Melting rheology in thermally stratified graphene-mineral oil reservoir (third-grade nanofluid) with slip condition

Zehba Raizah; Sadique Rehman; Anwar Saeed; Mohammad Akbar; Sayed M. Eldin; Ahmed M. Galal

doi:10.1515/ntrev-2022-0511

Article Open Access

Melting rheology in thermally stratified graphene-mineral oil reservoir (third-grade nanofluid) with slip condition

Zehba Raizah , Sadique Rehman , Anwar Saeed , Mohammad Akbar , Sayed M. Eldin and Ahmed M. Galal

Published/Copyright: April 18, 2023

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From the journal Nanotechnology Reviews Volume 12 Issue 1

Abstract

More effective and lengthy energy storage systems have been highly desired by researchers. Waste heat recovery, renewable energy, and combined heating and power reactors all utilize energy storage technologies. There are three techniques that are more effective for storing thermal energy: Latent heat storage is one type of energy storage, along with sensible heat storage and chemical heat storage. Latent thermal energy storage is far more efficient and affordable with these methods. A method of storing heat energy in a substance is melting. The substance is frozen to release the heat energy it had been storing. A ground-based pump’s heat exchanger coils around the soil freezing, tundra melting, magma solidification, and semiconducting processes are examples of melting phenomenon. Due to the above importance, the present study scrutinizes the behavior of third-grade nanofluid in a stagnation point deformed by the Riga plate. The Riga plate, an electromagnetic actuator, is made up of alternating electrodes and a permanent magnet that is positioned on a flat surface. Graphene nanoparticles are put in the base fluid (Mineral oil) to make a homogenous mixture. Mathematical modeling is acquired in the presence of melting phenomenon, quadratic stratification, viscous dissipation, and slippage velocity. Suitable transformations are utilized to get the highly non-linear system of ODEs. The remedy of temperature and velocity is acquired via the homotopic approach. Graphical sketches of various pertinent parameters are obtained through Mathematica software. The range of various pertinent parameters is 1 ≤ B 1 ≤ 4 , B 2 = 1 , 3 , 5 , 7 , B 3 = 0.1 , 0.5 , 0.9 , 1.3 , 0.8 ≤ A ≤ 1.2 , Re = 1 , 3 , 5 , 7 , S 1 = 1 , 3 , 5 , 7 , M 1 = 1 , 6 , 11 , 16 , 0.1 ≤ ϑ ≤ 0.4 , 0.1 ≤ Q ≤ 0.4 , Ec = 1 , 3 , 5 , 7 , 0.1 ≤ S ≤ 0.4 and Nr = 1 , 6 , 11 , 16 . Skin friction (drag forces) and Nusselt number (rate of heat transfer) are explained via graphs. The velocity is enhancing the function against melting parameter while temperature is the decelerating function as melting factor is amplified. The temperature field reduces with the accelerating estimations of stratified parameter. The energy and velocity profiles de-escalate with intensifying values of volume fraction parameter.

Keywords: third-grade nanofluid; mineral oil; thermal stratification; slip conditions; melting heat transport; Riga plate; HAM

Nomenclature

A: ratio parameter
a 1 , a 2: dimensional constants
B k ( k = 1 − 3 ): unitless material parameters
c s: solid surface heat capacity
( C p ) nf: specific heat of nanofluid ( J kg − 1 K − 1 )
d: width between magnets and electrode ( m )
Ec: Eckert number
f: base fluid
I: unit tensor
k 0: applied current within electrode ( A )
k nf: thermal conductivity of nanofluid ( W m − 1 K − 1 )
M 1: melting factor
m 0: magnetization of magnets ( A m − 1 )
N l: slip length ( m )
P: pressure ( kg m − 1 s − 2 )
Pr: Prandtl number
Q: modified Hartmann number
Re: Reynolds number
S: thermally stratified parameter
S 1: slip parameter
S 2: Cauchy stress tensor
s: solid nano-particles
T ∞: ambient temperature ( K )
T 0: reference temperature ( K )
T m: melting temperature ( K )
U w , U ∞: stretching and free stream velocity ( m s − 1 )
u , v: velocity component ( m s − 1 )
ν f: kinematic viscosity
x , y: space coordinate ( m )
ρ nf: nanofluid’s density ( kg m − 3 )
μ nf: nanofluid’s absolute viscosity ( kg m − 1 K − 1 )
η k ( k = 1 − 3 ): material parameters
ε: latent heat ( J )
ϑ: volume fraction parameter

1 Introduction

The process of stratification is crucial for the movement of mass and heat. The process of creating liquid layers with various densities is known as stratification. Thermal or concentration changes in the liquid flow are what create variations in density. Stratification has numerous uses in the natural, agricultural, medicinal, and industrial sciences. There are numerous ways in which stratification can occur, such as in reservoirs, air heterogeneous mixtures, and the ocean, to name a few. Additionally, it is very important to control the levels of oxygen and hydrogen in the atmosphere and the oceans, as it has a big effect on oxygen deprivation, algae populations, and water quality in the lower regions of ponds, rivers, and lakes. Rehman et al. [1] discussed the thermally stratified Powell–Eyring fluid with melting phenomenon. The homotopy analysis technique (HAM) is exploited to acquire the series solution to the problem. Farooq et al. [2] explored the impact of blood-magnesium nanofluid on thermally stratified Powell–Erying fluid. The energy profile is a de-escalating function against the intensifying behavior of the thermal stratification factor. Jabeen et al. [3] have constructed the physical significance of thermal and solutal stratifications on the Powell–Erying liquid by employing of Cattaneo–Christov heat flux. Simon and Mutuku [4] investigated the Powell–Eyring liquid with the idea of dual stratification through the existence of the magnetic effect. A famous numerical method called the predictor–corrector scheme is used to solve the issue. Sudarsana Reddy and Sreedevi [5] investigated the heat-mass transmission of nanoliquid toward the permeable media along the cumulative impact of thermal and solutal stratification. Utilizing the Galerkin finite element method, the liquid model is resolved. Kumar and Srinivas [6] talked about the impact of Joule heating and radiation on the Powell–Eyring nanoliquid over an inclined porous extended sheet.

