Thermal investigation and physiochemical interaction of H 2 O and C 2 H 6 O 2 saturated by Al 2 O 3 and γAl 2 O 3 nanomaterials

Abstract

Applications:

The interaction of nanoparticles and base solvents of different nature attained much interest of the researchers in the recent time. These use in medication, detection of cancer cells, applied thermal engineering, and electrical and mechanical engineering. Among the broad range of applications, investigation of nanofluid through converging/diverging channel is important which is of much interest in the field of medical sciences.

Purpose and methodology:

The core purpose of this study is to introduce a new heat transfer model for two natures of nanofluids with bi host solvents. The model in hand achieved through nanofluid expressions, similarity equations and induction of novel dissipation effects. At later stage, numerical treatment is performed to explore the actual behaviour of nanofluids inside the oblique walls which is very important.

Core findings:

From the drawn results, it is found that the motion could be controlled by expanding the channel walls ( $\alpha =5^o$ ) and high Re and in Al₂O₃-H₂O it is optimum. The nanofluids based on Al₂O₃ and C₂H₆O₂ have much ability to transmit heat than the other nanofluids. Moreover, dissipation effects ( $Ec=0.1,0.2,0.3,0.4$ ) played significant role and boosted the temperature while keeping $Re=70,\alpha =5^o$ and $\alpha =-5^o$ , respectively. Also, the study is validated and achieved good agreement between existing and the current study.

Nomenclature

α Converging/diverging parameter

$\alpha <0$  Converging parameter

$\alpha >0$ Diverging parameter

u Component of velocity

θ Polar angle

T Temperature

${\mathrm{T}}_{\mathrm{w}}$  Temperature at the wall

ϕ volume fraction

ρ_nf Nanofluid density

μ_nf Nanofluid dynamic viscosity

${\mathrm{k}}_{\mathrm{nf}}$  Nanofluid thermal conductivity

${\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{nf}}$  Heat capacity of the nanofluid

μ_f Dynamic viscosity of the base solvent

${\left(c_p\right)}_f$  Heat capacity of the fluid

${\rho }_{\mathrm{f}}$  Density of the base fluid

${\mathrm{k}}_{\mathrm{f}}$  Regular liquids thermal conductance

$F(\eta )$  dimensionless velocity

$\beta (\eta )$  dimensionless temperature

Nu Nusselt number

Pr Prandtl number

Re Reynolds number

Ec Eckert number

r Radial direction

Introduction

Thermal transport between two non-parallel walls is one of the potential research motives. Such type of flow is also called flow between converging/diverging channel, flow between oblique channel or Jeffery–Hamel flow. The analysis of such sort of flows^1,2 cannot be overlooked due to their wide uses in daily life as well as in industrial area. These comprised in mechanical, aerospace, chemical, environmental and civil engineering, etc. Also, flow bounded by two oblique walls occur naturally in rivers and oceans. The earlier contribution in this important direction gave by Jeffery ¹ and Hamel. ² That is why such flows named as Jeffery–Hamel flows.

It is a fact that common fluids have poor heat transfer characteristics. However, a huge amount of thermal transportation is essential to accomplish various industrial production processes. In such processes, regular fluids fail to provide considerable rate of heat transfer. The researcher’s thought that the heat transfer properties of conventional fluids can be increased if nano size of various metal particles and their oxides suspended in the common fluids.

To overcome the major problem of heat transfer enhancement, nanofluids developed which are colloidal suspensions of nano-sized particles dispersed in the conventional fluids stably and interruptedly. This newly developed class termed as Nanofluids. Choi ³ contributed earlier in this direction. The nanofluids are a colloidal mixture of metals like copper, Ag and Al₂O₃ in the host liquids. The dispersion of nanoparticles in the base solvent improves its thermal conductivity which is significant for thermal transport. Later on, many studies presented regarding the nanofluids heat transport in different scenarios. Ali etal. ⁴ and Sobamowo et al. ⁵ attracted towards the investigation of hybrid and third grade nanoliquids by considering the effects of stretchable surface, MHD, solar thermal radiations and squeezed parameters. The studies related to heat transport enhancement if cross nanoliquids with two sort of nanoparticles, Joule heating and moveable surface with changeable thickness are discussed by Chamkha et al. ⁶ and Wahed, ⁷ respectively.

