Applications of variable thermal properties in Carreau material with ion slip and Hall forces towards cone using a non-Fourier approach via FE-method and mesh-free study

This research highlights the utilization of two viscosity models to study the involvement of variable properties in heat and momentum transport in a rotating Carreau fluid past over a cone. The rheology of the Carreau material is assessed by the variable dynamic viscosity over the heating cone. The transport of momentum phenomenon is modeled by considering generalized Ohm’s law in Carreau liquid and thermal transport in derived by considering variable thermal conductivity, heat flux model. The considered model is derived in the form of nonlinear PDEs with boundary layer analysis. The nonlinear PDEs are converted into coupled ODEs by using approximate transformation and converted equations are solved numerically by finite element methodology. The impact of numerous parameters is displayed graphically, and their behavior is discussed in detail.

Introduction

The features of heat transfer are applicable in several industrial, chemical, and thermal processes. This type of phenomenon is applicable in polymers, gas engineering, cooling process, solar cells, water, thermal process, and oils. Recent work on heat transfer in several rheological fluids should be mentioned. Nazeer et al. (2022) estimated the thermal aspects of multiphase flow taking into account magnetohydrodynamic and thermal radiation over a stretching frame. Luan et al. (2022) discussed turbulator effects in heat transfer phenomena in an energy transfer mechanism. Imran et al. (2022) discussed enhancement in thermal energy using the concept of several shapes of nanoparticles involving hybrid nanoparticles over a horizontal plate. The characteristics of fluids exhibit a range of rheological behavior. Therefore, it is very complex to study rheological behaviors, taking into account rheological stress (strain relation). The Newtonian model contains only viscosity rheology and its utilization in non-Newtonian contexts provides inaccurate information with respect to mass diffusion, flow, and heat energy. Carreau fluid is type of non-Newtonian fluid. Mathematical model related to Carreau fluid was investigated by Carreau (1972). The stress tensor regarding Carreau fluid is defined as

$\mathit{\tau }=-pI+{\mu }_0{\left[1+{\mathrm{\Gamma }}^2{{\rm Y}}^2\right]}^{n-1/2}{\mathit{A}}_{\mathbf{1}.}(1)$

where $\mathit{\tau }$ is the stress tensor, $n$ is the power law index number, $I$ is the identity matrix, $A_1$ is the Rivlin Ericksen tensor, and ${{\rm Y}}^2$ is the strain rate tensor. Nazir et al. (2021) discussed the rheology equations of a Carreau liquid using the impacts of ion slip and Hall forces in a rotating cone. They used a numerical approach to find numerical consequences. Sohail et al. (2022) studied three kinds of mass species in Carreau liquid in the presence of activation energy over a stretching plate. Nabwey et al. (2022) investigated the thermal aspects of Carreau nanofluid taking into account the chemical species and bioconvection in a cylinder. Reedy et al. (2022) discussed a model for a Carreau fluid including entropy generation over a heated microchannel. This involved multiple impacts based on thermal radiation and viscous dissipation using no-slip conditions. Imran et al. (2021) estimated the features of a Carreau liquid in energy transfer using magnetic dipoles and radiation containing nanofluids and microorganisms past a wedge, numerically solved with the shooting approach. Song et al. (2022) discussed thermal aspects in Carreau fluid considering nanoparticles in a stretching cylinder using Marangoni boundary conditions (BCs). Farooq et al. (2021) used Cattaneo-Christov theory in energy equations, taking thermal radiation, bioconvection flow, and heat source into account.

Recent investigations have shown that mass diffusion and thermal conductivity during transport mass species and heat cannot increase and remain constant. During changes of thermal energy, viscosity and thermal conductivity are functions of heat energy during the transport of particles. Alhussain and Tassaddiq (2022) proposed a model of variable viscosity in the presence of hybrid nanomaterial in Casson fluid, considering thin film considering magnetic field in channel. They have utilized analytical approach to simulate numerical consequences. Kumawat et al. (2022) performed numerical results of entropy generation in MHD flow considering variable viscosity in the presence of variable viscosity. Nazir et al. (2020) simulated the numerical study of a Carreau liquid using the concept of variable properties in mass diffusion and heat energy involving non-Fourier’s law over a frame using finite element scheme. It was found that the highest heat energy was obtained for variable viscosity than for constant viscosity. Chaurasiya et al. (2022) discussed the influence of variable thermal conductivity taking into account semi-conductor sources using a finite element scheme. Wang et al. (2022) investigated the impacts of tri-hybridized nanofluid taking into account variable properties of fluid in the presence of non-Fourier’s law computed by the FE-method. Naseem et al. (2021) discussed the numerical consequences of Soret and Dufour effects involving the role of thermal radiation over a stretching frame, including variable thermal conductivity. Sohail et al. (2021) investigated the performance of entropy generation in a Casson fluid inserting the impact of Lorentz force. They discussed features of variable properties including various effects past a stretching frame. Akbar et al. (2022) performed investigations of unsteady flow in peristaltic transport using variable viscosity (function of temperature) simulating an exact solution approach. Akram et al. (2022a) estimated the thermal features of peristaltic flow involving electroosmotically adding a suspension of nanomaterial in a curved microchannel. Maraj et al. (2022) developed mathematically modeling of rotational MHD flow in the presence of Hall force inserting two types of nanoparticles using slip conditions in a vertical channel. Habib and Akbar (2021) investigated a new to nanofluids using sensitive Staphlococcus aureus and Staphlococcus aureus. They concluded that drug resistance can be reduced and overcome using drug conjugate and gold nanoparticles. Akram et al. (2021) introduced thermal transfer characterizations inserting hybrid nanofluid (Ag–Au) considering electroosmotic pumping in microchannel. Akram et al. (2022b) analyzed electroosmotic flow based on water-silver nanofluids considering peristalsis flow using two various approaches of nanofluid. Akram et al. (2022c) introduced investigation of electroosmosis in the presence of MHD peristaltic flow containing suspension of SWCNTs nanofluid filled in aqueous media. Multiple aspects of Carreau liquid applied to a magnetic field in a suspension of a nanofluid using slip effects across a slandering surface were studied by Raju et al. (2019). Khan and Sultan (2015) discussed thermal features of Dufour and Soret effects in Eyring-Powell liquid using a porous heated cone. Dawar et al. (2021) investigated flow configurations in Williamson fluid, taking into account non-isosolutal conditions inserting nanofluid over wedge and cone.

