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Entropy production simulation of second-grade magnetic nanomaterials flowing across an expanding surface with viscidness dissipative flux

  • Wasim Jamshed , Ramanahalli Jayadevamurthy Punith Gowda , Rangaswamy Naveen Kumar , Ballajja Chandrappa Prasannakumara , Kottakkaran Sooppy Nisar EMAIL logo , Omar Mahmoud , Aysha Rehman and Amjad Ali Pasha
Published/Copyright: September 17, 2022
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Nanotechnology Reviews
From the journal Nanotechnology Reviews Volume 11 Issue 1

Abstract

The principal focal point of the current review is the second-grade nanofluid (SGNF) stream with slanted magnetohydrodynamics and viscous disintegration impacts across a moving level flat surface with entropy investigation. Here, we have done a comparative study on copper–methanol and iron–methanol second-grade nanoliquids. The governing conditions of the SGNF model are changed into ordinary differential equations (ODEs) by using supportive changes. To tackle the non-straight ODEs, the Runge-Kutta Fehlberg-45 procedure is utilized. The result reveals that the velocity gradient of copper–methanol second-grade nanoliquid is strongly affected by suction, magnetic, and second-grade fluid parameters and declines faster when compared to iron–methanol second-grade nanoliquid. Copper–methanol SGNF shows improved heat transfer than iron–methanol SGNF for improved values of Eckert and Biot numbers.

Keywords: second-grade nanofluid; inclined magnetic effect; Joule heating; entropy generation

Nomenclature

A

unsteadiness parameter

A ς 1 , A ς 2

Rivlin–Ericksen tensors

Bi ς

Biot number

Br

Brinkman number

C f

skin friction

d/dt

material time derivative

Ec

Eckert number

f ( η )

non-dimensional velocity profiles

h f

coefficient of heat transfer rate

I

identity tensor

k

thermal conductivity

k 0

thermal conductivity of solid

k ⁎

mean absorption coefficient

K

porous medium parameter

M

magnetic parameter

N r

radiation parameter

Nu x

Nusselt number

p

pressure

Pr

Prandtl number

q r

radiative heat flux

Re f

local Reynolds number

S

mass transfer variable

T

temperature

T ∞

ambient temperature

( u , v )

velocity components

V

fluid velocity

V w

porous stretching surface

( x , y )

directions

α

second-grade parameter

α f

thermal diffusivity parameter

α 1 and α 2  

material variables

Γ

angle of inclination

η

dimensionless similarity coordinate

θ ( η )

dimensionless thermal profile

μ

dynamic viscosity

ν

kinematic viscosity

ρ

density

ρ C p

heat capacitance

σ

electrical conductivity

σ ⁎

Stefan–Boltzmann constant

Ω

dimensionless thermal gradient

Λ

velocity slip variable

ϕ

volumetric fraction coefficient

Subscripts

f

fluid

nf

nanofluid

s

solid nanoparticle

w

wall/surface

∞

ambient

1 Introduction

A few fluid dynamics problems have appreciated the attention given to the flow involving non-Newtonian liquids because of their application in industry and technology. It is well known that engineers, physicists, and mathematicians face a superior challenge in the dynamics of non-Newtonian fluids. In many fields, including food, drilling operations, and bio-engineering, non-linearity can manifest itself in a number of ways. For such fluids, the Navier–Stokes principle is insufficient and no single constitutive equation is available in the literature that exhibits the properties of all fluids. Many fluid models have been proposed as a consequence of the complex behavior of such fluids. Among these, viscoelastic-type liquids have received much attention. The subclass of viscoelastic liquids is the second-grade liquid (SGL) model that can reasonably be expected to have an analytical solution. Nowadays, major research is being carried out on the regime of non-Newtonian liquids due to their considerable functional efficiency. Sahoo et al. [1] quizzed the aspects of Joule heating and slip impact on the magnetohydrodynamic (MHD) stream of SGL past a sheet with viscous dissipation. Imtiaz et al. [2] pondered the Ohmic and magnetic effects to deliberate SGL flow provoked by the rotating disk. Hayat et al. [3] quizzed the SGL stream with nanoparticle suspension over a sheet with magnetic impact. Kalaivanan et al. [4] deliberated the activation energy effect on the SGL stream on a surface with stretching. Wakeel Ahmed et al. [5] quizzed the upshot of modified-Fourier heat flux on SGL with nanoparticle suspension past a stretchy geometry.

In recent years, the modeling and analysis of nanofluid flows have become a frequent area of study. Nanofluids are being developed as a breakthrough means of improving heat transmission. Nanoparticles may be used to overcome cooling issues in thermal frameworks by using heat transport fluids containing suspended nanoparticles. In connection with this, numerous investigators examined the stream of nanoliquids over dissimilar surfaces. Shafiq et al. [6] swotted the bioconvection steam of SGL with nanoparticle suspension past a surface with a chemical reaction effect. Gowda et al. [7] swotted the thermophoresis effect on liquid flow with twin nanoparticle suspensions past a poignant disk. Christopher et al. [8] scrutinized the chemical reaction power of hybrid nanoliquid flow on a cylinder. Jayadevamurthy et al. [9] examined a bioconvection stream of fluid with dual nanoparticle suspension on a poignant disk with spin. Hayat et al. [10] quizzed the magnetic field upshot on the SGL stream past the Riga wall.

Nuclear engineering relies heavily on the magnetic effect and Ohmic heating. In Joule or Ohmic heating, electricity flows into an item and creates heat at the same time. A number of scholars have explored the boundary layer stream issues with Joule heating on various geometric forms. Shashikumar et al. [11] studied the Brinkman–Forchheimer stream in a microchannel with numerous slips, viscous dissipation, and Ohmic heating. According to Hayat et al. [12], Ohmic heating and melting may have a major influence on the flow of viscous fluid over a stretchable plate. Dusty hybrid nanoliquid flow was studied by Radhika et al. [13] for the stimulation of the magnetic effect. The stream of a Sisko nanoliquid owing to a spinning disk was examined by Ijaz et al. [14] with radiation effects. The MHD flow of hybrid nanoliquids through an disk with activation energy was examined by Reddy et al. [15].