The flow between parallel rotating plates is very significant due to their variety of applications in electric engine generator, cooling thermal system, etc. In the light of these facts, very recently Khan et al. ⁸ presented a three-dimensional nanofluids flow bounded between two parallel sheets. The plates and nanofluid rotate together. Further, they assumed that nanoparticles and regular liquids must be in thermal equilibrium. They examined very significant trends of various flow quantities on the momentum and thermal transport. The effects of under consideration flow parameters also computed. The mechanism of heat transport in water/Cu and water/Ag between converging/diverging channel by implementing Least Square Method and thermal radiation, magnetic field effects in the presence of stretching/shrinking effects of the walls reported in Usman et al., ⁹ Dogonchi and Ganji, ¹⁰ respectively. Latest significant investigation regarding heat transport between converging/diverging channel (Akinshilo et al. ¹¹ ) under varying physical constraints like porous media effects, internal heat generation/absorption, the dynamics of blood suspended by nanoparticles ¹² inside the channel with flexible walls, analytical treatment of heat transfer model through ADM, ¹³ and different nanofluids prepared by variety of nanoparticles and host solvent with different characteristics are examined by Ullah et al. ¹⁴ and Kumar et al., ¹⁵ respectively. The researchers concluded that the hybrid nanoliquids have unique heat transport mechanism and very useful for future applications.

Recently, Rashidi et al. ¹⁶ examined the flow of nanofluids over a stretching sheet. They observed that the fluid temperature declines with effective Prandtl model and rises without an effective Prandtl model. Also, entropy generation was part of his discussion. A review analysis on magneto-nanofluid and behavior of different nondimensional physical quantities presented in Sheikholeslami and Rokni. ¹⁷ The investigation of thermal transport mechanism in tangent hyperbolic fluid by taking physical impact of activation energy and zero mass flux flow condition reported in Kumar et al. ¹⁸ The solution of the model calculated via a numerical technique and varied the governing flow parameters to investigate the fluid flow behaviour. Intensifications in the local heat transport mechanism examined against Nr and the fluid velocity rises by increasing the value of We. The analysis of thermal and mass transportation in 2D flow over a paraboloid of revolution described in Zeeshan et al. ¹⁹ They analysed that the boundary layer thickness rises subject to the thickness parameter.

In the present time, researchers attracted towards the analysis of nanofluids inspired by MHD and inclined surfaces, ²⁰ solar thermal radiation in the existence of no mass flux, ²¹ thermosolutal convection and cross diffusion gradients. ²² They observed interesting outcomes regarding the heat transport mechanism in nanoliquids under aforementioned conditions and suggested these would be useful for many industrial applications. In recent years, researchers contributed in the field of nanofluids and their applications and effectiveness in various disciplines under different scenarios.^23–25 The inspection of reversible and irreversible cycles under physical aspects is important for thermodynamical studies and Tayebi et al. ²⁶ covered the analysis in annular enclosure. Recently, scientists and engineers from different fields focused on the development of various improved thermal models in varying geometries. These analysed by Chamkha et al., ²⁷ Eshaghi et al., ²⁸ Dogonchi et al. ²⁹ and Afshar et al. ³⁰ and studies described therein will cover the literature.

