Significance of Cu-Fe₃O₄ on fractional Maxwell fluid flow over a cone with Newtonian heating

Abstract

The goal of this research is to investigate fractional Maxwell hybrid nanofluids utilizing partial differential equations in terms of Caputo time fractional derivatives. Specifically, the effect of Newtonian heating on the thermal performance of a fractional Maxwell hybrid nanofluid moving over a permeable cone in the presence of thermal radiation and heat generation is considered. The Crank–Nicolson method and L1 algorithmt of Caputo derivative are used to find numerical solutions to the considered nonlinear problem. The effects of significant flow factors on fluid properties are examined and illustrated in various graphs. According to the results, the thermal performance of the fluid raised by 0.4%, 6.1%, and 3.1% on adding 4% volume fraction of ${\mathrm{F}\mathrm{e}}_3{\mathrm{O}}_4$, $\mathrm{C}\mathrm{u}$, and ${\mathrm{C}\mathrm{u}-\mathrm{F}\mathrm{e}}_3{\mathrm{O}}_4$, respectively, in the base fluid.

Keywords:

• Nanofluid
• Maxwell fluid model
• fractional derivative
• finite difference method

Nomenclature

Roman letters

$(u,v)$	=	velocity components (ms${}^{-1}$)
${\mathcal{C}}_1-{\mathcal{C}}_5$	=	nanofluid constants
$C_f$	=	Skin friction
$C_p$	=	specific heat capacity (Jkg${}^{-1}$K${}^{-1}$)
g	=	gravitational acceleration (ms${}^{-2}$)
Gr	=	Grashof number
K	=	non-dimensional porosity parameter
k	=	thermal conductivity (Wm${}^{-1}$K${}^{-1}$)
$k_0$	=	permiability of porous medium (m${}^2$)
$k_b$	=	absorption parameter
L	=	reference length (m)
$N{u}_x$	=	Nusselt number
Q	=	non-dimensional heat generation/absorption parameter
$Q_0$	=	heat generation/absorption parameter (Jm${}^{-3}$K${}^{-1}$s${}^{-1}$)
$q_w$	=	heat flux at the surface of the cone (Wm${}^{-2}$)
r	=	radius of the cone
Rd	=	non-dimensional thermal radiation
Ri	=	Richardson number
T	=	temperature (K)
t	=	time (s)
Pr	=	Prandtl number

Greek letters

α	=	fractional order
${\text{β}}_T$	=	volumetric thermal expansion (K${}^{-1}$)
$\mathrm{\Delta }t$	=	time step
$\mathrm{\Delta }x$	=	grid size in x direction
$\mathrm{\Delta }y$	=	grid size in y direction
λ	=	relaxation time (s)
µ	=	dynamic viscosity (kgm${}^{-1}$s${}^{-1}$)
ν	=	kinematic viscosity (m${}^2$s${}^{-1}$)
ρ	=	density (kgm${}^{-3}$)
σ	=	electrical conductivity (Sm${}^{-1}$)
${\sigma }_b$	=	Stefan Boltzman coefficient (Wm${}^{-2}$K${}^{-4}$)
φ	=	nanoparticles volume fraction

Subscripts/Superscripts

$\ast$	=	non-dimensional
f	=	base fluid
hnf	=	hybrid nanofluid
i	=	grid point in x direction
j	=	grid point in y direction
k	=	time level
nf	=	nanofluid
p	=	nanoparticles

1. Introduction

Viscoelastic fluids are essential in every aspect of our lives. The Maxwell model is a classical model used to describe the viscoelastic behavior of materials, such as polymers and fluids, by combining a spring and a dashpot in series. The spring represents the elastic behavior of the material, while the dashpot represents the viscous behavior. In the Maxwell model, the spring and dashpot are both assumed to be characterized by a single relaxation time. Several studies have examined the behavior of Maxwell fluid in various contexts; just a few will be described here. The procedure of flooding the leftover oil in the dead end with viscoelastic fluid was studied by [1]. In this work, the conventional momentum equation is solved by using the fractional-order Maxwell. Additionally, the flow control equation for fractional-order viscoelastic fluids is given a semi-analytical solution. The findings demonstrate that velocity fractional-order derivative considerably influences the properties of polymer solutions and the fluid elasticity can significantly boost the effectiveness of oil repellency. Hanif [2,3] applied the Friedrich shear stress and Cattaneo heat flow model to the traditional upper convected Maxwell fluid model to account for nonlinear convection effects. The governing equations of the fractional Maxwell fluid are then solved under steady-state conditions, and the results are presented graphically and described in detail. According to the results of the study, the skin friction increases with an increase in the velocity relaxation time parameter, but the opposite effect is observed for the velocity fractional derivative parameter. This suggests that the fractional derivative parameter has a more significant effect on the flow behavior than the relaxation time parameter in this particular system. Ramesh et al. [4] three-dimensional flow of Maxwell nanofluid in a porous region limited by a bidirectional stretched sheet with non-linear thermal radiation and a heat source/sink. Convective conditions are established for temperature and nanoparticle concentration. It is mentioned that Brownian motion plays an important influence on temperature and heat transmission rate, and when the Deborah number increases, the concentration of nanoparticles and temperature drop. Ramesh and Gireesha [5] studied Maxwell fluid over a stretching sheet with convective boundary conditions in the presence of heat generation/absorption. Wang et al. [6] studied a generalized Maxwell fluid model of helical flow between two irrational coaxial circular cylinders. Cui et al. [7] investigated the petrophysical model for a field seismic survey. It was discovered that there are considerable differences in the characteristics of wave field propagation between regular wave equations and elastic wave equations in porous. The obtained results are useful to other related fields, such as seismic exploration of dense oil reservoirs, and micro seismic simulation in cracked reservoirs filled with fracturing fluid. Parvin et al. [8] presented a 2D-double diffusive layer flow model of Maxwell fluid in the presence of a magnetic field. Ferrás et al. [9] presented an interesting work by using distributed-order derivatives to develop a generalized viscoelastic model. In the paper referenced by [10], the authors used the Helmholtz eigenvalue problem and Laplace transforms to analytically solve the linear momentum and Poisson-Boltzmann equations in tandem in a triangular region to study the unsteady electroosmotic flow of generalized Maxwell fluid. The momentum equation using the fractional Maxwell model is formulated to evaluate unsteady stagnation-point flow based on the properties of pressure in the boundary layer by Bai et al. [11]. The newly designed finite difference scheme and L1 algorithm are used to solve the nonlinear fractional differential equations, and the convergence of the system is confirmed by creating a numerical example. Maxwell fluid performs the fastest slip transport under the isothermal condition, this was discussed by Anwar et al. [12]. They considered Caputo-Fabrizio and Atangana-Baleanu fractional derivatives to describe slip flow, shear stress, and heat transfer processes. To predict solutions of the momentum and heat equations in primary coordinates, the Laplace transformation is first applied to dimensionless fractional models and then Stehfest's numerical approach is used. Furthermore, the Laplace inversion algorithms developed by Durbin and Zakian serve to validate the computed solutions for the velocity and energy fields. Moosavi et al. [13] used the Maxwell fractional model to analyze viscoelastic non-Newtonian fluid natural convection heat flow on a vertical forward-facing step.