In the packaging sector, the flow over a stretched surface problem is utilized in numerous technical approaches. Examples of approaches include molding, wire drawing, fresh rolling, creation of glass fiber, making of rubber product sheets, melt-spinning, and the conditioning of a huge metal plate in an electrolytic solution. In the industry, the polymer is constantly extruded from a die to a windup roller that is positioned a specific distance away to create polymer sheets and filaments. The thin polymer sheet is a fluid-filled floor that is constantly shifting and flowing at an uneven speed [7]. According to experiments, the stretching surface’s pace is roughly equal to the separation from the orifice [8]. The extending of a flexible flat sheet, which works in its flight with a velocity that changes linearly with path length from a given factor due to uniform stress software, is studied by Crane [9] as a consistent two-dimensional incompressible shear layer float of a Newtonian fluid. Given that Crane [9] was able to precisely solve the two-dimensional Navier–Stokes equation, this topic is very intriguing. A significant amount of research has been published in the flow discipline over a stretched surface as a result of this ground-breaking work [10,11,12,13,14,15].

A nanofluid is a liquid that contains particles that are smaller than a nanometer. These suspensions of colloidal fluid consist of nanoparticles suspended in the main liquid. In nanofluids, metals, carbides, oxides, or nanotubes are frequently used as nanomaterials. Numerous common liquids have been used as active liquids to transport heat in a variety of procedures. Due to its accessibility, water is frequently utilized as an active liquid; yet, due to its weak thermal conductivity, it is not considered to be a reliable heat carrier. Many applications also use non-exchange liquids including ethylene glycol, engine oil, and others. However, their use in heat transfer techniques is restricted due to the high viscidity and toxic environment of these compounds [16]. The pace of engine cooling is boosted by adding nanoparticles to ethylene glycol and water mixture that is used as an automobile coolant all over the world. High-performance computers with a maximum power of 100–300 W/cm² have been produced by Rafati et al. [17] in recent years as silicon chip electronics conditioning and microcomputer circuits have advanced. On the outer surface of the laminar boundary, the heat transfer of Newtonian nanofluid free convective is carefully explored. It has been found that the free convective heat transport is not only determined by the active nanofluid of thermal conductivity but also that the viscidity model used is susceptible and has a significant effect on the behavior of heat transference [18]. Jafaryar and Sheikholeslami [19] explored the effectiveness of magnetic field on the achievement of the photovoltaic solar system using ferrofluid. The finite volume method is implemented for simulation. As a result, improvement in inlet velocity reduces the cell temperature, and due to this, the electrical and thermal performances of PV cells are improved by nearly 6.519 and 92.97%. Jafaryar et al. [20] scrutinized the impact of Kelvin force on laminar nanoliquid flow (Fe₃O₄-water). Improvement occurs in ferrofluid velocity with the amplification of Kelvin force. Chand et al. [21] depicted the thermal instability in a Prandtl nanoliquid via a permeable medium. This indicated that the Prandtl and Darcy numbers have a de-stabilizing impact, while the modified diffusivity ratio and Lewis number have a stabilizing impact on the stationary convection. The significance of nanoliquid flow with various geometry were investigated by Jafaryar et al. [22] and Sheikholeslami [23]. Ali et al. [24] scrutinized the rotating Oldroyd-B nanoliquid magnetohydrodynamic (MHD) flow with Soret and Dufour characteristics via the stretchable sheet utilizing the finite element technique. In this inspection, the rotational parameter and Deborah number reduce the nanoliquid’s motion while the thermal relaxation factor declines the temperature. The influence of heat source/sink on the nanofluid flow along with inclined stretchable sheets and walls was studied by Ali et al. [25]. Abbasi et al. [26] explored the blood-based hybrid nanoliquid flow through a tapered complex wavy channel with slip conditions. This shows that nanoliquid’s motion increments with the volume fraction factor while temperature reduces against the volume fraction factor.

One of the many non-Newtonian kinds of liquids identified by Rivlin and Ericksen [27] is third-grade liquids. Hayat et al. [28] explored third-grade liquid with the impacts of Dufour and Soret using three-layered coordinate frames. Waqas [29] deliberated the characteristics of variable conductivity and heat source/sink with the chemical reaction on non-Newtonian liquid. The liquid’s temperature improves with the augmented estimations of variable conductivity parameter. Nasir et al. [30] demonstrated mixed convected Maxwell nanoliquid with chemical reaction and thermal radiation along with an extended sheet. Higher thermal radiation and chemical reaction factors improved the temperature and concentration, individually. Turkyilmazoglu [31] concentrated on a range of expanding plate examples. Nanofluid streams of third-grade fluid under broad dispersion and Newtonian warming were studied by Shehzad et al. [32]. The fluid stream that was investigated in a permeable level plate by Sajid et al. [33] was taken using the finite element method. Time-subordinate crushing stream arrangements between two equal surfaces were found by Rashidi et al. [34]. The surface has lately been considerably increased by Shehzad et al.’s [35] recent attention on third-grade liquid streams above. The behavior of second-grade fluid in a revolving, three-layered shape was studied by Hussnain et al. [35]. The second-grade liquid on a moving level plate was explored by Bariş [36]. Second-grade liquid in three-aspect, an infinite flat plane divider with verbose attractions, was recently studied by Shoaib et al. [37]. Ramzan et al. [38] concentrated on the radioactive consequences of second-grade nanofluids. The focus of Mustafa et al. [39] was on second-grade nanofluid across a spreading sheet.