The above cited literature reveals that different researchers reported significant studies from various physical aspects in different nanofluids. However, no one examined a comparative heat transfer analysis in water and EG (ethylene glycol) dispersed by ${\mathrm{Al}}_2{\mathrm{O}}_3$ and $\gamma {\mathrm{Al}}_2{\mathrm{O}}_3$ nanoparticles between converging/diverging channel under viscous dissipation effects. This is significant research gap to detect a nanofluid with rich thermal characteristics. Therefore, this research will address the following novel openings and comparative thermal enhancement in H₂O and EG dispersed with ${\mathrm{Al}}_2{\mathrm{O}}_3$ and $\gamma {\mathrm{Al}}_2{\mathrm{O}}_3$ nanoparticles. After, successful completion of the study, the audience will aware to the answer of the following research questions.

• Comparative heat transport mechanism in ${\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$ , $\gamma {\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}{\mathrm{Al}}_2{\mathrm{O}}_3-\mathrm{EG}$ and $\gamma {\mathrm{Al}}_2{\mathrm{O}}_3-\mathrm{EG}$ .

• How Re and integrated dissipation effects will impact on the momentum and energy transport inside the channel.

• The local heat transfer rate at the channel walls by changing Eckert number and solid volume fraction of nanoparticles.

• How the fluid motion behaves near the channel walls.

Formulation of the models

Unidirectional laminar flow of nanofluids has crucial role in many applied research and in medical sciences. Such flows frequently occur in the practical life and flow between converging/diverging channel which exhibits unidirectional flow is of huge interest in the area of medical sciences and possess fascinating dynamics. The fluid moves with the velocity u around the middle line. Since the flow is only along the central line; therefore, only u component will be responsible for the nanoliquid motion. Moreover, θ indicates the polar angle and r is the radial direction. The channel walls are stationary and positioned at separating angle α which corresponds to converging ( $\alpha <0$ ) and diverging ( $\alpha >0$ ) walls. Further used nanomaterials uniformly dispersed in the host liquids and thermally in equilibrium. Figure 1 depicts the flow of particular nanofluids between two non-parallel walls.

Figure 1.

The flow of nanofluids between two non-parallel walls.

The particular set of PDEs that govern the flow of nanofluids between converging/diverging channel are as under ³¹ :

$${\mathrm{r}}^{-1}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{ru}\right)=0,$$

(1)

$${\rho }_{\mathrm{nf}}\left(\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{r}}\right)=-\frac{\partial \mathrm{p}}{\partial \mathrm{r}}+{\mu }_{\mathrm{nf}}\left(\frac{{\partial }^2\mathrm{u}}{\partial {\mathrm{r}}^2}+{\mathrm{r}}^{-1}\frac{\partial \mathrm{u}}{\partial \mathrm{r}}+{\mathrm{r}}^{-2}\frac{{\partial }^2\mathrm{u}}{\partial {\theta }^2}-{\mathrm{r}}^{-2}\mathrm{u}\right)$$

(2)

$$-\frac{1}{\mathrm{r}{\rho }_{\mathrm{nf}}}\frac{\partial \mathrm{p}}{\partial \theta }+\frac{2{\mathrm{r}}^{-2}{\mu }_{\mathrm{nf}}}{{\rho }_{\mathrm{nf}}}\frac{\partial \mathrm{u}}{\partial \theta }=0,$$

(3)

$$\begin{array}{l}{\left(\rho {\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{nf}}\mathrm{u}\frac{\partial \mathrm{T}}{\partial \mathrm{r}}={\mathrm{k}}_{\mathrm{nf}}\left(\frac{{\partial }^2\mathrm{T}}{\partial {\mathrm{r}}^2}+{\mathrm{r}}^{-1}\frac{\partial \mathrm{T}}{\partial \mathrm{r}}+{\mathrm{r}}^{-2}\frac{{\partial }^2\mathrm{T}}{\partial {\theta }^2}\right)\\ +{\mu }_{\mathrm{nf}}\left(4{\left(\frac{\partial \mathrm{u}}{\partial \mathrm{r}}\right)}^2+{\mathrm{r}}^{-2}{\left(\frac{\partial \mathrm{u}}{\partial \theta }\right)}^2\right)\end{array}$$