There are numerous industrial and engineering processes where the heat transfer phenomenon occurs. According to this framework, the most active topic of research in the scientific community is to increase the thermal performance of a system. In this context, nanofluids play a significant role and are used by engineers and scientists in different circumstances. For instance, Nimmy et al. [14] studied convection-radiation heat transfer of a partially wetted dovetail extended surface. Hanif and Shafie [15,16] investigated fractional Maxwell nanofluid flow over a horizontal surface and found that the nanoparticle volume fraction and conjugate parameter play in important role in enhancing the thermal efficiency of a system. The thermal performance of a constant hydromagnetic flow of ${\mathrm{T}\mathrm{i}}_6{\mathrm{A}\mathrm{l}}_4\mathrm{V}-{\mathrm{H}}_2\mathrm{O}$ nanofluid within two arbitrarily electrically conducting walls enclosing a Darcy-Brinkman porous medium under the effects of energy dissipation is explored by Singh et al. [17]. In the context of the Riga wedge, the flow of a mono nanofluid (${\mathrm{S}\mathrm{i}\mathrm{O}}_2$/Kerosene oil) and hybrid nanofluid (${\mathrm{T}\mathrm{i}\mathrm{O}}_2+{\mathrm{S}\mathrm{i}\mathrm{O}}_2$/Kerosene oil) is investigated by [18]. A heat transfer fractional Maxwell fluid along convective derivative oblique stagnation-point flow towards an oscillating tensile plate was observed by Bai et al. [19]. It is found that pressure encourages Maxwell fluid velocity. Kaleem et al. [20] considered Maxwell hybrid nanofluid flow over an infinite vertical surface using the Caputo time fractional model. The impact of velocity slip and radiative heat on the thin-film flow of time-dependent hydrodynamic Oldroyd-B and Maxwell nanofluids is investigated by Minnam Reddy et al. [21]. The flow of fluid embedded in porous media with engine oil as the base fluid and nanoparticles of molybdenum disulfide (${\mathrm{M}\mathrm{o}\mathrm{S}}_2$) and zinc oxide (ZnO) is presented by Sagheer et al. [22]. With the help of the MATLAB bvp4c method, the rebuilt mathematical model is simulated using the local non-similarity technique up to the second level of truncation. Vijatha and Reddy [23] studied Maxwell hybrid nanofluid on a cylinder in the presence of non-linear thermal radiation and Cattaneo-Christov heat flux. Ramesh et al. [24] has shown how thermophoretic particle deposition and heat generation/absorption affect the laminar flow of a Casson hybrid-type nanofluid via a nonlinear stretched surface. Khan et al. [25] presented a role of partial slip and temperature-dependent viscosity in the bioconvection assessment of Maxwell nanofluid confined by a stretched. The major impacts of Friedrich shear stress, Cattaneo heat flux, and hybrid nanoparticles on the heat transfer and flow properties of Maxwell hybrid nanofluid is investigated by Hanif and Shafie [26] in the presence of magnetic field and Ohmic heating. The numerical solutions for the fractional Maxwell model are successfully provided using an implicit finite difference technique with the Caputo fractional derivative. Madhukesh et al. [27] investigated the heat transfer in MHD Casson–Maxwell nanofluid between two porous disks with Cattaneo–Christov. Ramesh et al. [28] investigated the chemical reaction and activation energy effects in Maxwell nanofluid flow.

Motivated by the above discussion, this research is designed to evaluate the thermal performance of Maxwell hybrid nanofluid in the presence of Newtonian heating. Prior research has not yet investigated the flow of a Maxwell hybrid nanofluid past a permeable cone. To bridge this research gap, this study article analyzes the flow of Maxwell hybrid nanofluid under the effects of heat radiation, heat generation/absorption, and Newtonian heating. Moreover, a fractional model might describe the complicated properties of viscoelastic material simply and elegantly. For instance, the exponential relaxation module of traditional ordinary models cannot accurately describe the algebraic degradation that many materials experience throughout the relaxation process. Therefore, a fractional Maxwell model has been suggested. A finite difference method, namely, the Crank-Nicolson method is explored to get the linear discrete forms of non-dimensional systems. After that, the numerical solutions are obtained by implementing the Thomas algorithm in MATLAB software. The obtained numerical results are presented via graphs and tables.