The melting phenomenon has significance in different industrial applications including solidification of magma, storage technologies, preparation of semiconductors materials, tundra melting, etc. Also, thermal radiation is utilized to improve the heat transport rate having numerous applications in industry such as polymer production, removing toxic microorganisms from fluid, combustion reactors, etc. The literature that is currently available indicates that none of the studies that have been published separately have taken into account the Tiwari–Das nanoliquid scheme over the Riga plate in the proximity of stagnation point, as well as the influences of controlling parameters like thermal stratification, melting phenomenon, radioactivity heating flux, and slippage conditions. Graphene nanoparticles are suspended in mineral oil to make a nanofluid. The series solution is acquired via the convergence technique called the HAM [40–44]. Finally, the impact of various pertinent parameters is presented graphically and the physical meaning of each parameter is discussed. The quantity of physical interest i.e., skin friction and Nusselt number, is explained physically through graphs.

2 Mathematical statement of the problem

Consider the steady, incompressible nanofluid flow of third-grade fluid near the stagnation point with slip velocity through Riga plate. The melting phenomenon is taken into account in order to enhance heat transport. Viscous dissipation, slip velocity, and quadratic stratification are also taken in the flow model. The MHD phenomenon also controls the flow, due to the presence of the Riga plate. Characteristics of heat transport are scrutinized via thermal stratifications and radiation. Further, the melting surface temperature T m is chosen to be less than the ambient temperature T ∞ , i.e., ( T ∞ > T m ) . Moreover, graphene nanoparticles are suspended in base fluid mineral oils. Figure 1 shows the geometry of the flow problem. The following relations for incompressible third-grade nanofluid model all-inclusive body forces with momentum and continuity equations.

(1) div v = 0 ,

(2) ρ nf d v d t = div S 2 + ρ nf b ,

where v stands for the velocity, ρ nf denotes the nanofluid’s density, S 2 demonstrates the Cauchy stress tensor for third-grade nanofluid, b scrutinizes the body forces, and d d t portrays the material derivative.

(3) S 2 = − p I + μ nf H 1 ∗ + ε 1 H 1 ∗ + ε 2 H 2 ∗ 2 + γ 1 H 3 ∗ + γ 2 ( H 2 ∗ H 1 ∗ + H 1 ∗ H 2 ∗ ) + γ 3 ( T r H 1 ∗ 2 ) H 1 ∗ ,

where p denotes the pressure, μ nf depicts the nanofluid’s dynamic viscosity, I denotes the unit tensor, ε k , γ k denote the material constants, and H 1 ∗ , H k ∗ denotes the Rivlin–Erickson tensor. Here

(4) H 1 ∗ = grad v + ( grad v ) T ,

(5) H k ∗ = d H k − 1 ∗ d t + H k − 1 ∗ grad ( v ) + grad ( v ) T H k − 1 ∗ , ( k = 2 , 3 ) .

The Clausius–Duhem inequality is described as the relations follows;

(6) μ nf ≥ 0 , ∣ ε 1 + ε 2 ∣ ≤ 2 6 μ nf γ 3 , ε 1 ≥ 0 , γ 3 ≥ 0 , γ 1 = 0 = γ 2 ,

Implementing equation (6) on equation (3), we acquired the following:

(7) S 2 = − p I + μ nf H 1 ∗ + ε 1 H 1 ∗ + ε 2 H 2 ∗ 2 + + + γ 3 ( Tr H 1 ∗ 2 ) H 1 ∗ .

The governing equations are acquired utilizing the boundary layer approximations of third-grade nanoliquid as follows [45]:

(8) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(9) ρ nf u ∂ u ∂ x + v ∂ u ∂ y = U ∞ ∂ U ∞ ∂ x + μ n f ∂ 2 u ∂ y 2 + η 1 u ∂ 3 u ∂ x ∂ y 2 + v ∂ 3 u ∂ y 3 + ∂ u ∂ x ∂ 2 u ∂ y 2 + 3 ∂ u ∂ y ∂ 2 u ∂ x ∂ y + 2 η 2 ∂ u ∂ y ∂ 2 u ∂ x ∂ y + 6 η 3 ∂ u ∂ y 2 ∂ 2 u ∂ y 2 + π k 0 m 0 8 e − π d y ,

(10) ( ρ C p ) nf u ∂ T ∂ x + v ∂ T ∂ y = k nf ∂ 2 T ∂ y 2 + 16 σ ∗ T ∞ 3 3 k ∗ ∂ 2 T ∂ y 2 + η 1 u ∂ u ∂ y ∂ 2 u ∂ x ∂ y + v ∂ u ∂ y ∂ 2 u ∂ y 2 + μ nf ∂ u ∂ y 2 + 2 η 3 ∂ u ∂ y 4 ,

The boundary conditions are as follows:

(11) u ( y ) = U w + N l ∂ u ∂ y , T ( y ) = T m , k nf ∂ T ∂ y y = 0 = ρ nf ( ε + c s ( T m − T 0 ) ) v ( x , y ) , at y = 0 , u ( y ) = U ∞ ( x ) = b x , T ( y ) = T ∞ , y → ∞ ,

where

(12) ρ nf = ( 1 − ϑ ) ρ f + ϑ ρ s , U w = c x , μ nf = ( 1 − ϑ ) − 2.5 μ f , ( ρ C p ) nf = ( 1 − ϑ ) ( ρ C p ) f + ϑ ( ρ C p ) s , k nf k f = ( k s + 2 k f ) − 2 ϑ ( k f − k s ) ( k s + 2 k f ) + ϑ ( k f − k s ) , T m = T 0 + a 1 x 2 , T ∞ = T 0 + a 2 x 2 .

where ρ nf denotes the nanofluid’s density, μ nf represents the viscosity, η K ( K = 1 , 2 , 3 ) is the material parameter, U w demonstrates the stretching velocity, k 0 indicates the applied current within the electrodes, N l indicates the slip length, ε stands for the latent heat, d represents the width between magnets and electrodes, u and v stand for the velocity components in x and y directions, respectively, c s demonstrates the solid surface heat capacity, m 0 stands for the magnetization of permanent magnets, T 0 denotes the reference temperature, ϑ is the nanoparticle volumetric fraction coefficient, ρ f , μ f , k f , and ( C p ) f are the consistency, dynamic viscidness, thermal conductivity, and adequate heat capacity of the normal fluid individually, a 1 and a 2 are individually the dimensional constants and ρ s , k s and ( C p ) s are the consistency, thermal conductance, and adequate heat capacity of the nanoparticles, respectively. Table 1 lists the material properties of the nanomolecules employed in this study as well as the typical fluid mineral oil.