The utilization of variable properties to study momentum and thermal transport in Carreau model past over a cone has not been sufficiently thoroughly studied. This contribution covers this question, and the modeled equations are solved numerically via finite element approach, and the impact of different involved parameters on velocity and temperature is discussed.

Mathematically development and physical consequences

zThree-dimensional thermal features in terms of variable properties in a Carreau liquid in rotating cone are visualized. Two kinds of forces, namely, ion slip and Hall forces, are considered. The rheology of a Carreau liquid in the presence of variable viscosity is imposed into fluidic motion. Heat energy assessed using the non-Fourier approach. Lorentz force is inserted along the z-direction of a heated cone. Additionally, motion associated with particles is generated due to rotational movement of cone with angle $\Omega .$ The wall temperature is considered $T_w.$ Thermal conductivity related fluid is considered function of temperature. The rotating cone is captured by Figure 1. Boundary layer approximation (BLA) is utilized to develop model of PDEs. The reduced form of PDEs is given below. Gravitational acceleration acts downward in a cone. Figure 1 predicts the physical behavior of the model (Khan et al., 2014).

FIGURE 1

FIGURE 1. Rotating cone and coordinates system.

Conservation laws (Khan et al., 2014; Malik et al., 2016) are defined as follows:

$\nabla \bullet \stackrel{\sim }{\mathit{V}}=\mathbf{0},(2)$

${\rho }_{Thnf}\left(\frac{\mathit{d}\stackrel{\sim }{\mathit{V}}}{\mathit{d}\mathit{t}}\right)=-\nabla p+\nabla \bullet \mathit{\tau }+\mathbf{J}\times \mathbf{B},(3)$

${\nabla \bullet \mathit{B}=\mathbf{0},\mu }_{Thnf}\mathit{J}=\nabla \times \mathbf{B},,\nabla \times \mathbf{E}=\frac{\partial \mathit{B}}{\partial t},(4)$

$\mathit{J}=\left[\mathit{E}+\stackrel{\sim }{\mathit{V}}\times \mathbf{B}\right]{\mathit{\sigma }}_{\mathit{T}\mathit{h}\mathit{n}\mathit{f}}-\left(\mathit{J}\times \mathit{B}\right)\frac{B_e}{\left|\mathit{B}\right|}+\left(\mathit{J}\times \mathit{B}\right)\frac{{B_{i}B}_e}{{\left|\mathit{B}\right|}^{\mathbf{2}}}\times \mathit{B},(5)$

${\left(\rho C_p\right)}_{thnf}\left(\frac{\mathit{d}T}{\mathit{d}\mathit{t}}\right)=K_{Thnf}\left({\nabla }^{\mathbf{2}}T\right)-\nabla \bullet \mathit{q}+Q_0\left(T-T_{\infty }\right).(6)$

The reduced form of the PD equations (Khan et al., 2014; Malik et al., 2016) is given below:

$\frac{\partial \left(XU\right)}{\partial Y}+\frac{\partial \left(XV\right)}{\partial Z}=0,(7)$

$\begin{array}{l}U\frac{\partial U}{\partial X}+W\frac{\partial U}{\partial Z}=\frac{{\left(B_0\right)}^2\sigma }{\rho \left[{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2\right]}\left[V{B}_e-\left(1+B_i{B}_e\right)U\right],\\ +\frac{V^2}{X}+G\beta \left(T-T_{\infty }\right)\cos \alpha +\frac{\partial }{\partial z}\left[{\left\{1+{\mathrm{\Gamma }}^2{\left(\frac{\partial U}{\partial y}\right)}^2\right\}}^{n-1/2}\frac{\partial U}{\partial z}\right],\end{array}(8)$

