Comparative investigations of Ag/H2O nanofluid and Ag-CuO/H2O hybrid nanofluid with Darcy-Forchheimer flow over a curved surface

Wenjie Lu; Umar Farooq; Muhammad Imran; Wathek Chammam; Sayed M. El Din; Ali Akgül

doi:10.1515/ntrev-2023-0136

Article Open Access

Comparative investigations of Ag/H₂O nanofluid and Ag-CuO/H₂O hybrid nanofluid with Darcy-Forchheimer flow over a curved surface

Wenjie Lu , Umar Farooq , Muhammad Imran , Wathek Chammam , Sayed M. El Din and Ali Akgül

Published/Copyright: November 11, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Nanotechnology Reviews Volume 12 Issue 1

Abstract

Nanofluid performed well and produced good results in heat transport phenomena, attracting scientists to suspend other combinations of nanoparticles, called “hybrid nanofluid. Hybrid nanofluids are superior than nanofluids due to their thermal capabilities and emerging benefits that contribute to the boost in the rate of heat transmission. Applications for these nanoparticles, including sophisticated lubricants, are increasing in the fields of bioengineering and electricity. The main prospective of this research is to inquire about the water-based dual nature nanofluid stream numerical simulation through the irregular stretched sheet with heat transfer. In this perspective, silver with base fluid water is used as nanoparticles for nanofluid, and for making hybrid nanofluid, copper oxide and silver particles are used with water-based fluid. Modified Fourier and Fick’s model for heat flux utilized the above phenomenon and observed the heat and mass transport. Similarity variables are needed to transform the partial differential equations into associated nonlinear ordinary differential equations, which are then computationally resolved by the technique of bvp4c which is a built-in function in MATLAB mathematical software. Based on the concurrent approximations, reformations are performed to determine the impact of various quantities on flow variables. The predicted outcomes are depicted in velocity, temperature, and concentration profiles through graphical depiction. The factors indicate that the hybrid nanofluid is more powerful in the transfer of heat than a basic nanofluid because of its superior thermal characteristics. The velocity profile decays for the increasing values of Darcy-Forchheimer parameter. The thermal profile increases for the higher magnitude of Darcy-Forchheimer parameter. The velocity distribution profile increases for the higher values of curvature parameter, while the thermal profile decreases. This unique work might benefit nanotechnology and related nanocomponents. This safe size-controlled biosynthesis of Ag and CuO nanoparticles has resulted in a low-cost nanotechnology that may be used in a variety of applications. Biosynthesized Ag and CuO particles have been used successfully in a variety of applications, including biomedical, antibacterial agents, biological, food safety, and biosensing, to mention a few.

Graphical abstract

Keywords: hybrid nanofluid; Darcy-Forchheimer flow; Newtonian heating; heat source-sink; Cattaneo; Christov heat theory; curved stretching sheet; RKF-45 approach

Nomenclature

Ag: silver
CuO: copper oxide
Fr: Darcy-Forchheimer parameter
H 2 O: water
K hnf: thermal conductivity of hybrid nanofluid
K nf: thermal conductivity of nanofluid
Pr: Prandtl number
Q: heat source-sink parameter
Q t: thermal relaxation parameter
u , v: components
ρ nf: density of nanofluid
μ nf: viscosity of nanofluid
( ρ C p ) nf: heat capacity of nanofluid
( ρ C p ) hnf: heat capacity of hybrid nanofluid
ρ hnf: density of hybrid nanofluid
μ hnf: viscosity of hybrid nanofluid
β: curvature parameter
Π: Newtonian heating parameter
Φ 1 = Φ 2: volume fraction of nanoparticle

