Analytical Solution of a Dissipative Flow Conveying Ternary Hybrid Nanofluids Induced By a Porous Surface with Lorentz Forces

MAHARIQ, Ibrahim; KEZZAR, Mohamed; RAGUPATHI, Pachiyappan; KHAN, Umair; RASHID, Farhan Lafta; SHAABAN, Abeer; SARI, Mohamed Rafik

doi:10.2478/ama-2025-0061

Figures & Tables

Influence of α on f′, θ and φ when: φ1 = φ2 = φ3 = 0.01, Re = –1, M = 1, Sc = 1, Kr = 0.1 and Pr = 21

Influence of M on f′ and θ when: φ1 = φ2 = φ3 = 0.01, Re = –1, α = –1, Sc = 1, Kr = 0.1 and Pr = 21

Influence of fie on f′ when: φ1 = φ2 = φ3 = 0.01, α = –1, Sc = 1, Kr = 0.1 and Pr = 21

Influence of fie on θ(η) and φ(η) when: φ1 = φ2 = φ3 = 0.01, φ = –1, Sc = 1, Kr = 0.1 and Pr = 21

Influence of Ec on θ when: φ1 = φ2 = φ3 = 3 = 0.01, Re = –1, α = –1, Sc = 1 and Kr = 0.1

Influence of Sc on θ when: φ1 = φ2 = φ3 = 0.01, Re = –1, α = – 1, M = 1, Kr = 0.1 and Pr = 21

Influence of both Re and M on f″ when: φ1 = φ2 = φ3 = 0.01, Re = –1, α = –1, Kr = 0.1 and Pr = 21

Influence of both Re and M on θ′(–1) when: φ1 = φ2 = φ3 = 0.01, α = –1, Kr = 0.1 and Pr =21

Influence of both Re and Ec on –θ′(–1) when: φ1 = φ2 = φ3 = 0.01, α = –1, Kr = 0.1 M = 1 and Pr = 21

Influence of both Re and Ec on φ′(–1) when: φ1 = φ2 = φ3 = 0.01, α = –1, Kr = 0.1 M = 1 and Pr = 21

Influence of both φ1 and φ2 on φ′(–1) when: Re = –1, Ec = 0.01, φ3 = 0.01, α = –1, Kr = 0.1 M = 1 and Pr = 21

Comparison of f′, θ and φ with HAM-package when : Re = α = –1, Kr = M = Sc = 1 and Pr = 21

Comparison for f″(–1), θ′(–1) and φθ′(–1) when α = 1, φ = 0_06, Kr = 0_1, Ec = 0, M = 1, Sc = 1 and Pr = 6_2

Re	f″(–1)^[51]	f″(–1)^ADM	θ′(–1)^[51]	θ′(–1)^ADM	φ′(–1)^[51]	φ′(–1)^ADM
–1	2.00603	2.00603	–0.0041705	–0.0041705	–0.263183	–0.263182
0	1.83493	1.83492	–0.072609	–0.072607	–0.405111	–0.405111
+1	1.53430	1.53430	–0.730132	–0.730132	–0.610916	–0.610916

Explanation of the parameter control constraints

Symbol	Name	Formula
α	Time-dependent dimensionless parameter	a(t)a′(t)vf{{a(t){a^\prime }(t)} \over {{v_f}}}
Re	Reynolds number	Aαa′vf{{A\alpha a'} \over {{v_f}}}
Sc	Schmidt numbers	vfDB{{{v_f}} \over {{D_B}}}
Ec	Eckert number	v2ΔT(CP)bf{{{v^2}} \over {\Delta T{{\left( {{C_P}} \right)}_{bf}}}}
M	Hartmann number	σfa2B02μf\sqrt {{{{\sigma _f}{a^2}B_0^2} \over {{\mu _f}}}}
Pr	Prandtl number	ρf.CPf.fmaxkf{{{\rho _f}.{C_{Pf}}.{f_{max}}} \over {{k_f}}}
Br	Brinkman number	Pr ∗ Ec

j_ama-2025-0061_utab_001

Symbol	Description	Units (if applicable)
A	Permeability constant	---
Br	Brinkman number	---
C	Concentration	mol/m³
Cp	Specific heat capacity	J/kg·K
D	Mass diffusivity	m²/s
Ec	Eckert number	---
f	Dimensionless velocity function	---
M	Hartmann number	---
Pr	Prandtl number	---
Re	Reynolds number	---
Sc	Schmidt number	---
T	Temperature	K
u,v	Velocity components	m/s
α	Unsteadiness parameter	---
η	Similarity variable	---
θ	Dimensionless temperature	---
μ	Dynamic viscosity	Pa·s
ρ	Density	kg/m³
σ	Electrical conductivity	S/m
φ	Dimensionless concentration	---

