Irreversibility analysis in dissipative magnetohydromagnetic flow of non-Newtonian nanomaterials

Abstract

The theme of this article is to scrutinize the entropy rate in hydromagnetic flow of Reiner–Philippoff nanofluid by a stretching surface. Energy equation is developed through first law of thermodynamic with dissipation and Joule heating. Furthermore, random and thermophoretic motion is discussed. Additionally, binary reaction is discussed. Physical feature of irreversibility analysis is discussed. Nonlinear expression is obtained by suitable transformation. The obtained systems are solved through the numerical method (bvp4c). Variation of entropy rate, thermal field, velocity profile, and concentration against sundry variables are discussed. Computational outcomes of thermal and mass transport rate for influential parameters are studied in tabularized form. A reverse effect holds for thermal field and velocity through magnetic variable. Higher Bingham number leads to a rise in velocity field. An intensification in thermal field and concentration is noted for thermophoretic variable. An enhancement in fluid variable leads to augments velocity. An increment in entropy analysis is seen for magnetic effect. Larger estimation of diffusion variable improves entropy rate. A reduction in concentration is noticed for reaction variable.

Keywords

Reiner–Philippoff nanofluid joule heating Brownian and thermophoresis diffusions viscous dissipation and entropy analysis

Introduction

Non-Newtonian fluids have achieved much consideration due to their innovative application in the field of biosciences, physiological, technological, industrial, and engineering processes. Non-Newtonian liquids are distinguished due to two characteristic viscosity and normal stress differences behaviors. Cement, blood, paints, custard, shampoo, greases, ketchup, and honey are example of non-Newtonian liquids. One constitutive relation (Navier–Stokes) is insufficient to characterize these non-Newtonian liquids with their complicated characteristics. As a result, many models for examining the rheological features of non-Newtonian liquids have been proposed in the literature. Reiner–Philippoff fluid is most important model which is a three-variable liquid phase that relates shear stress to a gradient of velocity. Boundary layer flow analysis of Reiner–Philippoff liquid is studied in Refs. 1 and 2. Numerical analysis of Reiner–Philippoff liquid flow subject towards a stretched sheet is demonstrated in Refs. 3 and 4. Hayat et al.⁵ studied the thermal transport analysis for third grade nanofluid due to rotating disks. Cattaneo–Christov thermal flux in non-Newtonian (Reiner–Philippoff) liquid with Ohmic heating is illustrated by Kumar et al.⁶ Some appropriate studies about non-Newtonian (Reiner–Philippoff) liquid are given in Refs. 7–25.

The significant thermal energy losses as a result of heat transfer, molecular vibration, Joule Thomson effect, diffusion effects, and fluid friction occur in everyday applications including heat transfer in porous medium. This energy loss is known as entropy generation. The efficiency of energy can be measured in terms of entropy generation which depends on simultaneous application of thermodynamics first and second laws. Entropy production assessment is commonly employed to differentiate between irreversible and reversible processes. Entropy minimization is an important objective for scientists and researchers in recent era. This minimization principle is utilized in numerous daily life activities such as energy storage systems, nuclear reactor cooling, refrigeration, solar energy, hybrid-powered engines, air conditioners, bio sensors, micro-manufacturing, radiators, microelectronics, and many others. Initially, Bejan^26–28 discussed the entropy minimization in convective flow with thermal transportation. Khan et al.²⁹ explored the irreversibility analysis for radiative nanoliquid flow with melting effect. Entropy analysis for Casson liquid flow with radiation effect past a vertical cylinder is analyzed by Reddy et al.³⁰ Irreversibility analysis for a combined forced convection of a viscous fluid within the channel is discussed by Sicily and Nejmeh.³¹ Few relevant advancements regarding entropy analysis are presented in Refs. 32–49.

From the above study, we noticed that there is scarce research work existed in literature to scrutinize the entropy generation in Reiner–Philippoff nanofluid flow subject to stretched surface. In this communication, we deliberate the irreversibility analysis in hydromagnetic Reiner–Philippoff nanoliquid flow by a stretchable surface. Joule heating and viscous dissipation are addressed in heat equation. Effect of Brownian and thermophoresis motion are also discussed. Physical feature of irreversibility analysis is discussed. Additionally, binary chemical reaction is also accounted. Partial differential equations are transmitted to ordinary one by similarity variables. The obtained systems are solved by numerical technique (bv4c method). Impact of influential parameters on entropy rate, thermal field, concentration, and velocity are graphically studied. Performance of thermal and mass transport rates versus involved variables are numerically discussed through tables.