(4)

Here, law of conservation of mass represented by equation (1), equation (2–4) representing the momentum and energy equations, respectively. The feasible set of boundary conditions for the model are:

$$\begin{array}{c}\frac{\partial \mathrm{u}}{\partial \theta }=0\\ \frac{\partial \mathrm{T}}{\partial \theta }=0\\ u=U\end{array}\}$$

(5)

$$\begin{array}{c}u=0\\ T={\mathrm{T}}_{\mathrm{w}}\end{array}\}$$

(6)

The suitable similarity transformations defined as:

$$\mathrm{F}\left(\eta \right)=\frac{\mathrm{f}\left(\theta \right)}{{\mathrm{f}}_{\max }},\beta \left(\eta \right)=\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{w}}}\mathrm{and}=\frac{\theta }{\alpha }$$

(7)

Integrating equation (1), the following form:

$$\mathrm{f}\left(\theta \right)=\mathrm{ru}\left(\mathrm{r},\theta \right)$$

(8)

The transport properties of the nanofluids estimated by using the following empirical correlations related thermophysical values (Table 1)^32,33:

Table 1.

Nanofluids empirical correlations used in the models.^32,33

${\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$	$\gamma {\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$	$\gamma {\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$
${\mu }_{\mathrm{nf}}={\mu }_{\mathrm{f}}{\left(1-\varphi \right)}^{-2.5}$	${\mu }_{\mathrm{nf}}={\mu }_{\mathrm{f}}(123{\varphi }^2+7.3\varphi +1)$	${\mu }_{\mathrm{nf}}={\mu }_{\mathrm{f}}(306{\varphi }^2-0.19\varphi +1)$ ,
${\mathrm{k}}_{\mathrm{nf}}={\mathrm{k}}_{\mathrm{f}}\left\{\frac{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}-2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}+2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}\right\}$ .	${\mathrm{k}}_{\mathrm{nf}}={\mathrm{k}}_{\mathrm{f}}(4.97{\varphi }^2+2.72\varphi +1)$	${\mathrm{k}}_{\mathrm{nf}}={\mathrm{k}}_{\mathrm{f}}(28.906{\varphi }^2+2.8272\varphi +1)$ ,
Density and heat capacity
${\rho }_{\mathrm{nf}}=\left\{\left(1-\varphi \right)+\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}}\right\}{\rho }_{\mathrm{f}}$ , ${\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{nf}}=\left\{\left(1-\varphi \right)+\frac{\varphi {\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{s}}}{{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}\right\}{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}$
Model	$\rho (\mathrm{kg}/{\mathrm{m}}^3)$	${\mathrm{c}}_{\mathrm{p}}({\mathrm{kg}}^{-1}{\mathrm{K}}^{-1})$	$\mathrm{k}({\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1})$	$\mathrm{Pr}$
${\mathrm{H}}_2\mathrm{O}$	997.1	4179	0.613	6.96
${\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$	1116.6	2382	0.249	204
${\mathrm{Al}}_2{\mathrm{O}}_3$	3970	765	40	–

Using the self-similar variables embedded in equation (7) and the empirical correlations for the nanofluids, the following nonlinear and coupled nature of the flow model:

Final heat transfer models

The final version of the models is given as below:

