287 (2017).

Article CAS Google Scholar

Muhammad, T., Alsaedi, A., Shehzad, S. A. & Hayat, T. A revised model for DarcyForchheimer flow of Maxwell nanofluid subject to convective boundary condition. Chin. J. Phys. 55, 963–976 (2017).

Article CAS Google Scholar

Noor, N. A. M., Shafie, S. & Admon, M. A. Effects of viscous dissipation and chemical reaction on MHD squeezing flow of Casson nanofluid between parallel plates in a porous medium with slip boundary condition. Eur. Phys. J. Plus 135(10), 1–24 (2020).

Google Scholar

Noor, N. A. M., Shafie, S. & Admon, M. A. MHD squeezing flow of Casson nanofluid with chemical reaction, thermal radiation and heat Generation/Absorption. J. Adv. Res. Fluid Mech. Therm. Sci. 68(2), 94–111 (2020).

Article Google Scholar

Alqahtani, A. M. et al. Thermal analysis if a radiative nanofluid over a stretching/shrinking cylinder with viscous dissipation. Chem. Phys. Lett. 808, 140133 (2022).

Article CAS Google Scholar

Naduvinamani, N. B. & Shankar, U. Radiative squeezing flow of unsteady magneto-hydrodynamic Casson fluid between two parallel plates. J. Central South Univ. 26, 1184–1204 (2019).

Article CAS Google Scholar

Chu, Y. M. et al. Radiative thermal analysis for four types of hybrid nanoparticles subject to non-uniform heat source: Keller box numerical approach. Case Stud. Therm. Eng. 40, 102474 (2022).

Article Google Scholar

Javed, M., Qadeer, F., Imran, N., Kumam, P. & Sohail, M. Peristaltic mechanism of Ellis fluid in curved configuration with homogeneous and heterogeneous effects. Alex. Eng. J. 61, 10677–10688 (2022).

Article Google Scholar

Ahmed, M. F. et al. Numerical computation for gyrotactic microorganisms in MHD radiative Eyring–Powell nanomaterial flow by a static/moving wedge with Darcy–Forchheimer relation. Micromachines 13, 1768 (2022).

Article PubMed PubMed Central Google Scholar

Shahzad, A. et al. Thin film flow and heat transfer of Cu-nanofluids with slip and convective boundary condition over a stretching sheet. Sci. Rep. 12, 1–16 (2022).

Google Scholar

Abbasi, A. et al. Heat transport exploration for hybrid nanoparticle (Cu, Fe3O4)-based blood flow via tapered complex wavy curved channel with slip features. Micromachines 13, 1415 (2022).

Article PubMed PubMed Central Google Scholar

Sohail, M. et al. Modeling of three dimensional Prandtl hybrid nano-material over a heated rotating cone involving hall and ion slip currents via finite element procedure. Sci. Rep. 12, 1–12 (2022).

Article Google Scholar

Li, Z. et al. Analysis of surface pressure pulsation characteristics of centrifugal pump magnetic liquid sealing film. Front. Energy Res. 10, 937299 (2022).

Article Google Scholar

Waqas, H. et al. Gyrotactic motile microorganisms impact on pseudoplastic nanofluid flow over a moving riga surface with exponential heat flux. Crystals 12, 1308 (2022).

Article CAS Google Scholar

Qu, M. et al. Laboratory study and field application of amphiphilic molybdenum disulfide nanosheets for enhanced oil recovery. J. Pet. Sci. Eng. 208, 109695 (2022).

Article CAS Google Scholar

Wang, F. et al. An implication of magnetic dipole in Carreau Yasuda liquid influenced by engine oil using ternary hybrid nanomaterial. Nanotechnol. Rev. 11, 1620–1632 (2022).

Article CAS Google Scholar

Li, S., Khan, M. I., Alzahrani, F. & Eldin, S. M. Heat and mass transport analysis in radiative time dependent flow in the presence of Ohmic heating and chemical reaction, viscous dissipation: An entropy modeling. Case Stud. Therm. Eng. 42, 102722 (2023).

Article Google Scholar

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The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: (22UQU4400074DSR04).

School of Computer Science and Technology, Shandong Technology and Business University, Yantai, 264005, China

Shuguang Li

St John's College of Engineering and Technology, Yemmiganur, Kurnool District, Andhra Pradesh, 518360, India

Kodi Raghunath

College of Engineering, Industrial Engineering Department, Umm Al-Qura University, 28821, Al-Khalidiya District, Al-Qunfudhah City, Kingdom of Saudi Arabia

Ayman Alfaleh

Department of Mathematical Sciences, Federal Urdu University of Arts, Sciences & Technology, Gulshan-e-Iqbal Karachi, 7530, Pakistan

Farhan Ali & A. Zaib

Department of Mechanical Engineering, Lebanese American University, Beirut, Lebanon

M. Ijaz Khan

Department of Mathematics and Statistics, Riphah International University I-14, Islamabad, 44000, Pakistan

M. Ijaz Khan

Center of Research, Faculty of Engineering, Future University in Egypt, New Cairo, 11835, Egypt

Sayed M. ElDin

Department of Computational Science, CHRIST (Deemed to Be University), Ghaziabad, 201003, India

V. Puneeth

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All authors are equally contributed in the research work.

Correspondence to M. Ijaz Khan.

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Li, S., Raghunath, K., Alfaleh, A. et al. Effects of activation energy and chemical reaction on unsteady MHD dissipative Darcy–Forchheimer squeezed flow of Casson fluid over horizontal channel. Sci Rep 13, 2666 (2023). https://doi.org/10.1038/s41598-023-29702-w

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Received: 21 October 2022

Accepted: 09 February 2023

Published: 15 February 2023

DOI: https://doi.org/10.1038/s41598-023-29702-w

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Effects of activation energy and chemical reaction on unsteady MHD dissipative Darcy