Chemically reactive MHD fluid flow along with thermophoresis and Brownian effects

Abstract

In the current manuscript, the aim of the study is to analyze the Eyring-Powell nanofluid flow under the influence of chemical reaction and radiation effects in a slender cylinder in the presence of a non-linear heat source/sink. The flow of Eyring-Powell nanofluid through a slender cylinder along with chemical reaction and MHD effect is not studied yet. Which is the novelty of current research work. Flow analysis is taken near the stagnation point. MHD effect is considered for controlling turbulence. Rosseland model is applied for the estimation of heat flux. The governing system of PDE’S is converted into highly nonlinear ODE’S by utilizing suitable similarity transformations along with the boundary conditions. The resulting coupled systems of nonlinear ordinary differential equations (ODE), accompanied by boundary conditions, are solved using the powerful bvp4c method in MATLAB software. The physical significance of evolved parameters is investigated by graphs and tables. Nusselt number and skin friction are analyzed for several parameters. It is observed that the increase in the MHD effect rises the velocity but lessens the temperature profile and the concentration of the fluid declines when the chemical reaction parameter is incremented. The specific application of the study is that the Eyring-Powell is used in a variety of industrial applications, including lubrication, plastics processing, and oil drilling.

Keywords

Eyring-Powell fluid chemical reaction slender cylinder stagnation point flow MHD Rosseland model bvp4c

Introduction

Magneto hydrodynamic (MHD) contains a fluid, a magnet, and a field that collectively analyze the development of gadgets. MHD flow has several practical and hypothetical investigations due to their industrial applications. Because of several manufacturing applications like electric propulsion, aircraft formation, and cooling of nuclear reactors. MHD is a vigorous area of interest in biological engineering. Malik et al.¹ investigate that the blood behaves like an electric conductor containing hemoglobin in the iron molecules. In comparison to plate geometry, the magnitude of skin friction and the rate of heat transmission are much higher for cylindrical surfaces. Sheikholeslami,² examined the MHD two-phase nanofluid flow in a permeable medium, and numerical results were presented using a computer technique. Using CVFEM, a macroscopical simulation of the effects of magnetic forces on alumina convective flow in a permeable cavity is performed. Rehman et al.³ discussed the MHD stopping point flow of Nano-particles over a stretching surface. Many researchers have worked on MHD second-grade nanofluid flow via several geometries with boundary layer flows.^4–9

The colloidal suspension of particles with a size less than 100 nm is classified as nanofluids. In industries, nanofluids play an imperative role to enhance heat transfer rates. Thermal conductivity and heat transfer are widely used in the development of energy-saving devices. Regular fluids like oil and water with lower thermal conductivity are not more efficient. Fluids with higher heat transfer rates are needed to meet the requirement of the world. Sus¹⁰ analyzed nanoparticles magnifying the heat conduction rate. In manufacturing industries, the formation of microchips and in nuclear reactors nanofluids and chemical reactions in Casson fluid are extensively used to magnify the heat flux.^4,5,11,12 Nadeem et al.¹³ examined an innovative class of nanofluids and deduced that their structure may be tabular or rod-like. Parmar and Jain¹⁴ studied that cancer treatment can be done by the nanoparticles of gold rather than conventional drugs. Many researchers have worked on nanofluids.^15,16 Patil et al.¹⁷ studied the slipperation of thermal and flow speed impacts on the natural convective two-phase nanofluid model. The influence of Darcy law over nanofluids with Lorentz forces in a porous medium was evaluated by Sheikholeslami et al.¹⁸ Rashidi et al.¹⁶ studied the combined effects of nanoparticles with base fluids to enhance heat conduction. The combined thermal system’s thermal efficiency is significantly impacted by the geometrical characteristics of coil wires.