Explanation of the quantities control constraints

Symbol	Formula
A₁	(1−φTiO2)−2.5(1−φSiO2)−2.5(1−φAl2O3)−2.5{\left( {1 - {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}} \right)^{ - 2.5}}{\left( {1 - {\varphi _{Si{O_2}}}} \right)^{ - 2.5}}{\left( {1 - {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right)^{ - 2.5}}
A₂	(1−φTiO2)((1−φSiO2)((1−φAl2O3)+φAl2O3ρAl2O3ρf)+φMoS2.ρMoS2ρf)+φTiO2·ρTiO2ρf\left( {1 - {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}} \right)\left( {\left( {1 - {\varphi _{Si{O_2}}}} \right)\left( {\left( {1 - {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right) + {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}{{{\rho _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \over {{\rho _f}}}} \right) + {\varphi _{Mo{S_2}}}.{{{\rho _{Mo{S_2}}}} \over {{\rho _f}}}} \right) + {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}\cdot{{{\rho _{{\rm{Ti}}{{\rm{O}}_2}}}} \over {{\rho _f}}}
A₃	σthnfσf{{{\sigma _{thnf}}} \over {{\sigma _f}}}
A₄	kthnfkf{{{k_{thnf}}} \over {{k_f}}}
A₅	(1−φTiO2){ (1−φSiO2)[ (1−φAl2O3)+φAl2O3(ρCp)Al2O3(ρCp)f ]+φSiO2.(ρCp)SiO2(ρCp)f }+φTiO2.(ρCp)TiO2(ρCp)f\left( {1 - {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}} \right)\left\{ {\left( {1 - {\varphi _{Si{O_2}}}} \right)\left[ {\left( {1 - {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right) + {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}{{{{\left( {\rho {C_p}} \right)}_{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \over {{{\left( {\rho {C_p}} \right)}_f}}}} \right] + {\varphi _{Si{O_2}}}.{{{{\left( {\rho {C_p}} \right)}_{Si{O_2}}}} \over {{{\left( {\rho {C_p}} \right)}_f}}}} \right\} + {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}.{{{{\left( {\rho {C_p}} \right)}_{{\rm{Ti}}{{\rm{O}}_2}}}} \over {{{\left( {\rho {C_p}} \right)}_f}}}

Effects of φ on the f′(0) and θ(0) when Re = α = –1, Kr = M = Sc = 1 and Pr = 21

	φ_TiO₂	φ_SiO₂	φ_Al₂O₃	f″(–1)	θ′(–1)
N-F	0%	0%	0%	1.4067241	0.5258221
	2%	0%	0%	1.3993983	0.52438206
	0%	2%	0%	1.4016671	0.5247220
	0%	0%	2%	1.3987235	0.52430242
	φ_TiO₂	φ_SiO₂	φ_Al₂O₃	f″(–1)	θ′(–1)
HN-F	0%	0%	0%	1.4067241	0.5258221
	2%	2%	0%	1.3941953	0.52334135
	0%	2%	2%	1.39345981	0.5232619
	2%	0%	2%	1.39123039	0.522949
	φ_TiO₂	φ_SiO₂	φ_Al₂O₃	f″(–1)	θ′(–1)
THN-F	0%	0%	0%	1.4067241	0.5258221
	1%	1%	1%	1.3965176	0.52382605
	2%	2%	2%	1.3858312	0.52196548
	0%	0%	2%	1.3746924	0.52023433