$\begin{array}{c}U\frac{\partial V}{\partial X}+W\frac{\partial V}{\partial Z}=-\frac{{\left(B_0\right)}^2\sigma }{\rho \left[{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2\right]}\left[U{B}_e+\left(1+B_i{B}_e\right)V\right],\\ -\frac{UV}{X}+\frac{\partial }{\partial z}\left[{\left\{1+{\Gamma }^2{\left(\frac{\partial V}{\partial y}\right)}^2\right\}}^{n-1/2}\frac{\partial V}{\partial z}\right],\end{array}(9)$

$\begin{array}{c}U\frac{\partial T}{\partial X}+W\frac{\partial T}{\partial Z}=\frac{1}{\rho C_p}\frac{\partial }{\partial Z}\left(K\left(T\right)\frac{\partial T}{\partial Z}\right)+\frac{Q_0}{\rho C_p}\left(T-T_{\infty }\right)-\frac{{\lambda Q}_0}{\rho C_p}\left(U\frac{\partial T}{\partial X}+W\frac{\partial T}{\partial Z}\right)\\ -\lambda \left(U^2\frac{{\partial }^{2}T}{\partial X^2}+W^2\frac{{\partial }^{2}T}{\partial Z^2}+2UW\frac{{\partial }^{2}T}{\partial X\partial Z}\right)-\lambda \left(U\frac{\partial V}{\partial X}+W\frac{\partial V}{\partial Z}\right)\frac{\partial T}{\partial Z}-\lambda \left(U\frac{\partial U}{\partial X}+W\frac{\partial U}{\partial Z}\right)\frac{\partial T}{\partial X}.\end{array}(10)$

BCs (boundary conditions) (Malik et al., 2016) are

$U=0,V=\mathrm{\Omega }X\sin \alpha ,T=T_W,U\to 0,V\to 0,T\to T_{\infty }(11)$

Viscosity and thermal conductivity (Nazir et al., 2020) are functions of heat energy, as follows:

$\begin{array}{c}\frac{{\mu }_{\infty }}{\mu }=\left[1-\gamma \left(T_{\infty }-T\right)\right],\\ \frac{K\left(T\right)}{k_{\infty }}=\left[1-ϵ\left(\frac{T_{\infty }-T}{T_w-T_{\infty }}\right)\right]\end{array}(12)$

The similarity variables (Malik et al., 2016) of current the analysis are

$U=-\frac{\mathrm{\Omega }X\sin \alpha F^{\prime }}{2},V=\mathrm{\Omega }X\sin \alpha G,W={\left(\mathrm{\Omega }{\nu }_f\sin \alpha \right)}^{1/2}F,\Theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},\eta ={\left(\frac{\mathrm{\Omega }\sin \alpha }{{\nu }_f}\right)}^{1/2}Z(13)$

Eq. 12 is known as a transformation, and the selection of the transformation is adopted to satisfy the continuity equation. Transformations are used in Eqs. 7–9. The ODEs are derived as

$\begin{array}{c}\frac{{\Theta }_{\gamma }}{{\Theta }_{\gamma }-1}\left\{1+n{\left(W{F}^{″}\right)}^2\right\}{\left\{1+{\left(W{F}^{″}\right)}^2\right\}}^{\mathrm{n}-3/2}{F}^{‴}+\frac{{\Theta }_{\gamma }}{{\left({\Theta }_{\gamma }-1\right)}^2}{\left\{1+{\left(W{F}^{″}\right)}^2\right\}}^{\mathrm{n}-1/2}{F}^{″}{\Theta }^{\prime }\\ +\frac{M^2}{{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2}\left[\left(1+B_e{B}_i\right)F^{\prime }+2B_{e}G\right]+\left(\frac{1}{2}{F^{\prime }}^2-F{F}^{″}-2G^2-2\lambda \Theta \right)=0,\end{array}(14)$

$\begin{array}{c}\frac{{\Theta }_{\gamma }}{{\Theta }_{\gamma }-1}\left\{1+n{\left(W{G}^{\prime }\right)}^2\right\}{\left\{1+{\left(W{G}^{\prime }\right)}^2\right\}}^{n-3/2}{G}^{{}^{″}}+\frac{{\Theta }_{\gamma }}{{\left({\Theta }_{\gamma }-1\right)}^2}{\left\{1+{\left(W{G}^{\prime }\right)}^2\right\}}^{n-1/2}{G}^{\prime }\Theta \prime \\ -\frac{M^2}{{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2}\left[\left(1+B_e{B}_i\right)G-\frac{1}{2}{B}_e{F}^{{}^{\prime }}\right]+\left(G{F}^{{}^{\prime }}-F^{{}^{\prime }}{G}^{\prime }\right)=0,\end{array}(15)$

${\left(1+ϵ\Theta \right)\Theta }^{{}^{″}}+ϵ{\left({\Theta }^{{}^{\prime }}\right)}^2+{\mathit{Pr}}_{\infty }F{\theta }^{\prime }+{\gamma }_1{\mathit{Pr}}_{\infty }\left[F^2{\Theta }^{{}^{″}}-QF{\Theta }^{\prime }+\frac{1}{2}FG\prime \Theta \prime \right]+{\mathit{Pr}}_{\infty }{H}_t\Theta =0.(16)$

