EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6395 ISSN 1307-5543 – ejpam.com Published by New York Business Global Squeeze Flow of Ethylene Glycol-based Carbon Nanotube with the Influence of Joule Heating, Viscous Dissipation and Chemical Reactions Naseer Khan1, Muhammad Farooq1, Wajid Ullah Jan1, Berna Uzun2,3, Dragan Pamucar4,∗, Ilker Ozsahin2, Hijaz Ahmad2,5,6,∗ 1 Department of Mathematics, Abdul Wali Khan University Mardan, KP, Pakistan 2 Operational Research Center in Healthcare, Near East University, Nicosia/TRNC, 99138 Mersin 10, Turkey 3 Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai, Tamilnadu, India 4 Széchenyi István University, Gyor, Hungary 5 VSB–Technical University of Ostrava, CEET, ENET Centre, 17. listopadu 2172/15, 708 00 Ostrava–Poruba, Czech Republic 6 Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Republic of Korea Abstract. In this article, we discuss a hydrothermal model that describes the flow of Carbon Nanotube nanofluid between squeezing plates while accounting for joule heating, viscous dissipa- tion and homogeneous-heterogeneous chemical reactions. The (Alumina) and Carbon Nanotubes (single-wall carbon nanotube or multi-wall carbon nanotube) have been used as nanoparticles with a base fluid Ethylene glycol with water (Alumina + C2H6O2 − H2O). The homotopy analysis technique included into Mathematica and the MATLAB worked in BVP4c are used to approve and verify the results. We obtain that excellent agreement is observed between the semi-analytical Homotopy analysis method solutions and the numerical bvp4c results, demonstrating the high ac- curacy and reliability of the adopted methodologies. The residual errors of the Homotopy analysis method have been shown physically and numerically for single-wall carbon nanotube and multi-wall carbon nanotube. The other embedding parameters like Reynolds squeeze parameter Sr, Prandtl number P ∗ r , volume fraction ϕ1, ϕ2, schmit number S∗ r and Eckert numbers E1, E2 have been in- tended and discussed. The aim of this research is the improvement towards the consumption of energy in the field of engineering and industry. Furthermore, from the results it has been noted that the based on the data the multi-wall carbon nanotubes have a stronger effect on velocity, temperature distribution, homogeneous and heterogeneous chemical reactions profiles, which is shown by graphs and tables. 2020 Mathematics Subject Classifications: 76A02, 76D05, 76M55, 76V05, 80M40 ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6395 Email addresses: pamucar.dragan@sze.hu (D. Pamucar), hijaz.ahmad@neu.edu.tr (H. Ahmad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 2 of 34 Key Words and Phrases: Squeezing plates, nanofluid, homogeneous and heterogeneous chemical reaction 1. Introduction Industry and engineering require materials to cool quickly and to transfer heat quickly, both of which consume time and energy. The majority of researchers have employed nanofluid in the last ten years for improving cooling and heat transfer rates. Nanofluid is created when liquids and tiny particles (between 1 and 100 nm in size) are combined. These particles are often utilized in the nanofluid solution up to a 5 percent concentration. One significant family of carbon in both single- and multiple-walled forms is carbon nanotubes. The term ”ponder nanomaterial” is commonly used to describe carbon nanotubes, which have a large angle proportion and a very broad range of unique mechanical, chemical, ther- mal, and optical characteristics. Because of their unique qualities, carbon nanotubes are one of the most popular exchange materials in the materials sciences. Iijimain [1], Endo, and associates [2] deserve recognition for their pioneering work on the in-depth analysis of TEM images of the carbon nanotube in 1976. They demonstrated the arrangement of car- bon tubes that were nanoscale (with lengths ranging from 4 to 30 nm) and tube-like using the circular segment release vanishing approach. The researchers devised the process for adopting the single wall carbon nanotubes later in 1993 [3, 4]. Since their gradual discov- ery in the early 1990s, carbon nanotubes have sparked a frenzied interest among academics and modernists alike due to their fascinating characteristics and latent uses in various in- dustries, including aviation, electronics, automotive, optical, and vitality transformation [5, 6]. The heat transfer enhancement in shell and tube heat exchangers using a tri-hybrid nanofluid suspension consisting of copper (Cu), iron oxide (Fe3O4), and multiwall carbon nanotubes dispersed in water as the base fluid by F. Mebarek-Oudina [7]. The thermal transport and