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Abstract

Flow and heat distribution over a convergent or divergent channel plays a major role in aeronautical, pharmaceutical, dynamic, civil, climatic, and biomechanical engineering, as well as heat transfer through a porous medium, has piqued the interest of researchers due to its numerous aerospace and automotive manufacturing, including waste disposal, raw petroleum products, grain collection, porous coating, petroleum lagoons, groundwater pollution, packed-bed power plants etc. Further, addition of nanoparticles to base liquid will improve the thermal distribution. Based on the above affordable application the present study will focus on heat transfer enhancement in ternary nanofluids flows induced by stretched convergent or divergent channels. The mobility of ternary nanoparticles occurs in the porous zone. The uniform energy absorption and generation influence is included in the energy expression. Through the use of comparable variables, the flow expressions are transformed into a system of non-linear ODEs (ordinary differential equations). To tackle the problem, the Runge-Kutta-Fehlberg fourth and fifth order (RKF-45) scheme with shooting procedure is adopted. The nature of imposing limitations on physically significant quantities is explored and carried out. The augmentation in solid volume-fraction of both stretched/shrinked channels resulted a reduction in temperature. Furthermore, it is found that the increasing heat sink or source factor and Eckert number augmented the rate of energy transport in the diverging channel, but reverse nature is found in the converging channel. It is also noticed that the ternary nanofluid has a greater impact than the hybrid and mano-nanofluid.

Keywords

Ternary nanofluid heat source/sink porous medium convergent/divergent channel stretchable/shrinkable wall

Introduction

In recent days, the thermal transportation of nanofluids and nanomaterials play a critical role in the development of modern products. Oils, ethyl glycol, and water are poor heat transfer liquids; however, by dipping these liquids into nano-sized materials, the thermal conductivity of these liquids can be improved significantly. Choi and Eastman¹ experimented the idea of nano liquid for the first time in 1995. They found that the thermal efficiency is enhanced significantly when compared to the base liquid. Later, Kang et al.² experimentally checked these results. Khanafer et al.³ had examined the thermal distribution concert inside the closed cage containing nano liquid immersed in solid particles. Nanofluids have emerged as one of the significant pillar in the industrial revolution. Nanoliquids have wide range of applications in the field of physical sciences, biological sciences, electronics, transportation, nuclear controlling, environment as well as the security of the nation. Recently, Madhukesh et al.⁴ investigated the Bio-Marangoni convection flow of Casson nano liquid via a porous media with chemically reactive activation energy. Abbasi et al.⁵ deliberated the effectiveness of various temperature dependent nanoparticles along with Joule heating and mixed convection. Upreti et al.⁶ made a numerical investigation on flow of Sisko nanoliquid over a stretched surface in Darcy-Forchheimer porous medium with heat radiation. Wakif et al.⁷ conducted a meta-analysis on the thermal-migration of nano/tiny-sized particles in the movement of various fluids. As the industries demand more thermal distribution, existing nanoliquids are mixed with another diverse type of particles, which resulted new type of nanofluid called as hybrid nano liquid. These types of fluids exhibit more thermal conductivity when compared to normal nanoliquids. These fluids were later on replaced by nano liquid in large-scale industries. Ramesh⁸ reported the shape factor influences on hybrid nanofluid through viscous heating. Madhukesh et al.⁹ addressed the thermophoresis and convective heating condition influences on the hybrid nanomaterials flows induced by the movement of thin needle. Akbar et al.¹⁰ executed the ion-slip and Hall current aspects in hydromagnetized biologically inspired hybrid nanomaterial flows. Joshi et al.¹¹ explored into the heat and mass distribution investigation of magneto hybrid nanofluid motion with volumetric thermal production. In the 21^st century, some researchers shown interest in enhancing the thermal distribution, for this another nanoparticle is being added to the hybrid type nanofluid called as modified nanofluid. Basically, it is a mixture of three distinct nanoparticles in the base liquid. Nadeem and Abbas¹² studied the modified nanofluid flow over a stretching sheet in the presence of porous medium. Abbas et al.¹³ investigated the modified nanofluid under time-dependent viscosity induced by the Riga plate.

Many researchers paid attention to address the heat source or sink effect because it maintained the thermal distribution of the system. Thermal performance of the liquids is controlled with the help of heat source or sink during the liquid motion. This physical effect is widely used in the manufacturing of plastic as well as rubber sheets, removal of waste material contains radioactive nature. Recently, the evaluation of heat sink and source influences on versatile fluid flow problems is executed by distinct investigators. The nanofluid flow and thermal distribution in converging and diverging walls under energy source or sink effects is addressed by Dogonchi and Ganji.¹⁴ Pandey and Kumar¹⁵ examined the MHD flow over stretched and shrinked channels under heat source-sink by using cu-water nanoliquid. Ramesh et al.¹⁶ deliberated the heat transmission analysis of aluminum alloys and magnetic graphene oxide nanoparticles. Ramesh and Madhukesh¹⁷ analyzed the activation energy mechanism in hybrid carbon nanotubes with heat source/sink. Upreti et al.¹⁸ investigated the Ohmic heating and non-uniform heat sink/source roles on 3D Darcy-Forchheimer CNTs nanofluids flows. Singh et al.¹⁹ examined the slip micropolar liquid flow through a permeable wedge.