Table 1. Thermo-physical properties of water nanoparticles [29,30].

	ρ	σ	${\text{β}}_T$	$C_p$	k
Materials	${\mathrm{k}\mathrm{g}\mathrm{m}}^{-3}$	${\mathrm{S}\mathrm{m}}^{-1}$	${\mathrm{K}}^{-1}$	${\mathrm{J}\left(\mathrm{k}\mathrm{g}\mathrm{K}\right)}^{-1}$	${\mathrm{W}\left(\mathrm{m}\mathrm{K}\right)}^{-1}$
Pure water	997.1	0.05	$21\times {10}^{-5}$	4179	0.613
${\mathrm{F}\mathrm{e}}_3{\mathrm{O}}_4$	5200	$2.5\times {10}^4$	$1.3\times {10}^{-5}$	670	6
$\mathrm{C}\mathrm{u}$	8933	$5.96\times {10}^7$	$1.67\times {10}^{-5}$	385	401

2. Mathematical formulation

The detailed mathematical modeling of the Maxwell hybrid nanofluid with time-fractional derivative is the emphasis of this section.

2.1. Stress tensor

The extra stress tensor τ for Maxwell fluid is described mathematically as (1) $(\tau +{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha }\tau )=\mu {\rm Y},$(1) where ${\rm Y}$ represents shear stress, λ is time relaxation, µ is the viscosity of the fluid. The well-known Caputo derivative ${\mathrm{\partial }}_t^{\alpha }$ is defined as (2) $\begin{array}{rl}{\mathrm{\partial }}_t^{\alpha }f\left(t\right)& =\frac{{\mathrm{\partial }}^{\alpha }}{\mathrm{\partial }{t}^{\alpha }}f\left(t\right)\\ & =\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_0^t{(t-\tau )}^{n-\alpha -1}\frac{{\mathrm{\partial }}^n}{\mathrm{\partial }{\tau }^n}f\left(\tau \right)\mathrm{d}\tau ,\\ & n-1<\alpha <n,n\in \mathbf{N}.\end{array}$(2) The Gamma function $\mathrm{\Gamma }(\cdot )$ is (3) $\mathrm{\Gamma }\left(\xi \right)={\int }_R{\tau }^{\xi -1}{\mathrm{e}}^{-\tau }\mathrm{d}\tau ,\mathrm{\forall }\xi \in \mathbf{C},\mathrm{\Re }\left(\xi \right)>0.$(3)

2.2. Hybrid nanofluid properties

Let us add two distinct nanoparticles $p_1$, $p_2$ of volume fraction ${\phi }_{p_1}$ and ${\phi }_{p_1}$, respectively, in a water-based fluid. Then the thermal physical properties of the resultant fluid, i.e. hybrid nanofluid are modified as [30–32] (4) $\begin{array}{rl}{\rho }_{hnf}& =(1-{\phi }_{p_2}){\rho }_{nf}+{\phi }_{p_2}{\rho }_{p_2},\\ {\rho }_{nf}& =(1-{\phi }_{p_1}){\rho }_f+{\phi }_{p_1}{\rho }_{p_1},\\ {\mu }_{hnf}& =\frac{{\mu }_{nf}}{(1-{\phi }_{p_2}{)}^{2.5}},{\mu }_{nf}=\frac{{\mu }_f}{(1-{\phi }_{p_1}{)}^{2.5}},\\ (\rho {\text{β}}_T{)}_{hnf}& =(1-{\phi }_{p_2})(\rho {\text{β}}_T{)}_{nf}+{\phi }_{p_2}(\rho {\text{β}}_T{)}_{p_2},\\ (\rho {\text{β}}_T{)}_{nf}& =(1-{\phi }_{p_1})(\rho {\text{β}}_T{)}_f+{\phi }_{p_1}(\rho {\text{β}}_T{)}_{p_1},\\ (\rho C_p{)}_{hnf}& =(1-{\phi }_{p_2})(\rho C_p{)}_{nf}+{\phi }_{p_2}(\rho C_p{)}_{p_2},\\ (\rho C_p{)}_{nf}& =(1-{\phi }_{p_1})(\rho C_p{)}_f+{\phi }_{p_1}(\rho C_p{)}_{p_1},\\ \frac{k_{hnf}}{k_{nf}}& =\frac{(k_{p_2}+2k_{nf})+2{\phi }_{p_2}(k_{p_2}-k_{nf})}{(k_{p_2}+2k_{nf})-{\phi }_{p_2}(k_{p_2}-k_{nf})},\\ \frac{k_{nf}}{k_f}& =\frac{(k_{p_1}+2k_f)+2{\phi }_{p_1}(k_{p_1}-k_f)}{(k_{p_1}+2k_{1f})-{\phi }_{p_1}(k_{p_1}-k_f)}.\end{array}$(4)

2.3. Problem description

In this study, we consider that an incompressible Maxwell hybrid nanofluid $(\rho =constant)$ is moving along a vertical cone. Two distinct nanoparticles, namely, ${\mathrm{F}\mathrm{e}}_3{\mathrm{O}}_4$ and $\mathrm{C}\mathrm{u}$ are suspended in the base fluid, therir properties are given in Table 1. The surface of the cone is taken as the x-axis and the y-axis is considered normal to the cone, see Figure 1. Moreover, the viscous dissipation effects are supposed to be insignificant and no pressure gradient is applied. Bearing the above assumptions in mind, the boundary layer and Boussenisq approximations yield us to [33]

Figure 1. Graphical representation.