The Prandtl fluid at infinity is delivered as

${\mathit{Pr}}_{\infty }=\left(1+ϵ\Theta \right)\left(1-\frac{\Theta }{{\Theta }_{\gamma }}\right)\mathit{Pr}.(17)$

Substituting Eq. (16) into Eq. (15), we have

$\begin{array}{c}{\left(1+ϵ\Theta \right)\Theta }^{″}+ϵ{\left({\Theta }^{{}^{\prime }}\right)}^2+{\gamma }_1\left(1+ϵ\Theta \right)\left(1-\frac{\Theta }{{\Theta }_{\gamma }}\right)\mathit{Pr}\left[F^2{\Theta }^{″}-QF{\Theta }^{\prime }+\frac{1}{2}F{G}^{\prime }\Theta \prime \right]\\ +\left(1+ϵ\Theta \right)\left(1-\frac{\Theta }{{\Theta }_{\gamma }}\right)\mathit{Pr}F{\Theta }^{{}^{\prime }}+\left(1+ϵ\Theta \right)\left(1-\frac{\Theta }{{\Theta }_{\gamma }}\right)\mathit{Pr}{H}_t\Theta =0.\end{array}(18)$

${\theta }_{\gamma }=1/\gamma \left(T_w-T_{\infty }\right)$ is the fluid viscosity number. The case is known as constant viscosity for ${\theta }_{\gamma }\to \infty$ and case for liquids for ${\theta }_{\gamma }=-1.$

Wall stresses (skin friction coefficients) are formulated as

${{Re}^{1/2}C}_F=\frac{{\theta }_{\gamma }}{{\theta }_{\gamma }-1}{\left[1+W^2{\left(F^{″}\left(0\right)\right)}^2\right]}^{n-1/2}{F}^{″}\left(0\right)(19)$

${{Re}^{1/2}C}_G=\frac{{\theta }_{\gamma }}{{\theta }_{\gamma }-1}{\left[1+W^2{\left(G^{\prime }\left(0\right)\right)}^2\right]}^{n-1/2}{G}^{\prime }\left(0\right)(20)$

${Re}^{-1/2}Nu=\left(1+ϵ\right){\Theta }^{\prime }\left(0\right)(21)$

The heat transfer rate in terms of variable thermal conductivity is

Numerical methodology

The formulated ODEs are numerically simulated by a numerical scheme (FEM) [14, 16, and 31]. The description associated with FEM is mentioned below.

Discretization of domain

In this step, the desired ODEs are transformed into numbers of elements (300 elements). Residuals of the current problem are achieved. Eqs. 13–15 are strong forms with boundary conditions. A strong form can be converted into a weak form by collecting all terms on one side and multiplying it by weight functions and integrated it on whole domain. The weighted residuals of the problem implementation due to $F^{\prime }=S$ are as follows:

${\int }_{{\eta }_e}^{{\eta }_{e+1}}{w}_1\left(F^{\prime }-S\right)d\eta =0,(22)$

${\int }_{{\eta }_e}^{{\eta }_{e+1}}{w}_2\left[\begin{array}{c}\frac{{\Theta }_{\gamma }}{{\Theta }_{\gamma }-1}\left\{1+n{\left(W{F}^{″}\right)}^2\right\}{\left\{1+{\left(W{F}^{″}\right)}^2\right\}}^{n-3/2}{S}^{″}\\ +\frac{M^2}{{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2}\left[\left(1+B_e{B}_i\right)F^{\prime }+2B_{e}G\right]\\ \left(\frac{1}{2}{F^{\prime }}^2-F{F}^{″}-2G^2-2\lambda \Theta \right)\end{array}\right]d\eta =0,(23)$

${\int }_{{\eta }_e}^{{\eta }_{e+1}}{w}_3\left[\begin{array}{c}\frac{{\Theta }_{\gamma }}{{\Theta }_{\gamma }-1}\left\{1+n{\left(W{G}^{\prime }\right)}^2\right\}{\left\{1+{\left(W{G}^{\prime }\right)}^2\right\}}^{n-3/2}{G}^{″}\\ +\frac{{\Theta }_{\gamma }}{{\left({\Theta }_{\gamma }-1\right)}^2}{\left\{1+{\left(W{G}^{\prime }\right)}^2\right\}}^{n-1/2}{G}^{\prime }{\Theta }^{\prime }+\left(G{F}^{\prime }-F^{\prime }{G}^{\prime }\right)\\ -\frac{M^2}{{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2}\left[\left(1+B_e{B}_i\right)G-\frac{1}{2}{B}_e{F}^{\prime }\right]\end{array}\right]d\eta =0,(24)$