The effects of viscous dissipation are often ignored, but when the liquid viscosity is high, its presence becomes important. It alters the distribution of temperatures by playing a role as a source of energy that affects the rates of heat transfer. Hayat et al. [16] debriefed the viscous dissipation upshots in the nanoliquid stream initiated by a disk. Sithole et al. [17] quizzed the radiation effect on the MHD flow of SGL with suspended nanoparticles past a sheet. Hazarika et al. [18] reduced the solutions for the MHD stream of nanofluid past a sheet with viscous dissipation. The upshot of Joule heating in the radiative heat transport of non-Newtonian liquids like power-law, third-grade, and Jeffery fluids is an essential factor for scientists because of their dynamic applications in micro-fluidic devices, micro-electromechanical systems, aerospace, and chemical reactors. Enthused by these applications, several scholars scrutinized the stimulation of Joule heating and thermal radiation on diverse liquid streams. Gireesha et al. [19] deliberated the impact of radiation on the MHD flow of Jeffrey nanoliquid past a porous extending sheet. Shit and Mandal [20] quizzed the entropy creation on MHD flow of Casson nanoliquid on an extending surface with radiation effect.

During the past several decades, several researchers concentrating on the generation of entropy have conducted various studies on energy production in diverse fluid stream conditions. Thermodynamic performance is a thought-provoking characteristic of building the right apparatus when energy saving is a foremost issue. The relevance of entropy generation in liquid flow, as well as its importance in several industrial applications such as ACs, heat pumps, and fire engines, has encouraged many researchers. Bejan [21] was the first to bring up the topic of entropy management. He demonstrated that thermofluidic systems can be treated with characteristics of entropy and provided a calculation formulation for the production rate of entropy. Captivated by these indications, many researchers discuss the generation of entropy in several fluid streams through different surfaces. Yusuf et al. [22] quizzed the generation of entropy in an MHD Williamson nanoliquid stream passing over a sheet. Azam et al. [23] looked at the production of entropy in a Williamson fluid flow with nanoparticle suspensions while accounting for Ohmic heating. Bhatti et al. [24] quizzed the formation of entropy in a Williamson liquid stream with nanoparticle suspension while accounting for thermophoretic and Brownian motion. Alsaadi et al. [25] quizzed the paraphernalia of radiative heat flux on the MHD stream of Williamson nanomaterial liquid on a surface. Refs. [26–40] include new additions that consider conventional and nanofluids with heat and mass transmission in various physical circumstances.

The aforementioned writing features the basic elements of the liquid stream across different calculations. The progression of SGL fluid over a moving surface with a thermal radiation impact has not yet been explored. As a result, the primary goal of this study is to investigate the features of non-uniform stretching velocity with thermal radiation, Ohmic, and viscous dissipation effects in an SGL stream. In addition, graphical representations are used to explain the main effects of various non-dimensionless factors on fluid profiles.

2 Mathematical formulation

The mathematical representation of the moving flat horizontal surface with non-uniform stretching velocity is given as follows:

(2.1) U w ( x , t ) = b x 1 − ξ t ,

where b is the initial rate of stretching of the porous sheet. T w ( x , t ) = T ∞ + b ⁎ x 1 − ξ t is the temperature of the insulated sheet under consideration, and for the sake of ease, the sheet’s left end is supposed to be fixed at x = 0 . Furthermore, the temperature variation rate is represented by b*. The mathematical model is considered under the following conditions and assumptions (Figure 1):

  • Second-grade nanofluid (SGNF)

  • Tiwari and Das nanofluid model

  • Porous medium

  • Porous stretching flat surface

  • Laminar unsteady flow

  • Viscous dissipation

  • Radiative heat flux

  • Newton and slip boundary conditions.

Figure 1 Flow geometry.
Figure 1

Flow geometry.

The Cauchy stress tensor in an SGL is mathematically represented as follows (see, for details, Shah et al. [41]):

(2.2) S ⁎ = μ A ς 1 + α 1 A ς 2 + α 1 A ς 1 2 − p I ,

(2.3) A ς 1 = ( grad V ) + ( grad V ) T ,

(2.4) A ς 2 =   d A ς 1 dt + A ς 1 ( grad V ) + A ς 1 ( grad V ) T .

The Clausius–Duhem inequality is confirmed. Furthermore, we find that the Helmholtz free energy is minimal in equilibrium for the liquid is at rest when

(2.5) μ ≥ 0 , α 1 ≥ 0 , α 1 + α 2 = 0 .

If α 1 + α 2 = 0 , then the equation of SGL reduces to a viscous liquid.

The SGNF flow equations [41] under normal boundary layer assumptions, viscous dissipation, and joule heating with radiation heat flux are as follows:

(2.6) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2.7)   ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = α 1 ρ n f ∂ u ∂ x ∂ 2 u ∂ y 2 + ∂ 3 u ∂ t ∂ y 2 + u ∂ 3 u ∂ x ∂ y 2 + ∂ u ∂ y ∂ 2 v ∂ y 2 + v ∂ 3 u ∂ y 3 + μ n f ρ n f ∂ 2 u ∂ y 2 − σ n f B 2 ( t ) ρ n f u sin 2 ( Γ ) ,

(2.8) ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = k n f ( ρ C p ) n f ∂ 2 T ∂ y 2 − 1 ( ρ C p ) n f ∂ q r ∂ y + μ n f ( ρ C p ) n f ∂ u ∂ y 2 + σ n f ( ρ C p ) n f B 2 ( t ) u 2 sin 2 ( Γ ) .  

The relevant boundary conditions are as follows:

(2.9) u ( x , 0 ) = U w + N w ∂ u ∂ y , v ( x , 0 ) = V w ,   − k 0 ∂ T ∂ y = h f ( T w − T ) ,

(2.10) u → 0 ,   ∂ u ∂ y → 0 ,   T → T ∞ as y → ∞ .

The dispersion of nanoparticles into methanol liquid causes improved thermophysical features. Table 1 summarizes the material parameters for the SGNF (see, for example, refs. [42–46]).