Porous medium plays one of the major roles in the branches of medical and industrial sciences. In biological area, transportation process in the human kidneys and lungs, stone present in the glad bladder, arteries clogging and even in the blood vessels. We can also observe porous medium naturally in our nature for example wood, limestone, leakage of water in the river buds. From the above point of view, enormous application in the scientific, metallurgy and earth science researchers studied the porous medium. Kumar et al.²⁰ studied the oscillatory hydromagnetic convected energy transfer in converging/diverging channel immersed in porous medium. Akinshilo²¹ studied the nanofluid porous medium in converging/diverging channel using Akbari Ganji method. Ahmad et al.²² deliberated the time-dependent Walter-B nanoliquid flow in the presence of porous space and heat generation. Pandey and Kumar²³ investigated the effects of natural convection and heat radiation on nano liquid mobility in a porous media with viscous dissipation over stretching cylinder. Singh et al.²⁴ used the Keller-Box approach to study the micropolar fluid flow with melting heat transportation and chemical reaction.

Nowadays, growing technologies and industries require devices contain long life duration, better functioning and exact performance. The study over a converging and diverging wall has wide range of application due to its industrial, biological and physical implications. The converging-diverging channel has been widely used in flows over river, cannel, and streams. One of the amazing ways in the human body is the flow of blood through capillaries that are connected to arteries. Rashid et al.²⁵ studied the shape effect of nanoparticles over converging/ diverging channel. The Darcy-Forchheimer model of hybrid nanomaterial flow over stretched convergent-divergent channels is simulated by Ramesh et al..²⁶ Saifi et al.²⁷ analysed the thermal distribution in converging-diverging channel through Adomian decomposition scheme. Adnanet al.²⁸ examined the analytical and numerical calculation of thermal radiation effects over a stretchable converging-diverging channel. Dogonchi and Ganji²⁹ deliberated the MHD nanofluids flows over stretchable/shrinkable thermally radiated convergent-divergent channel. Mishra et al.³⁰ studied the nanoparticle role and energy generation/absorption on stretching shrinking nanofluid in the presence of nanoliquid.

Work	Porous medium	Heat source or sink	Stretching/shrinking	Fluid type
				Nano	Hybrid	Modified
Dogonchi and Ganji¹⁴	⨯	✓	✓	✓	⨯	⨯
Pandey and Kumar¹⁵	⨯	✓	✓	✓	⨯	⨯
Kumar et al.²⁰	✓	✓	⨯	⨯	⨯	⨯
Ramesh et al.²⁶	✓	⨯	✓	⨯	✓	⨯
Present work	✓	✓	✓	⨯	⨯	✓

The present work status is compared with the previously conducted works and it is shown in the above table. Based on the above-served literature, even though many researchers conducted work on convergent divergent channels in the presence of mano and hybrid nanofluids but no one studied the influence of modified nanofluid in the presence of the porous medium, heat source, or sink and stretching/shrinking effect. This is the first work conducted on modified nanofluid.

Mathematical modeling of the problem

Consider a steady, incompressible, two-dimensional flow of a modified nanofluid containing copper $(C u)$ , aluminum oxide $(A l_{2} O_{3})$ , silver $(A g)$ particles and water as a base liquid is saturated in the presence of porous medium over a stretchable shrinkable channel which makes an angle $2 α .$ Let us assume channels are radially stretching and shrinking. The velocity of the wall is stated as $u_{r} = U_{w} = s / r$ where, s represents the wall stretching/shrinking rate. To formulate the flow phenomenon polar coordinate system $(r, θ)$ is introduced is shown in the Figure 1 (see Basir et al..³¹⁾ Velocity dispersal is given as $V = (u_{r} (r, θ), 0, 0)$ and T represents the temperature. The temperature of surfaces in walls of the channels is represented by $T_{w} .$ Further heat source/sink is considered in the present study. Based on the above assumption, the governing equations are given by (see Dogonchi and Ganji,¹⁴ Rashid et al.²⁵ and Khan et al..³²⁾

Figure 1.

Geometry of the mathematical model.

Conservation of mass equation

\frac{1}{r} (\frac{\partial}{\partial r} (u_{r} r)) = 0.

(1)

Momentum equations

\begin{aligned} u_{r} \frac{\partial}{\partial r} (u_{r}) = & - \frac{1}{ρ_{m n f}} \frac{\partial p}{\partial r} \\ + \frac{μ_{m n f}}{ρ_{m n f}} [\frac{\partial^{2}}{\partial r^{2}} (u_{r}) + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} (u_{r}) + \frac{1}{r} \frac{\partial}{\partial r} (u_{r}) - \frac{(u_{r})}{r^{2}}] \\ - \frac{μ_{m n f}}{ρ_{m n f}} \frac{u_{r}}{K *}, \end{aligned}

(2)

- \frac{1}{r ρ_{m n f}} \frac{\partial p}{\partial θ} + \frac{2 ν_{m n f}}{r^{2}} \frac{\partial}{\partial θ} (u_{r}) = 0.

(3)

Energy equation

u_{r} \frac{\partial T}{\partial r} = \frac{k_{m n f}}{{(ρ C p)}_{m n f}} [\frac{\partial^{2} T}{\partial c} + \frac{1}{r^{2}} \frac{\partial^{2} T}{\partial θ^{2}} + \frac{1}{r} \frac{\partial T}{\partial r}] + \frac{μ_{m n f}}{{(ρ C p)}_{m n f}} [4 {(\frac{\partial}{\partial r} (u_{r}))}^{2} + \frac{1}{r^{2}} {(\frac{\partial}{\partial θ} (u_{r}))}^{2}] + \frac{Q_{0} T}{{(ρ C p)}_{m n f}} .

