Modeling and simulation for Cattaneo–Christov heat analysis of entropy optimized hybrid nanomaterial flow

Abstract

Here, the hydromagnetic entropy optimized flow of a hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanoliquid by a curved stretchable surface is addressed. The Darcy–Forchheimer model is utilized for porous space. Lead (Pb) and ferric oxide (Fe₂O₃) are considered the nanoparticles and ethylene glycol (C₂H₆O₂) as the base liquid. Thermal expression consists of dissipation and ohmic heating. Entropy generation is under consideration. The Cattaneo–Christov heat flux impact is discussed. Non-dimensional partial expressions by adequate transformations have been reduced to ordinary differential systems. The ND-solve technique is implemented for numerical solutions of dimensionless systems. Graphical illustrations of velocity, thermal field and entropy against influential variables for both nanoliquid (Pb/C₂H₆O₂) and hybrid nanoliquid (Pb + Fe₂O₃/C₂H₆O₂) are presented. Graphical illustrations of velocity, thermal field and entropy against sundry variables for both nanoliquid (Pb/C₂H₆O₂) and hybrid nanoliquid (Pb + Fe₂O₃/C₂H₆O₂) are presented. Influences of sundry variables on the Nusselt number and drag force for both nanoliquid (Pb/C₂H₆O₂) and hybrid nanoliquid (Pb + Fe₂O₃/C₂H₆O₂) are examined. A higher thermal relaxation time tends to intensify the heat transport rate and temperature. An increment in the magnetic variable leads to an enhancement of the entropy and thermal field. An improvement in liquid flow is seen for volume fraction variables. Velocity against the porosity variable and Forchheimer number is reduced. The Brinkman number leads to maximization of entropy generation.

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1. Introduction

During the past few decades, hybrid nano-fluids have become popular among researchers, scientists and engineers due to their superior heat transport properties. Hybrid nano-fluids are suspensions of two or more nano-particles in the base fluid. The heat conductive performance of convectional materials (like blood, water, engine oil, ethylene glycol, etc.) can be enhanced by inserting nano-particles (like silica, carbon nanotubes, silver alumina, gold, lead, ferric oxide, etc.). Hybrid nano-fluids possess higher thermal conductivity for heat transfer phenomena such as those relevant to engines, vehicle radiators, machining, biomedicine, warming procedure on buildings, hybrid powered engines, pharmaceutical industries, paper production and many others. The concept of heat transport enhancement of the base liquid through addition of nanoparticles was initially given by Choi and Eastman et al.^1,2 Hassan et al.³ examined the partial slip effect for the flow of hybrid ferro-liquids. Radiation and porous space effects were addressed. Chabani et al.⁴ examined the electrically conducting hybrid (Ag–Al₂O₃/H₂O) nanomaterial flow inside a modified trapezoidal permeable enclosure. Thermal transport analysis for hybrid nanoliquids with suction/injection in a rotating system was reported by Abdollahi et al.⁵ Cattaneo–Christov analysis in the magnetohydrodynamic flow of ternary nanomaterials with nonlinear radiation was studied by Alqawasmi et al.⁶ Few latest developments related to hybrid nano-fluid flow under different geometric configurations are given in ref. 7–15.

Heat transfer is induced in view of temperature differences and subsequent temperature changes and distribution in different bodies. Heat transfer plays a significant role in industries, and biomedical and engineering processes such as those relevant to heat exchangers, air conditioning, refrigeration, equipment power collectors, fuel cells, drug targeting, microelectronics, heat conduction in tissues, etc. The mechanism of heat transfer was initially given by Fourier.¹⁶ Fourier's law through parabolic expression predicts the infinite speed of heat waves. It is not acceptable physically. Therefore the thermal relaxation time was taken into account by Cattaneo.¹⁷ For material invariant formulation to the Cattaneo's frame work Christov¹⁸ introduced Oldroyd's upper convected derivative. Razaq et al.¹⁹ explored the convective flow of the Reiner–Rivlin fluid subject to Cattaneo–Christov fluxes and MHD. Radiation impact for the magnetized flow of Prandtl nanoliquid with Cattaneo–Christov fluxes theory was given by Salmi et al.²⁰ Haneef et al.²¹ analyzed heat transport for Oldroyd-B material flow with Cattaneo–Christov fluxes. Random motion and thermophoresis effects in magnetohydrodynamic Oldroyd-B nanoliquid flow were studied by Hayat et al.²² Latest articles for Cattaneo–Christov heat and mass fluxes are mentioned in ref. 23–28.