${\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$	$\mathrm{F}‴+{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}}\right)2\alpha \mathrm{ReFF}\prime +4{\alpha }^2\mathrm{F}\prime =0$
	${\beta }^{″}+\mathrm{PrEc}\frac{\left\{\left(1-\varphi \right)+\frac{\varphi {\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{s}}}{{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}\right\}}{\left\{\frac{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}-2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}+2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}\right\}{\left(1-\varphi \right)}^{2.5}}\left(4{\alpha }^2{\mathrm{F}}^2+{\mathrm{F}\prime }^2\right)=0$ ,
$\gamma {\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$	$\mathrm{F}‴+\frac{\left(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}}\right)}{(123{\varphi }^2+7.3\varphi +1)}2\alpha \mathrm{ReFF}\prime +4{\alpha }^2\mathrm{F}\prime =0,$ ,
	${\beta }^{″}+\mathrm{PrEc}\frac{\left\{\left(1-\varphi \right)+\frac{\varphi {\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{s}}}{{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}\right\}}{\left(4.97{\varphi }^2+2.72\varphi +1\right)}(123{\varphi }^2+7.3\varphi +1)\left(4{\alpha }^2{\mathrm{F}}^2+{\mathrm{F}\prime }^2\right)=0$
$\gamma {\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$	$\mathrm{F}‴+\frac{\left(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}}\right)}{(306{\varphi }^2-0.19\varphi +1)}2\alpha \mathrm{ReFF}\prime +4{\mathrm{\alpha }}^2\mathrm{F}\prime =0,$
	${\beta }^{″}+\mathrm{PrEc}\frac{\left\{\left(1-\varphi \right)+\frac{\varphi {\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{s}}}{{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}\right\}}{(28.906{\varphi }^2+2.8272\varphi +1),}(306\varphi {\varphi }^2-0.19\varphi +1)\left(4{\alpha }^2{\mathrm{F}}^2+{\mathrm{F}\prime }^2\right)=0$ ,
Boundary conditions	$\begin{array}{c}F\left(\eta \right)=1\\ \mathrm{F}\prime \left(\eta \right)=0\end{array}\}$ at $\eta =0$ $\mathrm{F}\left(\eta \right)=0$ , at $\eta =1$
	${\beta }^{\prime }\left(\eta \right)=0$ , at $\eta =0$ $\beta \left(\eta \right)=1$ , at $\eta =1$

The nondimensional physical parameters appeared in above models are $\mathrm{Pr}=\frac{{\mathrm{U}}^2}{{\left({\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}{\mathrm{T}}_{\mathrm{w}}}$ , $\mathrm{Ec}=\frac{{\mu }_{\mathrm{f}}{\left({\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}{{\mathrm{k}}_{\mathrm{f}}}$ and α is dimensionless converging/diverging parameter. The study of heat transfer at different locations inside the walls (Local Nusselt number) and shear stresses is important from various engineering point of view. For the current problem, these defined in the following manner:

$C_f={\tau }_w{\left[{\rho }_{nf}{U}^2\right]}^{-1}$ and $N_u=q_w{|}_{\theta =\alpha }r{\left[T_w{k}_f\right]}^{-1}=-\frac{\partial T}{\partial r}{|}_{\theta =\alpha }r{k}_{nf}{\left[T_w{k}_f\right]}^{-1}$ .

Further, necessary mathematical action reduced the above expression in the following form (Table 2).

Table 2.

The formulae for skin friction and local Nusselt number.^32,33

${\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$	${\mathrm{ReC}}_{\mathrm{F}}=\frac{1}{{\left(1-\varphi \right)}^{2.5}(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}})}\mathrm{F}\prime (1)$ ,	$\alpha \mathrm{Nu}=\left\{\frac{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}-2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}+2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}\right\}(-{\beta }^{\prime }(1))$
$\gamma {\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{H}}_2\mathrm{O}$	${\mathrm{ReC}}_{\mathrm{F}}=\frac{(123{\varphi }^2+7.3\varphi +1)}{(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}})}\mathrm{F}\prime (1)$ ,	$\alpha \mathrm{Nu}=(4.97{\varphi }^2+2.72\varphi +1)(-{\beta }^{\prime }(1))$
$\gamma {\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{C}}_2{\mathrm{H}}_6{\mathrm{O}}_2$	${\mathrm{ReC}}_{\mathrm{F}}=\frac{(306{\varphi }^2-0.19\varphi +1)}{(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}})}\mathrm{F}\prime (1)$ ,	$\alpha \mathrm{Nu}=\left(28.906{\varphi }^2+2.8272\varphi +1\right)\left(-{\beta }^{\prime }\left(1\right)\right)$