(4)

The appropriate boundary conditions are

u_{r} = u \frac{\partial u_{r}}{\partial θ} {\frac{\partial T}{\partial θ}}_{m a x} at θ = 0 and u_{r} = s r^{- 1}, T = T_{w} r^{- 2} as θ \to \pm α .

(5)

In the present study, it is considered only radial flow. For radial motion

f (θ) = r u_{r} (r, θ)

The following similarity variables are introduced:

f (η) = \frac{f (θ)}{f_{m a x}}, g (η) = \frac{r^{2} T}{T_{w}}, η = \frac{θ}{α} .

(6)

Here,

f_{m a x} = r u_{m a x}

denotes the central line of momentum.

The effective thermophysical properties of modified nanofluid are given by (see Usman et al..³³⁾

μ_{m n f} = \frac{μ_{f}}{{(1 - (ϕ_{1} + ϕ_{2} + ϕ_{3}))}^{2.5}},

(7)

ρ_{m n f} = (1 - (ϕ_{1} + ϕ_{2} + ϕ_{3})) ρ_{f} + ϕ_{1} ρ_{S 1} + ϕ_{2} ρ_{S 2} + ϕ_{3} ρ_{S 3},

(ρ C p)_{m n f} = (1 - (ϕ_{1} + ϕ_{2} + ϕ_{3})) (ρ C p)_{f} + ϕ_{1} (ρ C p)_{S 1} + ϕ_{2} (ρ C p)_{S 2} + ϕ_{3} (ρ C p)_{S 3},

\frac{k_{m n f}}{k_{b f}} = \frac{ϕ_{1} k_{1} + ϕ_{2} k_{2} + ϕ_{3} k_{3} + 2 (ϕ_{1} + ϕ_{2} + ϕ_{3}) k_{f} + 2 (ϕ_{1} + ϕ_{2} + ϕ_{3}) (ϕ_{1} k_{1} + ϕ_{2} k_{2} + ϕ_{3} k_{3}) - 2 {(ϕ_{1} + ϕ_{2} + ϕ_{3})}^{2} k_{f}}{ϕ_{1} k_{1} + ϕ_{2} k_{2} + ϕ_{3} k_{3} + 2 (ϕ_{1} + ϕ_{2} + ϕ_{3}) k_{f} - (ϕ_{1} + ϕ_{2} + ϕ_{3}) (ϕ_{1} k_{1} + ϕ_{2} k_{2} + ϕ_{3} k_{3}) + {(ϕ_{1} + ϕ_{2} + ϕ_{3})}^{2} k_{f}}

Thermophysical properties of nanoparticles and base fluid are given by (see Ramesh et al.²⁶ and Hayat et al..³⁴⁾

Material	$ρ (k g / m^{3})$	$C p (k g^{- 1} K^{- 1})$	$k (W m^{- 1} K^{- 1})$
Copper $(C u)$	8933	385.0	401
Aluminium oxide $(A l_{2} O_{3})$	3970	765.0	40
Silver $(A g)$	10500	235	429
Water $(H_{2} O)$	997.1	4 179	0.613

The following equations are obtained when substituting similarity variables (6) into (1) to (5) and by the elimination of pressure term,

\frac{f^{″'}}{A_{1} A_{2}} + 2 α Re f f^{'} + 4 α^{2} f^{'} - \frac{λ}{A_{1} A_{2}} α^{2} f^{'} = 0,

(8)

\frac{k_{m n f}}{k_{f} A_{3}} [g^{″} + 4 α^{2} g] + 2 α^{2} Pr f g + \frac{Pr E c}{Re A_{1} A_{3}} [4 α^{2} f^{2} + {(f^{'})}^{2}] + \frac{α^{2} H s g}{A_{3}} = 0.

(9)

Reduced boundary conditions are

f (0) = 1, f^{'} (0) = 0, g^{'} (0) = 0, f (1) = S, g (1) = 1.

(10)

Here,

A_{1} = (1 - (ϕ_{1} + ϕ_{2} + ϕ_{3}))^{2.5}, A_{2} = (1 - (ϕ_{1} + ϕ_{2} + ϕ_{3})) + ϕ_{1} \frac{ρ_{S 1}}{ρ_{f}} + ϕ_{2} \frac{ρ_{S 2}}{ρ_{f}} + ϕ_{3} \frac{ρ_{S 3}}{ρ_{f}},

A_{3} = (1 - (ϕ_{1} + ϕ_{2} + ϕ_{3})) + ϕ_{1} \frac{{(ρ C p)}_{S 1}}{{(ρ C p)}_{f}} + ϕ_{2} \frac{{(ρ C p)}_{S 2}}{{(ρ C p)}_{f}} + ϕ_{3} \frac{{(ρ C p)}_{S 3}}{{(ρ C p)}_{f}},

λ = \frac{r^{2}}{K *}

Porosity parameter,

Re = \frac{α f_{m a x}}{ν_{f}}

Reynolds number,

Pr = \frac{{(ρ C p)}_{f} f_{m a x}}{k_{f}}

Prandtl number,

E c = \frac{α f_{m a x}^{2}}{{(C p)}_{f} T_{w}}

Eckert number,

H s = \frac{r^{2} Q_{0}}{k_{f}}

Heat source/sink parameter,

S = \frac{s}{f_{m a x}}

is stretching/shrinking parameter.