Heat transfer and chemical reaction collectively play a significant role in many processes. Such kinds of problems have extensive applications like as the destruction of crops because of freezing, hot rolling, polymer industries, and swirling of fibers. Kumar et al.¹⁹ analyzed the micropolar fluid flow over a surface with a chemical reaction in a porous medium. The chemical reaction attribute has a proclivity to manage the concentration profile was investigated by Reddy et al.²⁰ MHD (second order) incompressible fluid flow on a sheet with chemical reaction was analyzed by Khan et al.²¹

The flows near the stagnation point are referred to as stagnation point flows. In the present era, stopping point flows have attained the interest of researchers due to significant rules in industrial areas. Mehmood et al.²² investigated the Casson fluid’s stopping point flow across a stretching body. Rehman et al.²³ analyzed the stopping point flow of a second-grade fluid across an exponentially stretching surface. Rehman et al.²⁴ studied the stopping point flow of nanofluids suspended in a porous medium. Nadeem et al.²⁵ analyzed the oscillating stopping flow of nanofluids for different geometries. Specific practical applications of this study are that it is commonly used to model the behavior of certain materials, such as polymers, molten plastics, and lubricants. It is a viscoplastic fluid, which means that it exhibits both solid-like and liquid-like behavior depending on the applied stress or strain rate. The Eyring-Powell fluid exhibits a yield stress, which means that it behaves like a solid until the applied stress exceeds the yield stress. Once the yield stress is exceeded, the material begins to flow, and its viscosity decreases with increasing shear rate. The plastic viscosity of the Eyring-Powell fluid is related to the activation energy required to initiate flow, and it can be used to predict the flow behavior of the material under different conditions. The Eyring-Powell fluid is used in a variety of industrial applications, including lubrication, plastics processing, and oil drilling. Its ability to model the behavior of complex fluids has made it a valuable tool in materials science and engineering.

It is a known fact that non-Newtonian fluids conferred a distinctive task to scientists and engineers. Blood, jelly, ketchup, etc. are non-Newtonian fluids. Abdelsalam et al.²⁶ analyzed that non-Newtonian fluids are more complex in various stages than physical fluids. Blood comprises many suspensions including, proteins, nutrients, leukocytes, gases, etc. Non-Newtonian fluids can’t be studied by linear relations due to their nonlinear behavior. Various models have been proposed to analyze these fluids, like the Eyring-Powell fluid model, Casson, and Jeffery fluid model. Various scientists examined these fluid models.^27–29 Khan et al.^30–32 examined how a second-grade fluid flowed in a thin film across a stretching sheet. Powel and Eyring first proposed the Powel-Eyring fluid model in 1944. Powel-Eyring fluids have a significant role in controlling environmental pollution, formulation of Anti-fog devices, etc. Nadeem and Rehman³³ considered the Powel-Eyring nanofluid flow in a moving cylinder. Sher Akbar et al.³⁴ examined MHD Powel-Eyring fluid and concluded for higher intensity resistance to flow increases. Salawu et al.³⁵ examined the nonlinear solar thermal radiation optimization for hybrid Prandtl–Eyring nano liquid in aircraft. Humane et al.³⁶ examined the magneto-micropolar fluid’s thermal convection using a porous stretching apparatus. Shamshuddin and Prakash Sharma³⁷ studied the thermal elaboration of ethylene glycol-based nanostructures via a heated vertical surface. In this particular study, the rheological aspects of magneto Nano liquid movement in a uniformly vertically porous sheet are investigated.

Literature shows that MHD non-Newtonian flow with chemical reaction and the nonlinear source is not studied yet. The purpose of this analysis is to investigate the MHD and chemical effects on Powell-Eyring fluid in the presence of a non-linear source\sink.

Mathematical problem formulation

We analyze the stagnation point flow of Eyring-Powell flow in a slender cylinder. The velocity is assumed to be independent of the circumferential component but depends upon $x and r$ , where the components and $r$ are considered to be along the boundary and radial direction. Chemical reaction and magnetic effects are considered for the flow along with the heat source. The resulting problem takes the form³⁸:

{(r w)}_{r} + r u_{x} = 0,

(1)

\begin{array}{l} u u_{x} + w u_{r} = U \frac{d U}{d x} + υ (1 + M_{1}) (u_{r r} + \frac{1}{r} u_{r}) \\ - \frac{2}{3 ρ β c^{3}} [\begin{matrix} \frac{w^{2}}{r^{2}} u_{r r} + u_{r} (\frac{2}{r^{2}} w w_{r} - \frac{w}{r^{3}} + u_{x} u_{x r} - \frac{1}{r} w u_{x r} \\ - w_{r r} u_{x} - w_{r} u_{x r} - \frac{1}{r} w_{r} u_{x}) - w_{r} u_{r r} u_{x} \end{matrix}] \\ + \frac{(ρ_{2}^{*} - ρ_{2}) (φ_{2} - φ_{\infty})}{ρ_{2}} - \frac{σ B_{°}^{2}}{ρ} u \end{array}

(2)

\begin{matrix} w T_{r} + u T_{x} = α (T_{rr} + \frac{1}{r} T_{r}) + \frac{υ}{c_{p}} (1 + M_{1}) (u_{r})^{2} \\ - \frac{4 γ M_{1}}{3 c_{p} c^{2}} (u_{r})^{2} [\frac{w^{2}}{r^{2}} - w_{r} u_{x}] + ρ_{2}^{*} c_{p}^{*} [D_{B} (φ_{r} T_{r} + φ_{x} T_{x}) \\ + \frac{D_{T}}{T_{\infty}} (T_{rr} + T_{xx})] - \frac{1}{r} \frac{\partial}{\partial r} [r q_{f}] + q^{*} = 0, \end{matrix}

(3)

w φ_{r} + u φ_{x} = D_{B} (φ_{rr} + \frac{1}{r} φ_{r}) + \frac{D_{T}}{T_{\infty}} (T_{rr} + T_{xx}) - R_{r} (φ - φ_{\infty}) .

(4)

Where the Rosseland model radiative heat flux $q_{f}$ is provided by,

q_{f} = \frac{- 4 σ_{1}^{*}}{3 k_{1}^{*}} T_{r}^{4} .

(5)

$T^{4}$ can be written as with the help of the Taylor series,

T^{4} = 4 T_{\infty}^{3} T - 3 T_{\infty}^{4} .

So $q_{f}$ is,

q_{f} = \frac{- 16 σ_{1}^{*}}{3 k_{1}^{*}} T_{r} .

(6)

Here $w$ and $u$ are considered as the velocity components along the $z$ and $x$ direction respectively, pressure is $p$ , g is gravitational acceleration, Eyring-Powel parameter is $M_{1}$ , $ρ$ denotes the density, T is temperature, $γ$ is curvature, $q^{*}$ is heat source, kinematic viscosity is $υ$ , $β^{*}$ is thermal expansion coefficient, $c_{p}$ represent specific heat using constant pressure, $U_{\infty}$ is the velocity at the surface, free stream velocity is defined as

U = U_{\infty} (\frac{x}{l}) .

The boundary conditions are,

u (x, a) \to U (x), u (x, a) = 0 for r \to \infty,

(7)

T (x, a) \to T_{\infty}, T (x, a) = T_{w} (x) as r \to \infty .

(8)

Geometry of the problem

Two-dimension, incompressible, and steady-state Eyring-Powell flow in the presence of a boundary layer approximation theory is studied (Figure 1).

Figure 1.

Flow geometry.

$w$ and $u$ are the velocity components. Flow is along r-direction and x is considered perpendicular to it. $U_{w}, T_{w}$ , and $φ_{w}$ are surface velocity, temperature, and concentration respectively. $U_{\infty}, T_{\infty}$ , and $φ_{\infty}$ are ambient fluid velocity, temperature, and concentration.

Similarity transformation

Utilizing appropriate similarity transformations³⁹:

u = \frac{x U_{\infty}}{l} f^{'} (η), w = \frac{- a}{r} (\frac{U_{\infty}}{l}) f (η),

(9)

θ = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, η = \frac{r^{2} - a^{2}}{2 a} (\frac{U_{\infty}}{υ l}) .