(5) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(ru\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(rv\right)=0,\end{array}$(5) (6) $\begin{array}{rl}& {\rho }_{hnf}(\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y})\\ & =\frac{\mathrm{\partial }{\tau }_{xy}}{\mathrm{\partial }y}-\frac{{\mu }_{hnf}}{k_0}u\\ & +g(\rho {\text{β}}_T{)}_{hnf}(T-T_{\mathrm{\infty }})\cos \text{β},\end{array}$(6) (7) $\begin{array}{rl}& {\left(\rho C_p\right)}_{hnf}(\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y})\\ & =k_{hnf}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}-\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y}+Q_0(T-T_{\mathrm{\infty }}),\end{array}$(7) where $k_0,g,q_r,Q_0,{\rho }_{hnf},{\mu }_{hnf},(\rho {\text{β}}_T{)}_{hnf},(\rho C_p{)}_{hnf},$ and $k_{hnf}$ are permeability of porous medium, gravitational acceleration, radiative heat flux, heat generation/absorption, density, dynamic viscosity, thermal expansion, heat capacitance, and thermal conductivity of hybrid nanofluid, respectively. The imposed initial boundary conditions [34] (8) $\begin{array}{rl}& u=0,v=0,T=T_{\mathrm{\infty }},t\le 0,\mathrm{\forall }x,y,\\ & u=0,v=0,T=T_{\mathrm{\infty }},t>0,x=0,\mathrm{\forall }y,\\ & u=u_w,{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}=-h_{s}T,t>0,x>0,y=0,}\\ & u=0,T=T_{\mathrm{\infty }},t>0,\mathrm{\forall }x,y\to \mathrm{\infty }.\end{array}$(8) It is simple to anticipate that Equation (1(1) $(\tau +{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha }\tau )=\mu {\rm Y},$(1) ) provides us (9) $(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha }){\tau }_{xy}=\mu \frac{\mathrm{\partial }u}{\mathrm{\partial }y}.$(9) Evaluating shear stress ${\tau }_{xy}$ from Equations (6(6) $\begin{array}{rl}& {\rho }_{hnf}(\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y})\\ & =\frac{\mathrm{\partial }{\tau }_{xy}}{\mathrm{\partial }y}-\frac{{\mu }_{hnf}}{k_0}u\\ & +g(\rho {\text{β}}_T{)}_{hnf}(T-T_{\mathrm{\infty }})\cos \text{β},\end{array}$(6) ) and (9(9) $(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha }){\tau }_{xy}=\mu \frac{\mathrm{\partial }u}{\mathrm{\partial }y}.$(9) ) leads us to (10) $\begin{array}{rl}& {\rho }_{hnf}(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha })(\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y})\\ & ={\mu }_{hnf}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}-\frac{{\mu }_{hnf}}{k_0}(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha })u\\ & +g(\rho {\text{β}}_T{)}_{hnf}(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha })(T-T_{\mathrm{\infty }})\cos \text{β},\end{array}$(10) Let us define the Rosseland approximation of the radiative heat flux $q_r$: (11) $q_r=-\frac{4{\sigma }_b}{k_b}\frac{\mathrm{\partial }{T}^4}{\mathrm{\partial }y},$(11) here ${\sigma }_b$ and $k_b$ denote the Stefan–Boltzmann coefficient and absorption coefficient, respectively. The Taylor expansion of $T^4$ about $T_{\mathrm{\infty }}$ is (12) $T^4\approxeq T_{\mathrm{\infty }}^4+4T_{\mathrm{\infty }}^3(T-T_{\mathrm{\infty }})+\cdots .$(12) Using approximation (12(12) $T^4\approxeq T_{\mathrm{\infty }}^4+4T_{\mathrm{\infty }}^3(T-T_{\mathrm{\infty }})+\cdots .$(12) ) in Equation (11(11) $q_r=-\frac{4{\sigma }_b}{k_b}\frac{\mathrm{\partial }{T}^4}{\mathrm{\partial }y},$(11) ) and differentiating w.r.t $‘‘y^{″}$ leads us to (13) $\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y}=-\frac{16{\sigma }_b{T}_{\mathrm{\infty }}^3}{3k_b}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}.$(13) The energy Equation (7(7) $\begin{array}{rl}& {\left(\rho C_p\right)}_{hnf}(\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y})\\ & =k_{hnf}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}-\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y}+Q_0(T-T_{\mathrm{\infty }}),\end{array}$(7) ) with thermal radiation (13(13) $\frac{\mathrm{\partial }{q}_r}{\mathrm{\partial }y}=-\frac{16{\sigma }_b{T}_{\mathrm{\infty }}^3}{3k_b}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}.$(13) ) is given as (14) $\begin{array}{rl}& {\left(\rho C_p\right)}_{hnf}(\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y})\\ & =(k_{hnf}+\frac{16{\sigma }_b{T}_{\mathrm{\infty }}^3}{3k_b})\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}+Q_0(T-T_{\mathrm{\infty }}).\end{array}$(14)

