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

Dynamic viscosity	$μ_{t h n f} = \frac{μ_{f}}{{(1 - φ_{{Tio}_{2}})}^{2.5} {(1 - φ_{S i O_{2}})}^{2.5} {(1 - φ_{{Al}_{2} O_{3}})}^{2.5}}$
Density	$ρ_{t h n f} = (1 - φ_{{TiO}_{2}}) ((1 - φ_{S i O_{2}}) ((1 - φ_{{Al}_{2} O_{3}}) ρ_{f} + φ_{{Al}_{2} O_{3}} ρ_{{Al}_{2} O_{3}}) + φ_{S i O_{2}} . ρ_{S i O_{2}}) + φ_{{TiO}_{2}} . ρ_{{TiO}_{2}}$
Specific heat	${(ρ C_{p})}_{t h n f} = (1 - φ_{{TiO}_{2}}) ((1 - φ_{S i O_{2}}) ((1 - φ_{{Al}_{2} O_{3}}) {(ρ C_{p})}_{f} + φ_{{Al}_{2} O_{3}} {(ρ C_{p})}_{{Al}_{2} O_{3}}) + φ_{S i O_{2}} \cdot {(ρ C_{p})}_{S i O_{2}}) + φ_{{TiO}_{2}} . {(ρ C_{p})}_{{TiO}_{2}}$
Electrical Conductivity	${\begin{array}{l} \frac{σ_{t h n f}}{σ_{h n f}} = 1 + \frac{3 (\frac{σ_{T i O_{2}}}{σ_{h n f}} - 1) φ_{T i O_{2}}}{(\frac{σ_{T i O_{2}}}{σ_{h n f}} + 2) - φ_{T i O_{2}} (\frac{σ_{T i O_{2}}}{σ_{h n f}} - 1)} \\ \frac{σ_{h n f}}{σ_{n f}} = 1 + \frac{3 (\frac{σ_{S i O_{2}}}{σ_{n f}} - 1) φ_{S i O_{2}}}{(\frac{σ_{S i O_{2}}}{σ_{n f}} + 2) - φ_{S i O_{2}} (\frac{σ_{S i O_{2}}}{σ_{n f}} - 1)} \\ \frac{σ_{n f}}{σ_{f}} = 1 + \frac{3 (\frac{σ_{{Al}_{2} O_{3}}}{σ_{f}} - 1) φ_{{Al}_{2} O_{3}}}{(\frac{σ_{{Al}_{2} O_{3}}}{σ_{f}} + 2) - φ_{{Al}_{2} O_{3}} (\frac{σ_{{Al}_{2} O_{3}}}{σ_{f}} - 1)} \end{array}$
Thermal Conductivity	${\begin{matrix} \frac{k_{t h n f}}{k_{h n f}} = \frac{k_{T i O_{2}} + 2 k_{h n f} - 2 φ_{T i O_{2}} (k_{h n f} - k_{T i O_{2}})}{k_{T i O_{2}} + 2 k_{h n f} - φ_{T i O_{2}} (k_{h n f} - k_{T i O_{2}})} \\ \frac{k_{h n f}}{k_{n f}} = \frac{k_{S i O_{2}} + 2 k_{n f} - 2 φ_{S i O_{2}} (k_{n f} - k_{S i O_{2}})}{k_{S i O_{2}} + 2 k_{n f} - φ_{S i O_{2}} (k_{n f} - k_{S i O_{2}})} \\ \frac{k_{n f}}{k_{f}} = \frac{k_{{Al}_{2} O_{3}} + 2 k_{f} - 2 φ_{{Al}_{2} O_{3}} (k_{f} - k_{{Al}_{2} O_{3}})}{k_{{Al}_{2} O_{3}} + 2 k_{f} - φ_{{Al}_{2} O_{3}} (k_{f} - k_{{Al}_{2} O_{3}})} \end{matrix}$

Tab. 2.

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

Tab. 3.

Explanation of the parameter control constraints

Symbol	Name	Formula
α	Time-dependent dimensionless parameter	$\frac{a (t) a^{'} (t)}{v_{f}}$
Re	Reynolds number	$\frac{A α a^{'}}{v_{f}}$
Sc	Schmidt numbers	$\frac{v_{f}}{D_{B}}$
Ec	Eckert number	$\frac{v^{2}}{Δ T {(C_{P})}_{b f}}$
M	Hartmann number	$\sqrt{\frac{σ_{f} a^{2} B_{0}^{2}}{μ_{f}}}$
Pr	Prandtl number	$\frac{ρ_{f} . C_{P f} . f_{m a x}}{k_{f}}$
Br	Brinkman number	Pr ∗ Ec

Tab. 4.