Table 1

Thermophysical features of second-grade nanoliquid

Properties Nanofluids
Dynamics viscosity μ n f = μ f ( 1 − ϕ ) − 2.5
Density ρ n f = ( 1 − ϕ ) ρ f + ϕ ρ s
Heat capacity ( ρ C p ) n f = ( 1 − ϕ ) ( ρ C p ) f + ϕ ( ρ C p ) s
Thermal conductivity k n f k f = ( k s + 2 k f ) − 2 ϕ ( k f − k s ) ( k s + 2 k f ) + ϕ ( k f − k s )
Electrical conductivity σ n f σ f = 1 + 3 σ s σ f − 1 ϕ σ s σ f + 2 − σ s σ f − 1 ϕ

The material features of the methanol and the nanoparticles being utilized in this work are given in Table 2 (see, for instance, refs. [47–49]).

Table 2

Primary properties of the base fluid and nanoparticles at a standard temperature

Thermophysical properties ρ (kg/m3) C p (J/kg K) k (W/m K) σ (S/m)
Copper (Cu) 8,933 385 401 5.96 × 107
Iron (Fe3O4) 5,180 670 9.7 0.74 × 106
Methanol (MeOH) 792 2,545 0.2035 0.5 × 10−6

Using the Roseland approximation, Brewster [50], one can write

(2.11) q r = − 4 σ ⁎ 3 k ⁎ ∂ ¥ 4 ∂ y .

3 Solution to the problem

By using the similarity approach to the governing partial differential equations, equations (2.1)–(2.3) of boundary value problems have been transformed into ordinary differential equations (ODEs). The stream function can be defined as

(3.1) u = ∂ ψ ∂ y ,   v = − ∂ ψ ∂ x ,

and similarity variables of the form

(3.2) χ ( x , y ) = b ν f ( 1 − ξ t ) y , ψ ( x , y ) = ν f b ( 1 − ξ t ) x f ( χ ) ,   θ ( χ ) = T − T ∞ T w − T ∞ ,

into equations (2.1)–(2.3). We obtain

(3.3) f ′ ″ + ϕ 1 ϕ 2 f f ″ − f ′ 2 − A χ 2 f ″ + f ′ + α ( 2 f ′ f ′ ″ + A 2 f ″ + χ 2 f i v − f ″ 2 − f f i v ) − ϕ 4 ϕ 2 M sin 2 ( Γ ) f ′ = 0 ,

(3.4) θ ″ 1 + 1 ϕ 4 Pr N r + Pr ϕ 3 ϕ 5 f θ ′ − f ′ θ − A θ + χ 2 θ ′ + Ec ϕ 1 ϕ 3 f ″ 2 +   ϕ 4 ϕ 3 M Ec sin 2 ( Γ ) f ′ 2 = 0 ,

with

(3.5) f ( 0 ) = S ,   f ′ ( 0 ) = 1 + Λ f ″ ( 0 ) ,   θ ′ ( 0 ) = − Bi ς ( 1 − θ ( 0 ) ) f ′ ( χ ) → 0 , f ″ ( χ ) → 0 ,   θ ( χ ) → 0 ,   a s   χ → ∞ ,

where ϕ i ′ s is 1 ≤ i ≤ 5 in equations (3.3) and (3.4) representing the following thermophysical properties for the SGNF:

(3.6) ϕ 1 = ( 1 − ϕ ) 2.5 , ϕ 2 = 1 − ϕ + ϕ ρ s ρ f , ϕ 3 = 1 − ϕ + ϕ ( ρ C p ) s ( ρ C p ) f , ϕ 4 = 1 + 3 ( σ s σ f − 1 ) ϕ ( σ s σ f + 2 ) − ( σ s σ f − 1 ) ϕ , ϕ 5 = ( k s + 2 k f ) − 2 ϕ ( k f − k s ) ( k s + 2 k f ) + ϕ ( k f − k s ) .

Equation (2.1) is satisfied identically. In the above equations, ′ takes derivatives w.r.t χ , where A = ξ b , α = α 1 b μ f and K = ν f ( 1 − ξ t ) b k , Pr = ν f α f , N r = 16 3 σ ⁎ ¥ ∞ 3 κ ⁎ ν f ( ρ C p ) f , α f = κ f ( ρ C p ) f , S = − V w 1 − ξ t ν f   b . Λ = b ν f ( 1 − ξ t ) μ f , Ec = U w 2 ( C p ) f ( T w − T ∞ ) , and Bi ς = h f k 0 ν f ( 1 − ξ t ) b .

The skin friction ( C f ) and the local Nusselt number (Nu x ) can be stated as follows (see, for example, Shah et al. [41]):

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

where τ w and q w represent the heat flux determined by

(3.8) τ w = μ n f ∂ u ∂ y + α 1 ∂ 2 u ∂ t ∂ y + u ∂ 2 u ∂ x ∂ y + 2 ∂ u ∂ y ∂ u ∂ x + v ∂ 2 u ∂ 2 y y = 0 ,      q w = − k n f 1 + 16 3 σ ⁎ T ∞ 3 κ ⁎ ν f ( ρ C p ) f ∂ T ∂ y y = 0 .

Applying the non-dimensional transformations (3.2), one obtains

(3.9) C f Re x 1 2 = f ″ ( 0 ) ( 1 − ϕ ) 2.5 + α 3 f ″ ( 0 ) f ′ ( 0 ) − f ′ ″ ( 0 ) f ( 0 ) + A 2 ( χ f ′ ″ ( 0 ) + 3 f ″ ( 0 ) ) , Nu x Re x − 1 2 = − k n f k f ( 1 + N r ) θ ′ ( 0 ) .  

4 Entropy generation minimization

There is a constant fear among scientists and engineers that valuable energy will be squandered. A thorough entropy generation study of the system causing irreversible useful energy is thus essential. MHD is a non-ideal phenomenon that results in a growth in the system’s entropy. Then, we have [51]

(4.1) E G = k n f T ∞ 2 ∂ T ∂ y 2 + 16 3 σ ⁎ T ∞ 3 κ ⁎ ν f ( ρ C p ) f ∂ T ∂ y 2 + μ n f T ∞ ∂ u ∂ y 2 + σ n f B 2 ( t ) sin 2 ( Γ ) u 2 T ∞ .