(10)

The PDE’s (1)–(4) are transformed into an ODE’s using transformations (9) and (10) as³:

\begin{matrix} - (1 + 2 γ η) f ″ + 2 γ f ″ + \frac{1}{1 + M_{1}} (1 + ff ″ - {f'}^{2}) \\ + \frac{K M_{1} γ}{1 + M_{1}} [(ff' f ″' - {f'}^{2} f ″) - H_{a} f^{'} - γ (1 + 2 γ η) (3 f {f ″}^{2} + {f'}^{2} f ″') \\ - \frac{γ}{(1 + 2 γ η)} (ff' f ″ + f^{2} f ″') + \frac{2 γ^{2}}{{(1 + 2 γ η)}^{2}} f^{2} f ″] + λ θ = 0, \end{matrix}

(11)

\begin{matrix} (1 + R_{d}) (1 + 2 γ η) θ ″ + 2 (1 + R_{d}) γ θ' + \Pr (f θ' - f' θ) \\ + (1 + 2 γ η) B_{P} θ' ψ' + T_{P} (1 + 2 γ η) {θ'}^{2} + (1 + M_{2}) \\ \Pr Ec (1 + 2 γ η) {f ″}^{2} + 2 K M_{1} \Pr Ec [γ ff' f ″ - (1 + 2 γ η) {f'}^{2} f ″ \\ - \frac{γ^{2}}{(1 + 2 γ η)} {f'}^{2} f ″] + (P^{*} f' + Q^{*} θ) = 0, \end{matrix}

(12)

\begin{matrix} (1 + 2 γ η) ψ ″ + 2 γ ψ' + Le \Pr (f ψ' - f' ψ) \\ + \frac{T_{P}}{B_{P}} ([(1 + 2 γ η) θ ″ + 2 γ θ'] - Q_{°} ψ = 0 . \end{matrix}

(13)

Where curvature parameter is $γ = (\frac{υ l}{U_{\infty} a^{2}})$ , the Buoyancy parameter $λ = \frac{g β^{*} Δ T x}{U_{\infty}^{2}}$ , $\Pr = \frac{υ}{α}$ is the Prandtl number, Eckert number $Ec = \frac{U_{\infty}^{2}}{c_{p} Δ T}$ , $K$ and $M_{1}$ represents Eyring-Powel parameters $K = \frac{2 U_{\infty}^{2}}{3 c^{2} l^{2}}, M_{1} = \frac{1}{β μ c}$ , $B_{P} = \frac{ρ^{*} c_{p}^{*} D_{B} (φ_{2} - φ_{\infty})}{ρ c_{p} α}$ is the Brownian parameter for motion, $T_{P} = \frac{ρ^{*} c_{p}^{*} D_{T} (T_{w} - T_{\infty})}{ρ c_{p} α T_{\infty}}$ is the thermophoresis parameter, and $Le = \frac{α}{D_{B}}$ is Lewis number.

Dimensionless boundary conditions are as

f (0) = 1, f' (0) = 0, f' \to 1, as η \to \infty,

(14)

θ (0) = 1, θ \to 0, as η \to \infty,

(15)

ψ (0) = 1, ψ \to 0, as η \to \infty .

(16)

$τ_{w}$ is stress over the cylinder, Nusselt number is $N_{u}$ , $q_{w}$ is heat flux, and skin friction is $_{C_{f}}$ . These are the basic physical quantities that need to be examined.

\frac{N_{u}}{{Re}_{x}^{\frac{1}{2}}} = - θ' (0), τ_{w} = (τ_{rx})_{r \to a}, q_{w} = - k {(\frac{\partial T}{\partial y})}_{y = 0},

(17)

\frac{1}{2} C_{f} {Re}_{x}^{\frac{1}{2}} = (1 + M_{1}) f ″ (0) - \frac{1}{3} M_{1} λ {f ″}^{3} (0) .

(18)

In $x$ -direction $τ_{rx}$ is the component of stress, ${Re}_{x}$ is the Reynold number.

Numerical procedure

Equations (11)–(13) with conditions (14–16) are highly nonlinear and be numerically solved by the faithful technique bvp4c of MATLAB. Regardless, we convert the administering issue into an arrangement in the governing system is substituted as follows:

f (η) = y_{1}, f' = y_{2}, f ″ = y_{3}, f ″' = y'_{3},

(19)

θ (η) = y_{4}, θ' = y_{5}, θ ″ = y'_{5},

(20)

ψ (η) = y_{6}, ψ' = y_{7}, ψ ″ = y'_{7} .