Entropy minimization (EM) optimizes thermal system performance by exploring associated irreversibility through heat and mass transfer, Joule heating and liquid friction. Entropy is a measure of uncertainty and disorderness of a system and its surroundings. All natural processes are thermodynamically irreversible. Entropy is inversely proportional to the temperature of a system. Entropy has a direct proportion to the reversible change in heat. Entropy generation is the loss of energy in thermodynamical systems due to diffusion processes, temperature difference, electric resistance, fluid mixing, chemical reaction, radiation and resistive forces. Applications of entropy generation are found in electronic cooling systems, geothermal reservoirs, thermal and nuclear reactors and heat exchanger pumps. Entropy optimization in thermally convection flow was studied in Bejan.^29–31 Iftikhar et al.³² analyzed entropy generation for non-Newtonian biviscosity fluid in a square cavity. Hayat et al.³³ addressed the magnetized entropy optimized flow of the Reiner–Rivlin material. Maiti et al.³⁴ reported entropy generation analysis for time-dependent hybrid nanoliquid flow by shrinking a disk. Latest investigations for entropy analysis are highlighted in studies.^35–42

This communication discusses the magnetohydrodynamic Darcy–Forchheimer flow of a hybrid nanomaterial. The Cattaneo–Christov heat relation is under consideration. Ferric oxide (Fe₂O₃) and lead (Pb) are used as nanoparticles. Ethylene glycol (C₂H₆O₂) is used as a conventional liquid. Joule heating and dissipation are considered in thermal relation. Entropy generation analysis is carried out. Governing nonlinear expressions are made dimensionless by implementation of suitable transformations. ND-solve is utilized for numerical solutions. Liquid flow, entropy and temperature for both nanoliquid (Pb/C₂H₆O₂) and hybrid nanoliquid (Fe₂O₃ + Pb/C₂H₆O₂) are explored. Surface drag force and heat transport rate against emerging variables for nanoliquid (Pb/C₂H₆O₂) and hybrid nanoliquid (Fe₂O₃ + Pb/C₂H₆O₂) are numerically discussed.

2. Modeling

Two-dimensional magnetized flow of the hybrid (Fe₂O₃ + Pb/C₂H₆O₂) nanoliquid by the curved stretching surface is addressed. Darcy–Forchheimer model analysis is carried out. Lead (Pb) and ferric oxide (Fe₂O₃) are used as nanoparticles and ethylene glycol (C₂H₆O₂) as the base liquid. Cattaneo–Christov heat flux with ohmic heating and dissipation is under consideration. Entropy analysis is carried out. The surface has stretching velocity (u_w = u₀e^s/L) with u₀ reference velocity. The magnetic field of constant strength (B₀) is applied. Fig. 1 comprises a physical model.⁴³

Fig. 1 Flow sketch.

The governing equations satisfy:^44–49

with^50–52In the above expressions (u,v) depict velocity components, μ_hnf the dynamic viscosity, ρ_hnf the density, (r,s) signify the Cartesian coordinates, σ_hnf the electrical conductivity, the inertia coefficient, k_p the porous medium permeability coefficient, R the radius of curvature, (ρc_p)_hnf the heat capacitance, C_b the drag force coefficient, L the reference length, T the temperature, k_hnf the thermal conductivity, c_p the specific heat, T_∞ the ambient temperature, δ_E the heat relaxation time, B₀ the magnetic field strength, T_w the wall temperature and p the pressure.

2.1. Thermophysical characteristics

Mathematical expressions for nanofluid and hybrid nanomaterials and numerical values of conventional fluid and nanoparticles have been given through Tables 1 and 2.^53,54

Table 1 Mathematical expressions for thermophysical characteristics^53,54

Properties	Nanoliquid
Viscosity
Density	ρ nf = (1 − ϕ1)ρf + ϕ1ρs1
Heat capacity	(ρcp)nf = (1 − ϕ1)(ρcp)f + ϕ1(ρcp)s1
Electrical conductivity
Thermal conductivity
Properties	Hybrid nanoliquid
Viscosity
Density	ρ hnf = (1 − ϕ2){(1 − ϕ1)ρf + ϕ1ρs1} + ϕ2ρs2
Heat capacity	(ρcp)hnf = (1 − ϕ2){(1 − ϕ1)(ρcp)f + ϕ1(ρcp)s1} + ϕ2(ρcp)s2
Electrical conductivity
Thermal conductivity

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Table 2 Thermophysical characteristics of ethylene glycol (C₂H₆O₂), lead (Pb) and ferric oxide (Fe₂O₃)^53,54