This equation’s first component reflects the irreversibility of heat transmission, the second term is related to fluid friction, and the third is related to inclined MHD phenomena.

NG represents the dimensionless entropy generation, which is defined as [52–56]:

(4.2) NG = T ∞ 2 b 2 E G k f ( T w − T ∞ ) 2 .

According to equation (3.2), the non-dimensional form of the entropy equation can be determined as follows:

(4.3) NG = Re ϕ 5 ( 1 + N r ) θ ′ 2 + 1 ϕ 1 Br Ω ( f ″ 2 + ϕ 1 ϕ 4 M sin 2 ( Γ ) f ′ 2 ) .

5 Computational procedure: shooting approach

The shooting method [57] is employed for finding modeled equation solutions. The localized solution of equations (3.3) and (3.4), subject to (3.5) constraints, is found via the shooting technique. The shooting methodology is given as follows (Figure 2).

Figure 2 Shooting technique methodology.
Figure 2

Shooting technique methodology.

Initial order ODEs are required for this method’s first step. To satisfy these criteria, conversion of (3.3)–(3.5) into first-order system yields

(5.1) z 1 = f ' ,

(5.2) z 2 = z 1 ' ,

(5.3) z 3 = z 2 ' ,

(5.4) z 4 = θ ' ,

(5.5) z 2 ' + ϕ 1 ϕ 2 f z 2 − z 1 2 − A χ 2 z 2 + z 1 + α ( 2 z 1 z 2 ' ) + A 2 z 2 + χ 2 z 3 ' − z 2 2 − f z 3 ' − ϕ 4 ϕ 2 M sin 2 ( Γ ) z 1 = 0 ,

(5.6) z 4 ' 1 + 1 ϕ 4 Pr N r + Pr ϕ 3 ϕ 4 f z 4 − z 1 θ − A θ + χ 2 z 4 + Ec ϕ 1 ϕ 3 z 2 2 + ϕ 4 ϕ 3 M Ecsin 2 ( Γ ) z 1 2 = 0 ,

(5.7) f ( 0 ) = S , z 1 ( 0 ) = 1 + Λ z 2 ( 0 ) , z 3 ( 0 ) = − B ς ( 1 − θ ( 0 ) ) , z 1 ( ∞ ) → 0 , z 2 ( ∞ ) → 0 , θ ( ∞ ) → 0 .

6 Code validation

Although this approach was tested by comparing its heat transfer rate findings to those found in earlier studies [58–61], it was shown to be valid. Comparing the levels of consistency found across the various research is summarized in Table 3. It is important to note that the findings of this study are absolutely correct.

Table 3

Comparison of − θ ′ ( 0 ) by variation in Prandtl number, ϕ = 0, A = 0, Λ = 0, N r = 0, M = 0, Ec = 0, S = 0, and Bi ς → ∞

Prandtl number (Pr) 10.0 7.0 3.0 1.0 0.72
Ishak et al. [58] 3.7207 3.0723 1.9237 1.0000 0.8086
Ishak et al. [59] 3.7006 3.0723 1.9236 1.0000 0.8086
Abolbashari et al. [60] 3.72067390 3.07225021 1.92368259 1.00000000 0.80863135
Das et al. [61] 3.72055436 3.07314679 1.92357431 1.00000000 0.80876122
Present results 3.72055429 3.07314651 1.92357420 1.00000000 0.80876181

7 Numerical results and discussions

The intention of this part is to display the encouragement of several dimensionless parameters on intricate profiles. For example, the current problem indicates the SGNF stream with inclined MHD, viscous dissipation effects, and entropy analysis. Here, we have done a comparative study on copper–methanol and iron–methanol second-grade nanoliquids. In this section, we analyzed the actions of frequent dimensionless parameters on the respective profiles by using appropriate graphs.

Figure 3 demonstrates the sway of M on f′ of both copper–methanol and iron–methanol second-grade nanoliquids. Here, the upsurge in M declines the f′. Physically, M corresponds to the Lorentz force that slows the movement of the liquid particles, resulting in the declination of f′. Furthermore, the f′ of copper–methanol SGNF is strongly triggered by the magnetic field and decays faster when compared to iron–methanol SGNF. The change in f′ of both nanoliquids for varied Λ is exemplified in Figure 4. Here, gain in Λ declines the f′. Furthermore, the f′ for copper–methanol SGNF is strongly triggered by Λ and declines faster when compared to iron–methanol SGNF. The provocation of S on f′ of both copper–methanol and iron–methanol SGNFs is demonstrated in Figure 5. The escalation in S drops the f′ for both nanoliquids. Here, the f′ of copper–methanol SGNF is strongly triggered by S and declines faster when compared to iron–methanol SGNF. Figure 6 displays the sway of α on f′ of both copper–methanol and iron–methanol SGNFs. The gain in α progresses the f′. Moreover, the f′ of iron–methanol SGNF is strongly exaggerated by α and inclines faster when compared to copper –methanol SGNF.

Figure 3 Sway of M on f′.
Figure 3

Sway of M on f′.

Figure 4 Sway of Λ \text{Λ} on f′.
Figure 4

Sway of Λ on f′.

Figure 5 Sway of S on f′.
Figure 5

Sway of S on f′.

Figure 6 Sway of α on f′.
Figure 6

Sway of α on f′.