${\int }_{{\eta }_e}^{{\eta }_{e+1}}{w}_4\left[\begin{array}{c}+\left(1+ϵ\Theta \right)\left(1-\frac{\Theta }{{\Theta }_{\gamma }}\right)\mathit{Pr}F{\Theta }^{\prime }+\left(1+ϵ\Theta \right)\left(1-\frac{\Theta }{{\Theta }_{\gamma }}\right)\mathit{Pr}{H}_t\Theta \\ {\gamma }_1\left(1+ϵ\Theta \right)\left(1-\frac{\Theta }{{\Theta }_{\gamma }}\right)\mathit{Pr}\left[F^2{\Theta }^{″}-QF{\Theta }^{\prime }+\frac{1}{2}F{G}^{\prime }{\Theta }^{\prime }\right]\\ {\left(1+ϵ\Theta \right)\Theta }^{″}+ϵ{\left({\Theta }^{\prime }\right)}^2\end{array}\right]d\eta =0,(25)$

Shape functions

Various shape functions are utilized in the current problem. In this approach, however, a linear shape function is implemented to obtain an approximate solution. Several shape functions are used in FEM. However, the linear type of shape function is utilized in this numerical approach. Shape functions are based on a linear type of polynomial. The values variables ($N,F,\Theta$) are defined as

$N={\displaystyle \sum }_{j=1}^2\left(N_j{\psi }_j\right),F={\displaystyle \sum }_{j=1}^2\left(F_j{\psi }_j\right),\Theta ={\displaystyle \sum }_{j=1}^2\left({\Theta }_j{\psi }_j\right).(26)$

Shape functions are

${\psi }_j={\left(-1\right)}^{j-1}\frac{\eta -{\eta }_{j-1}}{{\eta }_j-{\eta }_{j-1}}.(27)$

Developments of stiffness matrices using assemble process

The Galerkin approach is utilized to derive the stiffness matrices. The concept of the assembly process is implemented to find stiffness matrices and global stiffness matrix. The sizes of the boundary vectors, stiffness matrices, and source vectors are

$K_{ij}^{11}={\int }_{{\eta }_e}^{{\eta }_{e+1}}\left(\frac{d{\psi }_j}{d\eta }{\psi }_i\right)d\eta ,K_{ij}^{12}={\int }_{{\eta }_e}^{{\eta }_{e+1}}\left({\psi }_j{\psi }_i\right)d\eta ,b_i^1=0,K_{ij}^{13}=0,K_{ij}^{14}=0,(28)$

$K_{ij}^{22}={\int }_{{\eta }_e}^{{\eta }_{e+1}}\left[\begin{array}{c}-\frac{{\Theta }_{\gamma }}{{\Theta }_{\gamma }-1}\left\{1+n{\left(W\overline{S^{\prime }}\right)}^2\right\}{\left\{1+{\left(W\overline{S^{\prime }}\right)}^2\right\}}^{n-3/2}\frac{d{\psi }_i}{d\eta }\frac{d{\psi }_j}{d\eta }\\ \frac{{\Theta }_{\gamma }}{{\left({\Theta }_{\gamma }-1\right)}^2}{\left\{1+{\left(W{F}^{″}\right)}^2\right\}}^{n-1/2}\overline{{\Theta }^{\prime }}{\psi }_i\frac{d{\psi }_j}{d\eta }+\frac{1}{2}\overline{S}{\psi }_i{\psi }_j\\ -\overline{F}{\psi }_i\frac{d{\psi }_j}{d\eta }+\frac{M^2\left(1+B_e{B}_i\right)}{{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2}{\psi }_i{\psi }_j\end{array}\right]d\eta ,b_i^2=0,K_{ij}^{21}=0,(29)$

$K_{ij}^{23}={\int }_{{\eta }_e}^{{\eta }_{e+1}}\left[\frac{M^{2}2B_e}{{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2}{\psi }_i{\psi }_j-2\overline{G}{\psi }_i{\psi }_j\right]d\eta ,K_{ij}^{24}={\int }_{{\eta }_e}^{{\eta }_{e+1}}\left[-2\lambda \overline{G}{\psi }_i{\psi }_j\right]d\eta ,(30)$

$K_{ij}^{33}=\underset{{\eta }_e}{\overset{{\eta }_{e+1}}{{\displaystyle \int }}}\left[\begin{array}{c}-\frac{{\Theta }_{\gamma }}{{\Theta }_{\gamma }-1}\left\{1+n{\left(W{G}^{\prime }\right)}^2\right\}{\left\{1+{\left(W{G}^{\prime }\right)}^2\right\}}^{n-3/2}\frac{d{\psi }_i}{d\eta }\frac{d{\psi }_j}{d\eta }\\ \frac{{\Theta }_{\gamma }}{{\left({\Theta }_{\gamma }-1\right)}^2}{\left\{1+{\left(W{G}^{\prime }\right)}^2\right\}}^{n-1/2}\overline{{\Theta }^{\prime }}{\psi }_i\frac{d{\psi }_j}{d\eta }+\overline{G}{\psi }_i{\psi }_j+H{\psi }_i\frac{d{\psi }_j}{d\eta }\\ -\frac{M^2\left(1+B_e{B}_i\right)}{{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2}{\psi }_i{\psi }_j\end{array}\right]d\eta ,(31)$

$K_{ij}^{32}={\int }_{{\eta }_e}^{{\eta }_{e+1}}\left[\frac{M^{2}1/2}{{\left(1+B_e{B}_i\right)}^2+{\left(B_e\right)}^2}{\psi }_i{\psi }_j\right]d\eta ,K_{ij}^{34}=0,K_{ij}^{31}=0,b_i^3=0,(32)$