1 Introduction

Nanofluid flow is recently used in various fields of life like medicine, electronics, computing, refrigeration, and thermal engineering. Nanofluids consist of base fluids in which nanometer-sized particles exist that have a high rate of heat conductivity and efficient properties. Numerous articles on nanofluids concentrate on grasping their behavior to use them in applications where improving straight heat transmission is crucial, such as in various industrial settings, nuclear reactors, transportation, electronics, biology, and food. It has also been stated that nanofluid acts as a smart fluid with variable heat transmission. Biological polymer synthesis, eco-friendly uses, sustainable and renewable cell innovations, microbes-improved oil recovery, biosensors, and biotechnology, as well as ongoing improvements in mathematical modeling, are just a few examples of the many fields in which bioconvection is used. Porosity groups distributed the surface spacing vertically in the porosity-surface plot, which contains the porosity interval. Mahian et al. [1] analyzed how entropy forms in nanofluid flow. Koo and Kleinstreuer [2] examined the flow of turbulent nanofluid fluxes in microscopic heat pipes. Sheikholeslami and Ganji [3] examined the thermal performance among parallel plates of copper with base fluid water nanofluid flow. Huminic and Huminic [4] determined the generation of entropy in dual-natured nanofluid and base fluid streams in heat transfer arrangements. Sheikholeslami et al. [5] explored nanofluid in the presence of an attractive field over a rotating surface and heat transport through it. Li and Kleinstreuer [6] scrutinized the heat depiction of nanofluid flow in a microfluidics model. Ghalandari et al. [7] purported the computational study of the nano-fluid stream from the inner part of a root canal. One of the researchers’ favorite fields of study is a closed structure with the hybrid nanofluid flow and heat transfer mechanisms have several implementations in technologies and research. Dual nanoparticles nanofluids are a base fluid assemblage of two or more different nanoparticles. To obtain the desired thermal exchange rate, particular nanomaterials’ weight percentages are being used. Salman et al. [8] investigated the backward and forward stages of the cross nanofluid stream and heat transport. Wang et al. [9] considered the heat transport rate properties of silver with base water hybrid nano-fluid. Wang et al. [10] described the mechanism of the stream of a hybrid-based nanofluid across a stretching geometry. Wang et al. [11] investigated the effects of hybrid nanofluid with hybrid nanoparticles and thermal radiation. Huang et al. [12] looked into the control of a dual nanoparticle nanofluid combination in a heat exchanger system. Tlili et al. [13] inspected the 3D magnetohydrodynamics stream of the sliding behavior of a methanol hybrid nanofluid over an irregular thick medium. Hayat and Nadeem [14] investigated the improvement of heat transmission with an Ag water cross nanofluid. Heat transfer expansion is a modern engineering problem in an extensive variety of fields such as heat pumps, computers, biotic and bio-nuclear plants, and others. Nanofluids, as new heat transfer solvents, can be an important medium for enhancing energy transport. These advantages are acquired in response due to the increment in efficient thermal conductivity and a change in fluid flow dynamics. Engineers and scientists are currently working to improve the transport processes in various engineering systems such as boilers, electronic equipment, heat exchangers, and others. Kasaeian et al. [15] investigated the movement of nanofluids and heat allocation in permeable media. Sheikholeslami et al. [5] investigated the nanofluid stream and heat transport in the attendance of a magnetic atrocity in a rotating system. Salman et al. [8] investigated the backward and forward dual nanoparticles nanofluid stream and heat transport. Turkyilmazoglu [16] investigated the nanofluid stream and heat removal in the existence of a rotating disc. The KKL correlation was determined by Kandelousi [17] for the modeling of nanofluid movement and heat transference in a porous medium. Amani et al. [18] conducted a careful examination of current developments in boundary layer flow and heat exchange between parallel surfaces. The nanofluid stream and heat transport caused by a stretchable cylindrical medium in the attendance of a magnetic atrocity were determined by Ashorynejad et al. [19]. The superiority of nanofluids over conventional fluids will be determined in terms of thermal conductivity intensity. The inclusion of metallic nanoparticles is what causes the nanofluid to have a greater thermal conductivity. The concentration of nanoparticles directly affects the thermal conductivity of the nanofluid. Gbadeyan et al. [20] explored Casson nanofluid stream involving convective heating also with slippage velocity as a function of flexible stickiness and temperature conductivity. Lahmar et al. [21] examined the heat transfer that results from compressing a time-dependent nanofluid flow while being influenced by an angled magnetic field and a changeable thermal conductivity. Al-Hussain et al. [22] described a magneto-bio convective and temporal conductivity increment in nanofluid stream involving motile micro-organisms. Eid and Nafe [23] studied the effects of temperature conductivity change and warmth production on magneto-dual nature nanofluid stream in a permeable media with slipping restriction. Paul et al. [24] determined the methodologies for evaluating the thermophysical properties of nanofluid. Tawfik [25] reported the uses and practical investigations of nanofluids-improved thermal conductivity. Yasir et al. [26] explored the thermal conductivity performance in hybrid nanofluid flow. Han et al. [27] utilized the improved heat flux system of Fourier and Fick to inquire about coupled stream and warmth transmission in viscoelastic media. Hayat et al. [28] examined how the improved heat flux system of Fourier and Fick affected the movement of a fluid having varying temperature conductivity across a variably thicker layer. Mustafa [29] investigated the heat flux phenomenon of Modified Fourier and Fick’s approach for rotational stream and upper-convicted Maxwell fluid heat transfer. Merkin [30] investigated the impact of heating caused by Newtonian forces in a naturally convective flow with constraints over a perpendicular medium. Hayat et al. [31] used a permeable cylinder to determine the impact on nanofluid stream caused heating effect by Newtonian forces. Salleh et al. [32] reported the streaming of heat and bounded layered stream through a stretchable medium heated by Newtonian forces. Naveed et al. [33] calculated the MHD stream of micropolar fluid caused by a curved stretchable medium involving heat conduction. Sajid et al. [34] conducted the stretch curved surface in a viscous fluid. Gowda et al. [35] used the Koo-Kleinstreuer and Li relationship and improved the heat flux system of Fourier and Fick to compute the stream of nanofluids across an irregularly stretched medium. Raju et al. [36] investigated the effects of ternary hybrid nanofluid with Darcy Walls and thermal radiation. Eswaramoorthi et al. [37] studied the aspects of Darcy-Forchheimer flow of nanofluid via a Riga surface. Ramesh et al. [38] estimated the flow of nanofluid with thermal radiation and thermophoretic effect. Qureshi et al. [39] studied the impact on nanofluids flow with nanocomposite material. Raza et al. [40] analyzed the aspects of hybrid nanofluids with activation energy via porous surfaces. Rehman et al. [41] expressed the significance of hybrid nanofluid with magnetic field and heat flux model. Figure 1 illustrates the variety of applications of nanoparticles in various fields. Most importantly, these particles are used in biomedical, agricultural, industrial, and environmental domains for efficient results.

Figure 1

Applications of CuO nanoparticles.