Mathematical analysis

As closed solutions are incredible for nonlinear nature of the flow models such as appeared in the study. Therefore, numerical technique is followed for the solution purpose. Implemented technique is easy to use, accurate and has less computational cost. Primarily, it works on the reduction of highly nonlinear model into joint system of first order which then solve using the numerical technique. The detailed implementation procedure is given below:

$${\mathrm{z}}_1=\mathrm{F},{\mathrm{z}}_2=\mathrm{F}\prime ,{\mathrm{z}}_3=\mathrm{F}″$$

(9)

$${\mathrm{z}}_4=\beta ,{\mathrm{z}}_5={\beta }^{\prime }$$

(10)

Firstly, written the models in the following version:

$A{l}_2{O}_3$ model

$$\mathrm{F}‴=-{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}}\right)2\alpha \mathrm{ReFF}\prime -4{\alpha }^2\mathrm{F}\prime =0,$$

(11)

$$\begin{array}{l}{\beta }^{″}=-\mathrm{PrEc}\frac{\left\{\left(1-\varphi \right)+\frac{\varphi {\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{s}}}{{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}\right\}}{\left\{\frac{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}-2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}+2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}\right\}{\left(1-\varphi \right)}^{2.5}}\\ \left(4{\alpha }^2{\mathrm{F}}^2+{\mathrm{F}\prime }^2\right)=0\end{array}$$

(12)

Thereafter, substitutions given in equations (9) and (10) utilized and obtained the following mathematical model:

$$\left[\begin{array}{c}{\mathrm{z}}_1^{\prime }\\ {\mathrm{z}}_2^{\prime }\\ {\mathrm{z}}_3^{\prime }\\ {\mathrm{z}}_4^{\prime }\\ {\mathrm{z}}_5^{\prime }\end{array}\right]=\left[\begin{array}{c}{\mathrm{z}}_2\\ {\mathrm{z}}_3\\ -{\left(1-\varphi \right)}^{2.5}\left(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}}\right)2\alpha Re{\mathrm{z}}_1{\mathrm{z}}_2-4{\alpha }^2{\mathrm{z}}_2\\ {\mathrm{z}}_5\\ \begin{array}{l}-PrEc\frac{\left\{\left(1-\varphi \right)+\frac{\varphi {\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{s}}}{{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}\right\}}{\left\{\frac{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}-2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}{{\mathrm{k}}_{\mathrm{s}}+2{\mathrm{k}}_{\mathrm{f}}+2\varphi \left({\mathrm{k}}_{\mathrm{f}}-{\mathrm{k}}_{\mathrm{s}}\right)}\right\}{\left(1-\varphi \right)}^{2.5}}\\ \left(4{\alpha }^2{\mathrm{z}}_1^2+{\mathrm{z}}_2^2\right)\end{array}\end{array}\right]$$

(13)

$\mathrm{Re}$ model

$$\mathrm{F}‴=-\frac{\left(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}}\right)}{\left(123{\varphi }^2+7.3\varphi +1\right)}2\alpha \mathrm{ReFF}\prime -4{\alpha }^2\mathrm{F}\prime ,$$

(14)

$$\begin{array}{l}{\beta }^{″}=-\mathrm{PrEc}\frac{\left\{\left(1-\varphi \right)+\frac{\varphi {\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{s}}}{{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}\right\}}{\left(4.97{\varphi }^2+2.72\varphi +1\right)}\\ (123{\varphi }^2+7.3\varphi +1)\left(4{\alpha }^2{\mathrm{F}}^2+{\mathrm{F}\prime }^2\right)=0\end{array},$$

(15)

