Hybrid nanofluid flow in a deformable and permeable channel

1. Introduction

The inclined magnetic field effects on the hybrid nanofluid flow inside a permeable channel with two-folded deformations (dilation and stretching) are of paramount importance due to its enhanced control over the flow and heat movement characteristics.¹ The magnetohydrodynamic effects have been vastly utilized to study the magnetic properties and behavior of the electrically conducting fluids. In industrial processes such as the jet cooling system, nuclear reactors and geothermal processes, a normal magnetic field is usually exploited to dampen the fluid motion.² Raza et al.³ elucidated the impacts of a transverse magnetic field on the nanofluid flow inside a porous channel and revealed that the fluid velocity depreciated in the upper half of the channel but uplifted near the permeable lower wall. Their results unveiled the dominance of the wall permeability on the magnetic behavior of fluid. However, the inclination of the magnetic field can be alternatively utilized to control the flow field inside a permeable channel.⁴ Applications like magnetic resonance imaging, magnetic drug targeting recognize the role of an angled magnetic field.⁵ Recently, Su and Yin⁶ analyzed the modifications due to an angled magnetic field on the squeezing fluid flow over a permeable plate. It was implied that greater inclinations of a magnetic field are useful to control the flow field instead of using the higher strength of the magnetic field. Some efforts have been made to visualize the impacts of an angle magnetic field on the fluid flow inside deforming wavy wall channels.^7,8 But more rigorous efforts are required to understand the role of inclined magnetic field in controlling the flow mediated dilations in case of hybrid nanofluids.

The utility of hybrid nanofluids as a heat carrier medium in various processes of industrial and biomedical nature plays a pivotal role.⁹ Hybrid nanofluids were invented as a better alternate to the mono nanofluids in order to trade off the properties of two or more nano-additives through a single homogeneous medium.¹⁰ Characteristics like improved chemical stability, greater heat absorption and mechanical strength make them a preferred choice of scientists.¹¹ Hybrid nanofluids displayed superior heat transfer performance than the mono nanofluids in these studies. Ghalambaz et al.¹² recently synthesized a novel coupling of Ag-MgO hybrid nanofluids, which is endowed with the benefit of high thermal conductivity of Silver (Ag) nanoparticles and superior stability and anti bacterial properties of magnesium oxide (MgO) nanoparticles. This advantageous combination of hybrid nanofluid has been utilized to augment the convective heat transfer inside different kind of enclosures.^13,14 Some recent studies utilized the hybrid nanofluids as effective heat transfer medium to overcome the disadvantage of the altering conditions at solid boundaries.^15,16 The scrutiny of flow and heat transfer aspects of a hybrid nanofluids pertaining constant thermophysical properties inside deforming channel geometries recently gained attention.^17,18

Critical reviews of the effective thermo-physical properties of hybrid nanofluids remarked with experimental proofs revealed that the surrounding temperature brings a noticeable change in the viscosity and thermal conductivity.^19,20 The increased Brownian movements owning to a rising temperature enhances the thermal conductivity of the hybrid nanoliquids. Whereas the weak Van der Waal forces due to temperature augmentation are responsible for reduction in viscosity. Some theoretical models were developed and modified by the researchers to imbibe the linear temperature dependence on the effective nanofluid viscosity and thermal conductivity.^21,22 The joint effects of variable fluid thermal conductivity and viscosity in case of a viscous micropolar fluid were emphasized by Alzahrani et al.²³ In case of a peristaltic moving Jeffery fluid, the respective declination and increment in the flow velocity due to variable thermal conductivity and viscosity were reported by Nagathan et al.²⁴ Some recent studies mentioned the assessment of the temperature varying viscosity and thermal conductivity of the hybrid nanofluid flow saturating a permeable wall.^25,26 But the influence of temperature variable viscosity and thermal conductivity together on the hybrid nanofluid flow inside permeable deforming channel in presence of flow mediated dilations had not been attended yet.