In this paper we studied four cases, namely

Case 1: The channel is convergent, when the value of $α$ is lesser than zero,

Case 2: The channel is divergent, when the value of $α$ is greater than zero,

Case 3: The channel is stretching, when the values of S is positive,

Case 4: The channel is shrinking, when the values of S is negative.

The important engineering coefficients are given by

C_{f} = \frac{τ_{w}}{ρ_{f} u_{m a x}^{2}} and N u = \frac{r q_{w}}{k_{f} T_{w}}

(11)

Here,

{τ_{w} = μ_{m n f} (\frac{1}{r} \frac{\partial u_{r}}{\partial θ}) |}_{θ = α} and {q_{w} = - k_{m n f} (\frac{\partial T}{\partial r} + \frac{1}{r} \frac{\partial T}{\partial θ}) |}_{θ = α}

(12)

C f = \frac{f^{'} (1)}{Re A_{1}} and r^{2} N u = \frac{- k_{m n f}}{k_{f}} \frac{g^{'} (1)}{α} .

(13)

Numerical procedure with code validation

The reduced ODEs (8) and (9) with conditions (10)arenumerically solved with the help of well-known mathematical computing software Maple. RKF45method via shooting process is implemented.Since, the derived problem is two-point boundary value and higher order. The problem is firstly reframed into and initial value and first order problem.

f^{'} = p, p^{'} = q,

\frac{q^{'}}{A_{1} A_{2}} + 2 α Re f p + 4 α^{2} p - \frac{λ}{A_{1} A_{2}} α^{2} p = 0,

(14)

g^{'} = r,

\begin{aligned} \frac{k_{m n f}}{k_{f} A_{3}} & [r^{'} + 4 α^{2} g] + 2 α^{2} Pr f g \\ + \frac{Pr E c}{Re A_{1} A_{3}} [4 α^{2} f^{2} + {(r)}^{2}] + \frac{α^{2} H s g}{A_{3}} = 0, \end{aligned}

(15)

with

f (0) = 1, p (0) = 0, r (0) = 0, f (1) = S, g (1) = 1.

(16)

The above initial value problem is numerically simplified by guessing the missing value by implementing shooting technique by assigning parameters value. The error tolerance is taken as $10^{- 6}$ and step size $h 1 = 0.1$ .

The algorithm for RKF-45 order is given as (see^35–37)

Runge -Kutta technique 4^th order

y_{i + 1} = y_{i} + \frac{25}{216} c_{1} - \frac{1}{5} c_{5} + \frac{2197}{4104} c_{4} + \frac{1408}{2565} c_{3}

Runge-Kutta technique 5^th order

z_{i + 1} = y_{i} + \frac{16}{135} c_{1} + \frac{2}{55} c_{6} - \frac{9}{50} c_{5} + \frac{28561}{56430} c_{4} + \frac{6656}{12825} c_{3}

and 6 steps sizes as follows

c_{1} = f (x_{i}, y_{i}) h 1

c_{2} = f (x_{i} + \frac{1}{4} h 1, y_{i} + \frac{1}{4} c_{1}) h 1

c_{3} = f (x_{i} + \frac{3}{8} h 1, y_{i} + \frac{3}{32} c_{1} + \frac{9}{32} c_{2}) h 1

c_{4} = f (x_{i} + \frac{12}{13} h 1, y_{i} + \frac{1932}{2147} c_{1} - \frac{7200}{2147} c_{2} + \frac{7296}{2147} c_{3}) h 1

c_{5} = f (x_{i} + h 1, y_{i} + \frac{439}{216} c_{1} - \frac{845}{4104} c_{4} - 8 c_{2} + \frac{3680}{513} c_{3}) h 1

c_{6} = f (x_{i} + \frac{1}{2} h 1, y_{i} - \frac{8}{27} c_{1} + 2 c_{2} - \frac{11}{40} c_{5} - \frac{3544}{2565} c_{3} - \frac{1859}{4104} c_{4}) h 1

For numerical computations, the flow parameter values are constrained, such as $Re = 5, λ = 0.5, Pr = 6.2, E c = 3, H s = 0.05, ϕ_{1} = 0.05, ϕ_{2} = 0.05, ϕ_{3} = 0.01$ and $α = 0.4$ (divergent channel), $α = - 0.4$ (convergent channel), $S = 0.5$ (stretching) and $S = - 0.5$ (shrinking). The obtained numerical results are best match with the existing work of Turkyilmazoglu²⁷ when $Re = 50$ _, $λ, Q$ and nanoparticles are absent is shown in the Table 1. Also one can note that Ternary nanofluid $(C u - A l_{2} O_{3} - A g / w a t e r)$ , hybrid nanofluid $(C u - A l_{2} O_{3} / w a t e r)$ and mano-nanofluid $(C u / w a t e r)$ is considered and discussed in the following section.