Explanation of the quantities control constraints

Symbol	Formula
A₁	${(1 - φ_{{TiO}_{2}})}^{- 2.5} {(1 - φ_{S i O_{2}})}^{- 2.5} {(1 - φ_{{Al}_{2} O_{3}})}^{- 2.5}$
A₂	$(1 - φ_{{TiO}_{2}}) ((1 - φ_{S i O_{2}}) ((1 - φ_{{Al}_{2} O_{3}}) + φ_{{Al}_{2} O_{3}} \frac{ρ_{{Al}_{2} O_{3}}}{ρ_{f}}) + φ_{M o S_{2}} . \frac{ρ_{M o S_{2}}}{ρ_{f}}) + φ_{{TiO}_{2}} \cdot \frac{ρ_{{TiO}_{2}}}{ρ_{f}}$
A₃	$\frac{σ_{t h n f}}{σ_{f}}$
A₄	$\frac{k_{t h n f}}{k_{f}}$
A₅	$(1 - φ_{{TiO}_{2}}) {(1 - φ_{S i O_{2}}) [(1 - φ_{{Al}_{2} O_{3}}) + φ_{{Al}_{2} O_{3}} \frac{{(ρ C_{p})}_{{Al}_{2} O_{3}}}{{(ρ C_{p})}_{f}}] + φ_{S i O_{2}} . \frac{{(ρ C_{p})}_{S i O_{2}}}{{(ρ C_{p})}_{f}}} + φ_{{TiO}_{2}} . \frac{{(ρ C_{p})}_{{TiO}_{2}}}{{(ρ C_{p})}_{f}}$

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

Tab. 5.

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

1 *The table nanoparticles dispersed in the base fluid. These concentrations could vary from low to high values, representing different levels of nanoparticle loading in the nanofluid

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

Tab. 6.

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

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	---

Tab. A. 1.

The solution procedure based on the ERKM

Equations	Conditions
$𝒴_{1}^{'} = 𝒴_{2}$	𝒴₁(–1) = –1
$𝒴_{2}^{'} = 𝒴_{3}$	𝒴₂(–1) = 0
$𝒴_{3}^{'} = 𝒴_{4}$	𝒴₃(–1) = α₁
$𝒴_{4}^{'} = \frac{A_{2}}{A_{1}} 〈 + α (3 𝒴_{3} + η 𝒴_{3}^{'}) + Re (𝒴_{2} 𝒴_{3} - 𝒴_{1} 𝒴_{3}^{'}) + A_{2}^{- 1} A_{3} M 𝒴_{3} 〉$	𝒴₄(–1) = α₂
$𝒴_{5}^{'} = 𝒴_{6}$	𝒴₅(–1) = 1
$𝒴_{6}^{'} = \frac{1}{A_{4}} 〈 + A_{5} \Pr 𝒴_{6} (η_{α} - 𝒴_{1} Re) + A_{1} B r {Re}^{2} 𝒴_{3}^{2} 〉$	𝒴₆(–1) = α₃
$𝒴_{7}^{'} = 𝒴_{8}$	𝒴₇(–1) = 1
$𝒴_{7}^{'} = 〈 + S c 𝒴_{8} (η α - 𝒴_{1} R e) - K_{r} 𝒴_{7} 〉$	𝒴₈(–1) = α₄

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DOI: https://doi.org/10.2478/ama-2025-0061 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088

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Language: English

Page range: 525 - 538

Submitted on: Feb 15, 2025

Accepted on: Jun 10, 2025

Published on: Dec 19, 2025

Published by: Bialystok University of Technology

In partnership with: Paradigm Publishing Services

Keywords:

nanolayers,

ternary hybrid nanofluid,

non-Newtonian,

biological fluids,

ADM and HAM-based packages

Related subjects:

Engineering,

Mechanical engineering,

Mechanics,

Bioengineering and biomedical engineering,

Biomechanics,

Civil engineering,

Environmental engineering

© 2025 Ibrahim MAHARIQ, Mohamed KEZZAR, Pachiyappan RAGUPATHI, Umair KHAN, Farhan Lafta RASHID, Abeer SHAABAN, Mohamed Rafik SARI, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 4 (December 2025)

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

Figures & Tables

Fig. 1.

Tab. 1.

Tab. 2.

Tab. 3.

Tab. 4.

Fig. 2.

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Fig. 13.

Tab. 5.

Fig. 14.

Tab. 6.

Tab. A. 1.

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