The phenomenon of flow mediated dilations finds applications in uniformly distributed irrigation systems, exhaust nozzles, coolant circulation, fibrous insulation system, human respiratory system, blood flow in arterial and lympathic organs, etc.^27,28 In medical science, the flow mediated dilation of the artery exists when the blood flow is enhanced in the artery. Uchida and Aoki²⁹ explored the viscous fluid flow within an impermeable expanding or contracting pipe. They realized that the viscous effects are limited to the deforming boundaries. Injection effects along with the channel deformity were later on analyzed by Dauenhauer and Majdalani.³⁰ Many studies highlighted the impacts of small to large scale suction/injection effects on the permeating fluid flow inside deformable channels.^31,32,33 These studies were limited to the exploration of flow field alone, until heat transfer aspects were analyzed.³⁴ The channel deformations in the mentioned studies were solely due to the orthogonal contractions of the permeable walls. The stretchable nature of the walls adds to an additional deformity in the channel configuration.³⁵ Moreover, in the mentioned studies, the channel walls have been assumed to be of evenly permeable. In case of channels with highly deforming rates, these permeations may exceed the fluid injection or suction to an exponential order. Therefore, the exploration of exponential permeability on a two-folded deforming channel (such as squeezing and stretching) may lead to some novel results.

This paper is arranged as follows. Mathematical model for the concerned phenomenon is elaborated in Sec. 2. In Sec. 3, the procedure of spectral quasi-linearization scheme is applied to current model, and the numerical solutions obtained by this scheme are discussed in Sec. 4 by mentioning the convergence criterion. Lastly, the conclusions of the study are pointed in Sec. 5.

2. Mathematical Description of Problem

Consider an unsteady $2$-D flow of an electrically conducting and incompressible Ag-MgO/water hybrid nanofluid inside a deformable horizontal channel (see Fig. 1). The height of the channel is taken as $D(t)=\sqrt{\frac{{\nu }_f(1-bt)}{a}}$ which contracts and expands as the parallel walls moves with velocity $V_d=\frac{dD}{dt}$, where $t<1∕b$ and b stands for squeezing constant.¹⁸ The hybrid nanofluid is also allowed to either enter or exit the channel through the exponentially permeating walls during the expansion or contraction of the channel. The permeabilities of the upper and lower walls are taken as $K_2$ and $K_1$, respectively, with $K_2>K_1$. Also, both walls of the deforming channel are assumed to stretch unsteadily with the velocity $U_w=ax{(1-bt)}^{-1}$, where a denotes the stretching constant. A variable magnetic field $B_0(t)$ applied at an angle $\gamma$ is considered to control the flow environment. The magnetic Reynolds number of the flow is considered to small so that the induced magnetic effects can be ignored.³⁶ Moreover, the ion slip and Hall current are considered negligible due to low electron-atom collision frequency.³⁷ The lower channel wall temperature is taken as $T_1$, whereas the upper channel wall is heated at temperature $T_2(>T_1)$. The radiative heat term is added to the energy conservation equation. The hybrid nanofluid flow is defined in a homogeneous single phase using the Tiwari and Das model. The effective thermophysical properties except the thermal conductivity and viscosity of the hybrid nanofluid are assumed to be non-variable.

• The unsteady magnetic field inclined at an angle $\gamma$ (with x-axis) is considered as⁷: $B=(B_0(t)cos\gamma ,B_0(t)sin\gamma ,0)$, where $B_0(t)=B_0{(1-bt)}^{-1∕2}$.

• The thermal conductivity ($k_{\mathrm{\text{hnf}}}$) and viscosity (${\mu }_{\mathrm{\text{hnf}}}$) of the hybrid nanofluid are assumed to possess the following respective temperature dependence²¹:

  $$\begin{array}{l}{k}_{\mathrm{\text{hnf}}}(T)=k_{\mathrm{\text{hnf}}}\left(1+\frac{1}{k_0}{\left(\frac{dk}{dT}\right)}_0(T-T_1)\right),\\ {\mu }_{\mathrm{\text{hnf}}}(T)={\mu }_{\mathrm{\text{hnf}}}\left(1-\frac{1}{{\mu }_0}{\left(\frac{d\mu }{dT}\right)}_0(T-T_1)\right),\end{array}$$
  where subscript ‘0’ is taken to denote the reference constant value.³⁸

• The fluid is considered to a non-scattering medium supposed to absorb and emit the thermal radiation. The Rosseland approximation of the radiation heat flux is adopted as³⁹

  $$q_r=-\frac{16{\sigma }^{\ast }}{3K^{\ast }}\left(\frac{\partial T^4}{\partial x}+\frac{\partial T^4}{\partial y}\right).$$
  For low-temperature difference in channel, the fourth-order temperature is expanded about $T_0$ by using the Taylor series obtained as
  $$T^4=T_0(4T_0^{2}T-3T_0^3),$$
  where higher-order terms are ignored. Here, $K^{\ast }$ is Rosseland mean absorption coefficient, $T_0$ is the constant reference temperature of fluid, ${\sigma }^{\ast }$ is the Stefan–Boltzmann constant.