entropy generation in viscous flow over a radially stretching disk, incorporat- ing the effects of magnetohydrodynamics (MHD), viscous dissipation, Joule heating, and radiation was study by Tahir Naseem [8]. The influence of geometric parameters on free convective heat transfer in a zigzag-walled cavity filled with a hybrid nano-fluid composed of magnesium oxide (MgO) and single-walled carbon nanotubes suspended in water was studied by F. Mebarek-Oudina [9]. Many researchers used various experimental method- ologies in an attempt to increase the carbon nanotubes thermal efficiency. Qiu et al. [9] have investigated a unique quantitative comparison between the heat transfer contribu- tions from thermal contact resistance at the carbon nanotubes array-solid interface (RC) and those from the thermal conductivity of the carbon nanotubes array (kCNT). carbon nanotubes-based electrical and thermal control in nanodevices is incredibly powerful. To create practical strategies for lowering the resistance at the bundle-bundle contact, Qiu et al. [10] used an experimental method. Recent reviews and summaries of Chinese research breakthroughs in the area of nanoscale heat transport characteristics have been provided by Qiu et al. [11]. They examined developments in atomic-level simulations as well as experimental advancements for certain types of nanoscale materials and architectures. Us- N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 3 of 34 ing the convective physical state, Zaidi et al. [12] investigated the wall jet streaming of carbon nanotubes nanofluid. Using an optimum and numerical technique, they examined the effects of the embedded factors. The single-walled carbon nanotubes and multi-walled carbon nanotubes over an upright cone under convective physical circumstances have been examined by Sreedevi et al. [13]. Haq et al. [14] used engine oil as the basis fluid in their investigation of carbon nanotubes. They looked at the nanofluid that was flowing between the concentric tubes. Sheikholeslami and Seyednezhad [15] used a water-based nanofluid to study the rapid heat transfer caused by the nonlinear magnetic field. Mechanical engi- neering relies heavily on thin-layer diffusion, which has important applications for coating processes. To connect the liquid’s diffusion with various types of sheets, discs, wires, fibers, etc., these coating applications require comprehensive knowledge. Refs. [16, 17] provide the seminal work regarding the fundamental idea and mathematical modeling associated with thin layer streaming. The majority of studies first employed the thin layer’s constant thickness. However, when nanofluid was developed, researchers [18–22] presented the idea of variable thickness for a bit better heat transmission and cooling process. Rehman et al.’s study [23] used kerosene, engine oil, and water as base fluids to investigate how to improve the thermal conductivities of carbon nanotubes. Ellahi et al.’s [24] study of nanofluid flow used a combination of carbon nanotubes and salt water. The rubber and plastic industries employ varieties of extending sheets for everyday applications. These sheets are shaped like tubes, plates, ellipses, circles, cones, and spheres. The radially extended surfaces have several uses in bioengineering and are connected to bilateral symmetry, rubber tires, and the radial artery. A radial disc’s fluid flow along its expanding surface has been studied by Sajjad et al. [25]. The thin layer streaming across a nonlinear expanding surface has been studied by Gul [26]. We were sufficiently inspired to investigate this subject after reading about the aforementioned evaluation, in which many researchers examined dis- tinct logical liquids together with various nanoparticles and produced significant warmth credits. Above all, they focused only on the attractive and slight impacts on cross-breed nanofluid while analyzing the aluminum composites of AA7072 and AA7072 + AA7075 in methanol fluid. According to the Lorentz forces found in this study, a hybrid nanofluid is less effective than a nanofluid. According to the literature, no study has yet thoroughly investigated the combined consequences of unstable flow between two squeezing plates when carbon nanotube nanoparticles are present. Consequently, we examined the im- pacts of a diverse and uniform chemical reaction on the flow between two squeezing plates in the presence of carbon nanotubes nanoparticles, utilizing all of the data. The combined inclusion of Joule heating, viscous dissipation, and homogeneous–heterogeneous chemical reactions offers new insight into their interplay on heat and mass transfer behavior. We conducted a systematic exploration of squeeze flow using hybrid nanofluids (carbon nan- otubes + alumina nanoparticles), an area that has not been comprehensively addressed in prior studies. The Navier–Stokes equations, incorporating Joule heating, viscous dissi- pation, and homogeneous–heterogeneous processes, are solved using MATLAB (BVP4c) and the homotopy analysis method implemented in Mathematica. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 4 of 34 Figure 1: Problem Geometry 2. Mathematical Formulation We take a laminar, incompressible, and carbon nanotubes-based nanofluid flow between horizontally parallel and squeezing plates with homogeneous and heterogeneous chemical reactions; Joule heating and viscous dissipation are seen in Figure 1. Ethylene glycol with water is taken as the base fluid. The carbon nanotubes are used as nanoparticles in the base fluids. The distance between the plates is h(t) = l √ 1− αt, where α is the characteristic parameter of the squeezing motion and l represents the initial distance of the plate at t = 0. The plates are separated while α < 0, but when α > 0, they compress until they meet at t = 1 α . A homogenous magnetic field spread in the y-axis, B∗(t) = B∗ 0√ 1−αt , frequently affects the velocity field. The temperature of both top and lower plates remains constant, i.e., ξi and ξj recursively. An investigation of Joule heating and viscous dissipation is made in the presence of a carbon nanotubes nanofluid. For cubic autocatalysis, the homogeneous reaction is as follows: Ω1 + 2Ω2 → 3Ω2, (1) and the concentration rate is Kca ∗(b∗)2, where as heterogeneous reaction on the catalyst surface, is Ω1 → Ω2, (2) a∗ and b∗ respectively, represent the concentration of the chemical species Ω1 and Ω2, while k∗c represents the rate constants. As the following equations illustrate, an external flow, the reaction rate vanishes beyond the boundary layer edge. Mass, momentum, ther- mal energy, homogeneous, and heterogeneous reaction conservation that is time-dependent may be determined in the Cartesian coordinates (x, y) obtained at the bottom plate center [27–30]. Here we use for equation and tables the abbreviation like (CNT for carbon nan- N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 5 of 34 otube, single wall carbon nanotube (SWCNT), multi wall carbon nanotube (MWCNT), Homotopy analysis method(HAM)). Mass of conservation equation is expressed as, ∂ψ ∂x + ∂υ ∂y = 0. (3) Conservation equation of momentum as, ∂ψ ∂t + ψ ∂ψ ∂x + υ ∂ψ ∂y = − 1 ρ(f) ∂p ∂x + µ(CNT ) ρ(f) [ ∂2ψ ∂x2 + ∂2ψ ∂y2 ] − σ(CNT )(B ∗)2ψ. (4) ∂υ ∂t + ψ ∂υ ∂x + υ ∂υ ∂y = − 1 ρ(f) ∂p ∂y + µ(CNT ) ρ(f) [ ∂2υ ∂x2 + ∂2υ ∂y2 ] (5) Joule heating and viscous dissipation energy equations given as, ∂ξ ∂t + ψ ∂ξ ∂x + υ ∂ξ ∂y = κ∗(CNT ) (ρcp)(f) [ ∂2ξ ∂x2 + ∂2ξ ∂y2 ] + σ(CNT )(B ∗)2 (ρcp)(f) (ψ2 + υ2) + 2µ(CNT ) (ρcp)(f) [ ( ∂2ψ ∂x )2 + ( ∂υ ∂y )2 + ( ∂υ ∂x + ∂ψ ∂y )2 ] − 2 3 [ ∂ψ ∂x + ∂υ ∂y ]2 (6) Similarly equations for homogeneous and heterogeneous forms as, ∂a∗ ∂t + ψ ∂a∗ ∂x + υ ∂a∗ ∂y = DΩ ∂2a∗ ∂y2 −K∗ c a ∗(b∗)2. (7) ∂b∗ ∂t + ψ ∂b∗ ∂x + υ ∂b∗ ∂y = D∗ B ∂2b∗ ∂y2 +K∗ c a ∗(b∗)2. (8) For the following equations, we used ψ, υ shows the recursive components of the horizontal and vertical velocity components of nanofluid, where fluid pressure is p, temperature distribution is ξ and the homogeneous and heterogeneous response variables are a∗ and b∗. The effective fluid density is expressed in terms of ρ. The heat capacity of the fluid is ρcp, and its electrical conductivity for carbon nanotubes nanofluid is σCNT . The chemical species associated with B∗ and Ω, diffusion coefficients are displayed by the variables DΩ and D∗ B respectively. Thermal conductivity for carbon nanotubes nanofluid by κCNT . 3. Similarity Transformation Approach For Boundary Conditions The following criteria were established for the problem’s boundaries: ψ = 0, υ = 0, ξ = ξl, DΩ ∂a∗ ∂y = Γ2a ∗, D∗ B ∂b∗ ∂y = −Γ2a ∗, at y = 0. ψ = 0, υ = −αD 2 √ 1− αt , ξ = ξψ, a∗ = a∗0, b∗ = 0, at y = h(t). (9) N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 6 of 34 The following similarity transformations technique [31] was used to convert the differential equations system into the system of ordinary differential equations. ψ = αxΦ′(Υ) 2(1− αt) , υ = −αlΦ(Υ) 2 √ 1− αt , ξ = ϑ(Υ)ξj , Υ = y l √ 1− αt , and a∗ = a∗0Φ(Υ), b∗ = a∗0(Υ), B∗(t) = B∗ 0√ 1− αt , ϑ = ξ − ξi ξj − ξi . There is an identical satisfaction of the continuity equation (3). The equations of momen- tum, temperature distribution, homogeneous and heterogeneous chemical reactions, is as follows Φ′′′′ − Sr A4 A1 ( Φ′′Φ′ + 2Φ′′ − ΦΦ′′′ +ΥΦ′′′ ) − A5 A1 Φ(a∗)2Φ′′ = 0. (10) A3 A1 ϑ′′ −P ∗ r Sr ( Υϑ′ −Φϑ′ − A5 A2 ME1Φ ′2 − A1 A2 2NE2Φ ′2 − A1 A2 NE1δΦ ′′2 − A5 A2 ME2 δ Φ2 ) = 0. (11) φ′′ − 2SrS ∗ c ( Υφ′ − Φφ′ + Γ1φ 2 ) = 0. (12) ′′β − SrS ∗ c ( Υ′ − Φ′ − Γ1φ 2 ) = 0. (13) The following outcome will be obtained by substituting Ai for the dimensionless constant. A1 = (µ)CNT µf , A2 = (ρCp)CNT (ρCp)f , A3 = (κ)CNT κf , A4 = (ρ)CNT ρf , A5 = κCNT ρCp . Likewise, the similarities transformation of the boundary circumstances change to, Φ(0) = 0, Φ′(0) = 0, ϑ(0) = 1, φ′(0) = Γ2φ(0), β′(0) = −Γ2φ(0). Φ(1) = 1/2, Φ′(1) = 0, ϑ(1) = 0, φ(1) = 1 (1) = 0, (14) In the case of carbon nanotubes, the squeezed Reynolds number is Sr = αl2 2ν(f) , the Hartman parameter is φa∗ = lB∗ 0 √ σ(CNT ) µ(f) , the Prandtl parameter is P ∗ r = (µcp)(CNT ) κ(f) , Schmidt num- ber is S∗ c = v(CNT ) DΩ , Homogeneous reaction strength is Γ1 = 2(κc)(CNT )(a ∗ 0) 2(1−αt) α and the heterogeneous reaction strength is Γ2 = (κs)(CNT ) DΩ , local Eckert number isE1 = α2x2 ξl(1−αt)2 , Eckert number is E2 = α2 ξl(1−αt) , magnetic field parameter is M = σ(CNT )(B ∗ 0 ) 2 2(ρCp)(f)α , porosity number is N = µ(CNT ) (ρCp)(f)α , δ = 1 l2 is a small parameter, and β = DΩ D∗ B represents the dif- fusion coefficient ratio. Here, we presume that Ω and B∗ are the coefficients of diffusion for chemical species of identical sizes.The alternative theory states that DΩ and DB∗ are comparable; hence, β = DΩ D∗ B = 1; also, (Υ) + φ(Υ) = 1 [32], N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 7 of 34 Constants of Study In engineering, among the coefficients of importance are the skin friction coefficient (CΦ), the Sherwood number (Srh), and the local Nusselt number (Nψ).. C∗ Φ = Sr h CΦ = Φ′′(0) , −ϑ′(0) = Nψ, −′(0) = −φ′(0) = Srh. The outcome when Ai is used as the dimensionless constant is as follows: A1 = (µ)CNT µf = µf (1− ϕ1) 2.5 (1− ϕ2) 2.5 . (15) A2 = (ρCp)CNT (ρCp)f = (1− ϕ2) [(1− ϕ1)(ρCp)CNT + ϕ1(ρCp)1] + ϕ2(ρCp)2. (16) A3 = (κ)CNT κf = (κ1 + κ2) + (1− ϕ1)κf + 2ϕ1κ1 + 2ϕ2κ2 (κ1 + κ2) + (1 + ϕ1)κf + (ϕ1κ1 + ϕ2κ2) . (17) A4 = ρCNT ρf = (1− ϕ2) [(1− ϕ1)(ρf ) + ϕ1ρ1] + ϕ2ρ2. (18) A5 = σnf = (κ)CNT (ρCp)f . (19) The volume fractions of the carbon nanotubes nanoparticles are expressed by the symbols ϕ1 and ϕ2, while the density of carbon nanotubes, nanofluid Alumina and base fluids, are expressed by ρ1, ρ2 and ρf respectively, and electrical conductivity of carbon nanotubes and for base fluids of nanoparticales are expressed by σ1, σ2 and σf . Table 1: Thermo physical properties of carbon nanotubes nanoparticales (Alumina) and Ethylene-glycol with water at 20oC. Physical properties ρ ( kg m3 ) cp ( J kgK ) k ( W mK ) σ ( sm−1 ) C2H6O2 +H2O 1063.8 3630 0.387 1× 10−7 Single wall carbon nanotubes 2600 425 6600 0.00597 Multi wall carbon nanotubes 1600 796 300 2.3× 10−5 Alumina 3970 765 40 1.3× 10−5 4. Approximate Analytical Solution Homotopy analysis method analysis was utilized to solve the system of equations (10− 13). As a result of Homotopy analysis method, a collection of base functions Υc, c ≥ 0 may be used to organize the functions Φ(Υ), ϑ(Υ), φ(Υ), and (Υ) respectively. ΦΨ () = ∞∑ £=0 a∗£ £, (20) N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 8 of 34 ϑΨ () = ∞∑ £=0 b∗£ £, (21) φΨ () = ∞∑ £=0 c∗£ £, (22) Ψ () = ∞∑ £=0 d∗£ £, (23) That is, the constant coefficients that we need to be found are a∗£, b∗£, c∗£ and d∗£. We choose the following initial approximations: Φ0() = 3 2 2 − 3, (24) ϑ0() = 1− , (25) φ0() = (1 + Γ∗ 2)/(1 + Γ∗ 2), (26) 0() = (Γ∗ 2 − Γ∗ 2)/(1 + Γ∗ 2β). (27) The auxiliary operators are chosen as ℓΦ = ∂4 ∂4 , ℓϑ = ∂2 ∂2 , ℓφ = ∂2 ∂2 , ℓ = ∂2 ∂2 , (28) with the following properties ℓΦ(£1 3 +£2 2 +£3 + ξ4) = 0, (29) ℓϑ(£5 +£6) = 0, (30) ℓφ(£7 +£8) = 0, (31) ℓ(£9 +£10) = 0, (32) where £1, £2, £3, £4, £5, £6, £7, £8, £9 and £10 are arbitrary constants. Deformation problems of zeroth order as follow: (1;⋎)ℓΦ[Φ̄(;⋎)− Φ0()] = qℏΦNΦ[Φ̄(;⋎), ϑ̄(;⋎), m̄(;⋎), n̄(;⋎)], (33) (1;⋎)ℓϑ[ϑ̄(;⋎)− ϑ0()] = qℏϑNϑ[Φ̄(;⋎), ϑ̄(;⋎), m̄(;⋎), n̄(;⋎)], (34) (1;⋎)ℓφ[φ̄(;⋎)− φ0()] = qℏφNφ[Φ̄(;⋎), φ̄(;⋎),̄ (;⋎)], (35) (1;⋎)ℓ [̄(;⋎)− 0()] = qℏN[Φ̄(;⋎), φ̄(; q),̄ (;⋎)]. (36) Given Eqs. (24− 27), the nonlinear operators are identified as NΦ[Φ̄(;⋎), ϑ̄(;⋎)] = ∂4Φ̄(;⋎) ∂4 − Sr A4 A1 [ ∂Φ̄(;⋎) ∂ ∂2Φ̄(;⋎) ∂2 + 2 ∂2Φ̄(;⋎) ∂2 +Υ ∂3Φ̄(;⋎) ∂3 −Φ∂ 3Φ̄(;⋎) ∂3 ] − A5 A1 φ(a∗)2∂ 2Φ̄(;⋎) ∂2 . (37) N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 9 of 34 Nϑ[Φ̄(;⋎), ϑ̄(;⋎)] = A3 A1 ∂2ϑ̄(;⋎) ∂2 − PrSr [ Υ ∂ϑ̄(;⋎) ∂ − Φ ∂ϑ̄(;⋎) ∂ − A5 A2 ME1 [ ∂Φ̄(;⋎) ∂ ]2 − 2 A1 A2 NE2 [ ∂Φ̄(;⋎) ∂ ]2 − A1 A2 NE1δ [ ∂2Φ̄(;⋎) ∂2 ]2 − A5 A2 ME2 Φ2 δ ] , Nφ[Φ̄(;⋎), φ̄(;⋎),̄ (;⋎)] = ∂2φ̄(;⋎) ∂2 − 2S∗ cSr [ Υ ∂φ̄(;⋎) ∂ − Φ ∂φ̄(;⋎) ∂ + Γ1φ() 2 ] , (38) N[Φ̄(;⋎), φ̄(;⋎),̄ (;⋎)] = ∂2̄(;⋎) ∂2 δ − S∗ cSr [ Υ ∂̄(;⋎) ∂ − Φ ∂̄(;⋎) ∂ − ΓΦ ∂̄(;⋎) ∂ ] , (39) The embedding parameter is ⋎, the nonzero auxiliary parameters are ℏΦ, ℏϑ, ℏφ, and ℏ, and the nonlinear parameters are NΦ, Nϑ, Nφ, and N. Given ⋎ = 0 and 1, we obtain Φ̄(, 0) = Φo(), Φ̄(, 1) = Φ(), ϑ̄(, 0) = ϑo(), ϑ̄(, 1) = ϑ(), φ̄(, 0) = φo(), φ̄(, 1) = φ(), (̄, 0) = o(), (̄, 1) = (), (40) Thus, since ⋎ change between 0 and 1, we may state that Φ̄(, 0), ϑ̄(, 0), φ̄(, 0), (̄, 0) varies from initial guesses Φ0(), ϑ0(), φ0() and 0() to exact solution Φ(), ϑ(), φ() and () respectively. After expanding these functions in Taylor’s series, we obtain: Φ(;⋎) = Φ0() + ∞∑ Ψ=1 ⋎ΨΦΨ (), (41) ϑ(;⋎) = ϑ0() + ∞∑ Ψ=1 ⋎ΨϑΨ (), (42) φ(;⋎) = φ0() + ∞∑ Ψ=1 ⋎ΨφΨ (), (43) (;⋎) = 0() + ∞∑ Ψ=1 ⋎Ψ Ψ (), (44) ΦΨ () = 1 Ψ ! ∂ΨΦ(;⋎) ∂Ψ ∣∣∣∣ ⋎=0 , ϑΨ () = 1 Ψ ! ∂Ψϑ(;⋎) ∂Ψ ∣∣∣∣ ⋎=0 ,φΨ () = 1 Ψ ! ∂Ψφ(;⋎) ∂Ψ ∣∣∣∣ ⋎=0 , Ψ () = 1 Ψ ! ∂Ψ (;⋎) ∂Ψ ∣∣∣∣ ⋎=0 (45) N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 10 of 34 Note that ℏΦ, ℏϑ and other factors are very important for the convergence of the afore- mentioned series ℏφ and ℏ. Given that the nonzero auxiliary parameters are selected in a way that ensures the convergence of equations (41− 44) at ⋎ = 1. We obtain Φ() = Φ0() + ∞∑ Ψ=1 ΦΨ (), (46) ϑ() = ϑ0() + ∞∑ Ψ=1 ϑΨ (), (47) φ() = φ0() + ∞∑ Ψ=1 φΨ (), (48) () = 0() + ∞∑ Ψ=1 Ψ (), (49) Differentiating the deformation equations (33 − 36). After setting ⋎ = 0 and calculating Ψ − times with regard to ⋎, we have ℓΦ[ΦΨ ()− χΨΦΨ−1()] = ℏΦRΦ,Ψ (), (50) ℓϑ[ϑΨ ()− χΨϑΨ−1()] = ℏϑRg,Ψ (), (51) ℓφ[φΨ ()− χΨφΨ−1()] = ℏφRφ,Ψ (), (52) ℓ[Ψ ()− χΨΨ−1()] = ℏϕR,Ψ (), (53) regards with the boundary conditions, ΦΨ (0) = 0, Φ′ Ψ (0) = 0, ϑΨ (0) = 1, φ′ Ψ (0) = Γ2φ(0), β′ Ψ (0) = −Γ2φ(0), ΦΨ (1) = 1/2, Φ′ Ψ (1) = 0, ϑΨ (1) = 0, φΨ (1) = 1, ′ Ψ (1) = 0, (54) where RΦ,Ψ () =Φ ′′′′ Ψ−1()− Sr A4 A1 [ ΥΦ′′′ Ψ−1() + 2Φ′′ Ψ−1() + Ψ−1∑ j=0 Φ′ j()Φ ′′ Ψ−j−1()− Ψ−1∑ j=0 Φj()Φ ′′′ Ψ−j−1() ] − A5 A1 φ(a∗)2Φ′′ Ψ−1(), (55) Rϑ,Ψ () = ( A3 A1 )ϑ′′Ψ−1()− P ∗ r Sr [ Υ Ψ−1∑ j=0 ϑ′Ψ−j−1()− Ψ−1∑ j=0 Φj()ϑ ′ Ψ−j−1()− ( A5 A2 )ME1 Ψ−1∑ j=0 (Φ′)2Ψ−j−1() −( A1 A2 )2NE2 Ψ−1∑ j=0 (Φ′)2Ψ−j−1()− ( A1 A2 )NE1δ Ψ−1∑ j=0 (Φ′′)2Ψ−j−1()− ( A5 A2 ) ME2 δ Ψ−1∑ j=0 (Φ)2Ψ−j−1() ] , (56) N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 11 of 34 Rφ,Ψ () = φ′′ Ψ−1()− 2SrS ∗ c Υφ′ Ψ−j−1() + Γ1φΨ−1() 2 − Ψ−1∑ j=0 Φj()φΨ−j−1()  . (57) R,Ψ () = ′′ Ψ−1()β − S∗ cSr [ Υ′ Ψ−1()− Ψ−1∑ j=0 Φj() ′ Ψ−j−1()− Γ1 Ψ−1∑ j=0 φj() 2 Ψ−j−1() ] (58) Here χΨ = { 1, iΦ Ψ > 1, where 0, Ψ = 1. The general solution of (50− 53) can be expressed in the following way: ΦΨ () = ∫ 0 ∫ 0 ∫ 0 ∫ 0 ℏΦRΦ,Ψ (z)dzdzdzdzdz + χΨΦΨ−1 +£1 3 +£2 2 +£3 +£4, (59) ϑΨ () = ∫ 0 ∫ 0 ℏϑRϑ,Ψ (z)dzdz + χΨϑΨ−1 +£5 +£6, (60) φΨ () = ∫ 0 ∫ 0 ℏφRφ,Ψ (z)dzdz + χΨφΨ−1 +£11 +£12, (61) Ψ () = ∫ 0 ∫ 0 ℏR,Ψ (z)dzdz + χΨΨ−1 +£13 +£14, (62) Therefore, the actual solution Φ(), ϑ(), φ() and φ() becomes Φ() ≈ Ψ∑ n=0 Φn(), ϑ() ≈ Ψ∑ n=0 ϑn(), φ() ≈ Ψ∑ n=0 φn(), () ≈ Ψ∑ n=0 n(). (63) 4.1. Analysis of convergence control parameters It is significant to notice that the non-zero auxiliary parameters ℏΦ, ℏϑ, ℏφ, and ℏ in the series solutions (59− 62). Find out where the homotopy series solutions converge and how quickly they do so. To find the optimal values for ℏΦ, ℏϑ, ℏφ, and ℏ, Liao (2010)’s ”average residual error” was employed. εΦΨ = 1 £+ 1 £∑ j=0 [ NΦ ( Ψ∑ i=0 Φ̄(), Ψ∑ i=0 ϑ̄() )]2 d, (64) εϑΨ = 1 £+ 1 £∑ j=0 [ Nϑ ( Ψ∑ i=0 Φ̄(), Ψ∑ i=0 ϑ̄() ) n=jDun ]2 d, (65) εφΨ = 1 £+ 1 £∑ j=0 [ Nφ ( Ψ∑ i=0 Φ̄(), Ψ∑ i=0 φ̄(), Ψ∑ i=0 (̄) ) n=jDun ]2 d, (66) εΨ = 1 £+ 1 £∑ j=0 [ N ( Ψ∑ i=0 Φ̄(),⋎ Ψ∑ i=0 φ̄(), Ψ∑ i=0 (̄) ) n=jDun ]2 d, (67) N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 12 of 34 In light of Liao (2010) εtΨ = εΦΨ + εϑΨ + εφΨ + εΨ , (68) This expressed the total squared residual error as εtΨ . Using the Mathematica program, the total average squared residual error is reduced (BVPh 2.0, 2014). 5. Error Analysis To guarantee that the residual error is as small as possible while maintaining ana- lytical efficiency, an error analysis is carried out. A numerical and analytical solution is provided by Homotopy analysis method and BVP4C. We do the analysis with a 40th-order approximation. The Mathematica program BVPh 2.0 is also used in this investigation to assess the validity of Homotopy analysis method approaches for maximum residual error 10−40. Regarding the consistency and conformation of the Homotopy analysis method so- lution, the outcomes are contrasted with the numerical solution of BVP4c using Matlab. Error analyses are displayed in Figures 2-3 for single wall carbon nanotubes and multi wall carbon nanotubes and Tables (1-19) to examine the two approaches’ dependability for different levels of physical parameters. Figure 2-3 and Table 2-3 both demonstrate how the maximum average residual errors of carbon nanotubes on Φ(Υ), ϑ(Υ), φ(Υ), and (Υ) are almost gradually reduced up to the 20th transition series. 