$K_{ij}^{44}={\int }_{{\eta }_e}^{{\eta }_{e+1}}\left[\begin{array}{c}-\left(1+ϵ\overline{\Theta }\right)\frac{d{\psi }_i}{d\eta }\frac{d{\psi }_j}{d\eta }+ϵ\overline{\Theta }{\psi }_i\frac{d{\psi }_j}{d\eta }\\ +\left(1+ϵ\overline{\Theta }\right)\left(1-\frac{\overline{\Theta }}{{\Theta }_{\gamma }}\right)\mathit{Pr}\left(1-\frac{\overline{\Theta }}{{\Theta }_{\gamma }}\right)\overline{F}{\psi }_i\frac{d{\psi }_j}{d\eta }\\ +\left(1+ϵ\overline{\Theta }\right)\left(1-\frac{\overline{\Theta }}{{\Theta }_{\gamma }}\right)\mathit{Pr}{H}_t{\psi }_i{\psi }_j\\ +{\gamma }_1\left(1+ϵ\overline{\Theta }\right)\left(1-\frac{\overline{\Theta }}{{\Theta }_{\gamma }}\right)\mathit{Pr}\left[\overline{F^2}\frac{d{\psi }_j}{d\eta }\frac{d{\psi }_j}{d\eta }-Q\overline{F}{\Theta }^{\prime }+\frac{1}{2}\overline{F}{G}^{\prime }{\psi }_i\frac{d{\psi }_j}{d\eta }\right]\end{array}\right]d\eta ,(33)$

$K_{ij}^{41}=0,b_i^4=0,K_{ij}^{42}=0,K_{ij}^{43}=0.(34)$

Assembly method

The assembly method is used to assemble all elements. The residual is defined as

$\left[R\right]=\left[M\left(F^{\left(r-1\right)},S^{\left(r-1\right)},{\Theta }^{\left(r-1\right)},{\varphi }^{\left(r-1\right)}\right)\right]\left[\begin{array}{c}{F}^r\\ G^r\\ {\mathrm{S}}^r\\ {\Theta }^r\end{array}\right]=\left[F\right].(35)$

The convergence of the problem must existed under ${10}^{-5}$, which is delivered as

$\frac{{\left({\displaystyle \sum }_{i=1}^N{\left(\left|{\omega }^r-{\omega }^{r-1}\right|\right)}^2\right)}^{1/2}}{{\left({\displaystyle \sum }_{i=1}^N\left|{\omega }^r\right|\right)}^2}<{10}^{-5}.(36)$

Code development and validation

Code based on FEM is developed on MAPLE. The Galerkin finite element method is used to simulate the problem on MAPLE 18. The FEM code is verified in a published study (Malik et al., 2016), which is recorded in Table 1. The grid-independent analysis is shown in Table 2.

TABLE 1

TABLE 1. Validation of results in term of shear stresses and Nusselt number when $H_s=0.0,\mathit{Pr}=0.7,Ec=0.0,{\beta }_e=0.0,{\beta }_i=0.0,\mathrm{a}\mathrm{n}\mathrm{d}\beta =0.2.$

TABLE 2

TABLE 2. Grid-independent investigation for $G\left({\eta }_{\max }/2\right),\Theta \left({\eta }_{\max }/2\right)$, and $F^{\prime }\left({\eta }_{\max }/2\right).$

Outcomes and discussion

The three-dimensional thermal aspects of a Carreau liquid in the presence of variable viscosity in a rotating cone are developed. Features related to ion slip and Hall theory are added in momentum equations. The concept associated with Cattaneo-Christov model (CCM) is utilized, involving heat generation/heat absorption. Moreover, variable fluidic properties are utilized in terms of variable thermal conductivity and viscosity dependent heat energy. The developed model is numerically solved using the finite element approach. The graphical results in view of thermal and velocity field are shown below.