Figure 1

(a) Sketch of Riga plate and (b) geometry of the model.

Table 1

Main properties of base liquid and nanoparticles at room temperature [46,47]

Thermo-physical properties	ρ ( kg m − 3 )	c p ( J kg − 1 K − 1 )	k ( W m − 1 k − 1 )	μ ( Pa s )
Graphene	2,250	2,100	2,500	—
Mineral oil	861	1,860	0.157	0.01335

The corresponding transformations are utilized in order to transmute PDEs into ODEs.

(13) ξ = y c υ f , u = c x f ′ ( ξ ) , v = − c υ f f ( ξ ) , θ ( ξ ) = T − T m T ∞ − T 0 .

After utilizing equation (13) in equations (8)–(10) than equation (8) is satisfies and equations (9) and (10) are transformed into the following form:

(14) ϑ 1 f ′ ″ + ϑ 2 ( f f ″ − f ′ 2 ) + B 1 ( 2 f ′ f ′ ″ − f f i v ) + ( 3 B 1 + 2 B 2 ) f ″ 2 + 6 ϑ 1 B 3 Re f ″ 2 f ′ ″ + Q ∗ Exp ( − β ξ ) + A 2 = 0 ,

(15) 1 Pr ( ϑ 4 + Nr ) θ ″ + 2 S f ′ + f θ ′ − f ′ θ + ϑ 1 ϑ 3 Ec f ″ 2 + B 1 Ec ϑ 3 ( f ′ f ″ 2 − f f ″ f ′ ″ ) + 2 B 3 Ec Re f ″ 4 = 0 ,

The unitless boundary conditions are as follows:

(16) f ′ ( 0 ) = 1 + S 1 f ″ ( 0 ) , θ ( 0 ) = 0 , Pr f ( 0 ) + ϑ 4 ϑ 1 M 1 θ ′ ( 0 ) = 0 , f ′ = A , θ = 1 − S , as ξ → ∞ .

where

(17) ϑ 1 = ( 1 − ϑ ) − 2.5 , ϑ 2 = ( 1 − ϑ ) + ϑ ρ s ρ f , ϑ 3 = ( 1 − ϑ ) + ϑ ( ρ C p ) s ( ρ C p ) f , ϑ 4 = k nf k f , B 1 = c η 1 μ f , B 2 = c η 2 μ f , B 3 = c 2 η 3 μ f , A = b c , Q = π k 0 m 0 8 c 3 x ρ f , β = π d υ f c , Pr = ( μ C p ) f k f , Re = c x 2 υ f , Ec = U w 2 ( C p ) f ( T ∞ − T 0 ) , S = a 1 a 2 , S 1 = N l c υ f , M 1 = ( C p ) f ( T ∞ − T 0 ) ε + c s ( T m − T 0 ) , Nr = 16 σ ∗ T ∞ 3 3 k ∗ k f .

where B K ( K = 1 , 2 , 3 ) denotes the fluid material parameter, A scrutinizes the ratio parameter, Q represents the modified Hartmann number, Pr demonstrates the Prandtl number, Re denotes the Reynolds number, Ec stands for the Eckert number, S indicates the thermal stratified parameter, S 1 represents the rapidity slip factor, M 1 denotes the melting factor, and Nr is the radiation parameter.

The drag force and heat rate transportation are specified as follows:

(18) C f = τ w ρ U w 2 , Nu x = x q w k f ( T ∞ − T m ) ,

where the wall drag force and wall heat transfer are as follows:

(19) τ w = μ nf ∂ u ∂ y + η 1 u ∂ 2 u ∂ x ∂ y + 2 ∂ u ∂ x ∂ u ∂ y + v ∂ 2 u ∂ y 2 + 2 η 3 ∂ u ∂ y 3 y = 0 , q w = − k f + 16 σ ∗ T ∞ 3 3 k ∗ ∂ T ∂ y y = 0 .

The unitless form of equation (18) is given as follows:

(20) C f Re x = [ ϑ 1 f ″ + B 1 ( 3 f ′ f ″ − f f ″ ′ ) + 2 Re B 3 f ″ 3 ] ξ = 0 , Nu x ( Re x ) − 1 / 2 = − ( 1 + Nr ) 1 − S θ ′ ( 0 ) .

3 Homotopic approach

In 1992, Liao introduced the homotopy approach. It is desirable to use the homotopic technique to obtain nonlinear model solutions. Moreover, the homotopy evaluation methodology allows us to simply adapt and change the converging position and charges of approximations acquired as needed, unlike the prior perturbations and nonperturbative procedures. In a nutshell, the homotopy assessment has the following additional benefits: it is valid even if a given nonlinear problem contains no small or extensive parameters at all, it can give us a convenient way to change and manipulate the convergence place and charge of estimation collection when necessary, and it can be used to accurately guesstimate a nonlinear problem by selecting specific units of base functions. In Figure 2, the phases in the homotopy approach are given

Figure 2

Flow chart of homotopy approach.

Additionally, it offers a great deal of freedom in selecting the initial guesses and linear operators, which are denoted by the following:

(21) f ( ξ ) = A ξ + ( 1 − A ) ( 1 + S 1 ) × ( 1 − e − ξ ) − ϑ 4 ( 1 − S ) M 1 ϑ 4 Pr , θ ( ξ ) = ( 1 − S ) ( 1 − e − ξ ) ,

(22) L f ( f ) = d 3 f d ξ 3 − d f d ξ , L θ ( θ ) = d 2 θ d ξ 2 − θ ,

with

(23) L f ( D 1 + D 2 e ξ + D 3 e − ξ ) = 0 , L θ ( D 4 e ξ + D 5 e − ξ ) = 0 .

where D i ( i = 1 , 2 , … 5 ) signify the optional fixed constant.