Results and discussion

The graphical results of the physical parameters on velocity and temperature fields are elaborated in this section. Figure 2 established the comparison between nanofluid, hybrid nanofluid and ternary nanofluid for the velocity field and temperature field. The velocity curve for the nanofluid is slightly higher than the velocity curve of hybrid and ternary nanofluids. Because of the presence of nanoparticles, the thickness of the boundary layer will be increased. The accumulation of several nanoparticles gradually reduces the velocity profile but thermal distribution increases. Moreover, hybrid nanofluid and ternary nanofluid have similar influence on velocity. A significant rise in temperature for nanofluid is observed against the temperature curves of hybrid and ternary nanofluids. Furthermore, the temperature curve of hybrid nanofluid is higher than the ternary nanofluid temperature curve. Figure 3 illustrated the importance of $α$ (divergent/convergent channel) onto the velocity curves for stretching channel. It is perceived that in case of divergent channel $(α > 0)$ and convergent channel $(α < 0)$ , the velocity curves are delineated reducing nature. However, the profiles against the convergent channel are higher that the profiles obtained against the divergent channel scenario. This is due to increment in the values of $α$ will flatten the velocity and diminishes the thickness of boundary layer. Figure 4 presents the temperature field drawn for the two cases of $α (⟨ 0 & ⟩ 0)$ . For $α < 0$ , the channel walls are squeezed due to which a reducing tendency in temperature curves is noticed however, a rise in fluid temperature is observed in case of expanding channel walls $(α > 0)$ . The phenomenon of temperature field is quite similar for expanding/contracting stretching channel, but the temperature profiles are nature wise in opposite direction (see Figure 4). The phenomenon of stretching/shrinking parameter in case of convergent and divergent channel on velocity field is elaborated in Figure 5. The stretching/shrinking parameter has opposite influence on velocity field for convergent channel and divergent channel. In case of divergent channel, the velocity curves are increased by the parameter S, while the velocity profiles preserved decreasing phenomenon for channel $α > 0$ . As in the shrinking situation, the walls of the channel are reduced as a result, and as a result, fluid mobility is reduced; the opposite trend is evident in the stretching case.

Figure 2.

Comparison between nanofluid, hybrid nanofluid and ternary nanofluid on (a) velocity (b) temperature.

Figure 3.

Velocity curve for various $α$ for stretching channel.

Figure 4.

Temperature curve for various $α$ for stretching channel.

Figure 5.

Velocity curve for various S in divergent and convergent channel cases.

Table 1.

Validation of the present numerical method with the published method when $Re = 50,$ $λ, Q$ and nanoparticles are absent. .

$α$	$S$	Turkyilmazoglu³⁸ $f^{'} (1)$	Present results $f^{'} (1)$
5	−1	−3.508103	−3.508103
	−0.5	−2.173044	−2.173044
	0	0.000000	0.000000
	0.5	−0.361846	−0.361846
	1	0.000000	0.000000
−5	−1	−4.652169	−4.652169
	−0	−2.833951	−2.833951
	1	0.000000	0.000000

Figure 6 is reported to discuss the shrinking parameter influence onto temperature field against two different scenarios of convergent and divergent channel. Due to increase in the wall shrinking rate, the temperature field is reduced for both cases of convergent channel $(α < 0)$ and divergent channel $(α > 0)$ channel. As the shrinking parameter increases, the fluid velocity decreases, resulting in less temperature distribution. As a result, the temperature distribution narrows. Figure 7 determines the Reynolds number influence on velocity and temperature fields. In Figure 7(a) it is noticed that when the channel walls mover away from each other, the velocity curves are reduced due to increase in parameter Re however, an opposite behavior in velocity curves against Re is observed in case of converging walls. Figure 7(b) mentioned that the temperature is declined due to the modification in Reynolds number for both scenarios of the stretching convergent and divergent channel. It is also perceived that the temperature is larger for the convergent channel case in comparison to the divergent channel. Basically, Re is the ratio of inertial to viscous force. As inertial forces dominate due to increment in the values of, as a result, both the fluid velocity and the momentum barrier layer increase.

Figure 6.

Temperature curve for various S in divergent and convergent channel cases.

Figure 7.

(a) velocity (b) temperature curves for various $Re$ in stretching channel case.

Supplementary Fig. 8 represents the effect of Reynolds number on velocity and temperature fields for shrinking convergent and divergent channel scenarios. Similar kind of behavior of Reynolds number on velocity and temperature curves is observed for Supplementary Fig. 8 as in case of Figure 7. The significance of porous medium is characterized in supplementary Fig. 9 for divergent and convergent stretching channel. The porous medium offers larger resistance to the fluid movement. As a result, the velocity and temperature fields are enhanced when stretchable channel walls approaching and diverging from each other. However, the velocity and temperature curves for convergent channel case are larger as compared with the divergent channel. Supplementary Fig. 10 explains the porosity parameter influence on f and g for convergent and divergent channel with shrinking walls. Similar kind of observations for the curves (f and g) against $λ$ are observed as noticed in supplementary Fig. 9. Because of the presence of porous space, the frictional drag force opposes the velocity of the liquid flow, causing the velocity to decrease and the temperature to rise.