2.4. Non-dimensional problem

To completely comprehend the mechanics of flow, we require a non-dimensional version of our suggested model. Therefore, we will introduce a set of non-dimensional parameters [35]: (15) $\begin{array}{rl}{x}^{\ast }& =\frac{x}{L},y^{\ast }=y{\left(\frac{u_w}{{\nu }_{f}L}\right)}^{\frac{1}{2}},r^{\ast }=\frac{r}{L},t^{\ast }=\frac{u_{w}t}{L},\\ u^{\ast }& =\frac{u}{u_w},v^{\ast }=v{\left(\frac{u_w{\nu }_f}{L}\right)}^{\frac{-1}{2}},T^{\ast }=\frac{T-T_{\mathrm{\infty }}}{T_{\mathrm{\infty }}},\\ {\lambda }^{\ast }& =\frac{u_w\lambda }{L}.\end{array}$(15) Finally, we arrived at (16) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(ru\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(rv\right)=0,\end{array}$(16) (17) $\begin{array}{rl}& {\mathcal{C}}_1(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha })(\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y})\\ & ={\mathcal{C}}_2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathcal{C}}_2}{K}(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha })u\\ & +{\mathcal{C}}_{3}Ri\cos \text{β}(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha })T,\end{array}$(17) (18) $\begin{array}{rl}& {\mathcal{C}}_4(\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y})=\frac{{\mathcal{C}}_5+Rd}{Pr}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}+QT.\end{array}$(18) Here $Ri,Re,Gr,K,Pr,Rd,Q,$ and ${\mathcal{C}}_1-{\mathcal{C}}_4$ are Richardson number, Reynolds number, Grashof number, porosity parameter, Prandtl number, radiation parameter, heat generation/absorption parameter, and nanofluid constants, respectively, defined as: (19) $\begin{array}{rl}\frac{1}{K}& =\frac{{\nu }_{f}L}{u_w{k}_0},Ri=\frac{Gr}{R{e}^2},Re=\frac{u_{w}L}{{\nu }_f},\\ Gr& =\frac{g{\text{β}}_T{T}_{\mathrm{\infty }}{L}^3}{{\nu }_f^2},Pr=\frac{{\left(\mu C_p\right)}_f}{k_f},Rd=\frac{16{\sigma }_b{T}_{\mathrm{\infty }}^3}{3k_b{k}_f},\\ Q& =\frac{Q_{0}L}{{\left(\rho C_p\right)}_f{u}_w},{\mathcal{C}}_1=\frac{{\rho }_{hnf}}{{\rho }_f},{\mathcal{C}}_2=\frac{{\mu }_{hnf}}{{\mu }_f},\\ {\mathcal{C}}_3& =\frac{{\left(\rho {\text{β}}_T\right)}_{hnf}}{{\left(\rho {\text{β}}_T\right)}_f},{\mathcal{C}}_4=\frac{{\left(\rho C_p\right)}_{hnf}}{{\left(\rho C_p\right)}_f},{\mathcal{C}}_5=\frac{k_{hnf}}{k_f}.\end{array}$(19) subject to initial boundary conditions: (20) $\begin{array}{rl}& u=0,v=0,T=0,t\le 0,\mathrm{\forall }x,y,\\ & u=0,v=0,T=0,t>0,x=0,\mathrm{\forall }y,\\ & u=1,{\displaystyle \frac{\mathrm{\partial }T}{\mathrm{\partial }y}=-\gamma (1+T),t>0,x>0,y=0,}\\ & u=0,T=0,t>0,\mathrm{\forall }x,y\to \mathrm{\infty }.\end{array}$(20)

2.5. Physical quantities

It is significant to discuss some important physical quantities like skin friction and Nusselt number [32] (21) $C_f=\frac{{\tau }_w}{{\rho }_f{u}_w^2},N{u}_x=\frac{x{q}_w}{k_f(T_w-T_{\mathrm{\infty }})}.$(21) ${\tau }_w^{\ast }={\mu }_{hnf}(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}{)}_{y=0}$ and $q_w=-k_{hnf}(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}{)}_{y=0}$. For a Maxwell fluid, the ${\tau }_w$ define by the following relationship: (22) $(1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha }){\tau }_w={\mu }_{hnf}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}_{y=0}.$(22) Equation (15(15) $\begin{array}{rl}{x}^{\ast }& =\frac{x}{L},y^{\ast }=y{\left(\frac{u_w}{{\nu }_{f}L}\right)}^{\frac{1}{2}},r^{\ast }=\frac{r}{L},t^{\ast }=\frac{u_{w}t}{L},\\ u^{\ast }& =\frac{u}{u_w},v^{\ast }=v{\left(\frac{u_w{\nu }_f}{L}\right)}^{\frac{-1}{2}},T^{\ast }=\frac{T-T_{\mathrm{\infty }}}{T_{\mathrm{\infty }}},\\ {\lambda }^{\ast }& =\frac{u_w\lambda }{L}.\end{array}$(15) ) helps us to obtained the non-dimensional form of Equation (21(21) $C_f=\frac{{\tau }_w}{{\rho }_f{u}_w^2},N{u}_x=\frac{x{q}_w}{k_f(T_w-T_{\mathrm{\infty }})}.$(21) ) (23) $\begin{array}{rl}& (1+{\lambda }^{\alpha }{\mathrm{\partial }}_t^{\alpha })C_{f}R{e}^{1/2}\\ & ={\mathcal{C}}_2{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}_{y=0},N{u}_{x}R{e}^{-1/2}=-{\mathcal{C}}_{5}x{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}_{y=0}.\end{array}$(23)