Figure 7 reveals the upshot of Eckert number on the thermal profile representing both iron–methanol and copper–methanol SGNFs. The rise in Ec enhances the thermal profile. It is moderately clear from the description of Ec that its higher values give rise to the strong heat effect, which raises the temperature. Here, copper–methanol SGNF shows improved heat transfer than iron–methanol SGNF for improved values of Ec. The impact of Bi ς on thermal gradient for both copper–methanol and iron–methanol second-grade nanoliquids is depicted in Figure 8. The enhancement of Bi ς improves the thermal profile of both nanoliquids. Physically, the boom in Bi ς produces large heat transport via convection, which results in increased heat transport. Furthermore, copper–methanol SGNF shows improved heat transport than iron–methanol SGNF, for improved values of Bi ς . The influence of N r on heat transfer for both nanoliquids is typified in Figure 9. The upsurge in N r advances the heat transference of both nanoliquids. When the N r is inclined, internal heat is generated, resulting in increased heat transport. Furthermore, copper–methanol SGNF shows improved heat transport than iron–methanol SGNF, with improved values of N r . The influence of M on the thermal profile of both liquids is typified in Figure 10. The increase in M enriches the thermal profile in both the fluid flows representing copper–methanol and iron–methanol second-grade nanoliquid. An increase in the magnetic parameters intensifies the magnetic field. Here, the fluid temperature gradually increases due to the thermal radiation effect, which supplies the additional heat to the flow system. Figure 11 illustrates the impact of M on the entropy generation profile versus radiation parameter for both nanoliquids. The upsurge in magnetic parameters improves the entropy generation of both iron–methanol and copper–methanol SGNFs. Moreover, the entropy generation profile of copper–methanol SGNF is strongly affected and inclines faster when compared to iron–methanol SGNF, for improved values of both M and N r .

Figure 7 Sway of Ec on θ \theta .
Figure 7

Sway of Ec on θ .

Figure 8 Sway of Bi ς {\text{Bi}}_{\varsigma } on θ.
Figure 8

Sway of Bi ς on θ.

Figure 9 Sway of N r on θ \theta .
Figure 9

Sway of N r on θ .

Figure 10 Sway of M on θ \theta .
Figure 10

Sway of M on θ .

Figure 11 Sway of N r on NG.
Figure 11

Sway of N r on NG.

The encouragement of various non-dimensional parameters on Nusselt number and skin friction for both iron–methanol and copper–methanol second-grade nanoliquids are tabulated in Tables 4 and 5. The impact of S , N r , M , Λ , ϕ , α , Bi , γ , and A on surface drag force is displayed in Table 5.

Table 4

Influence of varied dimensional parameters on Nusselt number

M N r Ec Bi ς α A Λ γ S ϕ NuRe x − 1 / 2   Cu–methanol NuRe x − 1 / 2 Fe3O4–methanol
0.1 0.2 0.15 0.5 0.5 0.4 0.2 π / 2 0.5 0.02 0.51278 0.512147
    0.510233 0.509482
0.3     0.507687 0.506817
0.1       0.484522 0.483892
0.15       0.498614 0.497982
0.1         0.519784 0.518999
0.2       0.505777 0.505421
0.4         0.4239 0.478266
0.6         0.596058 0.451237
0.6       0.514054 0.513416
0.7       0.5152 0.514431
0.5         0.512653 0.51202
0.6         0.512526 0.511893
Table 5

Influence of varied dimensional parameters on skin friction

M S Λ ϕ A α Bi ς γ N r Ec C f Re x 1 2 Cu–methanol C f Re x 1 2 Fe3O4–methanol
0.1 0.5 0.2 0.02 0.4 0.5 0.5 π / 2 0.2 0.15 −2.15294 −2.0902
0.2 −2.19105 −2.13351
0.3 −2.22811 −2.17525
0.2 −1.94159 −1.8929
0.3 −2.01037 −1.95713
0.1 −2.49201 −2.40893
0.3 −1.89158 −1.84274
0.01 −2.12828 −2.09517
0.03 −2.17136 −2.08288
0.5 −2.18766 −2.12508
0.6 −2.2233 −2.16011
0.6 −2.34291 −2.27535
0.7 −2.52543 −2.45309

In this case, increasing the values of S , M , ϕ , α , Bi ς , γ , and A decreases the friction factor of both nanoliquids. But the conflicting trend is portrayed as boosting the values of Λ for both iron–methanol and copper–methanol second-grade nanoliquids. Table 4 displays the consequences of N r , M , Λ , ϕ , α , Bi ς , γ , and A on the rate of heat transference of both iron–methanol and copper–methanol SGNFs. It is interesting to note from the table that the escalation in N r , α , and Bi improves the rate of heat transport in both SGNFs. But the conflicting trend is seen to boost up the values of Λ.

8 Final remarks

In this study, relative requests of the stream for the copper and iron non-Newtonian methanol-based nanofluids were accomplished over a piercing level plane surface with a non-uniform extending speed. The examination was conducted in the event of various physical impacts. The primary comments of the present review are as follows:

  • The velocity of copper–methanol SGNF is sturdily triggered by the suction parameter and declines faster than iron–methanol SGNF.

  • The velocity of copper–methanol SGNF is sturdily triggered by a magnetic field and drops faster than iron–methanol SGNF.

  • The velocity of iron–methanol SGNF is strongly exaggerated by α and inclines faster than copper–methanol SGNF.

  • Copper–methanol SGNF shows improved heat transport than iron–methanol SGNF for improved values of Bi ς .

  • Copper–methanol SGNGF shows improved heat transport than iron–methanol SGNF for improved values of Ec.

  • The entropy generation profile of copper–methanol SGNF is strongly affected and inclines faster when compared to iron–methanol SGNF for improved values of both magnetic and thermal radiation parameters.

  • Copper–methanol SGNF shows an improved rate of heat transference than iron–methanol SGNF.

  • The shooting method could be applied to a variety of physical and technical challenges in the future [62–71].