From the current literature review, our study turns the direction towards the time-independent movement of hybrid nanofluid over a stretchable curved surface and implementation of modified Fourier and Fick’s model of heat and mass transport. In our hybrid model, we took copper oxide and silver nanoparticles with base fluid water. The comparative study of nanofluid and dual-nature nanofluid models has also been considered from the perspective of numerous physical constraints. Results are numerically computed by the use of Runge–Kutta–Fehlburg fourth-fifth order (RKF-45) through the use of MATLAB command, and graphical analysis for the visualization of results are also taken into account.

2 Mathematical description

Here we investigate the study of two-dimensional incompressible flow of nanofluid and hybrid nanofluid containing CuO and Ag nanoparticles with base fluid water through a stretchable curved sheet inside the circle having a radius R as shown in Figure 2.

Figure 2

Physical configuration of the current problem.

The key assumptions of current investigations are listed below:

A comparative framework is investigated between nano and hybrid nanofluid.
Darcy-Forchheimer flow passing through the carved sheet is studied.
The silver and copper oxide nanoparticles are considered.
The heat source-sink and Cattaneo–Christov heat theory are investigated
Numerical and graphical results are presented by using the well-known method RK-45 approach.

The stream is ruled by the following system of equations [42–44,49]:

(1) ∂ r { ( r + R ) υ } + R u s = 0 ,

(2) 1 R + r u 2 = 1 ρ hnf p r ,

(3) v u r + R r + R u u s + u v r + R = − 1 ρ hnf R r + R p s + v hnf u rr + 1 r + R u r − 1 ( r + R ) 2 u − 1 ρ hnf F u 2 ,

(4) v T r + R r + R u T s = k h n f ( ρ c p ) h n f T r r + 1 r + R T r + Q 0 ( ρ c p ) h n f ( T - T ∞ ) + ϒ v 2 T r r + u 2 R r + R 2 T s s + v v r + R r + R u v s T r + R r + R v u r + u R r + R 2 u s T s + 2 R r + R u v T r s .

Corresponding boundary conditions are as follows [49]:

(5) u = u w ( s ) = a s , v = 0 , T = T w , − T r = h s T at r = 0 , u → 0 , u r → 0 , T → T ∞ , as r → ∞ .

Similarity transformation utilized in governing equation as follows [34,49]:

(6) u ( = a s F ′ ( ζ ) ) , v = − R r + R a v f F ( ζ ) , ζ = a v f r , θ ( ζ ) = T − T ∞ T w − T ∞ , θ ( ζ ) = T − T ∞ T ∞ , p ( = ρ hnf a 2 s 2 p ( ζ ) ) .

The heat conductivity, specific heat capacitance, concentration, and dynamic thickness of the nanofluid and hybrid nanofluid are given in Table 1.

Upon the utilization of the above similarity symbols, equation (1) is fulfilled and the remaining equations (2)–(4) including limit points (5) and (6) are developed into the following forms:

(7) Γ 1 P ′ − F ′ 2 ζ + β = 0 ,

(8) Γ 1 2 β ζ + β P = Γ 2 F ‴ + 1 ζ + β F ″ − 1 ζ + β 2 F ′ + β ζ + β F F ″ − β ζ + β F ′ 2 + β ( ζ + β ) 2 F F ′ − β ( ζ + β ) 2 ( ρ ) hnf ( ρ ) f 2 Fr F 2 ,

(9) ( k ) hnf ( k ) f Γ 3 1 Pr Θ ″ + 1 ζ + β Θ ′ − Q t β ζ + β 2 F 2 Θ ″ + F ′ F Θ ′ − 1 ζ + β F 2 Θ ′ + β ζ + β F Θ ′ + ( ρ C p ) hnf ( ρ C p ) f Pr Q Θ = 0 ,

where

Γ 1 = 1 ( 1 − Φ 2 ) ( 1 − Φ 1 ) + Φ 1 ρ s 1 ρ f + Φ 1 ρ s 2 ρ f ,

Γ 2 = 1 ( 1 − Φ 1 ) 2.5 ( 1 − Φ 2 ) 2.5 ( 1 − Φ 2 ) ( 1 − Φ 1 ) + Φ 1 ρ s 1 ρ f + Φ 2 ρ s 2 ρ f ,

Γ 3 = 1 ( 1 − Φ 2 ) ( 1 − Φ 1 ) + Φ 1 ( ρ C p ) s 1 ( ρ C p ) f + Φ 2 ( ρ C p ) s 2 ( ρ C p ) f .

The consistent compact boundary situations are as follows:

(10) f ′ ( 0 ) = 1 , f ( 0 ) = 0 , θ ( 0 ) = 1 f ′ ( ∞ ) → 0 , f ′ ′ ( ∞ ) → 0 , θ ( ∞ ) → 0 .

Along with another case

(11) θ ′ ( 0 ) = − Π ( 1 + θ ( 0 ) ) .

Using equation (8), we get the value of pressure. Equations obtained after removing pressure P ( η ) from equations (7) and (8) are as follows:

(12) Γ 2 F i v + 2 ζ + β F ‴ − 1 ( ζ + β ) 2 F ″ + 1 ( ζ + β ) 3 F ′ + β ζ + β ( F F ″ − F ″ F ′ ) + β ( ζ + β ) 2 ( F F ″ − F ′ 2 ) − β ( ζ + β ) 3 F F ′ − β ( ζ + β ) 2 ( ρ ) hnf ( ρ ) f 2 Fr F 2 = 0 .