Properties	ρ (kg m^−3)	k (W m^−1 K^−1)	c p (J k^−1 g^−1 K^−1)	σ (Ω^−1 m^−1)
Ethylene glycol	1116.6	0.249	2382	1.07 × 10^−7
Lead (Pb)	11 343	35	130	4.55 × 10^6
Ferric oxide (Fe2O3)	5200	80.2	670	0.74 × 10^6

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Letting

one obtainsHere shows the material parameter, the Eckert number, theForchheimer number, the thermal relaxation time, the magnetic variable, the Prandtl number and the porosity parameter. Here A₁, A₂, A₃, A₁₁, A₂₂, A₃₃ and A₄₄ are expressed as

3. Interesting quantities

3.1. Surface drag force

One may writeHere τ_rs wall shear stress is

Dimensionless version is

3.2. Nusselt number

Consideringand heat flux q_wwe can express thatin which indicates the local Reynolds number.

4. Entropy formulation

Mathematical expression is given byorHere represents the heat ratio parameter, the Brinkman number and the entropy rate.

5. Solutions methodology

Considering the derivatives with respect to ξ in eqn (7)–(10) we arrive atwith A₅₅ and A₆₆ as

Elimination of pressure from eqn (22) and (23) yields

5.1. Numerical approach

Governing problems are solved by the ND-solve algorithm. The Mathematica software is utilized to develop the computational results. For this we converted the boundary value problem to the initial value situation as follows:with A₇₇ and A₈₈ given below:

6. Discussion

The influence of emerging variables on liquid flow, entropy rate and temperature for both nanoliquid (Pb/C₂H₆O₂) and hybrid nanoliquid (Fe₂O₃ + Pb/C₂H₆O₂) is discussed. Here dotted lines show nanoliquid (Pb/C₂H₆O₂) behaviors and solid lines characterize the hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanoliquid characteristics. Engineering quantities like the skin friction coefficient and heat transport rate are discussed through tabulated forms. Comparison of recent analysis with previous results in the literature is constructed in Table 3. Here we discussed the comparison of the coefficient of skin friction versus a higher approximation of the curvature variable with Okechi et al.⁵⁵ for a limiting case (viscous fluid flow). Clearly a good consensus is noticed.

Table 3 Coefficient of skin friction comparison with Okechi et al.⁵⁵

K 1	Okechi et al.^55	Present results
5	1.4196	1.41963
10	1.3467	1.34675
20	1.3135	1.31354
30	1.3028	1.30281

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6.1. Velocity

Fig. 2 illustrates the flow behavior against the magnetic variable (M). An increment in the magnetic variable leads to intensification of the Lorentz force which declines the liquid flow. The influence of (λ) on (f′(η)) is portrayed in Fig. 3. Higher porosity corresponds to diminished velocity in both nanoliquid (Pb/C₂H₆O₂) and hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanoliquid cases. Fig. 4 shows the velocity impact against the Forchheimer number (Fr). A reduction in liquid flow (f′(η)) is noticed versus the Forchheimer number for both nanoliquid (Pb/C₂H₆O₂) and hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanomaterials. Fig. 5 and 6 elucidate the impact for flow versus nanoparticle volume fractions (ϕ₁ and ϕ₂). It is noticed that there is an enhancement in liquid flow versus higher nanoparticle volume fractions (ϕ₁ and ϕ₂).

Fig. 2 f′(η) variation versus M.

Fig. 3 f′(η) variation versus λ.

Fig. 4 f′(η) variation versus Fr.

Fig. 5 f′(η) variation versus ϕ₁.

Fig. 6 f′(η) variation versus ϕ₂.

6.2. Thermal field

Fig. 7 presents the trend of temperature against the thermal relaxation time variable (β). A higher estimation of thermal relaxation time corresponds to an increase of the thermal field. Fig. 8 indicates the importance for (M) on θ(η). A larger approximation of the magnetic variable intensifies the Lorentz force which raises the resistance to flow. Therefore, the temperature is increased. The influence of the Eckert number on (θ(η)) is depicted in Fig. 9. The outcome for the Eckert number (Ec) leads to creation of an additional kinetic energy and consequently the thermal field is improved. Characteristics of nanoparticle volume fractions (ϕ₁ and ϕ₂) on the thermal field are disclosed in Fig. 10 and 11. Clearly, higher nanoparticle volume fractions (ϕ₁ and ϕ₂) enhance the thermal field for both nanoliquid (Pb/C₂H₆O₂) and hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanomaterials.