3. Numerical analysis

Assume the numerical solutions $(u_{i,j}^k,v_{i,j}^k)$ at $(x_i,y_j,t_k)$ with time step $\mathrm{\Delta }t$ and mesh size $(\mathrm{\Delta }x,\mathrm{\Delta }y)$ such that (24) $\{\begin{array}{ll}{T}_k=k\mathrm{\Delta }t,& k=0,1,\dots ,N\\ x_i=i\mathrm{\Delta }x,& i=1,2,\dots ,R,\\ y_j=j\mathrm{\Delta }y,& j=1,2,\dots ,S.\end{array}$(24) The L1 algorithm of the Caputo derivative is: (25) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{\alpha }\upsilon }{\mathrm{\partial }{t}^{\alpha }}\simeq \frac{\mathrm{\Delta }{t}^{-\alpha }}{2\mathrm{\Gamma }(2-\alpha )}\\ & [{\upsilon }_{i,j}^{k+1}+{\upsilon }_{i,j}^k+\sum _{m=1}^k{b}_m{\upsilon }_{i,j}^{k+1-m}+\sum _{m=1}^{k-1}{b}_m{\upsilon }_{i,j}^{k-m}].\end{array}$(25) Note: $b_m=(a_m-a_{m-1}),a_m=(m+1{)}^{1-\alpha }-m^{1-\alpha }$. Let us define a set of constants (26) $\begin{array}{rl}{R}_1& ={\lambda }_1^{\alpha }\frac{\mathrm{\Delta }{t}^{-\alpha }}{\mathrm{\Gamma }(2-\alpha )},{\mathcal{B}}_1={\mathcal{C}}_1(1+R_1),{\mathcal{B}}_2={\mathcal{C}}_1{R}_1,\\ {\mathcal{B}}_3& =\frac{{\mathcal{C}}_2}{K}(1+R_1),{\mathcal{B}}_4=\frac{{\mathcal{C}}_2}{K}{R}_1,\\ {\mathcal{B}}_5& ={\mathcal{C}}_3\cos \text{β}(1+R_1),{\mathcal{B}}_6={\mathcal{C}}_3\cos \text{β}{R}_1.\end{array}$(26) Utilizing the approximation (25(25) $\begin{array}{rl}& \frac{{\mathrm{\partial }}^{\alpha }\upsilon }{\mathrm{\partial }{t}^{\alpha }}\simeq \frac{\mathrm{\Delta }{t}^{-\alpha }}{2\mathrm{\Gamma }(2-\alpha )}\\ & [{\upsilon }_{i,j}^{k+1}+{\upsilon }_{i,j}^k+\sum _{m=1}^k{b}_m{\upsilon }_{i,j}^{k+1-m}+\sum _{m=1}^{k-1}{b}_m{\upsilon }_{i,j}^{k-m}].\end{array}$(25) ), constants (26(26) $\begin{array}{rl}{R}_1& ={\lambda }_1^{\alpha }\frac{\mathrm{\Delta }{t}^{-\alpha }}{\mathrm{\Gamma }(2-\alpha )},{\mathcal{B}}_1={\mathcal{C}}_1(1+R_1),{\mathcal{B}}_2={\mathcal{C}}_1{R}_1,\\ {\mathcal{B}}_3& =\frac{{\mathcal{C}}_2}{K}(1+R_1),{\mathcal{B}}_4=\frac{{\mathcal{C}}_2}{K}{R}_1,\\ {\mathcal{B}}_5& ={\mathcal{C}}_3\cos \text{β}(1+R_1),{\mathcal{B}}_6={\mathcal{C}}_3\cos \text{β}{R}_1.\end{array}$(26) ) and the Crank-Nicolson method in Equations (18(18) $\begin{array}{rl}& {\mathcal{C}}_4(\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y})=\frac{{\mathcal{C}}_5+Rd}{Pr}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}+QT.\end{array}$(18) )–(16(16) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(ru\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(rv\right)=0,\end{array}$(16) ) lead us to: (27) $\begin{array}{rl}& \frac{\begin{array}{c}{u}_{i,j-1}^{k+1}-u_{i-1,j-1}^{k+1}+u_{i,j}^{k+1}-u_{i-1,j}^{k+1}+u_{i,j-1}^k\\ -u_{i-1,j-1}^k+u_{i,j}^k-u_{i-1,j}^k\end{array}}{4\mathrm{\Delta }x}\\ & +\frac{v_{i,j}^{k+1}-v_{i,j-1}^{k+1}+v_{i,j}^k-v_{i,j-1}^k}{2\mathrm{\Delta }y}=0.\end{array}$(27) (28) $\begin{array}{rl}& {\mathcal{B}}_1[\frac{u_{i,j}^{k+1}-u_{i,j}^k}{\mathrm{\Delta }t}+u_{i,j}^k\frac{u_{i,j}^{k+1}-u_{i-1,j}^{k+1}+u_{i,j}^k-u_{i-1,j}^k}{2\mathrm{\Delta }x}\\ & +v_{i,j}^k\frac{u_{i,j+1}^{k+1}-u_{i,j-1}^{k+1}+u_{i,j+1}^k-u_{i,j-1}^k}{4\mathrm{\Delta }y}]\\ & +{\mathcal{B}}_2[\sum _{m=1}^k{b}_m\{\frac{u_{i,j}^{k+1-m}-u_{i,j}^{k-m}}{\mathrm{\Delta }t}\\ & +u_{i,j}^{k-m}\frac{u_{i,j}^{k+1-m}-u_{i-1,j}^{k+1-m}}{2\mathrm{\Delta }x}\\ & +v_{i,j}^{k-m}\frac{u_{i,j+1}^{k+1-m}-u_{i,j-1}^{k+1-m}}{4\mathrm{\Delta }y}\}\\ & +\sum _{m=1}^{k-1}{b}_m{u}_{i,j}^{k-m}\{\frac{u_{i,j}^{k-m}-u_{i-1,j}^{k-m}}{2\mathrm{\Delta }x}\\ & +v_{i,j}^{k-m}\frac{u_{i,j+1}^{k-m}-u_{i,j-1}^{k-m}}{4\mathrm{\Delta }y}\}]\\ & ={\mathcal{C}}_2\frac{u_{i,j+1}^{k+1}-2u_{i,j}^{k+1}+u_{i,j-1}^{k+1}+u_{i,j+1}^k-2u_{i,j}^k+u_{i,j-1}^k}{2\mathrm{\Delta }{y}^2}\\ & +{\mathcal{B}}_3\frac{u_{i,j}^{k+1}+u_{i,j}^k}{2}+\frac{{\mathcal{B}}_4}{2}[\sum _{m=1}^k{b}_m{u}_{i,j}^{k+1-m}\\ & +\sum _{m=1}^{k-1}{b}_m{u}_{i,j}^{k-m}]+{\mathcal{B}}_5\frac{T_{i,j}^{k+1}+T_{i,j}^k}{2}\\ & +\frac{{\mathcal{B}}_6}{2}[\sum _{m=1}^k{b}_m{T}_{i,j}^{k+1-m}+\sum _{m=1}^{k-1}{b}_m{T}_{i,j}^{k-m}].\end{array}$(28) (29) $\begin{array}{rl}& {\mathcal{C}}_4[\frac{T_{i,j}^{k+1}-T_{i,j}^k}{\mathrm{\Delta }t}+u_{i,j}^k\frac{T_{i,j}^{k+1}-T_{i-1,j}^{k+1}+T_{i,j}^k-T_{i-1,j}^k}{2\mathrm{\Delta }x}\\ & +v_{i,j}^k\frac{T_{i,j+1}^{k+1}-T_{i,j-1}^{k+1}+T_{i,j+1}^k-T_{i,j-1}^k}{4\mathrm{\Delta }y}]\\ & =\frac{{\mathcal{C}}_5+Rd}{Pr}\\ & \frac{T_{i,j+1}^{k+1}-2T_{i,j}^{k+1}+T_{i,j-1}^{k+1}+T_{i,j+1}^k-2T_{i,j}^k+T_{i,j-1}^k}{2\mathrm{\Delta }{y}^2}\\ & +Q\frac{T_{i,j}^{k+1}+T_{i,j}^k}{2}.\end{array}$(29) The limits of the rectangular domain are $x_{max}=1$ and $y_{max}=5$, where $y_{max}$ refers to $y\to \mathrm{\infty }$. The time level $\mathrm{\Delta }t$ and mesh sizes $(\mathrm{\Delta }x,\mathrm{\Delta }y)$ along $(x,y)$ direction are considered as $\mathrm{\Delta }t=0.001,\mathrm{\Delta }x=0.05,$ and $\mathrm{\Delta }y=0.05$.