The dimensionless parameters are listed as follows:

Parameter name	Parameter notations	Parameter values
Curvature parameter	( β )	β = a v f R
Prandtl number	( Pr )	Pr = ( ρ C p ) k f
Thermal relaxation parameter	( Q t )	Q t ( = ϒ a )
Newtonian heating parameter	( Π )	Π = h s v f a
Darcy-Forchheimer parameter	( Fr )	Fr = C b K ⁎ 1 / 2 s
Heat source-sink parameter	( Q )	Q = 2 Q 0 L U w ( ρ c p )

Physical quantities including Nusselt number and skin friction exist in dimensionless versions as follows:

(13) R e C f = f ″ ( 0 ) − f ′ ( 0 ) β ( 1 − Φ 1 ) 2.5 ( 1 − Φ 2 ) 2.5 ,

(14) Nu Re = − ( k hnf ) ( k f ) θ ′ ( 0 ) ,

Here Re = a s 2 k f θ ′ ( 0 ) .

3 Numerical approach

For the numerical treatment of the problem, we used RKF-45 to the transformed classification of ordinary differential equations received after the use of similarity transformations on governing partial differential equations. RKF-45 is utilized through MATLAB built-in function and gets the required results. Figure 3 shows the flow chart of the current problem.

Figure 3

Flow chart of the current investigation.

4 Result and discussion

In this section, we perceived the velocity and energy outlines of the problem by considering various physical parameters. The impacts of Newtonian heating on nanofluid and hybrid nanofluid movement through a curved shallow beside the Cattaneo–Christov heat theory are studied. The model under consideration compares a hybrid nanofluid with silver (Ag) and copper oxide (CuO) nanomaterials in conventional fluids, water, to a model with metal (Ag) nanoparticles in base fluid, water. The ranges of the physical flow parameters are ( 0.3 < Fr < 0.9 ) , ( 0.0 < Φ 1 = Φ 2 < 0.07 ) , ( 0.3 < β < 0.9 ) , ( 3.0 < Pr < 9.0 ) , ( 0.1 < Q T < 1.5 ) , and ( 0.1 < Q < 0.7 ) . Figure 4 shows the influence of the Darcy-Forchheimer constraint ( Fr ) on the velocity distribution profile ( F ′ ( ζ ) ) nanofluid and hybrid nanofluid. In both cases, the velocity distribution profile ( F ′ ( ζ ) ) decays with the increase in the Darcy-Forchheimer ( Fr ) . Figure 5 reveals the effect of the volume portion of nanoparticles Φ 1 = Φ 2 on the velocity distribution profile ( F ′ ( ζ ) ) of hybrid nanofluid and nanofluid and this causes the upward trend in their velocity profile ( F ′ ( ζ ) ) . Figure 6 indicates the boost up in the velocity distribution profile ( F ′ ( ζ ) ) with the increase in the rate of the curvature factor ( β ) . Physically, the inclination in the curvature parameter causes the larger radius, and the liquid moves faster over the sheet, increasing the velocity gradient. In the temperature distribution of the problem, Figure 7 justifies that by increasing the significance of the Prandtl number ( Pr ) , the flowing liquid becomes slow due to the rise in temperature distribution profile ( Θ ( ζ ) ) of nanofluid and hybrid nanofluid flows. Physically, the greater the Prandtl number, the lower the thermal conductivity, thereby decreasing the temperature gradient. Lower Prandtl numbers, on the other hand, has higher conductivity due to the increase in the temperature, which raises the temperature of the boundary layer flow. Furthermore, the rate of declination in the temperature gradient of the hypothesized hybrid nanofluid is quicker in the nitrogen hydroxide case than in the constant temperature gradient scenario. Figure 8 depicts that by increasing the Darcy-Forchheimer ( Fr ) the temperature distribution profile ( Θ ( ζ ) ) of nanofluid and hybrid nanofluid flow is boosted up and supports the fluid flow. Also, the volume fraction parameter ( Φ 1 and Φ 2 ) of nanoparticles Figure 9 dictates that in nanofluid and hybrid nanofluid temperature distribution profile ( Θ ( ζ ) ) decreases on growing the importance of the volume segment factor. Figure 10 shows the reduction in thetemperature distribution profile ( Θ ( ζ ) ) due to the increase in the thermal relaxation parameter ( Q T ) . Basically, in nanofluid and hybrid nanofluid flow, particles become attracted to nanoparticles due to the increase in thermal relaxation constraint ( Q T ) . Figure 11 shows the inspiration of the curvature factor on the temperature distribution profile ( Θ ( ζ ) ) of the flow of nanofluid, in which silver nanoparticles are included and water ( H 2 O ) is taken as base fluid, and hybrid nanofluid, in which particles of silver and copper oxide are mixed in the base fluid, which decreases the flow motion. Figure 12 reflects that the flow motion increases due to the existence of heat source/sink coefficient ( Q ) in both types of nanofluids whether it is simple or hybrid. Furthermore, in comparison to the Newtonian heating scenario, the amount of slope in heat flux of the postulated stream is higher for the comparable wall temperature scenario. Such substantial observations were made because the presence of nanomaterials in liquid raises the viscosity of the fluid, which in turn opposes the flow pattern. Water gains an excessive amount of heat when nanoparticles are present because they have a limited capacity to absorb heat. The thermal energy transition is accelerated in part by these factors. For extremely small transverse curvature parameters, the problem can be solved in two dimensions; however, for a cylindrical shape, the diameter order may be identical to the order of the boundary-layer width. Three-dimensional charting is used to visualize the effects of different dimensionless factors on skin friction and Nusselt number. Figure 13 demonstrates the effect of curvature constraint ( β ) and volumetric fractions of nanoparticles ( Φ 1 = Φ 2 ) on skin friction, an increase in volumetric fractions and curvature parameters increases the skin friction profile. The pictorial version of Nusselt number is shown in Figure 14, in which it can be seen that increasing the heat relaxation restriction parameter ( Q t ) and curvature parameter ( β ) improves the heat transfer rate of the fluid flow involving nanoparticles, and also increases the heat and mass transport in the case of hybrid nanofluid . Table 1 displays the thermophysical properties of nanofluids and hybrid nanofluids such as viscosity, thermal expansion, heat capacity, and velocity. Table 2 displays the thermophysical behavior of nanoparticles (silver and copper oxide) and base fluid (water). Table 3 analyses the good outcomes and agreement between old and current work.