The corresponding system ODEs for α model reduced into the following form:

$$\left[\begin{array}{c}{\mathrm{z}}_1^{\prime }\\ {\mathrm{z}}_2^{\prime }\\ {\mathrm{z}}_3^{\prime }\\ {\mathrm{z}}_4^{\prime }\\ {\mathrm{z}}_5’\end{array}\right]=\left[\begin{array}{c}{\mathrm{z}}_2\\ {\mathrm{z}}_3\\ -\frac{\left(1-\varphi +\frac{\varphi {\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{f}}}\right)}{\left(123{\varphi }^2+7.3\varphi +1\right)}2\alpha Re{\mathrm{z}}_1{\mathrm{z}}_2-4{\alpha }^2{\mathrm{z}}_2\\ {\mathrm{z}}_5\\ \begin{array}{l}-PrEc\frac{\left\{\left(1-\varphi \right)+\frac{\varphi {\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{s}}}{{\left(\rho {\mathrm{c}}_{\mathrm{p}}\right)}_{\mathrm{f}}}\right\}}{\left(4.97{\varphi }^2+2.72\varphi +1\right)}(123{\varphi }^2+7.3\varphi +1)\\ \left(4{\alpha }^2{\mathrm{z}}_1^2+{\mathrm{z}}_2^2\right)\end{array}\end{array}\right]$$

(16)

The system of first order ODEs is then computed by using Mathematica 10.0.

Results and discussion

The importance of various nondimensional physical flow parameters cannot be overlooked. Because these quantities play a very important role in the momentum and energy transport. In the present study, these are Reynolds number ‘Re’, convergent/divergent parameter ‘α’, Eckert number ‘Ec’, volumetric fraction of alumina $({\mathrm{Al}}_2{\mathrm{O}}_3)$ and γ-alumina nanoparticles $(\gamma {\mathrm{Al}}_2{\mathrm{O}}_3)$ and Prandtl number ‘Pr’. It is important to note that $\alpha >0$ corresponds to diverging channel (opening channel) and $\alpha <0$ represents converging channel (narrowing channel). Further, water and ethylene glycol considered as base liquids.

Velocity profile

The velocity of nanofluids alters significantly due to induction of nanoparticles and base solvent characteristics. Therefore, Figure 2((a)–(d)) summarizes the fluid velocity behaviour under variable converging/diverging parameter and Re values. The nanofluids move at high velocity while the channel converging and it declines for opening case. Physically, rapid movement of nanoliquid particles occurs due to decrement in the flow portion and the particles attained optimum motion at the middle line. Therefore, the particles gain momentum which directly enhance the fluid motion. On the other side, diverging case resists the fluid movement due to maximum flow region. The fluid particles dispersed inside the channel, as a consequent the motion drops. Moreover, for high Re values, yield back flow towards the channel walls for diverging case because the particles attract near the wall and follow backflow path. The nanoliquid backflow motion can be controlled for converging case. These effects along with comparative velocity analysis for Al₂O₃-H₂O and γAl₂O₃-H₂O depicted in Figure 2(c) and (d). Similarly, the results for Al₂O₃-C₂H₆O₂ and γAl₂O₃-C₂H₆O₂ nanoliquids decorated in Figure 3((a)–(d)) and this time quite dominant effects observed compared to Al₂O₃-H₂O and γAl₂O₃-H₂O in both opening and narrowing walls.

Figure 2.

The nanofluids velocity against α (a) converging (b) diverging channel Re (c) converging (d) diverging channel for Al₂O₃-H₂O and γAl₂O₃-H₂O.

Figure 3.

The nanofluids velocity against $\alpha$ : (a) converging and (b) diverging channel and $Re$ (c) converging and (d) diverging channel for Al₂O₃-C₂H₆O₂ and γAl₂O₃-C₂H₆O_2.