Supplementary Fig. 1 1 portrays the importance of $ϕ_{3}$ on velocity field f and temperature field g by considering flow scenario in stretchable convergent and divergent channel. Against increased values of $ϕ_{3}$ the velocity curves are reduced for divergent channel and are enhanced in case of convergent channel. The addition of solid volume fraction thickens the boundary layer. As a result, the fluid will flow more slowly due to widening of the walls of the channel. A small change in velocity curves against $ϕ_{3}$ values is observed (see supplementary Fig. 11(a)). In supplementary Fig. 11(b), the temperature field is declined by $ϕ_{3}$ in accordance with convergent and divergent channel. Moreover, $ϕ_{3}$ has significance impact onto the temperature field. In supplementary Fig. 12, the effect of $ϕ_{3}$ is discussed on f and g for shrinking case of the convergent and divergent channel. The velocity and temperature fields behave in similar manner against modified $ϕ_{3}$ as in the case of stretching channel scenario.

Supplementary Fig. 1 3 is elaborated the implementation of Eckert number on temperature field for convergent and divergent channel. In supplementary Fig. 13(a), the temperature field for stretchable channel (convergent and divergent) case is modified against the Eckert number. For converging channel walls, the temperature curves are higher than those obtained for the divergent channel. In supplementary Fig. 13(b), same behavior in temperature profiles is noticed for convergent and divergent channel with shrinking wall case. However, the temperature curves attained against shrinking walls are larger form the temperature curves obtained for the stretchable channel. The Eckert number is used to describe the influence of self-heating in a fluid as a result of dissipation effects. At high flow velocities, the temperature profile in a fluidic system is dominated not only by the temperature gradients existing in the system, but also by the effects of dissipation owing to fluid internal friction. This will cause self-heating and, as a result, a shift in the temperature profile.

The heat source/sink parameter influence onto the temperature field in convergent and divergent channel is illustrated through supplementary Fig. 14. In supplementary Fig. 14(a), the channel walls are stretchable while in supplementary Fig. 14(b), the walls of the channel are shrinkable. Here, $H s > 0, H s < 0 & H s = 0$ signifies the heat source-sink and absence of heat source-sink factor, respectively. Thermal distribution of the system augments for developing $H s$ values. The heat sink will operate as an exchanger, transferring the heat generated by the surface into the nanofluid. As a result, in the case of a heat sink, the thermal distribution is modest, whereas in the case of a heat source, the temperature is produced by the surface. For both stretching and shrinking channel, the temperature curves are enhanced by the heat source/sink parameter. Moreover, the temperature field against the convergent channel is higher than the case of divergent channel.

Supplementary Fig. 1 5 is plotted in order to estimate the skin-friction co-efficient and local Nusselt number against mano nanofluid, hybrid nanofluid and ternary nanofluid for different values of $λ$ and S. Skin-friction co-efficient is declined trough $λ$ against S (< 0 & > 0) for both convergent and divergent channel. Values of $C_{f}$ are larger in divergent channel case in comparison with the convergent channel. Ternary nanofluid has larger $C_{f}$ than hybrid nanofluid and mano nanofluid (see supplementary Fig. 15(a)). Local Nusselt number for different kinds of nanofluids (mano, hybrid and ternary) against $λ$ is discussed in supplementary Fig. 15(b). For convergent channel whether shrinking/stretching, the Nusselt number is decreased, however an opposite behavior in Nusselt number is seen in case of divergent channel. Furthermore, a sharp decrease and increase is noticed for the stretching convergent and divergent channel, respectively. Heat transfer rate is larger for the ternary nanofluid. Supplementary Fig. 1 6(a) illustrates the Reynolds number effect on $C_{f}$ for S < 0 and S > 0. Skin-friction is calculated for different nanofluids. Overall increasing trend of $C_{f}$ is noticed for convergent and divergent channel. The stretchable convergent and divergent channel has greater $C_{f}$ values against Re than shrinking convergent and divergent channel. The mano nanofluid has the lowest $C_{f}$ values among the others nanofluids (hybrid and mano). Reynolds number enhances the Nusselt number for stretching/shrinking convergent channel and reduces for the divergent channel with (S >0 & S < 0). Ternary nanofluid has the largest heat transfer rate for convergent channel while it has smallest heat transfer rate value in case of the divergent channel for increased Re values (see supplementary Fig. 16(b)). Supplementary Fig. 1 7 considers the effects of heat source/sink parameter and Eckert number on local Nusselt number for stretchable/shrinkable convergent and divergent channel. In either case S > 0 or S < 0, the physical parameters magnified thermal rate for divergent channel while in scenario of convergent channel an opposite phenomenon is seen in the thermal rate.

Table 1 presents the validation of numerical technique under limiting case with.²⁷ Shear stresses are calculated for stretching and shrinking channel with the channel walls approaching and moving away from each other. Excellent agreement of the values $f^{'} (1)$ is achieved with the published work.²⁷

Table 2 presents the computational values of skin-friction co-efficient $f^{'} (1)$ for numerous values of $R e & λ$ . The obtained values are compared for ternary, hybrid and mano type nanofluids. The drag force is improved more with improved values of $R e & λ .$ Rise in the porous parameter will oppose the fluid motion and exhibits drag force as a result, it reduces the surface drag force. The obtained table concludes that the achieved values of $f^{'} (1)$ are significantly high in case of ternary when compared with hybrid and mano nanofluids in the presence of $R e & λ .$

Table 2.

Numerical computational values of $f^{'} (1)$ for $R e & λ$ when $α = 0.4, E c = 3,$ $H s = 0.05 & S = 0.5$ .