Mathematical formulation

Consider two-dimensional hydromagnetic flow of Reiner–Philippoff nanofluid over a stretchable sheet. Heat attribution is scrutinized through Joule heating and dissipation. Furthermore, thermophoretic and random diffusion impacts are addressed. Entropy analysis is also to be accounted. Additionally, first order reaction is also addressed. Constant magnetic force of strength $(B_{0})$ is applied. We consider $(u = u_{w} = a x^{1 / 3})$ as stretching velocity with rate constant $(a > 0)$ . Flow sketch is mentioned in Figure 1.

Figure 1.

Flow sketch.

The shear stress and deformation rate relation for Reiner–Philippoff liquid model satisfy (2) and (3)

\frac{\partial u}{\partial y} = \frac{τ}{μ_{\infty} + \frac{μ_{0} - μ_{\infty}}{1 + (\frac{τ}{τ_{s}})}}

(1)

In above equation, $τ_{s}$ signify reference shear stress, $μ_{0}$ , $μ_{\infty}$ are dynamic viscosities, and $τ$ is the shear stress.

Governing equations satisfy (2)–(7)

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0

(2)

u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \frac{1}{ρ_{f}} \frac{\partial τ}{\partial y} - \frac{σ_{f} B_{0}^{2}}{ρ_{f}} u

(3)

\begin{matrix} u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k_{f}}{{(ρ c_{p})}_{f}} \frac{\partial^{2} T}{\partial y^{2}} + \frac{μ}{{(ρ c_{p})}_{f}} {(\frac{\partial u}{\partial y})}^{2} + \frac{σ_{f} B_{0}^{2}}{{(ρ c_{p})}_{f}} u^{2} \\ + τ D_{B} \frac{\partial T}{\partial y} \frac{\partial C}{\partial y} + τ \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial y})}^{2} \end{matrix}}

(4)

u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = D_{B} \frac{\partial^{2} C}{\partial y^{2}} + \frac{D_{T}}{T_{\infty}} \frac{\partial^{2} T}{\partial y^{2}} - k_{r} (C - C_{\infty})

(5)

with

\begin{matrix} u = u_{w} = a x^{\frac{1}{3}}, v = 0, T = T_{w}, C = C_{w}, y = 0 \\ u \to 0, T \to T_{\infty}, C \to C_{\infty} as y \to \infty . \end{matrix}}

(6)

Here, $(u, v)$ indicate the velocity components; $(x, y)$ , the Cartesian coordinates; $σ_{f}$ , the electrical conductivity; $k_{f}$ , the thermal conductivity; ${(ρ c_{p})}_{f}$ , the heat capacitance; $ρ_{f}$ , the density; $ν_{f}$ , kinematic viscosity; $μ_{f}$ , the dynamic viscosity; $T_{\infty}$ , the ambient temperature; $T_{w}$ , the wall temperature; $D_{B}$ , the Brownian motion coefficient; $k_{r}$ , reaction rate; $C$ , the concentration; $D_{T}$ , thermophoresis coefficient; $C_{\infty}$ , the ambient concentration; and $C_{w}$ , the wall concentration.

Considering

\begin{matrix} u = a x^{\frac{1}{3}} f^{'} (η), v = - \frac{2}{3} x^{- \frac{1}{3}} \sqrt{a ν} f (η) + \frac{1}{3} η x^{- \frac{1}{3}} \sqrt{a ν} f^{'} (η), τ = ρ \sqrt{ν a^{3}} g (η) \\ θ (η) = \frac{T - T_{\infty}}{T_{w - T \infty}}, ϕ (η) = \frac{C - C_{\infty}}{C_{w - C \infty}}, η = \sqrt{\frac{a}{ν}} \frac{y}{x^{\frac{1}{3}}} \end{matrix}}

(7)

we have

g = f^{″} \frac{g^{2} + λ α^{2}}{g^{2} + α^{2}}

(8)

g^{'} + \frac{2}{3} f f^{″} - \frac{1}{3} {f^{'}}^{2} + M f^{'} = 0

(9)

θ^{″} + \frac{2}{3} Pr f θ^{'} + B r {f^{″}}^{2} + M B r {f^{'}}^{2} + Pr N b θ^{'} ϕ^{'} + Pr N t {θ^{'}}^{2} = 0