Temperature profile

Figure 4((a)–(f)) deals with investigation of heat transport and Nusselt number by engaging the parameters under different ranges. The results demonstrated for Al₂O₃-H₂O, γAl₂O₃-H₂O, Al₂O₃-C₂H₆O₂ and γAl₂O₃-C₂H₆O₂ nanoliquids for converging and diverging cases. These results explored that γ nanoparticles based nanoliquids are very reliable for thermal improvement while H₂O and C₂H₆O₂ taken as host solvents. Physically, ultra-improved thermal conductivity of γ-nanoparticles enhances the temperature and is better than simple nanoparticles. Moreover, the enhancement in the temperature is very quick for narrowing case. Physically, in this case, the fluid motion is abrupt and collision phenomena between the particles is very rapid which potentially contributed in the temperature enhancement. These results demonstrated in Figure 4((a)–(d)), respectively.

Figure 4.

The nanofluids temperature against $Ec$ (a) converging and (b) diverging channel and $Ec$ (c) converging and (d) diverging channel for Al₂O₃-C₂H₆O₂ and γAl₂O₃-C₂H₆O₂ and Nusselt number for (e) Ec, (f) ϕ.

Nusselt number and skin friction

The graphical results for changing Nusselt number under Ec and concentration of the nanoparticles are furnished in Figure 4(e) and (f), respectively. The results plotted for four distinct nanofluids and examined that local heat transport gradient is very high for γAl₂O₃-C₂H₆O₂ nanofluid because of their rich thermal conductivity which enhance the heat storage ability. Further, for more dissipative fluid and increasing concentration of nanoparticles have positive impact on Nu. Further, changes in skin friction under increasing Re for both $\alpha >0$ and $\alpha <0$ are computed in Table 3. The results give the signal that skin friction rises for EG base nanofluid than water. The core reason behind this behaviour is the difference between densities of base solvent and nanoparticles.

Table 3.

Numerical computation for skin friction via Re.

Parameter	Water based nanofluid		Ethylene Glycol based nanofluid
$Re$	$\alpha =-5^o$	$\alpha =5^o$	$\alpha =-5^o$	$\alpha =5^o$
10	2.16797	1.81981	2.16399	1.82552
20	2.33848	1.64320	2.32749	1.65470
30	2.50610	1.46580	2.48990	1.48309
40	2.67056	1.28838	2.64939	1.31139

Validity of the study

This section is devoted to fine comparative analysis of the current study for $F^{″}(0)$ with previously published work. The study is restricted for zero fraction factor $\varphi =0$ and $\alpha =5^o$ while the computation is carried out for multiple Re values. The inspection of Table 4 reveals that the current is in well agreement with the results of Motsa et al. ³¹ This proves the authenticity of the conducted study.

Table 4.

Comparison for $F^{″}(0)$ , $\alpha =5^0$ , $\varphi =0$ .

Re	Present	Motsa et al. 31
20	−2.52719	−2.52719
40	−3.16971	−3.16971
60	−3.94214	−3.94214
80	−4.84507	−4.84507
100	−5.86916	−5.86916
120	−6.99705	−6.99705
140	−8.20733	−8.20733
160	−9.47855	−9.47855

Conclusions

The study for different nanofluids laminar flow inside the oblique slanted walls is examined. The proper formulation of the model is performed through nanofluid models and similarity rules. After that, numerical treatment of the model has been done. The following core outcomes of the study investigated:

• Increasing the flow region (diverging channel) and high strength of Re are better to control the fluid motion inside the channel, whereas opposite results inspected for converging case.

• Integrated effects of viscous dissipation found to be good for thermal enhancement in nanofluids.

• γ–aluminium base nanofluids have more ability to transmits heat and is observed very high in the surrounding of middle portion.

• The local heat transport inside the walls is dominant for γAl₂O₃-C₂H₆O₂ than rest of the nanofluids.

• The analysis and code validated through existing scientific data and deduced that the study is replicated.