(21)

Hence the equivalent first-order system is given below:

y'_{3} = \frac{\begin{matrix} [2 c y_{3} + \frac{1}{1 + M_{1}} (1 + y_{1} y_{2} - y_{2}^{2}) - \frac{K M_{1} C (y_{2}^{2} . y_{3})}{1 + M_{1}} - \frac{3 K M_{1} C^{2}}{1 + M_{1}} (1 + 2 C η) y_{1} y_{3}^{2} \\ - \frac{K M_{1} C^{2}}{(1 + M_{1}) (1 + 2 C η)} (y_{1} y_{2} y_{3}) + \frac{2 K M_{1} C^{3}}{(1 + M_{1}) {(1 + 2 C η)}^{2}} y_{3} y_{1}^{2} + λ_{n} (1 - φ_{\infty}) (y_{4} + N_{r} y_{6}] - H_{a} y_{2} \end{matrix}}{[(1 + 2 C η) - \frac{K M_{1} C (y_{1} . y_{2})}{1 + M_{1}} + \frac{K M_{1} C^{2} (1 + 2 C η)}{(1 + M_{1})} (y_{2}^{2}) + \frac{K M_{1} C^{2}}{(1 + M_{1}) (1 + 2 C η)}]}

(22)

y'_{5} = \frac{\begin{matrix} [- 2 C (1 + R_{d}) y_{5} - P_{r} (y_{1} y_{5} - y_{2} y_{4}) - (1 + 2 C η) B_{P} y_{5} y_{7} - T_{P} (1 + 2 C η) y_{5}^{2} - (1 + M_{1}) \\ (P_{r} E_{c}) (1 + 2 C η) y_{3}^{2} - 2 K M_{1} P_{r} E_{c} ((C y_{1} y_{2} y_{3}) + (1 + 2 C η) y_{2}^{2} y_{3} + \frac{C^{2}}{(1 + 2 C η)} y_{2}^{2} y_{3})] + P^{*} y_{2} + Q^{*} y_{4} \end{matrix}}{(1 + R_{d}) (1 + 2 C η)}

(23)

y'_{7} = \frac{[- 2 C y_{7} + Le \Pr (y_{1} y_{7} - y_{2} y_{6}) + \frac{T_{P}}{B_{P}} ((1 + 2 C η) {y'}_{5} + 2 C y_{5})) - Q_{°} y_{6}]}{(1 + 2 C η)}

(24)

The following section presents the solution’s numerical results.

Numerical results and discussion

The major goal is to examine the effects of various parameters on the profiles of fluid velocity, temperature, and concentration. Graphs and tables show how these parameters effects on fluid flow.

Velocity profile

The impact of several parameters on the flow is shown in Figures 2 to 5. The influence of $M_{1}$ on $F' (η)$ is presented in Figure 2. By raising the fluid parameter $M_{1}$ , the fluid’s velocity increases. It is observed that by enhancing the fluid parameter $M_{1}$ viscosity of fluid declined which causes the rise in velocity. The impact of curvature on velocity is presented in Figure 3, which shows that with the rise in curvature velocity increases. Because enhancement of curvature lessens the radius which results in to rise in velocity. From Figure 4 it is believed that raising the buoyancy parameter will increase the fluid’s mass density of nanoparticles while lowering its velocity. Figure 5 shows a fluid velocity together with a rise in the Prandtl number and a boundary layer thickness decline, resulting in the decrease in fluid velocity.

Figure 2.

Fluid parameter effects on velocity.

Figure 3.

Curvature’s effect on velocity.

Figure 4.

Buoyancy parameter effects on velocity.

Figure 5.

Prandtl number effects on velocity.