(10)

ϕ^{″} + \frac{2}{3} S c f ϕ^{'} + \frac{N t}{N b} θ^{″} - γ S c ϕ = 0

(11)

with

\begin{matrix} f (0) = 0, f^{'} (0) = 1, θ (0) = 1, ϕ (0) = 1 \\ f^{'} (\infty) \to 0, θ (\infty) \to 0, ϕ (\infty) \to 0 \end{matrix}}

(12)

in which, $M = (σ_{f} B_{0}^{2} / a ρ_{f})$ denotes the magnetic variable; $N b = (τ D_{B} (C_{w} - C_{\infty}) / ν_{f})$ , Brownian motion variable; $\Pr = (ν_{f} / α_{f})$ , the Prandtl number; $N t = (τ D_{T} (T_{w} - T_{\infty}) / ν_{f} T_{\infty})$ , the thermophoresis variable; $λ = (μ_{0} / μ_{\infty})$ , the Reiner–Philippoff fluid variable; $B r = (E c \Pr)$ , the Brinkman number; $α = (τ_{s} / ρ_{f} \sqrt{a^{3} ν_{f}})$ , the Bingham number; $S c = (ν_{f} / D_{B})$ , Schmidt number; and $γ = (k_{r} / a)$ , chemical reaction parameter.

Entropy generation

Here entropy generation created due to fluid friction irreversibility, thermal transport irreversibility, Joule heating irreversibility, and mass transport rate irreversibility subject to stretching sheet. Mathematically it can be written as^29–35

S_{G} = \frac{k_{f}}{T_{\infty}^{2}} {(\frac{\partial T}{\partial y})}^{2} + \frac{μ_{f}}{T_{\infty}} {(\frac{\partial u}{\partial y})}^{2} + \frac{σ_{f} B_{0}^{2}}{T_{\infty}} u^{2} + \frac{R D_{B}}{T_{\infty}} (\frac{\partial T}{\partial y} \frac{\partial C}{\partial y}) + \frac{R D_{B}}{C_{\infty}} {(\frac{\partial C}{\partial y})}^{2}

(13)

N_{G} = α_{1} {θ^{'}}^{2} + B r {f^{″}}^{2} + M B r {f^{'}}^{2} + L (θ^{'} ϕ^{'} + \frac{α_{2}}{α_{1}} {ϕ^{'}}^{2})

(14)

Here, $N_{G} = (S_{g} T_{\infty} ν_{f} / k_{f} a x^{- 2 / 3} (T_{w} - T_{\infty}))$ indicates the entropy rate; $α_{1} = ((T_{w} - T_{\infty}) / T_{\infty})$ , the temperature ratio variable; $L = (R D_{B} (C_{w} - C_{\infty}) / k_{f})$ , the diffusion variable; and $α_{2} = ((C_{w} - C_{\infty}) / C_{\infty})$ , the concentration ratio variable.

Quantities of interest

Physical quantities of interest (mass and thermal transport rates) are expressed as

N u_{x} = \frac{{q_{w} |}_{y = 0}}{k_{f} (T_{w} - T_{\infty})}, S h_{x} = \frac{{j_{w} |}_{y = 0}}{D_{B} (C_{W} - C_{\infty})}

(15)

heat and mass fluxes

(j_{w} and q_{w})

satisfy

q_{w} = - k_{f} (\frac{\partial T}{\partial y}), j_{w} = - D_{B} (\frac{\partial C}{\partial y})

(16)

we have

N u_{x} R e_{x}^{- 1 / 2} = - θ^{'} (0), S h_{x} R e_{x}^{- 1 / 2} = - ϕ^{'} (0)

(17)

Numerical scheme

The resultant equations (8)–(11) subject to boundary conditions (12) are numerically solved by numerical approach (bvp4c). First, we convert the nonlinear ordinary system to first order system through following variables

\begin{matrix} f = y_{1}, f^{'} = y_{2}, f^{″} = y_{3}, f^{‴} = {y^{'}}_{3} \\ θ = y_{4}, θ^{'} = y_{5}, θ^{″} = {y^{'}}_{5} \\ ϕ = y_{6}, ϕ^{'} = y_{7}, ϕ^{″} = {y^{'}}_{7} \end{matrix}}

(18)