Parameters		Ternary	Hybrid	Mano
$R e$	$λ$	$f^{'} (1)$	$f^{'} (1)$	$f^{'} (1)$
1		−0.8884	−0.8937	−0.8995
2		−0.8261	−0.8350	−0.8376
3		−0.7626	−0.7752	−0.7781
	0	−0.6212	−0.6433	−0.6440
	1	−0.6432	−0.6618	−0.6649
	2	−0.6650	−0.6802	−0.6990

Conclusions

The distribution of flow and heat through a convergent or divergent channel is important in aeronautical, pharmaceutical, dynamic, civil, climatic, and biomechanical engineering. The thermal phenomena of ternary nanomaterials in stretchable/shrinkable convergent and divergent channels was included in the current work. The movement of the ternary nanofluids is characterized by the porous zone. The impact of the energy source/sink was involved in the energy equation. RKF-45 provides a numerical solution to the flow issue as well as a shooting process. The primary consequences are as follows:

The temperature field is lowered in the case of a stretchy convergent channel, but the reverse behavior is observed in the case of a stretchable divergent channel. The thermal phenomenon is declined for both cases of shrinkable convergent and divergent channel.

For the stretching/shrinking convergent and divergent channels, the temperature curves are changed against the porosity parameter and lowered through the Reynolds number.

The Eckert number increases the thermal field for convergent/divergent stretching and shrinking channels.

For the stretching/shrinking channel, the porosity parameter reduces skin friction $C_{f}$ while the Reynolds number increases it.

Local Nusselt number is increased in the case of a divergent channel and decreased in the case of a convergent channel against the heat source/sink parameter.

Ternary nanofluid shows significant impact than hybrid and mano fluids in surface drag force and rate of heat transfer phenomenon.

Supplemental Material

sj-docx-1-pie-10.1177_09544089221081344 - Supplemental material for Ternary nanofluid with heat source/sink and porous medium effects in stretchable convergent/divergent channel

Supplemental material, sj-docx-1-pie-10.1177_09544089221081344 for Ternary nanofluid with heat source/sink and porous medium effects in stretchable convergent/divergent channel by G.K. Ramesh, J.K. Madhukesh, S.A. Shehzad and A. Rauf in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Footnotes

Declaration of conflicting interests

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

Funding

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

ORCID iDs

G.K. Ramesh

J.K. Madhukesh

S.A. Shehzad

A. Rauf

Supplemental material

Supplemental material for this article is available online.

References

Choi

SUS

Eastman

. Enhancing thermal conductivity of fluids with nanoparticles. IL (United States): Argonne National Lab., 1995. ANL/MSD/CP-84938; CONF-951135-29

Kang

Kim

. Estimation of thermal conductivity of nanofluid using experimental effective particle volume. Exp Heat Transf 2006; 19: 181–191.

Khanafer

Vafai

Lightstone

. Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transfer 2003; 46: 3639–3653.

Madhukesh

Ramesh

Prasannakumara

, et al. Bio-Marangoni convection flow of casson nanoliquid through a porous medium in the presence of chemically reactive activation energy. Appl. Math. Mech.-Engl. Ed 2021; 42: 1191–1204.

Abbasi

Gul

Shehzad

. Effectiveness of temperature-dependent properties of Au, Ag, FeO, Cu nanoparticles in peristalsis of nanofluids. Int Commun Heat Mass Transfer 2020; 116: 104651.

Upreti

Joshi

Pandey

, et al. Numerical solution for Sisko nanofluid flow through stretching surface in a Darcy–Forchheimer porous medium with thermal radiation. Heat Transfer 2021; 50: 6572–6588.

Wakif

Animasaun

Satya Narayana

, et al. Meta-analysis on thermo-migration of tiny/nano-sized particles in the motion of various fluids. Chin J Phys 2020; 68: 293–307.

Ramesh

. Influence of shape factor on hybrid nanomaterial in a cross-flow direction with viscous dissipation. Phys Scr 2019; 94: 105224.

Madhukesh

Ramesh

Alsulami

, et al. Characteristic of thermophoretic effect and convective thermal conditions on flow of hybrid nanofluid over a moving thin needle. Waves Random Complex MEdia 2021; 0: 1–23.

10.

Akbar

Abbasi

Shehzad

. Effectiveness of Hall current and ion slip on hydromagnetic biologically inspired flow of Cu-Fe₃O₄/H₂O hybrid nanomaterial. Phys Scr 2020; 96: 025210.

11.

Joshi

Upreti

Pandey

, et al. Heat and mass transfer assessment of magnetic hybrid nanofluid flow via bidirectional porous surface with volumetric heat generation. Int. J. Appl. Comput. Math 2021; 7: 64.

12.

Nadeem

Abbas

. Effects of MHD on modified nanofluid model with variable viscosity in a porous medium. Intech Open 2020. DOI: 10.5772/intechopen.84266

13.

Abbas

Nadeem

Saleem

, et al. Transportation of modified nanofluid flow with time dependent viscosity over a Riga plate: exponentially stretching. Ain Shams Engineering Journal 2021; 12: 3967–3973.

14.

Dogonchi

Ganji

. Study of nanofluid flow and heat transfer between non-parallel stretching walls considering Brownian motion. Journal of the Taiwan Institute of Chemical Engineers 2016; 69: 1–13.

15.

Pandey

Kumar

. MHD Flow inside a stretching/shrinking convergent/divergent channel with heat generation/absorption and viscous-ohmic dissipation utilizing cu-water nanofluid. Computational Thermal Science 2018; 10: 457–471.