Contrast among constant and variable viscosities in velocity field

In this section, Figures 2–9 are plotted reading secondary and primary velocity fields against several parameters. It is noted that dot curves are associated with constant viscosity for $1/{\Theta }_{\gamma }=0$, and solid curves are plotted for representation of variable viscosity for ${\Theta }_{\gamma }=-1.$ Figures 2, 3 are plotted to notice variation in secondary and $G\left(\eta \right)$ velocities against change in $We.$ The flow associated with secondary and primary velocity curves slows down against change in $We.$ The flow for the case of constant viscosity is higher than the flow for variable viscosity. Thickness of the momentum layer is decreased when $We$ is inclined. Physically, $We$ is known as the Weissenberg parameter, and it is defined as the ratio between elastic force and viscous force. From definition point of view, $We$ is inversely proportional against viscous force. Therefore, the viscosity of fluidic particles is enhanced when $We$ is increased. Figures 4–7 are plotted to determine the visualization of velocities in the presence of ion slip and Hall parameters. For this case, flow of Carreau liquid is enhanced when ion slip and Hall numbers are increased. This is because generalized Ohm’s theory is utilized in current analysis. Moreover, the directly proportional relation is predicted among ion slip and Hall parameters. Lorentz force is visualized against implication of $B_e$ and $B_i$. Therefore, Lorentz force is decreased when $B_e$ and $B_i$ are increased. Lorentz force is produced using a generalized Ohm’s law in momentum equations. The production of electron frequency and electron collision (time) is known as $B_e.$ In additionally, the production of ion cyclotron frequency and ion collision time is called $B_i.$ Fluidic velocity is enhanced when collision and frequency among particles are enhanced. The layers of momentum in terms of thickness is an increasing function versus the implication of $B_e$ and $B_i.$ Therefore, significant acceleration is produced among the particles. Thickness regarding MLs (momentum layers) for constant viscosity (${\Theta }_{\gamma }\to \infty$) is greater than for variable viscosity. The impact of ${\Theta }_{\gamma }$ on velocity fields (primary and secondary directions) is addressed by Figures 8, 9. ${\Theta }_{\gamma }$ is defined as $1/\gamma \left(T_w-T_{\infty }\right)$, while ${\Theta }_{\gamma }$ has inversely proportional against temperature difference. Temperature differences increase among fluidic particles. Therefore, thickness regarding momentum layers is increased versus change in ${\Theta }_{\gamma }.$ Mathematically, ${\Theta }_{\gamma }$ has directly proportional relation versus the impact of velocity fields. Therefore, flow declines when ${\Theta }_{\gamma }$ also decreases.

FIGURE 2

FIGURE 2. Variation in primary velocity field versus $We$.

FIGURE 3

FIGURE 3. Variation in secondary velocity field versus $We$.

FIGURE 4

FIGURE 4. Variation in primary velocity field versus ${\beta }_i$.

FIGURE 5

FIGURE 5. Variation in secondary velocity field versus ${\beta }_i$.

FIGURE 6

FIGURE 6. Variation in primary velocity field versus ${\beta }_e$.

FIGURE 7

FIGURE 7. Variation in secondary velocity filed versus ${\beta }_e$.

FIGURE 8

FIGURE 8. Variation in velocity field versus ${\Theta }_{\gamma }$.

FIGURE 9

FIGURE 9. Variation in secondary velocity field versus ${\Theta }_{\gamma }$.

Contrast among constant and variable viscosities in temperature field

This subsection contains graphical explanation associated with thermal impacts for two cases (variable viscosity and constant viscosity) against change in heat source, $ϵ$, ${\Theta }_{\gamma }$, and $\mathit{Pr}$. These results are shown in Figures 10–13. It is noticed that dot curves are plotted for constant viscosity (${\Theta }_{\gamma }\to \infty$), and solid curves are plotted for viscosity (function of temperature). Figure 10 is prepared to determine the impact of $H_t$ on the thermal profile. It is addressed that heat energy is boosted when $H_t$ is increased. Physically, this occurs due to external heat sources on the thermal profile. Hence, extra heat energy can be added using an external heat source. Additionally, heat energy for constant viscosity is greater than heat energy for case of variable viscosity. It can be noted that the behavior of $H_t$ is based on numerical values. For negative values, $H_t$ is called heat absorption while $H_t$ is called heat absorption for positive values. An effect for $H_t>0$ on temperature performance is greater than effect for $H_t<0$. Thermal layers thickness can be adjusted trough variation in $H_t$ whereas $H_t$ is directly proportional against temperature difference. Therefore, heat energy is significantly increased when $H_t$ is enhanced. The role of $\mathit{Pr}$ on thermal profile is addressed in Figure 11. Thermal profile decreases against change in $\mathit{Pr}.$ Physically, it refers to division among thermal thickness and momentum thickness. It is observed that inverse proportional relation is experienced between $\mathit{Pr}$ and thermal layers. Consequently, thickness for thermal layers is significantly enhanced when $\mathit{Pr}$ is increased. Therefore, heat energy and TBL (thermal boundary layer) can be controlled using numerical values of $\mathit{Pr}.$ Figure 12 captures the impact of $ϵ$ on thermal profile. Mathematically, a directly proportional relationship can be seen among $ϵ$ and temperature difference. Consequently, heat energy is boosted when $ϵ$ is increased. Figure 13 is based on the relationship between heat energy and ${\Theta }_{\gamma }.$ $ϵ$ is dimensionless parameter, which is appeared using concept of non-uniform thermal conductivity. $ϵ$ is appeared in Eq. 11, which has direct proportional relation against temperature dependence (thermal conductivity). Therefore, the temperature difference among particles is increased when $ϵ$ is increased. Physically, temperature increases among particles due to temperature differences. Thermal energy is seen against change in ${\Theta }_{\gamma }.$ It can be noted that thermal energy is decreased against decline values of ${\Theta }_{\gamma }.$ Mathematically, ${\Theta }_{\gamma }$ is termed as $1/\gamma \left(T_w-T_{\infty }\right)$, which is inversely proportional versus temperature difference $\left(T_w-T_{\infty }\right)$. Hence, temperature difference is decreased when ${\Theta }_{\gamma }$ is increased.