Based on all these assumptions, the conservation equations governing the unsteady and incompressible hybrid nanofluid flow are given as^32,33

$$\begin{array}{l}{\rho }_{\mathrm{\text{hnf}}}\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+\frac{\partial }{\partial x}\left({\mu }_{\mathrm{\text{hnf}}}(T)\frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial y}\left({\mu }_{\mathrm{\text{hnf}}}(T)\frac{\partial u}{\partial y}\right)\\ -\frac{{\sigma }_{\mathrm{\text{hnf}}}{B}_0^2{sin}^2\gamma }{(1+bt)}u,\end{array}$$(2)

$$\begin{array}{l}{\rho }_{\mathrm{\text{hnf}}}\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+\frac{\partial }{\partial x}\left({\mu }_{\mathrm{\text{hnf}}}(T)\frac{\partial v}{\partial x}\right)+\frac{\partial }{\partial y}\left({\mu }_{\mathrm{\text{hnf}}}(T)\frac{\partial v}{\partial y}\right)\\ +\frac{{\sigma }_{\mathrm{\text{hnf}}}{B}_0^2{cos}^2\gamma }{(1+bt)}v,\end{array}$$(3)

$$\begin{array}{l}{(\rho c_p)}_{\mathrm{\text{hnf}}}\left(\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=\frac{\partial }{\partial x}\left(k_{\mathrm{\text{hnf}}}(T)\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(k_{\mathrm{\text{hnf}}}(T)\frac{\partial T}{\partial y}\right)\\ -\frac{\partial q_r}{\partial x}-\frac{\partial q_r}{\partial y},\end{array}$$(4)

with the boundary restrictions as³⁴ where u and v are the x-direction and y-direction velocity, respectively, p is the pressure, t stands for time, T is fluid temperature, $\gamma$ is magnetic field angle, $B_0$ is magnetic field strength, ${\lambda }_1$ and ${\lambda }_2$ are constant stretching rates of lower and upper wall, respectively. The symbols $\rho$, $\mu$, $c_p$, k and $\sigma$ stand for density, viscosity, heat capacity, thermal conductivity and electrical conductivity, respectively, with the subscript ‘hnf’ denoting the hybrid nanofluid. The thermophysical properties and the effective models for hybrid nanofluid are mentioned in Tables 1 and 2, respectively, where the sub-indices f, Ag and MgO stand for the base fluid, Ag nanoparticle and MgO nanoparticle, respectively.^12,13 Also, $\varphi$ denotes total nanoparticle volume fraction i.e. $\varphi ={\varphi }_{\mathrm{\text{Ag}}}+{\varphi }_{\mathrm{\text{MgO}}}$.

Now to obtain the similar solutions, the pressure is eliminated from Eqs. (2) and (3), and on utilizing the similarity transformations, we have¹⁸

The modified dimensionless form of the governing equations is obtained as

$$\begin{array}{l}(1-{\mu }_T\theta )f^{iv}-{\mu }_T(2{\theta }^{\prime }{f}^{\prime \prime \prime }+{\theta }^{\prime \prime }{f}^{\prime \prime })+\frac{{\rho }_{\mathrm{\text{hnf}}}}{{\rho }_f}\frac{{\mu }_f}{{\mu }_{\mathrm{\text{hnf}}}}[f{f}^{\prime \prime \prime }-f^{\prime }{f}^{\prime \prime }-A(3f^{\prime \prime }+\eta f^{\prime \prime \prime })]\\ -\frac{{\sigma }_{\mathrm{\text{hnf}}}}{{\sigma }_f}\frac{{\mu }_f}{{\mu }_{\mathrm{\text{hnf}}}}M{sin}^2\gamma f^{\prime \prime }=0,\end{array}$$(7)

with where $\prime$ signifies derivative with respect to $\eta$.

The non-dimensional parameters generated in the obtained equations are given as follows:

Note that ${\mathrm{\text{Re}}}_k>0$ indicates injection through the permeable walls, whereas ${\mathrm{\text{Re}}}_k<0$ stands for suction. Also, $A>0$ denotes channel contraction, and $A<0$ stands for channel dilation.

Engineering Coefficients

The surface drag and the heat transfer at the walls of the channel can be characterized by skin friction coefficient (${\mathrm{\text{Sf}}}_x$) and Nusselt number (${\mathrm{\text{Nu}}}_x$), respectively, as⁴⁰

with the following wall shear stress (${\tau }_w$) and heat transfer coefficient (h) : Using the similarity transformations as defined in (6), the dimensionless form of the skin friction coefficient (Sf) and Nusselt number (Nu) is obtained as

3. Numerical Scheme and Convergence Criterion

The leading dimensionless governing equations (7) and (8) together with boundary restrictions (9) are unraveled by adopting Chebyshev pseudo-spectral quasi-linearization method (CPQM). CPQM is a memory and time optimization procedure that provides highly convergent solutions within a small number of iterations. This method is successful to achieve accurate solutions of coupled nonlinear ODEs.^41,42 In this method, first, the nonlinear components of the governing equations are linearized using the quasi-linearization technique⁴³ and then the corresponding linearized form of leading equations are solved by utilizing the Chebyshev pseudo-spectral collocation method.⁴⁴

To apply this technique on current equations (7–9), take $f_r$ and ${\theta }_r$ as the approximate solutions at iteration step r, and $f_{r+1}$ and ${\theta }_{r+1}$ as the improved solutions of the system at next iteration $r+1$. On choosing

the quasi-linearization form of Eqs. (7)–(9) is written as

$$\begin{array}{l}{a}_{1,r}{f}_{r+1}^{iv}+a_{2,r}{f}_{r+1}^{\prime \prime \prime }+a_{3,r}{f}_{r+1}^{\prime \prime }+a_{4,r}{f}_{r+1}^{\prime }+a_{5,r}{f}_{r+1}+a_{6,r}{\theta }_{r+1}^{\prime \prime }+a_{7,r}{\theta }_{r+1}^{\prime }\\ +a_{8,r}{\theta }_{r+1}=a_{9,r},\end{array}$$(10)

with where the subscript $r=0(1)m$ symbolizes the iteration step and the variable coefficients are defined in Appendix A.

Let the unknown functions in terms of interpolating polynomials be approximated as

where ${\eta }_k=cos\frac{\pi k}{N}$ stands for the Chebyshev–Gauss–Lobatto collocation points with N as number of collocation points, and ${\mathrm{\Psi }}_k(\eta )$ is the interpolation polynomial (Lagrange polynomial for Chebyshev–Lobatto points) defined as⁴⁴

which is of cardinal nature i.e. ${\mathrm{\Psi }}_k({\eta }_i)={\delta }_{ik}$ (Kronecker delta). Here, $T_N$ is the Nth Chebyshev polynomial defined by $T_N(\eta )=cos[N{cos}^{-1}\eta ]$ and $d_k=0$ if $k=0$ or N, otherwise $d_k=1$ if $k=1,2,\dots ,N-1$.

Also, the derivatives of the interpolated functions at the ${\eta }_0,{\eta }_1,\dots ,{\eta }_N$ collocation points are defined as

where p is a positive integer denoting the order of derivative, and entries of $D_{i,k}$ are defined as⁴⁴

where $d_0=d_N=0$, otherwise $d_k=1$.

Now, using the expressions (13) and (14) at interior collocation points (${\eta }_i$) for $i=1,2,\dots ,N-1$, the discretized form of the coupled equations (10) and (11) is obtained as

with the boundary restriction (12) at ${\eta }_0$ and ${\eta }_N$ as The discretized equations (15) and (16) together with boundary conditions (17) form an iterative system of $(2N+2)\times (2N+2)$ linear equations, which are solved iteratively in the MATLAB environment by choosing the following initial guesses (at $r=0$):

$$\begin{array}{l}{f}_0(\eta )=\frac{{\lambda }_1+{\lambda }_2-(e^{-K_d}+1){\mathrm{\text{Re}}}_k}{4}{\eta }^3-\frac{{\lambda }_1-{\lambda }_2}{4}{\eta }^2-\frac{{\lambda }_1+{\lambda }_2-3(e^{-K_d}+1){\mathrm{\text{Re}}}_k}{4}\eta \\ -\frac{{\lambda }_1-{\lambda }_2-2(e^{-K_d}+1){\mathrm{\text{Re}}}_k}{2},\\ {\theta }_0(\eta )=\frac{1+\eta }{2}.\end{array}$$

Here, the accuracy of numerical solutions obtained by the application of Chebyshev pseudo-spectral quasi-linearization method is ensured and the criterion for tolerable error is fixed. For this purpose, the convergence norms of the difference between the flow variables $f_r$ and ${\theta }_r$ at consecutive iteration steps are evaluated as follows:

The accuracy of the solution or the tolerable error is fixed as $1e-7$, which is easily achieved at fourth iteration step by taking 80 collocation points (see Table 3).

4. Discussion of the Numerical Solution

The solutions are reported in the form of tables and figures to discuss the physical aspects of the variations in solutions (velocity field and temperature field) due to the alteration in parametric values of the involved dimensionless parameters. In each figure, solid and dotted lines are used to present hybrid nanofluid and nanofluid, respectively. The default values of these parameters are taken as ${\mathrm{\text{Re}}}_k=4$ (injection), ${\mathrm{\text{Re}}}_k=-4$ (suction), $A=-1$ (channel dilation), $A=1$ (channel contraction), $\gamma =45^{\circ }$, $M=10$, ${\mu }_T=0.5$, $k_T=2$, $K_d=4$, $\mathrm{\text{Rd}}=10$, $\mathrm{\text{Pr}}=7$, ${\lambda }_1=1$, ${\lambda }_2=1$, ${\varphi }_{\mathrm{\text{Ag}}}=0.05$ and ${\varphi }_{\mathrm{\text{MgO}}}=0.05$. The comparison table, Table 4, exhibits the strong similarity of results as a reducible case.¹⁸

The simultaneous analysis is carried out by comparing the velocity and temperature fields for Ag-MgO/water hybrid nanofluid (HNF) and Ag/water nanofluid (NF). The novel inter-dependence of squeezing/dilation and injection/suction is explored by choosing two physical cases. In Case I (dilation-injection), the volume of flow is maintained by injecting the same fluid into the flow field during the expansion of the width of the channel, whereas the fluid is pressed out by suction in Case II (contraction-suction) when the squeezing forces exist on the permeable walls of the channel. In both these cases, the walls of the channels are considered as stretchable and permeable to describe the pressed and elongational flow behavior. This type of flow exists in food processing, hot plate welding and rheological testing. The streamlines are plotted in Figs. 2 and 3 against permeation Reynolds number (${\mathrm{\text{Re}}}_k$) and flow mediated dilations/contractions (A) to exhibit the flow pattern changes with these parameters. It is evident from Fig. 2 that circulatory pattern disappears as injection/suction raises. The flow rate is elevated with $A<0$ (channel dilation) but it is depressed with $A>0$ (channel contractions) as seen in Fig. 3.

4.1. Variations in velocity distribution

Figures 4 and 5 show the variations in velocity field ($f^{\prime }(\eta )$) with permeation Reynolds number (${\mathrm{\text{Re}}}_k$) and dilation/compression parameter (A). Figure 4 exhibits that magnitude of velocity is enhanced with the incremented values of permeation Reynolds number in each case. However, profiles are inverted in Case II when compared with Case I. The results are true because lifted values of ${\mathrm{\text{Re}}}_k$ allow the more fluid to enter/leave the flow field through the pores of the walls and thus velocity is raised. Figure 5 portrays the configuration of the velocity profiles ($f^{\prime }(\eta )$) with the dilation/compression parameter (A). Almost two inflection points are detected in each case, and it is inferred that the magnitude of velocity is raised with the widening of the channel width in case of applied fluid injections. Though velocities are reduced near the walls, but this phenomenon is physically true because compression/expansion velocities create inflection points and the widening of the channel provides more space for fluid to flow in the mid portion. However, reverse trend is noticed when compression forces are applied on the walls of the channel.