{y^{'}}_{3} = \frac{g^{2} + α^{2}}{g^{2} + λ α^{2}} [\frac{1}{3} y_{2}^{2} - \frac{2}{3} y_{1} y_{3} - M y_{2}]

(19)

{y^{'}}_{5} = [- \frac{2}{3} Pr y_{1} y_{5} - B r y_{3}^{2} - M B r y_{2}^{2} - Pr N b y_{7} y_{5} - Pr N t y_{5}^{2}]

(20)

{y^{'}}_{7} = [- \frac{2}{3} y_{1} y_{7} + γ S c y_{6} - \frac{N t}{N b} {y^{'}}_{5}]

(21)

with

\begin{matrix} y_{1} = 0, y_{2} = 1, y_{4} = 1, y_{6} = 1 at η = 0 \\ y_{2} = 0, y_{4} = 0, y_{6} = 0 as η \to \infty \end{matrix}}

(22)

Discussion

Impact of velocity, thermal field, concentration, and entropy rate versus influential variables are discussed. Numerical outcomes of thermal and mass transport rates are deliberated through tables.

Velocity

Impact of Bingham number on $(f^{'} (η))$ is depicted in Figure 2. Larger approximation of Bingham number decays the fluid flow $(f^{'} (η))$ . Figure 3 sketches impact of velocity versus magnetic effect. An improvement in Lorentz force occurs with variation in magnetic variable, which decays velocity. Influence of $(λ)$ on $(f^{'} (η))$ is illustrated in Figure 4. Larger estimation of fluid variable corresponds to augments $(f^{'} (η))$ .

Figure 2.

$f' (η)$ against $α$ .

Figure 3.

$f' (η)$ against $M$ .

Figure 4.

$f' (η)$ against $λ$ .

Temperature

Influence of $N t$ and $N b$ on thermal field is portrayed in Figures 5 and 6. A reverse effect holds for temperature through random and thermophoretic variables. Higher approximation of Prandtl number decays heat diffusivity, and as a result, thermal field diminished (see Figure 7). Performance of thermal field $(θ (η))$ for magnetic variable is displayed in Figure 8. An intensification in magnetic effect boosts up resistive force to the fluid particles, and consequently, thermal field increased.

Figure 5.

$θ (η)$ against $N t$ .

Figure 6.

$θ (η)$ against $N b$ .

Figure 7.

$θ (η)$ against $\Pr$ .

Figure 8.

$θ (η)$ against $M$ .

Concentration

Performance of $ϕ (η)$ via $(N b)$ is illustrated in Figure 9. Clearly, concentration is decreasing function of random motion variable. Impact of concentration against $(N t)$ is illuminated in Figure 10. Here, concentration boosts up with variation in thermophoretic variable. Influence of $(γ)$ on $(ϕ (η))$ is revealed in Figure 11. A reduction occurs in concentration for reaction parameter. An amplification in fluid variable corresponds to reduce the concentration (see Figure 12).

Figure 9.

$ϕ (η)$ against $N b$ .

Figure 10.

$ϕ (η)$ against $N t$ .

Figure 11.

$ϕ (η)$ against $γ$ .

Figure 12.

$ϕ (η)$ against $λ$ .

Entropy optimization

Impact of entropy rate $(N_{G} (η))$ against magnetic effect is illustrated in Figure 13. An enhancement in magnetic parameter improves resistive force which boosts up collision amongst liquid particles. Thus, entropy rate is augmented. Influence of Bingham number on entropy rate is portrayed in Figure 14. Clearly, higher estimation of Bingham number decays the entropy rate. Influence of $(B r)$ on $(N_{G} (η))$ is demonstrated in Figure 15. Higher Brinkman number augments the kinetic energy, which augments viscous force. As a result, $(N_{G} (η))$ is boosted. An intensification in entropy analysis is noticed through diffusion parameter (see Figure 16).

Figure 13.

$N_{G} (η)$ against $M$ .

Figure 14.

$N_{G} (η)$ against $α$ .

Figure 15.

$N_{G} (η)$ against $B r$ .

Figure 16.

$N_{G} (η)$ against $L$ .

Engineering quantities

Variation of physical quantities $(N u_{x} and S h_{x})$ versus flow variable are discussed through Tables.

Nusselt number

Impact of Nusselt number (Nu_x) against involved parameters are discussed in Table 1. Clearly, an improvement in heat transport rate

(N u_{x})

is seen through random motion and Prandtl number. A reduction occurs in Nusselt number (

N u_{x}

) for thermophoresis variable.