$Figure 4 Upshot of Fr \text{Fr} on F ′ ( ζ ) F^{\prime} (\zeta ) .$

Figure 4

Upshot of Fr on F ′ ( ζ ) .

$Figure 5 Upshot of Φ 1 = Φ 2 {{\Phi }}_{1}={{\Phi }}_{2} on F ′ ( ζ ) F^{\prime} (\zeta ) .$

Figure 5

Upshot of Φ 1 = Φ 2 on F ′ ( ζ ) .

$Figure 6 Upshot of β \beta on F ′ ( ζ ) F^{\prime} (\zeta ) .$

Figure 6

Upshot of β on F ′ ( ζ ) .

$Figure 7 Upshot of Pr \Pr on Θ ( ζ ) \Theta (\zeta ) .$

Figure 7

Upshot of Pr on Θ ( ζ ) .

$Figure 8 Upshot of F r Fr on Θ ( ζ ) \Theta (\zeta ) .$

Figure 8

Upshot of F r on Θ ( ζ ) .

$Figure 9 Upshot of Φ 1 = Φ 2 {{\Phi }}_{1}={{\Phi }}_{2} on Θ ( ζ ) \Theta (\zeta ) .$

Figure 9

Upshot of Φ 1 = Φ 2 on Θ ( ζ ) .

$Figure 10 Upshot of Q T {Q}_{T} on Θ ( ζ ) \Theta (\zeta ) .$

Figure 10

Upshot of Q T on Θ ( ζ ) .

$Figure 11 Upshot of β \beta on Θ ( ζ ) \Theta (\zeta ) .$

Figure 11

Upshot of β on Θ ( ζ ) .

$Figure 12 Upshot of Q Q on Θ ( ζ ) \Theta (\zeta ) .$

Figure 12

Upshot of Q on Θ ( ζ ) .

$Figure 13 Upshot of β \beta and Φ 1 = Φ 2 {{\Phi }}_{1}={{\Phi }}_{2}\hspace{.25em} on skin friction.$

Figure 13

Upshot of β and Φ 1 = Φ 2 on skin friction.

$Figure 14 Upshot of β \beta and Q t {Q}_{\text{t}} on Nusselt number.$

Figure 14

Upshot of β and Q t on Nusselt number.

Table 1

Thermophysical characteristics of nanofluid and hybrid nanofluid [45,46]

Properties		Nanofluid
Density ( ρ nf )		ρ nf ( = ρ f ( 1 ‒ Φ 1 ) + Φ 1 ( ρ s 1 ) )
Viscosity ( μ nf )		μ nf = μ f ( 1 ‒ Φ 1 ) 2.5
Heat capacity ( ρ C p ) nf		( ρ C p ) nf = ( ρ C p ) f ( 1 ‒ Φ 1 ) + Φ 1 ( ρ C p ) s 1
Thermal conductivity ( K nf )		K nf K f = K s 1 + 2 K f ‒ 2 ( K f ‒ K s 1 ) Φ 1 K s 1 + 2 K f + Φ 1 ( K f ‒ K s 1 )
Properties	Hybrid nanofluid
Density ( ρ hnf )	ρ hnf = ( 1 ‒ Φ 2 ) ( 1 ‒ Φ 1 ) ρ f + Φ 1 ρ s 1 + Φ 2 ρ s 2
Viscosity ( μ hnf )	μ hnf = μ f ( 1 ‒ Φ 1 ) 2.5 ( 1 ‒ Φ 2 ) 2.5
Heat capacity ( ρ C p ) hnf	( ρ C p ) hnf = ( 1 ‒ Φ 2 ) ( 1 ‒ Φ 1 ) ( ρ C p ) f + Φ 1 ( ρ C p ) s 1 + Φ 2 ( ρ C p ) s 2
Thermal conductivity ( K hnf )	K hnf K nf = K s 2 + 2 K nf ‒ 2 ( K nf ‒ K s2 ) Φ 2 K s 2 + 2 K nf + Φ 2 ( K nf ‒ K s 2 ) , Here K nf K f = K s1 + 2 K f ‒ 2 ( K f ‒ K s 1 ) Φ 1 K s 1 + 2 K f + Φ 1 ( K f ‒ K s 1 )

Table 2

Thermophysical aspects of nanoparticles and base fluid [47,48]

Nanoparticles and base fluid	Properties
Nanoparticles and base fluid	Thermal conductivity K ( W m − 1 K − 1 )	Heat capacity C p ( J − 1 kg − 1 K − 1 )	Density ρ ( kg m − 3 )
Water	0.613	4,179	997.1
	0.613	4,179	997.1
Silver	429.0	235.00	10,500.0
	429.0	235.00	10,500.0
Copper oxide	765	5,318	6.32
	765	5,318	6.32