3.1 Zeroth deformation equations

(24) ( 1 − ϖ ) ℑ f [ f ( ξ ; ϖ ) − f 0 ( ξ ) ] = ϖ h f N f [ f ( ξ ; ϖ ) ] ,

(25) ( 1 − ϖ ) ℑ θ [ θ ( ξ ; ϖ ) − θ 0 ( ξ ) ] = ϖ h θ N θ [ θ ( ξ ; ϖ ) , f ( ξ ; ϖ ) ] ,

(26) f ( 0 ; ϖ ) = 1 + S 1 f ″ ( 0 ; ϖ ) , Pr f ( 0 ; ϖ ) + ϑ 4 ϑ 1 M 1 θ ( 0 ; ϖ ) = 0 , θ ( 0 ; ϖ ) = 0 , f ′ ( ∞ ; ϖ ) = A , θ ( ∞ ; ϖ ) = 1 − S ,

(27) N f [ f ( ξ ; ϖ ) ] = ϑ 1 ∂ 3 f ( ξ ; ϖ ) ∂ ξ 3 + ϑ 2 f ( ξ ; ϖ ) ∂ 2 f ( ξ ; ϖ ) ∂ ξ 2 − ∂ f ( ξ ; ϖ ) ∂ ξ 2 + B 1 2 ∂ f ( ξ ; ϖ ) ∂ ξ ∂ 3 f ( ξ ; ϖ ) ∂ ξ 3 − f ( ξ ; ϖ ) ∂ 4 f ( ξ ; ϖ ) ∂ ξ 4 + ( 3 B 1 + 2 B 2 ) ∂ 2 f ( ξ ; ϖ ) ∂ ξ 2 2 + 6 ϑ 1 B 3 Re ∂ 2 f ( ξ ; ϖ ) ∂ ξ 2 2 ∂ 3 f ( ξ ; ϖ ) ∂ ξ 3 + Q ∗ Exp ( − β ξ ) + A 2 ,

(28) N θ [ θ ( ξ ; ϖ ) , f ( ξ ; ϖ ) ] = 1 Pr ( ϑ 4 + Nr ) ∂ 2 θ ( ξ ; ϖ ) ∂ ξ 2 + 2 S ∂ f ( ξ ; ϖ ) ∂ ξ + f ( ξ ; ϖ ) ∂ θ ( ξ ; ϖ ) ∂ ξ − ∂ f ( ξ ; ϖ ) ∂ ξ θ ( ξ ; ϖ ) + ϑ 1 ϑ 3 Ec ∂ 2 f ( ξ ; ϖ ) ∂ ξ 2 2 + B 1 Ec ϑ 3 ∂ f ( ξ ; ϖ ) ∂ ξ ∂ 2 f ( ξ ; ϖ ) ∂ ξ 2 2 − f ( ξ ; ϖ ) ∂ 2 f ( ξ ; ϖ ) ∂ ξ 2 ∂ 3 f ( ξ ; ϖ ) ∂ ξ 3 + 2 B 3 Ec Re ∂ 2 f ( ξ ; ϖ ) ∂ ξ 2 4 ,

where ϖ ∈ [ 0 , 1 ] demonstrates the values of anchoring factor, while the auxiliary parameters are individually depicted by h f and h θ having non-zero estimations.

3.2 qth deformation equations

(29) h f R q f ( ξ ) = ℑ f [ f q ( ξ ) − χ q f q − 1 ( ξ ) ] ,

(30) h θ R q θ ( ξ ) = ℑ θ [ θ q ( ξ ) − χ q θ q − 1 ( ξ ) ] ,

(31) f q ′ ( 0 ) = 1 + S 1 f q ' ' ( 0 ) , Pr f ( 0 ) + ϑ 4 ϑ 1 M 1 θ q ' ( 0 ) = 0 , θ q ( 0 ) = 0 , f q ' ( ξ ) = 0 , θ q ( ξ ) = 0 , as ξ → ∞ .

(32) R q f ( ξ ) = ϑ 1 f ′ ″ q − 1 + ϑ 2 ∑ k = 0 q − 1 f ″ q − 1 − k f k − ∑ k = 0 q − 1 f ′ q − 1 − k f ′ k + B 1 2 ∑ k = 0 q − 1 f ′ q − 1 − k f ′ ″ k − ∑ k = 0 q − 1 f ″ ″ q − 1 − k f k + ( 3 B 1 + 2 B 2 ) ∑ k = 0 q − 1 f ″ q − 1 − k f ″ k + 6 ϑ 1 Re B 3 ∑ k = 0 q − 1 ∑ l = 0 n f ″ l − n f ″ n f ′ ″ q − 1 − k + Q ∗ Exp ( − β ξ ) + A 2 ( 1 − χ q ) ,

(33) R q θ ( ξ ) = 1 Pr ( ϑ 4 + Nr ) θ ″ q − 1 − ∑ k = 0 q − 1 f ′ q − 1 − k θ k + 2 S f ′ q − 1 + ∑ k = 0 q − 1 θ ′ q − 1 − k f k + ϑ 1 ϑ 3 Ec ∑ k = 0 q − 1 ( f ″ q − 1 − k f ″ k ) + B 1 Ec ϑ 3 ∑ k = 0 q − 1 ∑ l = 0 n f ″ n − l f ″ n f ′ q − 1 − k − ∑ k = 0 q − 1 ∑ l = 0 n f ″ n − l f ′ ″ n f q − 1 − k + 2 B 3 Ec Re ( f ″ ″ ) 2 ,

(34) χ q = 0 , q ≤ 1 1 , q > 1 .