Table 1.

Numerical outcomes of Nusselt number.

$\Pr$	$N t$	$N b$	$N u_{x}$
0.7	0.3	0.4	0.875571
1.0			0.983421
1.7			1.034512
1.0	0.2	0.4	1.79865
	0.6		1.87352
	0.9		1.93471
1.0	0.3	0.2	1.83475
		0.5	1.69756
		0.8	1.45871

Sherwood number

Performance of concentration gradient

(S h_{x})

versus flow variables is portrayed in Table 2. An increase in mass transport rate

(S h_{x})

is noted through Schmidt number and random motion variable. Larger thermophoretic variable reduces the gradient of concentration

(S h_{x})

Table 2.

Numerical results of mass transport rate via flow variable.

$N b$	$N t$	$S c$	$S h_{x}$
0.3	0.4	3.0	1.73456
0.6			1.85767
0.9			1.96898
0.3	0.2	3.0	1.34586
	0.5		1.23476
	0.8		1.17567
0.3	0.4	3.0	1.95495
		6.0	2.34567
		9.0	3.75478

Conclusions

Key observations of present analysis are given below.

An opposite trend holds for velocity through Bingham number and fluid variable.

Larger magnetic variable reduces the velocity field.

Temperature is boosted for increasing thermophoresis parameter, while opposite behavior is observed for thermophoresis variable.

An opposite effect is noticed for thermal field through Prandtl number and magnetic effect.

Concentration profile decays for random motion variable, while reverse trend is noted for thermophoresis variable.

Concentration decays for both Reiner–Philippoff liquid and reaction variables.

Reduction in entropy rate is observed through Bingham number.

An augmentation occurs in entropy rate for magnetic and diffusion variables.

Larger Brinkman number improves entropy generation.

An increment in thermal transport rate is noted through Prandtl number and random motion variable, while opposite effect is observed for thermophoresis variable.

An opposite effect holds for mass transport rate through thermophoretic and random motion variables.

Higher Schmidt number rises the mass transport rate.

Footnotes

Declaration of conflicting interests

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

Funding

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

ORCID iD

Sohail A Khana

Nomenclature

References

Ghoshal

. Dispersion of solutes in non-Newtonian flows through a circular tube. Chem Eng Sci 1971; 26: 185–188.

. Boundary layer flow of Reiner-Philippoff fluids. Int J Non-linear Mech 1994; 29: 871–877.

Yam

Harris

Ingham

, et al. Boundary-layer flow of Reiner-Philippoff fluids past a stretching wedge. Int J Non-linear Mech 2009; 44: 1056–1062.

Ahmad

Qasim

Ahmed

. Flow of Reiner-Philippoff fluid over a stretching sheet with variable thickness. J Braz Soc Mech Sci Eng 2017; 39: 4469–4473.

Hayat

Khan

, et al. Impact of activation energy in nonlinear mixed convective chemically reactive flow of third grade nanomaterial by a rotating disk. Int J Chem Reac Eng 2018; 17, DOI: 10.1515/ijcre-2018-0170.

Kumar

Reddy

Sudharani

MVVNL

, et al. Cattaneo-Christov heat diffusion phenomenon in Reiner-Philippoff fluid through a transverse magnetic field. Physica A Stat Mech i Appl 2020; 541: 123330, DOI: 10.1016/j.physa.2019.123330.

Kapur

Gupta

. Two-dimensional flow of Reiner-Philipp off fluids in the inlet length of a straight channel. Appl Sci Res Sec A 1965; 14: 13–24.

Doh

Cho

Ramya

, et al. Cattaneo-Christov heat flux model for inclined MHD micropolar fluid flow past a non-linearly stretchable rotating disk. Case Stud Ther Eng 2019; 14, DOI: 10.1016/j.csite.2019.100496.

Maninaga Kumar

Kavitha

. Three-dimensional flow of Jeffrey fluid between a rotating and stationary disks with suction. Ain Shams Eng J 2018; 9: 2351–2356.

10.

Oyelakin

Mondal

Sibanda

. Unsteady Casson nanofluid flow over a stretching sheet with thermal radiation, convective and slip boundary conditions. Alexandria Eng J 2016; 55: 1025–1035.

11.

Hayat

Saif

Ellahi

, et al. Simultaneous effects of melting heat and internal heat generation in stagnation point flow of Jeffrey fluid towards a nonlinear stretching surface with variable thickness. Int J Therm Sci 2018; 132: 344–354.