Table 3

Validation of numerical approach for ( − R e C f ) with ( β ) when ( Φ 1 = Φ 2 = 0 )

( β )	Sajjid et al. [34]	Madhukesh et al. [49]	Current work
5	0.75763	0.754505	0.754505
10	0.87349	0.872445	0.872445

5 Conclusion

The main aim of the current research work is to look at the characteristics of silver/water nanofluid and silver-copper oxide/water hybrid nanofluids with Cattaneo–Christov heat flux model passing through a stretchable curved sheet. The main governing system of equations is fixed numerically and graphically using the computational tool MATLAB with bvp4c solver. The promising impact of hybrid nanofluid has prompted many manufacturers, technologists, and engineers to incorporate this type of heat transfer medium in a variety of heat transfer systems such as heat exchangers, electronics, generators, transformers, cooling, biomedical, thermal conductors, and air conditioning. The following are the major outcomes of this study:

When the Darcy-Forchheimer parameter is augmented, the fluid velocity decreases.
The fluid velocity profile is increased with the increase in the magnitude of curvature parameter and volume fraction of nanoparticles.
The temperature profile increased with increment in the values of volume fraction nanoparticles.
The temperature profile decreases with the increasing estimations of Prandtl number.
The thermal filed is decreased with the increase in the variations of curvature parameter and thermal relaxation parameter.

Acknowledgments

The authors would like to thank Deanship of Scientific Research at Majmaah University for supporting this work under Project No. R-2023-616.

Funding information: The Deanship of Scientific Research at Majmaah University for supporting this work under Project No. R-2023-616.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Mahian O, Kianifar A, Kleinstreuer C, Moh’d A AN, Pop I, Sahin AZ, et al. A review of entropy generation in nanofluid flow. Int J Heat Mass Transf. 2013;65:514–32.10.1016/j.ijheatmasstransfer.2013.06.010Search in Google Scholar

[2] Koo J, Kleinstreuer C. Laminar nanofluid flow in micro heat-sinks. Int J Heat Mass Transf. 2005;48(13):2652–61.10.1016/j.ijheatmasstransfer.2005.01.029Search in Google Scholar

[3] Sheikholeslami M, Ganji DD. Heat transfer of Cu-water nanofluid flow between parallel plates. Powder Technol. 2013;235:873–9.10.1016/j.powtec.2012.11.030Search in Google Scholar

[4] Huminic G, Huminic A. Entropy generation of nanofluid and hybrid nanofluid flow in thermal systems: A review. J Mol Liq. 2020;302:112533.10.1016/j.molliq.2020.112533Search in Google Scholar

[5] Sheikholeslami M, Hatami M, Ganji DD. Nanofluid flow and heat transfer in a rotating system in the presence of a magnetic field. J Mol Liq. 2014;190:112–20.10.1016/j.molliq.2013.11.002Search in Google Scholar

[6] Li J, Kleinstreuer C. Thermal performance of nanofluid flow in microchannels. Int J Heat Fluid Flow. 2008;29(4):1221–32.10.1016/j.ijheatfluidflow.2008.01.005Search in Google Scholar

[7] Ghalandari M, MirzadehKoohshahi E, Mohamadian F, Shamshirband S, Chau KW. Numerical simulation of nanofluid flow inside a root canal. Eng Appl Comput Fluid Mech. 2019;13(1):254–64.10.1080/19942060.2019.1578696Search in Google Scholar

[8] Salman S, Talib AA, Saadon S, Sultan MH. Hybrid nanofluid flow and heat transfer over backward and forward steps: A review. Powder Technol. 2020;363:448–72.10.1016/j.powtec.2019.12.038Search in Google Scholar

[9] Wang F, Saeed AM, Puneeth V, Shah NA, Anwar MS, Geudri K, et al. Heat and mass transfer of Ag–H2O nano-thin film flowing over a porous medium: A modified Buongiorno’s model. Chin J Phys. 2023;84:330–42.10.1016/j.cjph.2023.01.001Search in Google Scholar

[10] Wang F, Asjad MI, Zahid M, Iqbal A, Ahmad H, Alsulami MD. Unsteady thermal transport flow of Casson nanofluids with generalized Mittag–Leffler kernel of Prabhakar’s type. J Mater Res Technol. 2021;14:1292–300.10.1016/j.jmrt.2021.07.029Search in Google Scholar

[11] Wang F, Khan SA, Gouadria S, El-Zahar ER, Khan MI, Khan SU, et al. Entropy optimized flow of Darcy-Forchheimer viscous fluid with cubic autocatalysis chemical reactions. Int J Hydrog Energy. 2022;47(29):13911–20.10.1016/j.ijhydene.2022.02.141Search in Google Scholar

[12] Huang D, Wu Z, Sunden B. Effects of the hybrid nanofluid mixture in plate heat exchangers. Exp Therm Fluid Sci. 2016;72:190–6.10.1016/j.expthermflusci.2015.11.009Search in Google Scholar

[13] Tlili I, Nabwey HA, Ashwinkumar GP, Sandeep N. 3-D magnetohydrodynamic AA7072-AA7075/methanol hybrid nanofluid flow above an uneven thickness surface with slip effect. Sci Rep. 2020;10(1):1–13.10.1038/s41598-020-61215-8Search in Google Scholar PubMed PubMed Central