Acknowledgments

The author Rangaswamy Naveen Kumar is thankful to Department of OBC and minority cell, Davangere University, Davangere, Karanataka, INDIA for financial support Fellowship 1/2021-22/6998.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2022-02-23
Revised: 2022-04-12
Accepted: 2022-07-10
Published Online: 2022-09-17

© 2022 Wasim Jamshed et al., published by De Gruyter

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

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  89. Effect of nanoadditives on the novel leather fiber/recycled poly(ethylene-vinyl-acetate) polymer composites for multifunctional applications: Fabrication, characterizations, and multiobjective optimization using central composite design
  90. Design selection for a hemispherical dimple core sandwich panel using hybrid multi-criteria decision-making methods
  91. Improving tensile strength and impact toughness of plasticized poly(lactic acid) biocomposites by incorporating nanofibrillated cellulose
  92. Green synthesis of spinel copper ferrite (CuFe2O4) nanoparticles and their toxicity
  93. The effect of TaC and NbC hybrid and mono-nanoparticles on AA2024 nanocomposites: Microstructure, strengthening, and artificial aging
  94. Excited-state geometry relaxation of pyrene-modified cellulose nanocrystals under UV-light excitation for detecting Fe3+
  95. Effect of CNTs and MEA on the creep of face-slab concrete at an early age
  96. Effect of deformation conditions on compression phase transformation of AZ31
  97. Application of MXene as a new generation of highly conductive coating materials for electromembrane-surrounded solid-phase microextraction
  98. A comparative study of the elasto-plastic properties for ceramic nanocomposites filled by graphene or graphene oxide nanoplates
  99. Encapsulation strategies for improving the biological behavior of CdS@ZIF-8 nanocomposites
  100. Biosynthesis of ZnO NPs from pumpkin seeds’ extract and elucidation of its anticancer potential against breast cancer
  101. Preliminary trials of the gold nanoparticles conjugated chrysin: An assessment of anti-oxidant, anti-microbial, and in vitro cytotoxic activities of a nanoformulated flavonoid
  102. Effect of micron-scale pores increased by nano-SiO2 sol modification on the strength of cement mortar
  103. Fractional simulations for thermal flow of hybrid nanofluid with aluminum oxide and titanium oxide nanoparticles with water and blood base fluids
  104. The effect of graphene nano-powder on the viscosity of water: An experimental study and artificial neural network modeling
  105. Development of a novel heat- and shear-resistant nano-silica gelling agent
  106. Characterization, biocompatibility and in vivo of nominal MnO2-containing wollastonite glass-ceramic
  107. Entropy production simulation of second-grade magnetic nanomaterials flowing across an expanding surface with viscidness dissipative flux
  108. Enhancement in structural, morphological, and optical properties of copper oxide for optoelectronic device applications
  109. Aptamer-functionalized chitosan-coated gold nanoparticle complex as a suitable targeted drug carrier for improved breast cancer treatment
  110. Performance and overall evaluation of nano-alumina-modified asphalt mixture
  111. Analysis of pure nanofluid (GO/engine oil) and hybrid nanofluid (GO–Fe3O4/engine oil): Novel thermal and magnetic features
  112. Synthesis of Ag@AgCl modified anatase/rutile/brookite mixed phase TiO2 and their photocatalytic property
  113. Mechanisms and influential variables on the abrasion resistance hydraulic concrete
  114. Synergistic reinforcement mechanism of basalt fiber/cellulose nanocrystals/polypropylene composites
  115. Achieving excellent oxidation resistance and mechanical properties of TiB2–B4C/carbon aerogel composites by quick-gelation and mechanical mixing
  116. Microwave-assisted sol–gel template-free synthesis and characterization of silica nanoparticles obtained from South African coal fly ash
  117. Pulsed laser-assisted synthesis of nano nickel(ii) oxide-anchored graphitic carbon nitride: Characterizations and their potential antibacterial/anti-biofilm applications
  118. Effects of nano-ZrSi2 on thermal stability of phenolic resin and thermal reusability of quartz–phenolic composites
  119. Benzaldehyde derivatives on tin electroplating as corrosion resistance for fabricating copper circuit
  120. Mechanical and heat transfer properties of 4D-printed shape memory graphene oxide/epoxy acrylate composites
  121. Coupling the vanadium-induced amorphous/crystalline NiFe2O4 with phosphide heterojunction toward active oxygen evolution reaction catalysts
  122. Graphene-oxide-reinforced cement composites mechanical and microstructural characteristics at elevated temperatures
  123. Gray correlation analysis of factors influencing compressive strength and durability of nano-SiO2 and PVA fiber reinforced geopolymer mortar
  124. Preparation of layered gradient Cu–Cr–Ti alloy with excellent mechanical properties, thermal stability, and electrical conductivity
  125. Recovery of Cr from chrome-containing leather wastes to develop aluminum-based composite material along with Al2O3 ceramic particles: An ingenious approach
  126. Mechanisms of the improved stiffness of flexible polymers under impact loading
  127. Anticancer potential of gold nanoparticles (AuNPs) using a battery of in vitro tests
  128. Review Articles
  129. Proposed approaches for coronaviruses elimination from wastewater: Membrane techniques and nanotechnology solutions
  130. Application of Pickering emulsion in oil drilling and production
  131. The contribution of microfluidics to the fight against tuberculosis
  132. Graphene-based biosensors for disease theranostics: Development, applications, and recent advancements
  133. Synthesis and encapsulation of iron oxide nanorods for application in magnetic hyperthermia and photothermal therapy