0 10 20 30 40 10-32 10-22 10-12 m E rr or gr ap h of Φ (ϒ ) 0 10 20 30 40 10-26 10-16 10-6 m E rr or gr ap h of ϑ (ϒ ) 0 10 20 30 40 10-30 10-20 10-10 m E rr or gr ap h of φ (ϒ ) 0 10 20 30 40 10-32 10-22 10-12 m E rr or gr ap h of  (ϒ ) Figure 2: Error analysis of Φ(Υ), ϑ(Υ), φ(Υ) and (Υ) for Alumina in case of single wall carbon nanotubes with P ∗ r = 4, M = 0.2, N = 0.3, β = 0.5, Sr = 0.1, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.2, Γ2 = 0.3, φa∗ = 0.1, ϕ1 = 0.1, ϕ2 = 0.2. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 13 of 34 0 10 20 30 40 10-31 10-21 10-11 m E rr or gr ap h of Φ (ϒ ) 0 10 20 30 40 10-26 10-16 10-6 m E rr or gr ap h of ϑ (ϒ ) 0 10 20 30 40 10-31 10-21 10-11 m E rr or gr ap h of φ (ϒ ) 0 10 20 30 40 10-33 10-23 10-13 m E rr or gr ap h of  (ϒ ) Figure 3: Error analysis of Φ(Υ), ϑ(Υ), φ(Υ) and (Υ) for Alumina in case of multi wall carbon nanotubes with P ∗ r = 4, M = 0.2, N = 0.3, β = 0.5, Sr = 0.1, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.2, Γ2 = 0.3, φa∗ = 0.1, ϕ1 = 0.1, ϕ2 = 0.2. Table 2: Total residual error of Φ(Υ), ϑ(Υ), φ(Υ) and (Υ) in case of single wall carbon nanotubes with P ∗ r = 4, M = 0.2, N = 0.3, β = 0.5, Sr = 0.1, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.2, Γ2 = 0.3, φa∗ = 0.1, ϕ1 = 0.1, ϕ2 = 0.2. n ϵΦn ϵϑn ϵφn ϵn 1 2.86053× 10−10 0.0000668542 7.0533× 10−12 3.98875× 10−13 5 3.05126× 10−23 2.76859× 10−16 3.66382× 10−31 2.21904× 10−32 10 6.30657× 10−39 2.50636× 10−29 4.74327× 10−37 8.3729× 10−39 15 7.57023× 10−39 1.58037× 10−32 4.47996× 10−37 8.3729× 10−39 20 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 25 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 30 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 35 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 40 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 14 of 34 Table 3: Total residual error of Φ(Υ), ϑ(Υ), φ(Υ) and (Υ) in case of multi wall carbon nanotubes with P ∗ r = 4, M = 0.2, N = 0.3, β = 0.5, Sr = 0.1, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.2, Γ2 = 0.3, φa∗ = 0.1, ϕ1 = 0.1, ϕ2 = 0.2. n ϵΦn ϵϑn ϵφn ϵn 1 3.19065× 10−10 0.0000570536 7.05213× 10−12 3.98808× 10−13 5 1.81537× 10−23 1.14989× 10−16 2.59573× 10−31 1.57021× 10−32 10 2.52521× 10−37 3.95875× 10−30 1.72533× 10−37 2.97937× 10−39 15 2.38733× 10−37 3.97163× 10−33 1.56546× 10−37 2.97937× 10−39 20 2.38733× 10−37 3.97163× 10−33 1.08337× 10−37 2.24561× 10−39 25 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 30 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 35 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 40 7.57023× 10−39 1.58037× 10−32 4.61396× 10−37 8.11943× 10−39 Table 4: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for Φ(Υ), ϑ(Υ), φ(Υ) and (Υ) with P ∗ r = 0.1, M = 0.5, N = 0.1, β = 0.7, Sr = −0.2, δ = 0.5, E1 = 0.1, E2 = 0.2, S∗c = 1.1, Γ1 = 0.5, Γ2 = −1.3, φa∗ = 0.1, ϕ1 = 0.1, ϕ2 = 0.2. MWCNT HAM Results MWCNT Numerical Results Υ Φ(Υ) ϑ(Υ) φ(Υ) (Υ) Φ(Υ) ϑ(Υ) φ(Υ) (Υ) 0 0 1.0000 0.3679 -0.6717 0 1.0000 0.3679 -0.6717 0.1001 0.0139 1.1193 0.4066 -0.6032 0.0139 1.1193 0.4066 -0.6032 0.2002 0.0517 1.2368 0.4494 -0.5347 0.0517 1.2368 0.4494 -0.5347 0.3003 0.1076 1.3481 0.4967 -0.4663 0.1076 1.3481 0.4967 -0.4663 0.4004 0.1755 1.4471 0.5490 -0.3981 0.1755 1.4471 0.5490 -0.3981 0.5005 0.2496 1.5241 0.6068 -0.3302 0.2496 1.5241 0.6068 -0.3302 0.6006 0.3238 1.5639 0.6707 -0.2628 0.3238 1.5639 0.6707 -0.2628 0.7007 0.3920 1.5371 0.7413 -0.1959 0.3920 1.5371 0.7413 -0.1959 0.8008 0.4481 1.3834 0.8194 -0.1297 0.4481 1.3834 0.8194 -0.1297 0.9099 0.4862 0.9682 0.9057 -0.0641 0.4862 0.9682 0.9057 -0.0641 1.0000 0.5000 0 1.0000 0 0.5000 0 1.0000 0 As Υ increases, the values of ϑ(Υ), and (Υ) drop. Whereas the value of Φ(Υ), φ(Υ) increases as Υ values increase, which is shown in Table 4. Table 5 shows the values of Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) for various values of Υ. These values are related to the local coefficient of skin friction, the local Nusselt and shewored numbers (homogeneous-heterogeneous reactions). 