Figures 6 and 7 elaborate the velocity behavior with the inclination angle ($\gamma$) and strength of the magnetic field (M), respectively. The depressing behavior of velocity profiles is noticed in each case with the enhancement of the angle of the magnetic field and magnetic field strength. Here, inflection points are also detected with the magnetic field parameter. Physically, the nanoparticles constituting the hybrid nanofluid get magnetized and form a chain-like structure in presence of magnetic field which eventually obstruct the fluid flow. However, this phenomenon is prominent near the larger permeating upper wall as magnetic field can easily penetrate the channel. It turns out that instead of using the enhanced Lorentz force for the reduction of velocity, the angled magnetic field of uniform strength can be alternatively used to obtain the similar results.

Figures 8 and 9 exhibit the variations in velocity field with permeability difference parameter ($K_d$) and variable viscosity parameter (${\mu }_T$). The velocity is decreased with $K_d$. In physical sense, the boundary layer is decreased on increasing permeability differences between the channel walls which reduced the flow in both the cases. The profiles of velocity are also reduced with variable viscosity parameter in the lower portion, but opposite is the trend in upper portion due to the appearance of a turning point in the flow field.

Figures 10 and 11 explain the role of stretching forces (${\lambda }_1$, ${\lambda }_2$) in the management of velocity field. In Case I, the stretching parameter of both the plates substantially changes the structure of velocity profiles, whereas the stretching parameter is unable to significantly alter the profile structure in Case II, though magnitude of velocity is raised here. Physically, it means that the channel dilation and stretching forces accelerate the effect of injection velocity which further inverts the velocity profiles significantly. However, the compression and stretching forces are unable to change the structure of profiles in Case II due to their opposite nature.

4.2. Variations in temperature distribution

Figures 12 and 13, respectively, report the effects of injection/suction parameter (${\mathrm{\text{Re}}}_k$) and dilation/compression parameter (A) on the temperature field. In Case I, the increasing injection parameter raises the temperature because more heat is carried into the flow field by the injecting fluid. The temperature of the flow field is reduced in Case II because the heated fluid is removed from the flow field by suction. It is noted from Fig. 13(a) that the temperature of hybrid nanofluid is lowered if width of the channel is enhanced, and it is physically true because wider space is generated by dilations which allow the heat to spread, thus the fluid temperature is reduced. The temperature is also reduced when channel is compressed and the fluid is removed by suction (see Fig. 13(b)).

Figures 14 and 15 explain the influence of inclination angles and the strength of magnetic field on the temperature field and it is found that temperature is reduced with heightened magnetic field. Similar trend is also seen for the magnitude of temperature at different angles of the magnetic field. Figures 16 and 17 depict that $\theta (\eta )$ is raised with the stretching parameter of the lower plate in each case, however it is hampered with the stretching parameter of the upper plate. In Case II, the hybrid nanofluid temperature is significantly larger than the temperature of the nanofluid.

Figures 18–20 elaborate the respective influence of variable thermal conductivity parameter ($k_T$), radiation parameter ($\mathrm{\text{Rd}}$) and permeability difference parameter ($K_d$) on the temperature profiles. It is noticeable that temperature is depressed with $k_T$ and $\mathrm{\text{Rd}}$ in Case I, but is augmented in Case II with these parameters. Physically, the enlarged thermal conductivity raises the thermal exchange of the hybrid nanofluid with surrounding walls which resulted in enhanced temperature. However, permeation difference parameter ($K_d$) has opposite effects on the temperature field and cooling effects are attained with $K_d$ in Case II. The hybrid nanofluid temperature is larger in Case II in relation to nanofluid temperature.