Appropriating to ϖ = 0 and ϖ = 1 , we can write

(35) f ( ξ ; 0 ) = f 0 ( ξ ) , f ( ξ ; 1 ) = f 0 ( ξ ) ,

(36) θ ( ξ ; 0 ) = θ 0 ( ξ ) , θ ( ξ ; 1 ) = θ 0 ( ξ ) .

The initial iteration begins from f 0 ( ξ ) and θ 0 ( ξ ) to the definitive solutions with the volatility of ϖ from zero to one. By implementing ϖ = 1 and Taylor’s expansion, we get

(37) f ( ξ ) = f 0 ( ξ ) + ∑ q = 1 ∞ f q ( ξ ) ,

(38) θ ( ξ ) = θ 0 ( ξ ) + ∑ q = 1 ∞ θ q ( ξ ) .

The general solutions ( f q , θ q ) are denoted by

(39) f q ( ξ ) = f q • ( ξ ) + D 1 + D 2 e ξ + D 3 e − ξ ,

(40) θ q ( ξ ) = θ q • ( ξ ) + D 4 e ξ + D 5 e − ξ ,

where f q • ( ξ ) and θ q • ( ξ ) scrutinize the unique solutions.

3.3 Code authenticity

The accuracy of the numerical method was evaluated by comparing the results of the heat transfer rate produced using the current methodology with that of previous works [48,49]. Table 2 summarizes the comparison of the outcomes of the most recent concurrence examination with those from earlier results.

Table 2

Contrasting estimations of heat transport rate − θ ′ ( 0 ) for various Prandtl numbers, when M 1 = 0 , S = 0 , S 1 = 0 , A = 0 and ϑ = 0 .

Pr	Ref. [48]	Ref. [49]	Present
0.72	0.1691	0.1690	0.1690
1.0	0.4539	0.4537	0.4537
3.0	0.9114	0.9113	0.9113
7.0	1.8954	1.8958	1.8958

4 Parametric study

Third-grade liquids are a subset of non-Newtonian liquids that can experience normal stresses even in rigid boundaries as well as non-Newtonian phenomena including shear thinning and thickening. It also has fluid characteristics that are viscid and stretchy. The non-Newtonian 3-grade nanofluids graphene-mineral oil is used to determine the physical characteristics.

4.1 Velocity profile

When the physical characteristics operating on the system are taken into account, as mentioned above, it is crucial to monitor the behavior of the velocity profile.

4.1.1 Impact of material parameter

When various B 1 values were employed to represent the behavior of nanofluids, the outcome was as displayed in Figure 3. The improvement of B 1 leads to a boost in the flow velocity of nanofluid. Similarly, Figure 4 shows that when different estimations of B 2 are intensifying, the nanofluid’s velocity also increases.

$Figure 3 Velocity profile vs B 1 {B}_{1} .$

Figure 3

Velocity profile vs B 1 .

$Figure 4 Velocity profile vs B 2 {B}_{2} .$

Figure 4

Velocity profile vs B 2 .

4.1.1.1 Physical interpretations

The easy changing of the basic fluid by B 1 , a substantial component is responsible for this outcome. Physically, this is because of the enhancement in the normal stress variations which results in amplifying the nanofluid’s velocity. The same physical behavior is achieved like B 1 . This is the only thing due to this velocity becoming high.

4.1.2 Impact of third-grade parameter

On the other hand, as the third-grade parameter B 3 increases, the velocity also improves, as seen in Figure 5. This is due to the inverse relationship between third-grade parameter B 3 and the nanofluid’s viscosity.

$Figure 5 Velocity profile vs B 3 {B}_{3} .$

Figure 5

Velocity profile vs B 3 .

4.1.2.1 Physical interpretations

Physically, when the third-grade parameter rises, it gives strength to the shear thinning impact which results in reducing the nanofluid’s viscosity with an enhanced rate of shear stress. Thus, velocity rises.

4.1.3 Impact of ratio parameter

Figure 6 emphasizes the consequence of the ratio parameter on the nanofluid. Investigation reveals that the velocity outline is counting up. Additionally, the thickness of the velocity restriction layer behaves similar to the free stream velocity in terms of stretching and receding.

Figure 6

Velocity profile vs A .

4.1.3.1 Physical interpretations

Furthermore, since the fluid and the wall are moving at the same speed, there is no boundary layer when A = 1 . For A > 1 , the velocity is greater far from the wall, but at, A < 1 the velocity is greater at the wall’s surface.

4.1.4 Impact of Reynolds number

As seen in Figure 7, inertial forces boost the system’s velocity, when boosting the Reynolds number. With their high kinetic energy, the nanofluid particles move more swiftly and display a more significant number of Reynolds number abnormalities.

Figure 7

Velocity profile vs Re.

4.1.4.1 Physical interpretations

Reynolds number determines the ratio of inertial forces to viscous forces, which aids in the prediction of flow patterns in a variety of fluid flow scenarios. Physically, this trend results from a drop in the viscous forces of nanofluid as the Reynolds number increases. Thus, velocity accelerates.

4.1.5 Impact of slip parameter

The velocity slip parameter defines the variations in velocity between the particles being carried by air and those being expelled. The velocity slip condition is computed along the walls. As seen in Figure 8, the velocity slip parameter caused the flow rate to slow down.

$Figure 8 Velocity profile vs S 1 {S}_{1} .$

Figure 8

Velocity profile vs S 1 .

4.1.5.1 Physical interpretations

When the slip parameter increases, the nanofluid’s speed also increases. This happens because of the greater adhesive force between the wall and fluid particles which improves the resistance for the transfer of stretching velocity to the fluid. As a result, the nanofluid’s velocity reduces.

4.1.6 Impact of melting parameter

The impact of the melting factor is displayed in Figure 9. It depicts that velocity is the enhancing function against the greater estimations of the melting parameter. Also, the thickness of the limit layer is dominant.

$Figure 9 Velocity profile vs M 1 {M}_{1} .$

Figure 9

Velocity profile vs M 1 .