  134. Contemporary nano-architectured drugs and leads for ανβ3 integrin-based chemotherapy: Rationale and retrospect
  135. State-of-the-art review of fabrication, application, and mechanical properties of functionally graded porous nanocomposite materials
  136. Insights on magnetic spinel ferrites for targeted drug delivery and hyperthermia applications
  137. A review on heterogeneous oxidation of acetaminophen based on micro and nanoparticles catalyzed by different activators
  138. Early diagnosis of lung cancer using magnetic nanoparticles-integrated systems
  139. Advances in ZnO: Manipulation of defects for enhancing their technological potentials
  140. Efficacious nanomedicine track toward combating COVID-19
  141. A review of the design, processes, and properties of Mg-based composites
  142. Green synthesis of nanoparticles for varied applications: Green renewable resources and energy-efficient synthetic routes
  143. Two-dimensional nanomaterial-based polymer composites: Fundamentals and applications
  144. Recent progress and challenges in plasmonic nanomaterials
  145. Apoptotic cell-derived micro/nanosized extracellular vesicles in tissue regeneration
  146. Electronic noses based on metal oxide nanowires: A review
  147. Framework materials for supercapacitors
  148. An overview on the reproductive toxicity of graphene derivatives: Highlighting the importance
  149. Antibacterial nanomaterials: Upcoming hope to overcome antibiotic resistance crisis
  150. Research progress of carbon materials in the field of three-dimensional printing polymer nanocomposites
  151. A review of atomic layer deposition modelling and simulation methodologies: Density functional theory and molecular dynamics
  152. Recent advances in the preparation of PVDF-based piezoelectric materials
  153. Recent developments in tensile properties of friction welding of carbon fiber-reinforced composite: A review
  154. Comprehensive review of the properties of fly ash-based geopolymer with additive of nano-SiO2
  155. Perspectives in biopolymer/graphene-based composite application: Advances, challenges, and recommendations
  156. Graphene-based nanocomposite using new modeling molecular dynamic simulations for proposed neutralizing mechanism and real-time sensing of COVID-19
  157. Nanotechnology application on bamboo materials: A review
  158. Recent developments and future perspectives of biorenewable nanocomposites for advanced applications
  159. Nanostructured lipid carrier system: A compendium of their formulation development approaches, optimization strategies by quality by design, and recent applications in drug delivery
  160. 3D printing customized design of human bone tissue implant and its application
  161. Design, preparation, and functionalization of nanobiomaterials for enhanced efficacy in current and future biomedical applications
  162. A brief review of nanoparticles-doped PEDOT:PSS nanocomposite for OLED and OPV
  163. Nanotechnology interventions as a putative tool for the treatment of dental afflictions
  164. Recent advancements in metal–organic frameworks integrating quantum dots (QDs@MOF) and their potential applications
  165. A focused review of short electrospun nanofiber preparation techniques for composite reinforcement
  166. Microstructural characteristics and nano-modification of interfacial transition zone in concrete: A review
  167. Latest developments in the upconversion nanotechnology for the rapid detection of food safety: A review
  168. Strategic applications of nano-fertilizers for sustainable agriculture: Benefits and bottlenecks
  169. Molecular dynamics application of cocrystal energetic materials: A review
  170. Synthesis and application of nanometer hydroxyapatite in biomedicine
  171. Cutting-edge development in waste-recycled nanomaterials for energy storage and conversion applications
  172. Biological applications of ternary quantum dots: A review
  173. Nanotherapeutics for hydrogen sulfide-involved treatment: An emerging approach for cancer therapy
  174. Application of antibacterial nanoparticles in orthodontic materials
  175. Effect of natural-based biological hydrogels combined with growth factors on skin wound healing
  176. Nanozymes – A route to overcome microbial resistance: A viewpoint
  177. Recent developments and applications of smart nanoparticles in biomedicine
  178. Contemporary review on carbon nanotube (CNT) composites and their impact on multifarious applications
  179. Interfacial interactions and reinforcing mechanisms of cellulose and chitin nanomaterials and starch derivatives for cement and concrete strength and durability enhancement: A review
  180. Diamond-like carbon films for tribological modification of rubber
  181. Layered double hydroxides (LDHs) modified cement-based materials: A systematic review
  182. Recent research progress and advanced applications of silica/polymer nanocomposites
  183. Modeling of supramolecular biopolymers: Leading the in silico revolution of tissue engineering and nanomedicine
  184. Recent advances in perovskites-based optoelectronics
  185. Biogenic synthesis of palladium nanoparticles: New production methods and applications
  186. A comprehensive review of nanofluids with fractional derivatives: Modeling and application
  187. Electrospinning of marine polysaccharides: Processing and chemical aspects, challenges, and future prospects
  188. Electrohydrodynamic printing for demanding devices: A review of processing and applications
  189. Rapid Communications
  190. Structural material with designed thermal twist for a simple actuation
  191. Recent advances in photothermal materials for solar-driven crude oil adsorption
https://doi.org/10.1515/ntrev-2022-0463
Keywords for this article
second-grade nanofluid; inclined magnetic effect; Joule heating; entropy generation
Creative Commons
BY 4.0