4. Results and discussion

In this research, Maxwell hybrid nanofluid flow over a cone enclosed in a porous media is investigated. In addition, the effects of thermal radiation, heat generation/absorption, and Newtonian boundary conditions are considered. The velocity, temperature, Nusselt number, and skin friction have been analyzed for the range of governing parameters: $0.1\le \alpha \le 0.9,0.1\le \lambda \le 0.9,0.1\le K\le 0.5,0\le Rd\le 0.3,0\le Q\le 1,$ and $0.1\le \gamma \le 0.9$. The Nusselt number and skin friction of fluid flow over a cone with no slip velocity and constant wall temperature are calculated and compared with those reported by [36]. It is revealed that the results are in great agreement, see Table 2.

Table 2. Comparison of nusselt number ($N{u}_{x}R{e}^{-1/2}$) and skin friction ($C_{f}R{e}^{1/2}$) with previously published work when ${\phi }_{hnf}=\lambda =1/K=Rd=Q=0$ and Ri = 1.

	$N{u}_{x}R{e}^{-1/2}$		$C_{f}R{e}^{1/2}$
Pr	Sambath [36]	Present	Sambath [36]	Present
0.01	0.0797	0.0797	1.3356	1.3557
0.1	0.2115	0.2111	1.0911	1.0943
1	0.5125	0.5126	0.7688	0.7680
10	1.0356	1.0343	0.4856	0.4859

In Figures 2–5, show the subjectivity of the fractional parameter, relaxation time parameter, porosity parameter, and Richardson number on the velocity profile, respectively. It is discovered that the velocity increases for the increasing values of α in Figure 2. While in the case of relaxation time velocity decreases with the higher values of relaxation time parameter λ, see Figure 3. It was expected physically because the relaxation time corresponds to an increase in fluid viscoelasticity, a decrease in the velocity. It has been noted that the velocity increases for the increasing values of K, shown in Figure 4. In general, the permeability of the porous media or porosity indicates a medium's ability to enable fluid to pass through it, and increasing the porosity parameter allows the fluid to move swiftly, resulting in increased velocity. The effects of Richardson number Ri on the velocity profile is drawn in Figure 5. This figure shows that the velocity of the fluid increases against Ri. The Richardson number quantifies the influence of buoyancy in contrast to the inertia of the external force. For higher values of Ri, there is a favorable pressure gradient due to buoyancy forces, which causes the flow to accelerate. Figures 6–8 show the fluctuations in the temperature of Maxwell hybrid nanofluid caused by active governing parameters. The temperature profile for various values of radiation parameter Rd is plotted in Figure 6. The pattern of this graph illustrates how the temperature profile of the fluid increases with increasing assumptions of the radiation parameter Rd. Generally, increasing values of the radiation parameter promote radiative transmission of heat, which improves radiative flux; as a result, the temperature profile steadily rises as the value of Rd increases. Figure 7 shows that the temperature of the fluid increases for increasing values of heat generation parameter Q. This is because the heat generation term is a source of thermal energy, and the temperature of the fluid is directly related to the amount of thermal energy present in the system. In Figure 8 temperature profile is drawn for the conjugate heat parameter. The temperature of the fluid is related to the conjugate heat parameter. Specifically, as the value of the conjugate heat parameter increases, the temperature of the fluid also increases. This is because the conjugate heat parameter represents the efficiency with which the fluid can dissipate thermal energy. When this parameter is high, the fluid can more effectively convert mechanical energy into thermal energy, which raises the temperature of the fluid. The decreasing temperature profile against Richardson number is illustrated in Figure 9. Temperature distributions approach a linear decrease from the wall to the free stream with low Richardson numbers, which correlate to weak buoyant forces (low Grashof numbers). The profiles indicate monotonic deterioration at larger Ri values.