4.1.6.1 Physical interpretations

When the melting factor amplifies, the nanofluid’s velocity also amplifies. This is due to higher convective flow toward a cooling surface being produced physically by the greater melting parameter. So, the velocity field expands.

4.1.7 Impact of volume fraction parameter

Ideal solutions have a volume concentration and volume fraction that are the same when all of the components’ volumes are summed together (the solvent size is equivalent to the total of its components’ volumes). A metal nanoparticle’s volume fraction is computed before mixing by dividing the volume of the component (metal nanoparticle) by the mixture (based fluid). Particle velocity is influenced by nanoparticle volume fraction, as seen in Figure 10 fluid dynamics speed falls as the volume fraction of nanoparticles in nanofluid increases.

$Figure 10 Velocity profile vs ϑ {\vartheta } .$

Figure 10

Velocity profile vs ϑ .

4.1.7.1 Physical interpretations

This effect is caused by the reduction in fluid viscosity that comes from the accumulation of nanofluid concentration and the widening of friction.

4.1.8 Impact of modified Hartmann number

The characteristic of the modified Hartmann number is highlighted in Figure 11. It scrutinizes that the nanofluid’s speed is amplified with the increasing estimations of the modified magnetic factor.

Figure 11

Velocity profile vs Q .

4.1.8.1 Physical interpretations

Physically, a higher estimation of the modified Hartmann number boosts the internal and external forces such as electric forces, adhesive forces, etc. As a result of these forces, the momentum of fluid enhances and thus, velocity accelerates.

4.2 Temperature profile

4.2.1 Impact of melting parameter

Figure 12 illustrates how the melting factor affects the temperature outline for a nanofluid based on graphene and mineral oil. The thermal field turns into a lowering function as the melting parameter rises. When the melting parameter is improved, the thermal boundary layer’s thickness rises.

$Figure 12 Temperature profile vs M 1 {M}_{1} .$

Figure 12

Temperature profile vs M 1 .

4.2.1.1 Physical interpretations

Physically, a greater melting parameter causes heat to pass from the heated liquid to the cold wall more rapidly, boosting heat transmission to the surroundings and ultimately causing a low temperature. Thus, temperature profile decelerates.

4.2.2 Impact of Eckert number

Figure 13 illustrates the impact of Eckert number on the nanofluid known as graphene-mineral oil on the temperature field. The distribution of temperature steadily gets better as the Eckert number rises.

Figure 13

Temperature profile vs Ec.

4.2.2.1 Physical interpretations

Drag forces between liquid particles, which convert mechanical energy of fluid particles into thermal energy, are the cause of this behavior in terms of physics. The temperature field enlarges as a result.

4.2.3 Impact of volume fraction parameter

Figure 14 demonstrates the consequence of the volume friction parameter on temperature distribution. It depicts that temperature is a reducing function against greater estimates of volume fraction parameter.

$Figure 14 Temperature profile vs ϑ {\vartheta } .$

Figure 14

Temperature profile vs ϑ .

4.2.3.1 Physical interpretations

From the physics point of view, the decrease occurs due to the low thermal conductivity of the base fluid (mineral oil).

4.2.4 Impact of thermal stratification parameter

The properties of thermal stratification are scrutinized in Figure 15. It is evident that when the magnitude of the stratified parameter increases, the temperature outline decelerates.

Figure 15

Temperature profile vs S .

4.2.4.1 Physical interpretations

Physically, as the stratification parameter improves, different density areas form and impede the mineral oil’s ability to transmit heat. Additionally, the lower convective potential between the sheet surface and ambient temperature is to blame for this effect. As a result, the nanofluid’s temperature drops.

4.2.5 Impact of radiation parameter

The relationship between the temperature and radiation parameter is highlighted in Figure 16. It can be seen in Figure 16 that the thermal field of graphene-mineral oil-based nanofluid falls.

Figure 16

Temperature profile vs Nr.

4.2.5.1 Physical interpretations

Physically, the intensifying estimations of radiation parameter increase the rate of mean absorption coefficient in such a situation. Therefore, temperature is reduced.

4.3 Quantities of physical interests

4.3.1 Nusselt number

Figure 17 shows that changing the volume fraction parameter enhances the heat rate transport. This outcome is due to the lower thermal conductivity of mineral oil, which results in lower temperature of nanofluids that enhances heat transfer rate.

$Figure 17 Nusselt number vs ϑ {\vartheta } .$

Figure 17

Nusselt number vs ϑ .

4.3.2 Skin friction

Figure 18 depicts the consequence of the volume fraction parameter on skin friction. It highlights de-escalating behavior against the volume friction parameter.

$Figure 18 Skin friction vs ϑ {\vartheta } .$

Figure 18

Skin friction vs ϑ .

4.4 Isotherms

Figure 19(a) and (b) is plotted in order to scrutinize the isotherms when ratio factor A < 1 and A > 1 , individually, for nanofluid’s velocity. Figure 19(a) demonstrates the profile of isotherms when the ratio factor A < 1 , i.e., A = 0.5 . The higher estimation of isotherms in such a case is 0.95, whereas the lower isotherms is 0.55. Figure 19(b) scrutinizes the outline of isotherms when ratio parameter A > 1 , i.e., A = 1.5 . The greater estimation of isotherms in this case is 1.45, while the minimum is 1.05.

$Figure 19 (a) Isotherms when A < 1 A\lt 1 . (b) Isotherms when A > 1 . A\gt 1.$

Figure 19

(a) Isotherms when A < 1 . (b) Isotherms when A > 1 .

4.5 Comparative study of present work and existing work

The comparison between the present work and existing work [45] is highlighted in Figure 20(a) and (b). The present work shows good agreement with the existing work velocity and temperature outlines. If the modified Hartmann number, volume fraction parameter, slip parameter, thermal stratification, and radiation parameter vanished in the present work, then we have obtained the solution of existing work [45].

Figure 20

(a) Velocity profile of present work and existing work [45]. (b) Temperature profile of present work and existing work [45].