The solution procedure based on the ERKM

Equations	Conditions
𝒴1′=𝒴2y_1^\prime = {y_2}	𝒴₁(–1) = –1
𝒴2′=𝒴3y_2^\prime = {y_3}	𝒴₂(–1) = 0
𝒴3′=𝒴4y_3^\prime = {y_4}	𝒴₃(–1) = α₁
𝒴4′=A2A1〈 +α(3𝒴3+η𝒴3′)+Re(𝒴2𝒴3−𝒴1𝒴3′)+A2−1A3M𝒴3 〉{\cal Y}_4^\prime = {{{A_2}} \over {{A_1}}}\left\langle { + \alpha \left( {3{{\cal Y}_3} + \eta {\cal Y}_3^\prime } \right) + {\mathop{\rm Re}\nolimits} \left( {{{\cal Y}_2}{{\cal Y}_3} - {{\cal Y}_1}{\cal Y}_3^\prime } \right) + A_2^{ - 1}{A_3}M{{\cal Y}_3}} \right\rangle	𝒴₄(–1) = α₂
𝒴5′=𝒴6{\cal Y}_5^\prime = {{\cal Y}_6}	𝒴₅(–1) = 1
𝒴6′=1A4〈 +A5Pr𝒴6(ηα−𝒴1Re)+A1BrRe2𝒴32 〉{\cal Y}_6^\prime = {1 \over {{A_4}}}\left\langle { + {A_5}\Pr {{\cal Y}_6}\left( {{\eta _\alpha } - {{\cal Y}_1}{\mathop{\rm Re}\nolimits} } \right) + {A_1}Br{{{\mathop{\rm Re}\nolimits} }^2}{\cal Y}_3^2} \right\rangle	𝒴₆(–1) = α₃
𝒴7′=𝒴8y_7^\prime = {y_8}	𝒴₇(–1) = 1
𝒴7′=〈 +Sc𝒴8(ηα−𝒴1Re)−Kr𝒴7 〉{\cal Y}_7^\prime = \left\langle { + Sc{{\cal Y}_8}\left( {\eta \alpha - {{\cal Y}_1}Re} \right) - {K_r}{{\cal Y}_7}} \right\rangle	𝒴₈(–1) = α₄

Thermophysical characteristics of a ternary hybrid nanofluid

Physical properties	ρ (kg/m³)	C_p(J/kg. °K)	k(W/m.°K)	σ(S/m)
Blood	1063.8	3594	0.492	0.8
TiO₂ → φ₁	4250	397.2	8.9538	2.4×10⁺⁶
SiO₂ → φ₂	2200	765	1.4013	3.5×10⁺⁶
Al₂O₃ → φ₃	3970	686	40	36.9×10⁺⁶

A characteristic made between the physical features of ternary hybrid nanofluids_