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  98. A comparative study of the elasto-plastic properties for ceramic nanocomposites filled by graphene or graphene oxide nanoplates
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  101. Preliminary trials of the gold nanoparticles conjugated chrysin: An assessment of anti-oxidant, anti-microbial, and in vitro cytotoxic activities of a nanoformulated flavonoid
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  106. Characterization, biocompatibility and in vivo of nominal MnO2-containing wollastonite glass-ceramic
  107. Entropy production simulation of second-grade magnetic nanomaterials flowing across an expanding surface with viscidness dissipative flux
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  109. Aptamer-functionalized chitosan-coated gold nanoparticle complex as a suitable targeted drug carrier for improved breast cancer treatment
  110. Performance and overall evaluation of nano-alumina-modified asphalt mixture
  111. Analysis of pure nanofluid (GO/engine oil) and hybrid nanofluid (GO–Fe3O4/engine oil): Novel thermal and magnetic features
  112. Synthesis of Ag@AgCl modified anatase/rutile/brookite mixed phase TiO2 and their photocatalytic property
  113. Mechanisms and influential variables on the abrasion resistance hydraulic concrete
  114. Synergistic reinforcement mechanism of basalt fiber/cellulose nanocrystals/polypropylene composites
  115. Achieving excellent oxidation resistance and mechanical properties of TiB2–B4C/carbon aerogel composites by quick-gelation and mechanical mixing
  116. Microwave-assisted sol–gel template-free synthesis and characterization of silica nanoparticles obtained from South African coal fly ash
  117. Pulsed laser-assisted synthesis of nano nickel(ii) oxide-anchored graphitic carbon nitride: Characterizations and their potential antibacterial/anti-biofilm applications
  118. Effects of nano-ZrSi2 on thermal stability of phenolic resin and thermal reusability of quartz–phenolic composites
  119. Benzaldehyde derivatives on tin electroplating as corrosion resistance for fabricating copper circuit
  120. Mechanical and heat transfer properties of 4D-printed shape memory graphene oxide/epoxy acrylate composites
  121. Coupling the vanadium-induced amorphous/crystalline NiFe2O4 with phosphide heterojunction toward active oxygen evolution reaction catalysts
  122. Graphene-oxide-reinforced cement composites mechanical and microstructural characteristics at elevated temperatures
  123. Gray correlation analysis of factors influencing compressive strength and durability of nano-SiO2 and PVA fiber reinforced geopolymer mortar
  124. Preparation of layered gradient Cu–Cr–Ti alloy with excellent mechanical properties, thermal stability, and electrical conductivity
  125. Recovery of Cr from chrome-containing leather wastes to develop aluminum-based composite material along with Al2O3 ceramic particles: An ingenious approach
  126. Mechanisms of the improved stiffness of flexible polymers under impact loading
  127. Anticancer potential of gold nanoparticles (AuNPs) using a battery of in vitro tests
  128. Review Articles
  129. Proposed approaches for coronaviruses elimination from wastewater: Membrane techniques and nanotechnology solutions
  130. Application of Pickering emulsion in oil drilling and production
  131. The contribution of microfluidics to the fight against tuberculosis
  132. Graphene-based biosensors for disease theranostics: Development, applications, and recent advancements
  133. Synthesis and encapsulation of iron oxide nanorods for application in magnetic hyperthermia and photothermal therapy
  134. Contemporary nano-architectured drugs and leads for ανβ3 integrin-based chemotherapy: Rationale and retrospect
  135. State-of-the-art review of fabrication, application, and mechanical properties of functionally graded porous nanocomposite materials
  136. Insights on magnetic spinel ferrites for targeted drug delivery and hyperthermia applications
  137. A review on heterogeneous oxidation of acetaminophen based on micro and nanoparticles catalyzed by different activators
  138. Early diagnosis of lung cancer using magnetic nanoparticles-integrated systems
  139. Advances in ZnO: Manipulation of defects for enhancing their technological potentials
  140. Efficacious nanomedicine track toward combating COVID-19
  141. A review of the design, processes, and properties of Mg-based composites
  142. Green synthesis of nanoparticles for varied applications: Green renewable resources and energy-efficient synthetic routes
  143. Two-dimensional nanomaterial-based polymer composites: Fundamentals and applications
  144. Recent progress and challenges in plasmonic nanomaterials
  145. Apoptotic cell-derived micro/nanosized extracellular vesicles in tissue regeneration
  146. Electronic noses based on metal oxide nanowires: A review
  147. Framework materials for supercapacitors
  148. An overview on the reproductive toxicity of graphene derivatives: Highlighting the importance
  149. Antibacterial nanomaterials: Upcoming hope to overcome antibiotic resistance crisis
  150. Research progress of carbon materials in the field of three-dimensional printing polymer nanocomposites
  151. A review of atomic layer deposition modelling and simulation methodologies: Density functional theory and molecular dynamics
  152. Recent advances in the preparation of PVDF-based piezoelectric materials
  153. Recent developments in tensile properties of friction welding of carbon fiber-reinforced composite: A review
  154. Comprehensive review of the properties of fly ash-based geopolymer with additive of nano-SiO2
  155. Perspectives in biopolymer/graphene-based composite application: Advances, challenges, and recommendations
  156. Graphene-based nanocomposite using new modeling molecular dynamic simulations for proposed neutralizing mechanism and real-time sensing of COVID-19
  157. Nanotechnology application on bamboo materials: A review
  158. Recent developments and future perspectives of biorenewable nanocomposites for advanced applications
  159. Nanostructured lipid carrier system: A compendium of their formulation development approaches, optimization strategies by quality by design, and recent applications in drug delivery
  160. 3D printing customized design of human bone tissue implant and its application
  161. Design, preparation, and functionalization of nanobiomaterials for enhanced efficacy in current and future biomedical applications
  162. A brief review of nanoparticles-doped PEDOT:PSS nanocomposite for OLED and OPV
  163. Nanotechnology interventions as a putative tool for the treatment of dental afflictions
  164. Recent advancements in metal–organic frameworks integrating quantum dots (QDs@MOF) and their potential applications
  165. A focused review of short electrospun nanofiber preparation techniques for composite reinforcement
  166. Microstructural characteristics and nano-modification of interfacial transition zone in concrete: A review
  167. Latest developments in the upconversion nanotechnology for the rapid detection of food safety: A review
  168. Strategic applications of nano-fertilizers for sustainable agriculture: Benefits and bottlenecks
  169. Molecular dynamics application of cocrystal energetic materials: A review
  170. Synthesis and application of nanometer hydroxyapatite in biomedicine
  171. Cutting-edge development in waste-recycled nanomaterials for energy storage and conversion applications
  172. Biological applications of ternary quantum dots: A review
  173. Nanotherapeutics for hydrogen sulfide-involved treatment: An emerging approach for cancer therapy
  174. Application of antibacterial nanoparticles in orthodontic materials
  175. Effect of natural-based biological hydrogels combined with growth factors on skin wound healing
  176. Nanozymes – A route to overcome microbial resistance: A viewpoint
  177. Recent developments and applications of smart nanoparticles in biomedicine
  178. Contemporary review on carbon nanotube (CNT) composites and their impact on multifarious applications
  179. Interfacial interactions and reinforcing mechanisms of cellulose and chitin nanomaterials and starch derivatives for cement and concrete strength and durability enhancement: A review
  180. Diamond-like carbon films for tribological modification of rubber
  181. Layered double hydroxides (LDHs) modified cement-based materials: A systematic review
  182. Recent research progress and advanced applications of silica/polymer nanocomposites
  183. Modeling of supramolecular biopolymers: Leading the in silico revolution of tissue engineering and nanomedicine
  184. Recent advances in perovskites-based optoelectronics
  185. Biogenic synthesis of palladium nanoparticles: New production methods and applications
  186. A comprehensive review of nanofluids with fractional derivatives: Modeling and application
  187. Electrospinning of marine polysaccharides: Processing and chemical aspects, challenges, and future prospects
  188. Electrohydrodynamic printing for demanding devices: A review of processing and applications
  189. Rapid Communications
  190. Structural material with designed thermal twist for a simple actuation
  191. Recent advances in photothermal materials for solar-driven crude oil adsorption
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