4.6 Quadratic regression analysis: Approximation of skin friction and Nusselt number

The quadratic regression analysis is a statistical technique utilized to find the relation among two or more pertinent flow factors. Regression approximation is specifically used to examine how the characteristics of pertinent flow factors alter as a result of the alteration in another flow factor, while the other flow factors are maintained constant. This examination offers an analysis of approximate quadratic regression formulas for the Nusselt number and skin friction coefficients. A quadratic regression approximation model for the reduced skin friction coefficients is disclosed for 100 varying estimations of the ratio parameter and velocity slip factor, randomly selected within the ranges [0.1, 1.0] and [0, 0.5] for the improving estimations of the modified Hartmann number, respectively. In addition, a quadratic regression approximation model for the Nusselt number has been described for 100 various estimations of thermal stratification parameter and radiation factor estimations that were randomly selected from the intervals [0.1, 0.7] and [0.05, 0.80], respectively, for the greater values of Eckert number. During the approximation procedure, the other persisting parameters are taken into consideration as constants as stated in the preceding section.

For the predicted Cf x and Nu x , due to changes in the ratio parameter, velocity slip parameter, thermal stratification parameter, and radiation parameter, individually, the quadratic regression approximation model is provided by

(41) Cf x ( pre ) = Cf x + p 1 A + p 2 S 1 + p 3 A 2 + p 4 S 1 2 + p 5 A S 1 ,

(42) Nu x ( pre ) = Nu x + g 1 S + g 2 Nr + g 3 S 2 + g 4 Nr 1 2 + g 5 S Nr ,

where, p F and g F ( F = 1 , 2 , … 5 ) are individually the coefficients of quadratic regression approximation for the Cf x and Nu x .

Tables 3 and 4 depict the coefficients of quadratic regression estimations of Cf x and Nu x individually for various pertinent parameters. The error bounds κ and κ 1 for Cf x and Nu x have also been demonstrated by the relations κ = ∣ Cf x ( pre ) − Cf x ∣ Cf x and κ 1 = ∣ Nu x ( pre ) − Nu x ∣ Nu x , individually. It is important to note that the coefficients of the ratio parameter or radiation factor turn negative as the modified Hartmann number or the Eckert number increases, as seen in Tables 3 and 4, respectively. This discovery shows that the approximate skin friction coefficients and Nusselt number, respectively, are adversely affected by the ratio parameter and radiation factor. Furthermore, it can be seen from the tabulated estimations that the coefficients of the velocity slip parameter are larger in magnitude than those of the ratio parameter. This confirms the findings that, in contrast to the ratio parameter, a small alteration in the velocity slip parameter can cause a significant alteration in the shear stress function. Similar to the thermal effect, a small improvement in the thermal stratification factor results in a significant difference in the heat transfer rate. Additionally, it is noted that the quadratic regression approximation’s optimum relative error for the Nusselt number is roughly zero and that the quadratic regression approximation for the skin friction coefficients approaches this admirable accuracy level more quickly than the former.

Table 3

List of quadratic regression coefficients of Cf x due to the variations in A and S 1 and error bound

Q	Cf x	p 1	p 2	p 3	p 4	p 5	κ
0.1	−0.8773	−0.3900	1.1575	−0.0241	−0.9204	0.4412	0.0035
0.5	−0.9311	−0.4093	1.2778	−0.0274	−1.1066	0.5391	0.0094
1.0	−0.9992	−0.3914	1.3950	−0.0007	−1.1313	0.4293	0.0041
1.5	−1.0737	−0.3801	1.5645	0.0059	−1.2922	0.4128	0.0032

Table 4

List of quadratic regression coefficients of Nu x due to the variations in S and Nr and error bound

Ec	Nu x	g 1	g 2	g 3	g 4	g 5	κ 1
0.1	−1.3110	1.8333	−0.4247	−1.5844	0.0737	−0.0835	0.0025
0.2	−1.0545	1.4809	−0.3868	−1.2820	0.0585	−0.0441	0.0031
0.3	−0.7971	1.1277	−0.3557	−0.9902	0.0490	−0.0202	0.0039
0.4	−0.5391	0.7623	−0.3214	−0.6692	0.0332	−0.0010	0.0045

5 Conclusion

The thermal and melting properties of the third-grade nanofluid are explored in depth. The controlling differential equations are solved analytically utilizing the HAM. The results have unquestionably been thoroughly investigated and discussed by all parties. The following conclusions from the study may be illustrated:

Higher third-grade parameters boost the nanofluid’s velocity due to inverse relations between fluid viscosity and third-grade parameters.
Reynolds number is the ratio between inertial forces to viscous forces. Accelerating the Reynolds number amplifies the velocity due to small viscous forces among the particles.
Improving the estimations of modified magnetic factor increases the nanofluid’s speed due to high internal and external forces.
By boosting the amount of melting parameter, the flow’s velocity can be improved due to greater convective flow toward the cooling surface.
Alternatively, the flow’s velocity will be forced to decelerate due to the fractional size of nanoparticles and the velocity slip factor.
De-amplification occurs in temperature outline due to amplifications of thermal stratification.
Temperature is de-escalating function against the higher estimations of melting parameter because of the transport of heat from heated fluid toward the cold Riga plate.
The volume fraction of the nanoparticles parameter and the radiation parameter aid in cooling the system.
The temperature profile in the system only increases when the Eckert number is high, because of the drag forces among the particles.
The rate of heat transport increases with increasing values of the volume fraction parameter, while the drag forces de-escalate against the volume fraction.

Funding information: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia, for funding this work through the Research Group Project under Grant Number (RGP.2/300/44).
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.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2022-10-11

Revised: 2022-11-27

Accepted: 2023-01-08

Published Online: 2023-04-18

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

Articles in the same Issue

https://doi.org/10.1515/ntrev-2022-0511

Keywords for this article

third-grade nanofluid; mineral oil; thermal stratification; slip conditions; melting heat transport; Riga plate; HAM

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