Dynamic viscosity	μthnf=μf(1−φTio2)2.5(1−φSiO2)2.5(1−φAl2O3)2.5{\mu _{thnf}} = {{{\mu _f}} \over {{{\left( {1 - {\varphi _{{\rm{Ti}}{{\rm{o}}_2}}}} \right)}^{2.5}}{{\left( {1 - {\varphi _{Si{O_2}}}} \right)}^{2.5}}{{\left( {1 - {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right)}^{2.5}}}}
Density	ρthnf=(1−φTiO2)( (1−φSiO2)((1−φAl2O3)ρf+φAl2O3ρAl2O3)+ φSiO2.ρSiO2 )+φTiO2.ρTiO2{\rho _{thnf}} = \left( {1 - {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}} \right)\left( {\left( {1 - {\varphi _{Si{O_2}}}} \right)\left( {\left( {1 - {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right){\rho _f} + {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}{\rho _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right) + } \right.\left. {{\varphi _{Si{O_2}}}.{\rho _{Si{O_2}}}} \right) + {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}.{\rho _{{\rm{Ti}}{{\rm{O}}_2}}}
Specific heat	(ρCp)thnf=(1−φTiO2)( (1−φSiO2)( (1−φAl2O3)(ρCp)f+ φAl2O3(ρCp)Al2O3 )+φSiO2·(ρCp)SiO2 )+φTiO2.(ρCp)TiO2{\left( {\rho {C_p}} \right)_{thnf}} = \left( {1 - {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}} \right)\left( {(1 - {\varphi _{Si{O_2}}})\left( {\left( {1 - {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right){{\left( {\rho {C_p}} \right)}_f} + } \right.} \right.\left. {\left. {{\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}{{\left( {\rho {C_p}} \right)}_{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right) + {\varphi _{Si{O_2}}}\cdot{{\left( {\rho {C_p}} \right)}_{Si{O_2}}}} \right) + {\varphi _{{\rm{Ti}}{{\rm{O}}_2}}}.{\left( {\rho {C_p}} \right)_{{\rm{Ti}}{{\rm{O}}_2}}}
Electrical Conductivity	{ σthnfσhnf=1+3(σTiO2σhnf−1)φTiO2(σTiO2σhnf+2)−φTiO2(σTiO2σhnf−1)σhnfσnf=1+3(σSiO2σnf−1)φSiO2(σSiO2σnf+2)−φSiO2(σSiO2σnf−1)σnfσf=1+3(σAl2O3σf−1)φAl2O3(σAl2O3σf+2)−φAl2O3(σAl2O3σf−1) \left\{ {\matrix{ {{{{\sigma _{thnf}}} \over {{\sigma _{hnf}}}} = 1 + {{3\left( {{{{\sigma _{Ti{O_2}}}} \over {{\sigma _{hnf}}}} - 1} \right){\varphi _{Ti{O_2}}}} \over {\left( {{{{\sigma _{Ti{O_2}}}} \over {{\sigma _{hnf}}}} + 2} \right) - {\varphi _{Ti{O_2}}}\left( {{{{\sigma _{Ti{O_2}}}} \over {{\sigma _{hnf}}}} - 1} \right)}}} \hfill \cr {{{{\sigma _{hnf}}} \over {{\sigma _{nf}}}} = 1 + {{3\left( {{{{\sigma _{Si{O_2}}}} \over {{\sigma _{nf}}}} - 1} \right){\varphi _{Si{O_2}}}} \over {\left( {{{{\sigma _{Si{O_2}}}} \over {{\sigma _{nf}}}} + 2} \right) - {\varphi _{Si{O_2}}}\left( {{{{\sigma _{Si{O_2}}}} \over {{\sigma _{nf}}}} - 1} \right)}}} \hfill \cr {{{{\sigma _{nf}}} \over {{\sigma _f}}} = 1 + {{3\left( {{{{\sigma _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \over {{\sigma _f}}} - 1} \right){\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \over {\left( {{{{\sigma _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \over {{\sigma _f}}} + 2} \right) - {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}\left( {{{{\sigma _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \over {{\sigma _f}}} - 1} \right)}}} \hfill \cr } } \right.
Thermal Conductivity	{ kthnfkhnf=kTiO2+2khnf−2φTiO2(khnf−kTiO2)kTiO2+2khnf−φTiO2(khnf−kTiO2)khnfknf=kSiO2+2knf−2φSiO2(knf−kSiO2)kSiO2+2knf−φSiO2(knf−kSiO2)knfkf=kAl2O3+2kf−2φAl2O3(kf−kAl2O3)kAl2O3+2kf−φAl2O3(kf−kAl2O3) \left\{ {\matrix{ {{{{k_{thnf}}} \over {{k_{hnf}}}} = {{{k_{Ti{O_2}}} + 2{k_{hnf}} - 2{\varphi _{Ti{O_2}}}\left( {{k_{hnf}} - {k_{Ti{O_2}}}} \right)} \over {{k_{Ti{O_2}}} + 2{k_{hnf}} - {\varphi _{Ti{O_2}}}\left( {{k_{hnf}} - {k_{Ti{O_2}}}} \right)}}} \cr {{{{k_{hnf}}} \over {{k_{nf}}}} = {{{k_{Si{O_2}}} + 2{k_{nf}} - 2{\varphi _{Si{O_2}}}\left( {{k_{nf}} - {k_{Si{O_2}}}} \right)} \over {{k_{Si{O_2}}} + 2{k_{nf}} - {\varphi _{Si{O_2}}}\left( {{k_{nf}} - {k_{Si{O_2}}}} \right)}}} \cr {{{{k_{nf}}} \over {{k_f}}} = {{{k_{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}} + 2{k_f} - 2{\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}\left( {{k_f} - {k_{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right)} \over {{k_{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}} + 2{k_f} - {\varphi _{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}\left( {{k_f} - {k_{{\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}}}} \right)}}} \cr } } \right.

Analytical Solution of a Dissipative Flow Conveying Ternary Hybrid Nanofluids Induced By a Porous Surface with Lorentz Forces

Figures & Tables

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Comparison for f″(–1), θ′(–1) and φθ′(–1) when α = 1, φ = 0_06, Kr = 0_1, Ec = 0, M = 1, Sc = 1 and Pr = 6_2

Explanation of the parameter control constraints

j_ama-2025-0061_utab_001

Explanation of the quantities control constraints

Effects of φ on the f′(0) and θ(0) when Re = α = –1, Kr = M = Sc = 1 and Pr = 21

The solution procedure based on the ERKM

Thermophysical characteristics of a ternary hybrid nanofluid

A characteristic made between the physical features of ternary hybrid nanofluids_

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