6. Results and Discussions This section has been structured to ascertain the effects of various important pa- rameters on the local rates of heat transfer at the squeezing plate surface and the lo- cal skin-fraction coefficient, both of which are computed and very significant in terms of physical properties. Equations (14) are obtained by numerically solving the system of non- N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 15 of 34 Table 5: Computational of multi wall carbon nanotube nanoparticales (Alumina+Ethylene-glycol with water) for Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with P ∗ r = 0.1, M = 0.5, N = 0.1, β = 0.7, Sr = −0.2, δ = 0.5, E1 = 0.1, E2 = 0.2, S∗c = 1.1, Γ1 = 0.5, Γ2 = −1.3, φa∗ = 0.1, ϕ1 = 0.3, ϕ2 = 0.4. MWCNT HAM Results MWCNT Numerical Results Υ Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′′(0) −ϑ′(0) −φ′(0) −′(0) 0 2.9796 0.2800 0.4782 -0.6832 2.9796 0.2800 0.4782 -0.6832 0.1001 2.3881 0.2843 0.4807 -0.6847 2.3881 0.2843 0.4807 -0.6847 0.2002 1.7963 0.3124 0.4813 -0.6843 1.7963 0.3124 0.4813 -0.6843 0.3003 1.2031 0.3819 0.4802 -0.6825 1.2031 0.3819 0.4802 -0.6825 0.4004 0.6076 0.5064 0.4780 -0.6795 0.6076 0.5064 0.4780 -0.6795 0.5005 0.0094 0.6970 0.4747 -0.6757 0.0094 0.6970 0.4747 -0.6757 0.6006 -0.5917 0.9639 0.4704 -0.6710 -0.5917 0.9639 0.4704 -0.6710 0.7007 -1.1961 1.3155 0.4652 -0.6654 -1.1961 1.3155 0.4652 -0.6654 0.8008 -1.8043 1.7563 0.4590 -0.6589 -1.8043 1.7563 0.4590 -0.6589 0.9099 -2.4171 2.2802 0.4515 -0.6511 -2.4171 2.2802 0.4515 -0.6511 1.0000 -3.0294 2.8540 0.4426 -0.6420 -3.0294 2.8540 0.4426 -0.6420 linear equations (10)-(13) for different values of physical parameters. Homotopy analysis method and BVP4C are used in the numerical approach. The current problem has a large number of physical characteristics, which allows for a broad variety of results. The distri- butions of velocity, temperature distribution, homogeneous and heterogeneous chemical reactions,Joule heating and viscous dissipation are all obtained by solving equations (10)- (13) and are shown in figures 2-13 for different physical parameters. The effect of single wall carbon nanotubes nanoparticles is shown by dashed lines, and the effect of multi wall carbon nanotubes particles is shown by solid lines in the graphs. The Reynolds squeeze parameter governs the strength of squeezing motion, where higher values accelerate the fluid motion and enhance skin friction, directly affecting energy dissipation and lubrica- tion efficiency. The Prandtl number reflects the relative thickness of velocity and thermal boundary layers; higher Pr reduces thermal diffusivity, leading to steeper temperature gradients and stronger heat transfer rates, which is crucial for cooling applications. The Schmidt number characterises the competition between momentum and mass diffusivity; larger values suppress solute penetration into the flow, thereby reducing the thickness of the concentration boundary layer and affecting reaction efficiency. The Eckert number, representing viscous dissipation effects, converts kinetic energy into internal energy, which elevates the fluid temperature and can hinder cooling performance if not controlled. The Lewis number links thermal and mass diffusivities; higher values favour thermal diffusion over mass transport, influencing both heat transfer enhancement and chemical reaction rates. These interpretations are now explicitly connected to practical implications. For ex- ample, optimising the squeeze Reynolds number and Prandtl number can enhance cooling performance in microelectromechanical systems and lubrication technologies, while careful control of Schmidt and Lewis numbers can improve mass transfer and reaction efficiency in chemical processing. The objective of tables (5-7) and (8-19) is to perform a numerical impact test of N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 16 of 34 different physical parameters. The tables demonstrate that all of the results agree well with the BVP4c and Homotopy analysis method results. It has been shown that the influences of skin friction coefficients, velocity, temperature, Nusselt and Sherwood numbers, and both homogeneous and heterogeneous factors produce an increase in the mass transfer rate. An increase in the internal heat production parameter results in a decrease in the skin friction coefficient and the rate of heat transmission. Increasing the squeezing parameter has been shown to provide a set of outcomes that are comparable. The local Nusselt number and the friction factor decrease when the squeezing parameter is decreased. The skin friction coefficient demonstrated a declining trend, indicating that the floor was drawing the fluid. Tables 2-19 shows the results for Φ(Υ), ϑ(Υ), φ(Υ) and (Υ) , Φ′′(0), −ϑ′(0), −φ′(0) and −′(0). Table 6: Computational for single wall carbon nanotubes nano-particales and multi wall carbon nanotube hybrid nano-particales in case of skin friction Φ′′(0). Various fluid model parameters SWCNT Nano particals MWCNT Hybrid nano particals Sr φa∗ ϕ1 ϕ2 Φ′′(0) Φ′′(0) 0.1 0.01 0.1 0.2 3.0471 3.0214 -0.1 3.0048 2.9786 -0.3 2.9618 2.9352 0.01 2.9834 2.9570 0.05 