Temperature profile

Figures 6 to 9 shows how several parameters affect temperature. The thickness of the thermal boundary layer and temperature profile both decrease as the Prandtl number increases, as seen in Figure 6. This demonstrates the concept that as thermal diffusivity increases, less energy can be transferred, which results in a decrease in the thermal boundary layer and temperature. In Figure 7, which is a sketched analysis of the consequence of curvature on temperature profile, it is shown that as curvature increases, heat conduction rate decreases, which increases temperature behavior. As the buoyancy parameter increases, the temperature rises as shown in Figure 8. Temperature increases with the increase in $Nr$ presented in Figure 9.

Figure 6.

Temperature profile for several values of $P_{r}$ .

Figure 7.

Temperature profile for several values of $γ$ .

Figure 8.

Temperature profile for several value of $λ$ .

Figure 9.

Temperature profile for several values of $Nr$ .

Concentration profile

Figures 10 to 13 are sketched to examine the influence of several parameters over the concentration. Figure 10 shows by enhancing the thermophoresis parameter concentration increase because the particles of nanofluid travels far away from the hotter surface to the cold. A thermophoretic force causes the nanoparticles to move toward areas of greater or lower temperature when there is a temperature disparity in the nanofluid. Prandtl number increment raises the concentration of the nanofluid is depicted in Figure 11.

Figure 10.

Concentration profile for several value of $Tp$ .

Figure 11.

Concentration profile for several value of Pr.

Figure 12.

Concentration profile for several value of $B_{P}$ .

Figure 13.

Concentration profile for several value of $Le$ .

Figure 12 is a piece of good evidence that increase $Bp$ decreases the concentration. Nanoparticle collisions and random motion are enhanced with an increase in the Brownian motion parameter, which lowers fluid concentration. Figure 13 indicates by increasing the Lewis number concentration decreases, because increasing $Le$ mass diffusivity declines and reduces the concentration.

Impact of MHD

Figure 14 analyzes the influence of magnetic parameter over the velocity profile and numerical results shows the increase in velocity with the rise in magnetic parameter. MHD increases the movements of nanoparticles which causes in rise of Lorentz forces and resulting the enhancement of velocity. Figure 15 shows the impact of MHD over temperature profile and depicts that with increase in magnetic parameter temperature decreases.

Figure 14.

MHD’s influence over velocity profile.

Figure 15.

MHD’s influence over temperature profile.

Table 1 is formulated to compare the results of previous researches for several values of $P_{r}$ when other parameters are taken to be zero. Nusselt number and the Skin friction are calculated for some parameters as shown in Table 2.

Table 1.

Comparison of numerical approximations $- θ' (0)$ with several values of $\Pr$ .

PrGrubka and Bobba⁴⁰		Chen⁴¹	Present
0.75	1.0775	1.07753	1.0775331
1.25	1.4444	1.44443	1.4444445
3.5	2.4087	2.40872	2.4087281
10.5	4.7879	4.78796	4.7879627

Table 2.

Numerical values for Nusselt number against several parameters.

$Rd$	$\Pr$	$γ$	$R \overset{- \frac{1}{2}}{e_{x}} N u_{x}$
0.25	1.5	0.2	0.533107
0.75			0.773319
1.25			1.020951
2.0			1.436571
0.25	1.6		0.532107
	1.8		0.775505
	2.0		0.833413
	1.5		0.578143
		0.4	0.965525
		0.8	0.346056
		1.0	0.385449
		0.2	0.388200

Concluding remarks

We analyzed the numerical results by considering MHD, non-linear heat source/sink, and chemical reaction. Flow analysis is made through a vertical cylinder. The following results are worth citing:

Increase in MHD effect rises the velocity but lessens the temperature profile because when the MHD effect is increased, it accelerates the fluid motion, leading to higher velocities. At the same time, the dissipation of kinetic energy caused by the Lorentz forces reduces the fluid temperature, resulting in a lower temperature profile.

Temperature profile increases or decreases with the positive or negative values $λ$ accordingly.

Concentration of the fluid declines when the chemical reaction parameter is incremented because as the reaction rate increases, the consumption of reactants becomes more efficient, causing their concentrations to decrease.

Skin friction should be lowered when the unsteady parameter is incremented. The magnitude of skin friction is influenced by various factors, including the flow velocity, viscosity of the fluid, and the characteristics of the boundary layer.