12.

Jusoh

Nazar

Pop

. Magnetohydrodynamic rotating flow and heat transfer of ferrofluid due to an exponentially permeable stretching/shrinking sheet. J Magnetism Magn Mater 2018; 465: 365–374.

13.

Wang

. Review of similarity stretching exact solutions of the Navier-Stokes equations. Eur J Mech B Fluids 2011; 30: 475–479.

14.

Tahmooressi

Kasaeian

Tarokh

, et al. Numerical simulation of aggregation effect on nanofluids thermal conductivity using the lattice Boltzmann method. Int Commun Heat Mass Tran 2020; 110, DOI: 10.1016/j.icheatmasstransfer.2019.104408.

15.

Rauf

Abbas

Shehzad

, et al. Numerical simulation of chemically reactive Powell-Eyring liquid flow with double diffusive Cattaneo-Christov heat and mass flux theories. Appl Math Mech 2018; 39: 467–476.

16.

Reddy

Hiremath

Kumar

. Computational modeling of unsteady third-grade fluid flow over a vertical cylinder: a study of heat transfer visualization. Results Phys 2018; 8: 671–682.

17.

Jayachandra Babu

Sandeep

. UCM flow across a melting surface in the presence of double stratification and cross-diffusion effects. J Mol Liquids 2017; 232: 27–35.

18.

Abiev

RSh

. Mathematical model of two-phase Taylor flow hydrodynamics for four combinations of non-Newtonian and Newtonian fluids in microchannels. Chem Eng Sci 2022; 247, DOI: 10.1016/j.ces.2021.116930.

19.

Silva

BPD

Borges

RFO

Ferraz

IFR

, et al. Effect of previous sedimentation time in filtration rate for shear-thinning suspensions. J Pet Sci Eng 2022; 208, DOI: 10.1016/j.petrol.2021.109453.

20.

Guo

Zhou

Wong

RCK

. Numerical investigation of particle cloud sedimentation in power-law shear-thinning fluids for moderate Reynolds number. Chem Eng Sci 2022; 247, DOI: 10.1016/j.ces.2021.117066.

21.

Kalakonda

Banne

. Studies of electrical, thermal, and mechanical properties of single-walled carbon nanotube and polyaniline of nanoporous nanocomposites. Nanomater Nanotechnol 2021; 11: 1–8.

22.

Aly

Mohamed

Oztop

, et al. Rotating cylinder and magnetic field on solid particles diffusion inside a porous cavity filled with a nanofluid. Nanomater Nanotechnol 2021; 11: 1–17.

23.

Hajatzadeh Pordanjani

Aghakhani

Karimipour

, et al. Investigation of free convection heat transfer and entropy generation of nanofluid flow inside a cavity affected by magnetic field and thermal radiation. J Therm Anal Calorim 2019; 137: 997–1019.

24.

Bagherzadeh

Jalali

Sarafraz

, et al. Effects of magnetic field on micro cross jet injection of dispersed nanoparticles in a microchannel. Int J Numer Meth Heat Fluid Flow 2020; 30: 2683–2704.

25.

Johari

Sirat

Mohamed

, et al. Effects of post-annealing on MoS2 thin films synthesized by multi-step chemical vapor deposition. Nanomater Nanotechnol 2021; 11: 1–7.

26.

Bejan

. Second law analysis in heat transfer. Energy Int. J 1980; 5: 721–732.

27.

Bejan

. A study of entropy generation in fundamental convective heat transfer. J Heat Transfer 1979; 101: 718–725.

28.

Bejan

. Entropy Generation Minimization. New York, NY: CRC Press, 1996.

29.

Khan

Hayat

Alsaedi

, et al. Melting heat transportation in radiative flow of nanomaterials with irreversibility analysis. Renew Sustain Energ Rev 2021; 140, DOI: 10.1016/j.rser.2021.110739.

30.

Janardhana Reddy

Kethireddy

Kumar

, et al. A molecular dynamics study on transient non-Newtonian MHD Casson fluid flow dispersion over a radiative vertical cylinder with entropy heat generation. J Mol Liquids 2018; 252: 245–262.

31.

Sakly

Nejma

. Heat and mass transfer of combined forced convection and thermal radiation within a channel: entropy generation analysis. App Ther Eng 2020; 171, DOI: 10.1016/j.applthermaleng.2020.114903.