[14] Hayat T, Nadeem S. Heat transfer enhancement with Ag–CuO/water hybrid nanofluid. Results Phys. 2017;7:2317–24.10.1016/j.rinp.2017.06.034Search in Google Scholar

[15] Kasaeian A, Daneshazarian R, Mahian O, Kolsi L, Chamkha AJ, Wongwises S, et al. Nanofluid flow and heat transfer in porous media: a review of the latest developments. Int J Heat Mass Transf. 2017;107:778–91.10.1016/j.ijheatmasstransfer.2016.11.074Search in Google Scholar

[16] Turkyilmazoglu M. Nanofluid flow and heat transfer due to a rotating disk. Comput Fluids. 2014;94:139–46.10.1016/j.compfluid.2014.02.009Search in Google Scholar

[17] Kandelousi MS. KKL correlation for simulation of nanofluid flow and heat transfer in a permeable channel. Phys Lett A. 2014;378(45):3331–9.10.1016/j.physleta.2014.09.046Search in Google Scholar

[18] Amani M, Amani P, Bahiraei M, Ghalambaz M, Ahmadi G, Wang LP, et al. Latest developments in nanofluid flow and heat transfer between parallel surfaces: A critical review. Adv Colloid Interface Sci. 2021;294:102450.10.1016/j.cis.2021.102450Search in Google Scholar PubMed

[19] Ashorynejad HR, Sheikholeslami M, Pop I, Ganji DD. Nanofluid flow and heat transfer due to a stretching cylinder in the presence of the magnetic field. Heat Mass Transf. 2013;49(3):427–36.10.1007/s00231-012-1087-6Search in Google Scholar

[20] Gbadeyan JA, Titiloye EO, Adeosun AT. Effect of variable thermal conductivity and viscosity on Casson nanofluid flow with convective heating and velocity slip. Heliyon. 2020;6(1):e03076.10.1016/j.heliyon.2019.e03076Search in Google Scholar PubMed PubMed Central

[21] Lahmar S, Kezzar M, Eid MR, Sari MR. Heat transfer of squeezing unsteady nanofluid flow under the effects of an inclined magnetic field and variable thermal conductivity. Phys A: Stat Mech Appl. 2020;540:123138.10.1016/j.physa.2019.123138Search in Google Scholar

[22] Al-Hussain ZA, Renuka A, Muthtamilselvan M. A magneto-bioconvective and thermal conductivity enhancement in a nanofluid flow containing gyrotactic microorganism. Case Stud Therm Eng. 2021;23:100809.10.1016/j.csite.2020.100809Search in Google Scholar

[23] Eid MR, Nafe MA. Thermal conductivity variation and heat generation effects on magneto-hybrid nanofluid flow in a porous medium with slip condition. Waves Random Complex Media. 2022;32(3):1103–27.10.1080/17455030.2020.1810365Search in Google Scholar

[24] Paul G, Chopkar M, Manna I, Das PK. Techniques for measuring the thermal conductivity of nanofluids: a review. Renew Sustain Energy Rev. 2010;14(7):1913–24.10.1016/j.rser.2010.03.017Search in Google Scholar

[25] Tawfik MM. Experimental studies of nanofluid thermal conductivity enhancement and applications: A review. Renew Sustain Energy Rev. 2017;75:1239–53.10.1016/j.rser.2016.11.111Search in Google Scholar

[26] Yasir M, Hafeez A, Khan M. Thermal conductivity performance in hybrid (SWCNTs-CuO/Ethylene glycol) nanofluid flow: Dual solutions. Ain Shams Eng J. 2022;13(5):101703.10.1016/j.asej.2022.101703Search in Google Scholar

[27] Han S, Zheng L, Li C, Zhang X. Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model. Appl Math Lett. 2014;38:87–93.10.1016/j.aml.2014.07.013Search in Google Scholar

[28] Hayat T, Khan MI, Farooq M, Alsaedi A, Waqas M, Yasmeen T. Impact of Cattaneo–Christov heat flux model in the flow of variable thermal conductivity fluid over a variable thicked surface. Int J Heat Mass Transf. 2016;99:702–10.10.1016/j.ijheatmasstransfer.2016.04.016Search in Google Scholar

[29] Mustafa M. Cattaneo-Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid. Aip Adv. 2015;5(4):047109.10.1063/1.4917306Search in Google Scholar

[30] Merkin JH. Natural-convection boundary-layer flow on a vertical surface with Newtonian heating. Int J Heat Fluid Flow. 1994;15(5):392–8.10.1016/0142-727X(94)90053-1Search in Google Scholar

[31] Hayat T, Khan MI, Waqas M, Alsaedi A. Newtonian heating effect in nanofluid flow by a permeable cylinder. Results Phys. 2017;7:256–62.10.1016/j.rinp.2016.11.047Search in Google Scholar

[32] Salleh MZ, Nazar R, Pop I. Boundary layer flow and heat transfer over a stretching sheet with Newtonian heating. J Taiwan Inst Chem Eng. 2010;41(6):651–5.10.1016/j.jtice.2010.01.013Search in Google Scholar

[33] Naveed M, Abbas Z, Sajid M. MHD flow of micropolar fluid due to a curved stretching sheet with thermal radiation. J Appl Fluid Mech. 2015;9(1):131–8.10.18869/acadpub.jafm.68.224.23967Search in Google Scholar

[34] Sajid M, Ali N, Javed T, Abbas Z. Stretching a curved surface in a viscous fluid. Chin Phys Lett. 2010;27(2):024703.10.1088/0256-307X/27/2/024703Search in Google Scholar