Nusselt number quickly mounted by enlarging the unsteady parameter which results in maximum heat transfer rate.

Footnotes

Appendix

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Saquib Ul Zaman

Azad Hussain

References

Malik

Bilal

Bibi

. Numerical analysis for MHD thermal and solutal stratified stagnation point flow of Powell-Eyring fluid induced by cylindrical surface with dual convection and heat generation effects. Results Phys 2017; 7: 482–492.

Sheikholeslami

. Application of Darcy law for nanofluid flow in a porous cavity under the impact of Lorentz forces. J Mol Liq 2018; 266: 495–503.

Rehman

Achakzia

Nadeem

, et al. Stagnation point flow of Eyring Powell fluid in a vertical cylinder with heat transfer. J Power Technol 2016; 96: 57.

Ibrahim

Shankar

Nandeppanavar

. MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet. Int J Heat Mass Transf 2013; 56: 1–9.

Ellahi

. The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions. Appl Math Model 2013; 37: 1451–1467.

Zuhra

Khan

Islam

. Magnetohydrodynamic second-grade nanofluid flow containing nanoparticles and gyrotactic microorganisms. Comput Appl Math 2018; 37: 6332–6358.

Khan

. Bioconvection in second grade nanofluid flow containing nanoparticles and gyrotactic microorganisms. Braz J Phys 2018; 48: 227–241.

Palwasha

Khan

Shah

, et al. Study of two-dimensional boundary layer thin film fluid flow with variable thermo-physical properties in three dimensions space. AIP Adv 2018; 8: 105318.

Khan

. Mixed convection in MHD second grade nanofluid flow through a porous medium containing nanoparticles and gyrotactic microorganisms with chemical reaction. Filomat 2019; 33: 4627–4653.

10.

Sus

. Enhancing thermal conductivity of fluids with nanoparticles, developments and applications of non-Newtonian flows. ASME, FED, MD 1995; 231: 99–105.

11.

Patil

Shamshuddin

, et al. Eyring-Powell nano liquid flow through permeable elongated sheet conveying inclined magnetic field subject to constructive chemical reaction and multiple slip effects. Int J Model Simul 2023; 43: 533–548.

12.

Shamshuddin

Mebarek-Oudina

Salawu

, et al. Thermophoretic movement transport of reactive Casson nanofluid on Riga plate surface with nonlinear thermal radiation and uneven heat sink/source. J Nanofluids 2022; 11: 833–844.

13.

Nadeem

Ahmad

Muhammad

. Computational study of Falkner-Skan problem for a static and moving wedge. Sens Actuators B Chem 2018; 263: 69–76.

14.

Parmar

Jain

. MHD Powell–Eyring fluid flow with non-linear radiation and variable thermal conductivity over a permeable cylinder. Int J Heat Technol 2018; 36: 56–64.

15.

Sheikholeslami

Seyednezhad

. Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM. Int J Heat Mass Transf 2018; 120: 772–781.

16.

Rashidi

Eskandarian

Mahian

, et al. Combination of nanofluid and inserts for heat transfer enhancement: gaps and challenges. J Therm Anal Calorim 2019; 135: 437–460.

17.

Patil

Shamshuddin

Ramesh

, et al. Slipperation of thermal and flow speed impacts on natural convective two-phase nanofluid model across Riga surface: computational scrutinization. Int Commun Heat Mass Transf 2022; 135: 106135.

18.

Sheikholeslami

Shafee

. Lorentz forces effect on NEPCM heat transfer during solidification in a porous energy storage system. Int J Heat Mass Transf 2018; 127: 665–674.

19.

Kumar

Sandeep

Kumar

, et al. A comparative study of chemically reacting 2D flow of Casson and Maxwell fluids. Alex Eng J 2018; 57: 2027–2034.

20.

Reddy

SRR

Bala Anki Reddy

Bhattacharyya

. Effect of nonlinear thermal radiation on 3D magneto slip flow of Eyring-Powell nanofluid flow over a slendering sheet with binary chemical reaction and Arrhenius activation energy. Adv Powder Technol 2019; 30: 3203–3213.