32.

Khan

Hayat

Alsaedi

. Entropy optimization in passive and active flow of liquid hydrogen based nanoliquid transport by a curved stretching sheet. Int Communi Heat Mass Trans 2020; 119, DOI: 10.1016/j.icheatmasstransfer.2020.104890.

33.

Khan

Alzahrani

. Free convection and radiation effects in nanofluid (Silicon dioxide and Molybdenum disulfide) with second order velocity slip, entropy generation, Darcy-Forchheimer porous medium. Int J Hydrogen Energ 2021; 46: 1362–1369.

34.

Jarray

Mazgar

Ben Nejma

. Numerical analysis of entropy generation through non-grey gas radiation in a cylindrical annulus. Int J Hydrogen Energ 2017; 42: 8795–8803.

35.

Rashidi

Kavyani

Abelman

. Investigation of entropy generation in MHD and slip flow over a rotating porous disk with variable properties. Int J Heat Mass Transfer 2014; 70: 892–917.

36.

Abdel-Aty

Abdalla

Obada

A-SF

. Quantum information and entropy squeezing of a two-level atom with a non-linear medium. J Phys A Math Gen 2001; 34: 9129–9141.

37.

Kiyani

Hayat

Ahmad

, et al. Bidirectional Williamson nanofluid flow towards stretchable surface with modified Darcy’s law. Surf Inter 2020; 23 DOI: 10.1016/j.surfin.2020.100872.

38.

Salas

Cuevas

Haro

MLd

. Entropy generation analysis of magnetohydrodynamic induction devices. J Phys D: Appl Phys 1999; 32: 2605–2608.

39.

Salimath

Ertesvåg

. Local entropy generation and entropy fluxes of a transient flame during head-on quenching towards solid and hydrogen-permeable porous walls. Int J Hydrogen Energ 2021; 46: 26616–26630.

40.

Aïboud

Saouli

. Entropy analysis for viscoelastic magnetohydrodynamic flow over a stretching surface. Int J Non-linear Mech 2010; 45: 482–489.

41.

Kumar

Manoj

Moharana

. Second law analysis of recharging microchannel using entropy generation minimization method. Int J Mech Sci 2021; 193, DOI: 10.1016/j.ijmecsci.2020.106174.

42.

Hayat

Muhammad

Momani

. Numerical study of entropy generation in Darcy-Forchheimer (F-D) Bodewadt flow of CNTs. Int J Hydrogen Energ 2021, DOI: 10.1016/j.ijhydene.2021.08.013.

43.

Liu

Jian

Tan

. Entropy generation of electromagnetohydrodynamic (EMHD) flow in a curved rectangular microchannel. Int J Heat Mass Transfer 2018; 127: 901–913.

44.

Amniyeh Vatanparast

Hossainpour

Keyhani-Asl

, et al. Numerical investigation of total entropy generation in a rectangular channel with staggered semi-porous fins. Int Commun Heat Mass Transfer 2020; 111: 104446, DOI: 10.1016/j.icheatmasstransfer.2019.104446.

45.

Khan

Saeed

Khan

, et al. Entropy optimized CNTs based Darcy-Forchheimer nanomaterial flow between two stretchable rotating disks. Int J Hydrogen Energ 2019; 44: 31579–31592.

46.

Ullah

Shah

Kumam

, et al. Viscoelastic MHD nanofluid thin film flow over an unsteady vertical stretching sheet with entropy generation. Processes 2019; 7, DOI: 10.3390/pr7050262.

47.

Ullah

Hafeez

Mashwani

, et al. Non-linear thermal radiations and mass transfer analysis on the processes of magnetite carreau fluid flowing past a permeable stretching/shrinking surface under cross diffusion and Hall effect. Coatings 2020; 10, DOI: 10.3390/coatings10060523.

48.

Shah

Ullah

Bonyah

, et al. Hall effect on Titania nanofluids thin film flow and radiative thermal behavior with different base fluids on an inclined rotating surface. AIP Adv 2019; 9, DOI: 10.1063/1.5099435.

49.

Shah

Alzahrani

Dawar

, et al. Influence of Cattaneo-Christov model on Darcy-Forchheimer flow of micropolar ferrofluid over a stretching/shrinking sheet. Int Commun Heat Mass Trans 2020; 110, DOI: 10.1016/j.icheatmasstransfer.2019.104385.