4.3. Variations in skin-friction and Nusselt number

Figure 21 depicts the relative influence of magnetic field inclination angle ($\gamma$) and variable viscosity parameter (${\mu }_T$) on skin-friction as well as the influence of $\gamma$ and variable thermal conductivity parameter ($k_T$) on heat transfer. It is dictated that $\mathrm{\text{Sf}}$ at both the walls is augmented with $\gamma$ but depressed with ${\mu }_T$ in both the cases. The thermal conductivity parameter accelerates $\mathrm{\text{Nu}}$ at both the plates in each case. However, $\mathrm{\text{Nu}}$ of lower wall is decreased with $\gamma$ but it is lifted at the upper wall in each case.

Tables 5 and 6 are provided to interpret the significance of A, ${\mathrm{\text{Re}}}_k$, $K_d$, M, ${\lambda }_1$, ${\lambda }_2$ and $\varphi$ in controlling the flow transfer rates ($\mathrm{\text{Sf}}$) and heat movement rate ($\mathrm{\text{Nu}}$) at both sides of the channel. The dilation-injection case is considered in Table 5 whereas compression-suction case is reckoned in Table 6. It is analyzed from Table 5 that magnitude of skin-friction is augmented at both walls with ${\mathrm{\text{Re}}}_k$, M, $\mathrm{\text{Rd}}$ and $\varphi$ in Case I, however it is depreciated with dilations and permeability difference parameter. Physically stretching and injection environment develop slippery flow field near the walls that is why $\mathrm{\text{Sf}}$ is depressed with dilation parameter ($A<0$) and permeability difference parameter ($K_d$). The stretching parameter of the corresponding wall substantially influences the skin-friction due to the inversion of the velocity profiles. The Nusselt number is hampered with channel dilations but is augmented with M and $\varphi$ at both ends. $\mathrm{\text{Nu}}$ at the lower plate is raised with ${\mathrm{\text{Re}}}_k$, $K_d$ and ${\lambda }_1$, but reduced with $\mathrm{\text{Rd}}$ and ${\lambda }_2$, however this trend is reversed at the upper plate.

Table 6 exhibits that the trends for $\mathrm{\text{Sf}}$ with $A>0$, ${\mathrm{\text{Re}}}_k<0$, $K_d$, M and $\varphi$ in Case II are similar to the trends in Case I (Table 5), whereas opposite trend is observed with $\mathrm{\text{Rd}}$ at the upper channel wall. The stretching forces in Case II enhance the coefficients of skin-friction at both ends as the inverted profiles maintain their shape. In Case II, the trends for $\mathrm{\text{Nu}}$ with ${\lambda }_1$, ${\lambda }_2$ and $\varphi$ are similar to the trends in Case I, but reverse trends in $\mathrm{\text{Nu}}$ noted with $A>0$, ${\mathrm{\text{Re}}}_k<0$ and $K_d$. However, the heat transfer at both walls is augmented with M but is depressed with $\mathrm{\text{Rd}}$ in Case II.

5. Conclusion

The flow of hybrid nanofluid inside the channel of exponentially permeable stretching walls was considered under the assumption of flow mediated dilations/contractions. The flow field is exposed to angled magnetic field of variable strength. The temperature-dependent viscosity/thermal conductivity was considered due to the presence of radiative heat term in energy equation, as it is can largely influence the heat movement characteristics. Due to the substantial applications of this study to science, engineering and technology, the following findings are presented:

• Channel dilation/compression, magnetic field and stretching parameter develop inflection points in the flow field.

• Hybrid nanofluid temperature is larger by 18.51% in comparison to mono-nanofluid in Case II (contraction-suction) whereas in Case I (dilation-injection) smaller deviation in temperature is seen.

• In both cases, enlarged permeability differences ($K_d$) reduced by 47.85% of the skin-friction coefficients at both the walls.

• Greater inclination angles of magnetic field ($\gamma$) enhanced the heat movements by 10.05% of the hotter wall in both cases.

• The Case II is more suitable from engineering point of view as skin-friction is reduced and heat movement rate is enhanced with channel compression.

• The skin friction decreased by 52.18% with large viscosity variable parameter, whereas larger variable thermal conductivity parameters urged by 46.55% of the heat transfer rates in both cases.

The reported flow mediated dilation/contraction of the study can be used in targeted drug delivery (especially cardiovascular drugs), assessment of endothelial function, measurement of brachial artery dilation and blood vessel damage. The angled magnetic field can be utilized to successfully control the environmental skin-friction and heat transfer.