21.

Khan

Malik

, et al. Mixed convection flow of Eyring-Powell nanofluid over a cone and plate with chemical reactive species. Results Phys 2017; 7: 3716–3722.

22.

Mehmood

Rana

Akbar

, et al. Non-aligned stagnation point flow of radiating Casson fluid over a stretching surface. Alex Eng J 2018; 57: 939–946.

23.

Rehman

Ghulam

Israr

, et al. Boundary layer stagnation-point flow of second grade fluid over an exponentially stretching sheet. Am J Appl Math Stat 2015; 3: 211–219.

24.

Rehman

Nadeem

Malik

. Stagnation flow of couple stress nanofluid over an exponentially stretching sheet through a porous medium. Power Tech 2013; 93: 122–132.

25.

Nadeem

Khan

Saleem

. A comparative analysis on different nanofluid models for the oscillatory stagnation point flow. Eur Phys J Plus 2016; 131: 1–14.

26.

Abdelsalam

Bhatti

Zeeshan

, et al. Metachronal propulsion of a magnetised particle-fluid suspension in a ciliated channel with heat and mass transfer. Phys Scr 2019; 94: 115301.

27.

Liu

Zheng

Lin

, et al. Anomalous diffusion in rotating Casson fluid through a porous medium. Physica A 2019; 528: 121431.

28.

Aghighi

Ammar

Metivier

, et al. Rayleigh-Bénard convection of Casson fluids. Int J Therm Sci 2018; 127: 79–90.

29.

Nawaz

Naz

Awais

. Magnetohydrodynamic axisymmetric flow of Casson fluid with variable thermal conductivity and free stream. Alex Eng J 2018; 57: 2043–2050.

30.

Khan

Islam

Gul

, et al. Thin film flow of a second grade fluid in a porous medium past a stretching sheet with heat transfer. Alex Eng J 2018; 57: 1019–1031.

31.

Khan

Gul

Islam

, et al. Thermophoresis and thermal radiation with heat and mass transfer in a magnetohydrodynamic thin-film second-grade fluid of variable properties past a stretching sheet. Eur Phys J Plus 2017; 132: 20.

32.

Khan

Gul

Khan

, et al. Mixed convection in gravity-driven thin film non-Newtonian nanofluids flow with gyrotactic microorganisms. Results Phys 2017; 7: 4033–4049.

33.

Nadeem

Rehman

. Axisymmetric stagnation flow of a nanofluid in a moving cylinder. Comput Math Model 2013; 24: 293–306.

34.

Sher Akbar

Ebaid

Khan

. Numerical analysis of magnetic field effects on Eyring-Powell fluid flow towards a stretching sheet. J Magn Magn Mater 2015; 382: 355–358.

35.

Salawu

Obalalu

Shamshuddin

. Nonlinear solar thermal radiation efficiency and energy optimization for magnetized hybrid Prandtl–Eyring nanoliquid in aircraft. Arab J Sci Eng 2023; 48: 3061–3072.

36.

Humane

Patil

, et al. Buongiorno modelled nanoliquid consequence of thermal and solutal convection on the magneto-micropolar fluid inside an inclined porous stretching device. J Nanofluids 2023; 12: 211–222.

37.

Shamshuddin

Prakash Sharma

. Thermal elaboration of ethylene glycol-based magnetized nanostructures via a convective permeable heated vertical surface employing modified Buongiorno model. J Magn Magn Mater 2023; 571: 170588.

38.

Ashraf

Siddique

Hussain

. Impact of thermophoresis and Brownian motion on non-Newtonian nanofluid flow with viscous dissipation near stagnation point. Phys Scr 2020; 95: 055217.

39.

Ishak

Nazar

Pop

. The effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder. Int J Non Linear Mech 2007; 42: 1010–1017.

40.

Grubka

Bobba

. Heat transfer characteristics of a continuous stretching surface with variable temperature. J Heat Trans 1985; 107(1): 248–250.

41.

Chen

. Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat and Mass Trans 1998; 33: 471–476.