Figure 2. Velocity profile for α when $\lambda =0.9,\phi =0.01,K=0.5,Ri=0.1,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

Figure 3. Velocity profile for λ when $\alpha =0.5,\phi =0.01,K=0.5,Ri=0.1,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

Figure 4. Velocity profile for K when $\alpha =0.5,\lambda =0.9,\phi =0.01,Ri=0.1,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

The numerical results for the skin friction of different nanofluids are deputed in Figures 10–12. The results show that the fractional parameter has a positive correlation with skin friction, see Figure 10. A remarkable change in skin friction due to the relaxation time parameter is noted in Figure 11, that when it increases the skin friction decreases. In Figure 12, an identical behavior is observed for increasing values of the porosity parameter. The behavior of the Nusselt number for different nanofluids is studied in Figures 13–16. Increasing values of nanoparticle concentration increase the Nusselt number as shown in Figure 13. It was predicted, because nanoparticles have a propensity to enhance the thermal conductivity of weak convectional fluids, and therefore the Nusselt number or heat transfer rate of a system. However, this trend is reversed for increasing values of radiation and heat generation parameters as shown in Figures 14 and 15. The reduced convective heat transmission is owing to the increased radiative heat transfer. The Nusselt number decreases because it measures the proportion of convective to conductive heat transfer. On the other hand, the conjugate parameter γ increases the heat transfer efficiency, see Figure 16. It is obvious because the Nusselt number is directly proportional to the temperature gradient at the wall which is in direct correlation with the conjugate heat parameter. For this reason, the Nusselt number grows when γ increases.

Figure 5. Velocity profile for Ri when $\alpha =0.5,\lambda =0.9,\phi =0.01,K=0.5,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

Figure 6. Temperature profile for Rd when $\alpha =0.5,\lambda =0.9,\phi =0.01,K=0.5,Ri=0.1,Q=0.1,$ and $\gamma =0.9$.

Figure 7. Temperature profile for Q when $\alpha =0.5,\lambda =0.9,\phi =0.01,K=0.5,Ri=0.1,Rd=0.3,$ and $\gamma =0.9$.

Figure 8. Temperature profile for γ when $\alpha =0.5,\lambda =0.9,\phi =0.01,K=0.5,Ri=0.1,Rd=0.3,$ and $Q=0.1$.

Figure 9. Temperature profile for Ri when $\alpha =0.5,\lambda =0.9,\phi =0.01,K=0.5,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

Figure 10. Skin friction for α when $\lambda =0.9,\phi =0.01,K=0.5,Ri=0.1,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

Figure 11. Skin friction for λ when $\alpha =0.5,\phi =0.01,K=0.5,Ri=0.1,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

Figure 12. Skin friction for K when $\alpha =0.5,\lambda =0.9,\phi =0.01,Ri=0.1,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

Figure 13. Nusselt number for φ when $\alpha =0.5,\lambda =0.9,K=0.5,Ri=0.1,Rd=0.3,Q=0.1,$ and $\gamma =0.9$.

Figure 14. Nusselt number for Rd when $\alpha =0.5,\lambda =0.9,\phi =0.01,K=0.5,Ri=0.1,Q=0.1,$ and $\gamma =0.9$.

Figure 15. Nusselt number for Q when $\alpha =0.5,\lambda =0.9,\phi =0.01,K=0.5,Ri=0.1,Rd=0.3,$ and $\gamma =0.9$.

Figure 16. Nusselt number for γ when $\alpha =0.5,\lambda =0.9,\phi =0.01,K=0.5,Ri=0.1,Rd=0.3$, and $Q=0.1$.

5. Conclusions

The boundary layer flow of a fractional Maxwell hybrid nanofluid with Newtonian heating is examined in the presence of a porous medium. The Crank–Nicolson technique and the L1 algorithm are utilized to generate numerical solutions for the suggested problem. The impacts of pertinent flow parameters on fluid velocity, heat transfer, wall shear stress, and Nusselt number are studied in a variety of illustrations. The major outcomes of the research are listed below:

• The thermal conductivity boosts as the φ increases and raises the heat transfer of the fluid by 3.1% at 4% vol. of ${\mathrm{C}\mathrm{u}-\mathrm{F}\mathrm{e}}_3{\mathrm{O}}_4$ nanoparticles.

• The fluid motion slows down when λ rises, indicating that the viscoelasticity rises, and the fluid motion and wall shear stress both decrease.

• The temperature profiles indicate monotonic deterioration at larger Ri values.

• The results demonstrated that raising the Rd leads the thermal boundary layer to thicken and lower the Nusselt number.

Acknowledgements

The authors would like to acknowledge the financial support from Universiti Teknologi Malaysia for the funding under UTM Fundamental Research (UTMFR: Q.J130000.3854.23H22).

Disclosure statement

The authors wish to confirm that there are no known conflicts of interest associated with this publication.