FIGURE 10

FIGURE 10. Variation in temperature field versus ${\mathrm{H}}_t$.

FIGURE 11

FIGURE 11. Variation in temperature field versus $\mathit{Pr}$.

FIGURE 12

FIGURE 12. Variation in temperature field versus $ϵ$.

FIGURE 13

FIGURE 13. Variation in temperature field versus ${\Theta }_{\gamma }$.

Measurement of shear stresses and temperature gradient

Table 3 demonstrates the impacts of shear stresses and Nusselt numbers against variation in $\mathit{Pr},M,H_t$, and ${\Theta }_{\gamma }$. From this table, it can be seen that shear stresses are declined versus implication of $H_t.$ However, heat transfer rate is reduced when $H_t$ is increased. $M$ is magnetic parameter, which is appeared using concept of Lorentz force. Divergent forces (skin friction coefficients) are decreased versus the implications of the Lorentz force. The Nusselt number (temperature gradient) is enhanced against the implications of higher-impact Lorentz forces. Velocity is increased when the Lorentz force is enhanced Shear stresses are significantly declined when the Lorentz force is enhanced. These outcomes are presented in Table 3. Additionally, the Prandtl number arguments thermal rate (Nusselt number), and shear stresses are decreased when $\mathit{Pr}$ is enhanced. The division between momentum and thermal layers is represented by $\mathit{Pr}.$ Therefore, skin friction coefficients are enhanced when $\mathit{Pr}$ is increased.

TABLE 3

TABLE 3. Numerical impacts of shear stresses and temperature gradient versus $\mathit{Pr},M,H_t$, and ${\Theta }_{\gamma }.$

Prime consequences of performed research

The thermal features of 3D-Carreau model are developed under considerations of ion slip and Hall forces in rotating cone. Variable thermal properties in term of viscosity, mass diffusion, and thermal conductivity are addressed. Chemical species and external heat sources are utilized in the presence of a non-Fourier law. The following are the most important outcomes from this investigation:

The computations of the problem become grid independent at 300 elements;

Heat energy diffuses faster into non-Newtonian liquids for temperature-dependent viscosity than for the diffusion of heat energy for constant viscosity;

Motion into fluidic particles is enhanced against higher values of ion slip and Hall currents parameters;

Heat energy is boosted versus higher values of $H_t$, but heat energy is decreased when the Prandtl number is increased;

An enhancement in thermal energy occurs according to $ϵ$, utilized in the production of solar energy, cooking, baking, cooling, drying, and heating;

The thermal transfer rate is applicable in metal grinding, wire paintings, glasses melting, chemical products, solar collectors, machine cutting, food processing, electronic components, nuclear reaction, solar systems, glass fiber, and so on.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

UN: Conceptualization; Investigation; Software; Validation; Writing—review and editing. MS: Data curation; Writing—original draft; Writing—review and editing; Visualization; Methodology. KM: Formal analysis; Funding acquisition; Software. AS: Methodology; Project administration; Resources; Funding acquisition. RA: Project administration; Formal analysis. AG: Visualization; Supervision; Funding acquisition. MS: Software; Validation; Visualization; Supervision; Funding acquisition.

Funding

This work was partially funded by the research center of the Future University in Egypt, 2022.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Nomenclature

$Z,Y,X$ space coordinates

$B_0$ magnitude of magnetic field

$B_e$ Hall parameter

$G$ gravitational acceleration

$T_{\infty }$ ambient temperature

$n$ power law index number

$\lambda$ mixed convection number

$C_p$ specific heat capacitance

$\Omega$ angular velocity

$\mu$ fluidic viscosity

$\gamma F,\Theta$ dimensionless velocity and temperature

$W$ Weissenberg number

$\mathit{Pr}$ Prandtl number

$H_t$ heat source parameter

${Sc}_{\infty }$ Schmidt number at infinity

$Re$ Reynolds number

$Nu$ Nusselt number

PDEs partial differential equations

$w_2,w_1,w_3$ weight functions

${\psi }_j$ shape function

$\stackrel{\sim }{V}$ fluid velocity vector

$\nabla$ gradient operator

$\tau$ Cauchy stress tensor

$B$ magnetic induction

$V,W,U$ velocity components

$\sigma$ electrical conductivity

$B_i$ ion slip number

$T$ fluid temperature

$\beta$ coefficient of thermal expansion

$\Gamma$ time relaxation number

$\rho$ fluid density

$K$ thermal conductivity

$T_w$ wall temperature

${\mu }_{\infty }$ ambient viscosity

$ϵ$ very small number

${\Theta }_{\gamma }$ fluidic viscosity

$M$ magnetic parameter

${\mathit{Pr}}_{\infty }$ Prandtl number at infinity

$Sc$ Schmidt number

${\gamma }_1$ thermal relaxation number

$C_f$ skin friction coefficient

ODEs ordinary differential equations

FEM finite element method

$\eta$ independent variable

${\eta }_e,{\eta }_{e+1}$ stiffness elements

$E$ electrical field vector

$d/dt$ material derivative

$J$ vector current density

$Q$ heat flux vector