Fig. 7 θ(η) versus β.

Fig. 8 θ(η) versus M.

Fig. 9 θ(η) versus Ec.

Fig. 10 θ(η) versus ϕ₁.

Fig. 11 θ(η) versus ϕ₂.

6.3. Entropy rate

Fig. 12 reveals the impact of the magnetic variable on N_G(η). Increasing values of the magnetic variable enhance the resistive force in the flow region which produces an additional energy in the system and therefore entropy is maximized. Fig. 13 depicts the importance of entropy for the Brinkman number. The entropy rate rises versus a higher Brinkman number due to the viscous force. Fig. 14 and 15 show entropy characteristics against nanoparticle volume fractions (ϕ₁ and ϕ₂). A higher estimation of volume fractions (ϕ₁ and ϕ₂) leads to augmentation of the entropy rate for both nanoliquid (Pb/C₂H₆O₂) and hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanomaterials.

Fig. 12 N _G(η) versus M.

Fig. 13 N _G(η) versus Br.

Fig. 14 N _G(η) versus ϕ₁.

Fig. 15 N _G(η) versus ϕ₂.

6.4. Physical quantities

Engineering quantities like the skin friction coefficient (Re_s^1/2C_fs) and heat transport rate (Re_s^−1/2Nu_s) for nanoliquid (Pb/C₂H₆O₂) and hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanomaterials are discussed through tabulated forms.

6.4.1. Skin friction. Table 4 highlights the characteristics of the skin friction (Re_s^1/2C_fs) coefficient against sundry parameters for both nanoliquid (Pb/C₂H₆O₂) and hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanomaterials. It is found that larger Forchheimer number (Fr), porosity (λ), material (K₁) and magnetic (M) parameters result in skin friction reduction.

Table 4 Numerical results of the skin friction (Re_s^1/2C_fs) coefficient

M	K 1	λ	Fr	Res^1/2Cfs
				Pb/C2H6O2	Pb + Fe2O3/C2H6O2
0.4	0.1	0.1	0.2	8.4631	19.9922
0.8				7.35445	17.5033
0.12				6.57457	15.7212
0.3	0.45	0.1	0.2	0.475582	1.04693
	0.45			0.34613	0.754253
	0.51			0.231708	0.496568
0.3	0.45	0.2	0.2	0.344601	0.738767
		0.3		0.223838	0.453805
		0.4		0.11172	0.188597
			0.1	0.564833	1.20704
			0.4	0.313042	0.748393
			0.7	0.0991132	0.343877

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6.4.2. Heat transport rate. Table 5 shows the heat transport rate (Re_s^−1/2Nu_s) variation versus influential variables for both nanoliquid (Pb/C₂H₆O₂) and hybrid (Pb + Fe₂O₃/C₂H₆O₂) nanomaterials. Decay in the thermal transport rate is witnessed for larger Eckert number (Ec), thermal relaxation time (β) and magnetic (M) variables. An increment in the temperature gradient is detected for the Prandtl number.

Table 5 Computational values for Re_s^−1/2Nu_s

β	M	Pr	Ec	Res^−1/2Nus
				Pb/C2H6O2	Pb + Fe2O3/C2H6O2
0.01	0.1	5.0	0.4	2.94351	3.10761
0.03				2.81501	2.44288
0.05				2.68726	2.09427
0.01	0.4	5.0	0.4	2.59787	2.34593
			0.5	2.48648	1.82663
			0.6	2.37692	1.59412
0.01	0.1	3.5	0.4	2.45597	2.38122
		4.0		2.62835	2.52681
		4.5		2.79038	2.6638
0.01	0.1	5.0	0.2	3.23399	3.85736
			0.7	2.50778	1.71628
			0.11	1.92682	0.280292

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7. Closing remarks

Main findings are listed below.

• A higher magnetic parameter decays fluid flow whereas temperature enhances.

• Velocity for the Forchheimer number and porosity parameter is the same.

• A larger thermal relaxation time variable raises temperature.

• A larger estimation of nanoparticle volume fractions corresponds to an amplified fluid flow and thermal field.

• Entropy shows increasing behavior due to the Brinkman number.

• A Higher magnetic parameter leads to entropy rate enhancement.

• Surface drag force shows decreasing behavior for Forchheimer, porosity, material and magnetic parameters.

• The Prandtl number leads to Nusselt number enhancement.

• Nusselt number improvement against thermal relaxation time and magnetic parameters is ensured.

Conflicts of interest

There are no conflicts to declare.

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