16.

Ramesh

Shehzad

Rauf

, et al. Heat transport analysis of aluminum alloy and magnetite graphene oxide through permeable cylinder with heat source/sink. Phys Scr 2020; 95: 095203.

17.

Ramesh

Madhukesh

. Activation energy process in hybrid CNTs and induced magnetic slip flow with heat source/sink. Chin J Phys 2021; 73: 375–390.

18.

Upreti

Pandey

Kumar

, et al. Ohmic heating and Non-uniform heat source/sink roles on 3D Darcy–Forchheimer flow of CNTs nanofluids over a stretching surface. Arab J Sci Eng 2020; 45: 7705–7717.

19.

Singh

Pandey

Kumar

. Slip flow of micropolar fluid through a permeable wedge due to the effects of chemical reaction and heat source/sink with Hall and ion-slip currents: an analytic approach. Propulsion and Power Research 2020; 9: 289–303.

20.

Kumar

STD

Nath

Dinesh

DPA

. Oscillatory hydro magnetic convective heat transfer through a porous medium in a converging channel, international journal of innovative research in science. Engineering and Technology 2017; 6: 7.

21.

Akinshilo

. Investigation of nanofluid conveying porous medium through non-parallel plates using the Akbari Ganji method. Phys Scr 2020; 95: 125702.

22.

Ahmad

Faisal

Javed

. Unsteady flow of walters-B magneto-nanofluid over a bidirectional stretching surface in a porous medium with heat generation. Special Topics & Reviews in Porous Media: An International Journal 2021; 12: 49–70.

23.

Pandey

Kumar

. Natural convection and thermal radiation influence on nanofluid flow over a stretching cylinder in a porous medium with viscous dissipation. Alexandria Eng J 2017; 56: 55–62.

24.

Singh

Pandey

Kumar

. Numerical solution of micropolar fluid flow via stretchable surface with chemical reaction and melting heat transfer using keller-Box method. Propulsion and Power Research 2021; 10: 194–207.

25.

Rashid

Iqbal

Liang

, et al. Dynamics of water conveying zinc oxide through divergent-convergent channels with the effect of nanoparticles shape when Joule dissipation are significant. Plos One 2021; 16: e0245208.

26.

Ramesh

Shehzad

Tlili

. Hybrid nanomaterial flow and heat transport in a stretchable convergent/divergent channel: a Darcy-Forchheimer model. Appl Math Mech 2020; 41: 699–710.

27.

Saifi

Sari

Kezzar

, et al. Heat transfer through converging-diverging channels using adomian decomposition method. Engineering Applications of Computational Fluid Mechanics 2020; 14: 1373–1384.

28.

Adnan Asadullah

Khan

, et al. Analytical and numerical investigation of thermal radiation effects on flow of viscous incompressible fluid with stretchable convergent/divergent channels. J Mol Liq 2016; 224: 768–775.

29.

Dogonchi

Ganji

. Investigation of MHD nanofluid flow and heat transfer in a stretching/shrinking convergent/divergent channel considering thermal radiation. J Mol Liq 2016; 220: 592–603.

30.

Mishra

Pandey

Chamkha

, et al. Roles of nanoparticles and heat generation/absorption on MHD flow of Ag–H₂O nanofluid via porous stretching/shrinking convergent/divergent channel. Journal of Egyptian Mathematical Society 2020; 28: 17.

31.

Basir

MFM

Mabood

Satya Narayana

, et al. Significance of viscous dissipation on the dynamics of ethylene glycol conveying diamond and silica nanoparticles through a diverging and converging channel. J Therm Anal Calorim 2020; 147: 661–674.

32.

Khan

Adnan Ahmed

, et al. A novel hybrid model for Cu-Al₂O₃/H₂O nanofluid flow and heat transfer in convergent/divergent channels. Energies 2020; 13: 1686.

33.

Usman

Hamid

Zubair

, et al. Cu-Al₂O₃/water hybrid nanofluid through a permeable surface in the presence of nonlinear radiation and variable thermal conductivity via LSM. Int J Heat Mass Transfer 2018; 126: 1347–1356.

34.

Hayat

Kiran

Imtiaz

, et al. Hydromagnetic mixed convection flow of copper and silver water nanofluids due to a curved stretching sheet. Results in Physics 2016; 6: 904–910.

35.

Kumaraswamy Naidu

Harish Babu

Harinath Reddy

, et al. Radiation and partial slip effects on magnetohydrodynamic jeffrey nanofluid containing gyrotactic microorganisms over a stretching surface. J Therm Sci Eng Appl 2020; 13: 031011.

36.

Satya Narayana

Tarakaramu

Sarojamma

, et al. Numerical simulation of nonlinear thermal radiation on the 3D flow of a couple stress casson nanofluid Due to a stretching sheet. J Therm Sci Eng Appl 2021; 13: 021028.

37.

Satya Narayana

Tarakaramu

Harish Babu

. Influence of chemical reaction on MHD couple stress nanoliquid flow over a bidirectional stretched sheet. Int J Ambient Energy 2021; 0: 1–11.

38.

Turkyilmazoglu

. Extending the traditional Jeffery-Hamel flow to stretchable convergent/divergent channels. Comput Fluids 2014; 100: 196–203.

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