[35] Gowda RP, Al-Mubaddel FS, Kumar RN, Prasannakumara BC, Issakhov A, Rahimi-Gorji M, et al. Computational modeling of nanofluid flow over a curved stretching sheet using Koo–Kleinstreuer and Li (KKL) correlation and modified Fourier heat flux model. Chaos Solit Fractals. 2021;145:110774.10.1016/j.chaos.2021.110774Search in Google Scholar

[36] Raju CSK, Ahammad NA, Sajjan K, Shah NA, Yook SJ, Kumar MD. Nonlinear movements of axisymmetric ternary hybrid nanofluids in a thermally radiated expanding or contracting permeable Darcy walls with different shapes and densities: Simple linear regression. Int Commun Heat Mass Transf. 2022;135:106110.10.1016/j.icheatmasstransfer.2022.106110Search in Google Scholar

[37] Eswaramoorthi S, Loganathan K, Faisal M, Botmart T, Shah NA. Analytical and numerical investigation of Darcy-Forchheimer flow of a nonlinear-radiative non-Newtonian fluid over a Riga plate with entropy optimization. Ain Shams Eng J. 2023;14(3):101887.10.1016/j.asej.2022.101887Search in Google Scholar

[38] Ramesh GK, Madhukesh JK, Shah NA, Yook SJ. Flow of hybrid CNTs past a rotating sphere subjected to thermal radiation and thermophoretic particle deposition. Alex Eng J. 2023;64:969–79.10.1016/j.aej.2022.09.026Search in Google Scholar

[39] Qureshi MZA, Faisal M, Raza Q, Ali B, Botmart T, Shah NA, et al. Morphological nanolayer impact on hybrid nanofluids flow due to dispersion of polymer/CNT matrix nanocomposite material. AIMS Math. 2023;8(1):633–56.10.3934/math.2023030Search in Google Scholar

[40] Raza Q, Qureshi MZA, Khan BA, Kadhim Hussein A, Ali B, Shah NA, et al. Insight into dynamic of mono and hybrid nanofluids subject to binary chemical reaction, activation energy, and magnetic field through the porous surfaces. Mathematics. 2022;10(16):3013.10.3390/math10163013Search in Google Scholar

[41] Rehman SU, Fatima N, Ali B, Imran M, Ali L, Shah NA, et al. The Casson dusty nanofluid: Significance of Darcy–Forchheimer law, magnetic field, and non-Fourier heat flux model subject to stretch surface. Mathematics. 2022;10(16):2877.10.3390/math10162877Search in Google Scholar

[42] Olatundun AT, Makinde OD. Analysis of Blasius flow of hybrid nanofluids over a convectively heated surface. In Defect and Diffus Forum. Vol. 377, Trans Tech Publications Ltd; 2017. p. 29–41.10.4028/www.scientific.net/DDF.377.29Search in Google Scholar

[43] Hayat T, Qayyum S, Imtiaz M, Alsaedi A. Jeffrey fluid flow due to curved stretching surface with Cattaneo-Christov heat flux. Appl Math Mech. 2018;39(8):1173–86.10.1007/s10483-018-2361-6Search in Google Scholar

[44] Hayat T, Ullah I, Ahmad B, Alsaedi A. The radiative flow of Carreau liquid in presence of Newtonian heating and chemical reaction. Results Phys. 2017;7:715–22.10.1016/j.rinp.2017.01.019Search in Google Scholar

[45] Salahuddin T, Bashir AM, Khan M, Chu YM. A comparative analysis of nanofluid and hybrid nanofluid flow through the endoscope. Arab J Sci Eng. 2022;47(1):1033–42.10.1007/s13369-021-05968-ySearch in Google Scholar

[46] Waqas H, Farooq U, Liu D, Abid M, Imran M, Muhammad T. Heat transfer analysis of hybrid nanofluid flow with thermal radiation through a stretching sheet: A comparative study. Int Commun Heat Mass Transf. 2022;138:106303.10.1016/j.icheatmasstransfer.2022.106303Search in Google Scholar

[47] Dinarvand S. Nodal/saddle stagnation-point boundary layer flow of CuO–Ag/water hybrid nanofluid: a novel hybridity model. Microsyst Technol. 2019;25(7):2609–23.10.1007/s00542-019-04332-3Search in Google Scholar

[48] Sharma RP, Gorai D, Das K. Comparative study on the hybrid nanofluid flow of Ag–CuO/H2O over a curved stretching surface with Soret and Dufour effects. Heat Transf. 2022;51(7):6365–83.10.1002/htj.22595Search in Google Scholar

[49] Madhukesh JK, Kumar RN, Gowda RP, Prasannakumara BC, Ramesh GK, Khan MI, et al. Numerical simulation of AA7072-AA7075/water-based hybrid nanofluid flow over a curved stretching sheet with Newtonian heating: A non-Fourier heat flux model approach. J Mol Liq. 2021;335:116103.10.1016/j.molliq.2021.116103Search in Google Scholar

Received: 2023-06-06

Revised: 2023-09-18

Accepted: 2023-09-25

Published Online: 2023-11-11

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ntrev-2023-0136

Keywords for this article

hybrid nanofluid; Darcy-Forchheimer flow; Newtonian heating; heat source-sink; Cattaneo; Christov heat theory; curved stretching sheet; RKF-45 approach

Creative Commons

BY 4.0