3.5653 2.9596 0.1 4.9633 2.9679 0.1 2.9834 2.9813 0.2 2.9924 2.9813 0.3 3.0064 2.9812 0.1 2.9816 2.9570 0.4 2.9962 2.9571 0.7 3.0444 2.9579 N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 17 of 34 Table 7: Computational for single wall carbon nanotubes nano-particales and multi wall carbon nanotube hybrid nano-particales in case of Nustlt Number −ϑ′(0). Various fluid model parameters SWCNT Nano particals MWCNT Hybrid nano particals Sr P ∗ r ϕ1 ϕ2 N E1 E2 −ϑ′(0) −ϑ′(0) 0.1 0.1 0.1 0.2 0.1 0.1 0.2 2.3096 2.3093 -0.1 −0.3586 −0.3595 -0.3 −1.5130 −1.5159 0.1 −1.1924 −1.1945 0.2 −1.6217 −1.6250 0.3 −1.7025 −1.7061 0.1 −1.6584 −1.1945 0.3 −1.6375 −0.4360 0.5 −1.5889 0.3422 0.1 −1.4184 −1.1945 0.4 −0.2952 −0.2960 0.7 0.8333 0.8332 0.1 −1.1924 −1.1945 0.5 −1.1923 −1.1944 1 −1.1922 −1.1943 0.1 −1.1924 −1.1945 0.4 −1.1923 −1.1943 0.8 −1.1921 −1.1942 0.2 −1.1924 −1.1945 0.4 −1.1923 −1.1944 0.7 −1.1922 −1.1943 Table 8: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of Sr on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with P ∗ r = 0.1, M = 0.5, N = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, φa∗ = 0.01, ϕ1 = 0.1, ϕ2 = 0.2 MWCNT HAM Results MWCNT Numerical Results Sr Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) −0.2 2.9570 -1.1945 0.4782 -0.6832 2.9570 -1.1945 0.4782 -0.6832 −0.4 2.9133 -1.6300 0.4782 -0.6832 2.9133 -1.6300 0.4782 -0.6832 −0.6 2.8689 -1.7166 0.4782 -0.6832 2.8689 -1.7166 0.4782 -0.6832 -0.8 2.8238 -1.7613 0.4782 -0.6832 2.8238 -1.7613 0.4782 -0.6832 N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 18 of 34 Table 9: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of P ∗ r on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, M = 0.5, N = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, φa∗ = 0.01, ϕ1 = 0.1, ϕ2 = 0.2 MWCNT HAM Results MWCNT Numerical Results P ∗ r Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.1 2.9570 -1.1945 0.4782 -0.6832 2.9570 -1.1945 0.4782 -0.6832 0.2 2.9570 -1.6250 0.4782 -0.6832 2.9570 -1.6250 0.4782 -0.6832 0.3 2.9570 -1.7061 0.4782 -0.6832 2.9570 -1.7061 0.4782 -0.6832 0.4 2.9570 -1.7455 0.4782 -0.6832 2.9570 -1.7455 0.4782 -0.6832 Table 10: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of E1 on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, N = 0.1, β = 0.7, δ = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, φa∗ = 0.01, ϕ1 = 0.1, ϕ2 = 0.2. MWCNT HAM Results MWCNT Numerical Results E1 Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.1 2.9570 -1.1945 0.4782 -0.6832 2.9570 -1.1945 0.4782 -0.6832 0.2 2.9570 -1.1944 0.4782 -0.6832 2.9570 -1.1944 0.4782 -0.6832 0.3 2.9570 -1.1944 0.4782 -0.6832 2.9570 -1.1944 0.4782 -0.6832 0.4 2.9570 -1.1943 0.4782 -0.6832 2.9570 -1.1943 0.4782 -0.6832 Table 11: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of E2 on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, N = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, φa∗ = 0.01, ϕ1 = 0.1, ϕ2 = 0.2. MWCNT HAM Results MWCNT Numerical Results E2 Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.2 2.9570 -1.1945 0.4782 -0.6832 2.9570 -1.1945 0.4782 -0.6832 0.3 2.9570 -1.1944 0.4782 -0.6832 2.9570 -1.1944 0.4782 -0.6832 0.5 2.9570 -1.1943 0.4782 -0.6832 2.9570 -1.1943 0.4782 -0.6832 0.8 2.9570 -1.1942 0.4782 -0.6832 2.9570 -1.1942 0.4782 -0.6832 Table 12: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of δ on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, N = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, φa∗ = 0.01, ϕ1 = 0.1, ϕ2 = 0.2. MWCNT HAM Results MWCNT Numerical Results δ Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.1 2.9570 -1.1945 0.4782 -0.6832 2.9570 -1.1945 0.4782 -0.6832 −2 2.9570 -1.1946 0.4782 -0.6832 2.9570 -1.1946 0.4782 -0.6832 −4 2.9570 -1.1947 0.4782 -0.6832 2.9570 -1.1947 0.4782 -0.6832 −6 2.9570 -1.1948 0.4782 -0.6832 2.9570 -1.1948 0.4782 -0.6832 N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 19 of 34 Table 13: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of N on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, δ = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, φa∗ = 0.01, ϕ1 = 0.1, ϕ2 = 0.2. MWCNT HAM Results MWCNT Numerical Results N Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.1 2.9570 -1.1945 0.4782 -0.6832 2.9570 -1.1945 0.4782 -0.6832 0.3 2.9570 -1.1944 0.4782 -0.6832 2.9570 -1.1944 0.4782 -0.6832 0.8 2.9570 -1.1943 0.4782 -0.6832 2.9570 -1.1943 0.4782 -0.6832 1.4 2.9570 -1.1942 0.4782 -0.6832 2.9570 -1.1942 0.4782 -0.6832 Table 14: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of φa∗ on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, δ = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, N = 0.1, ϕ1 = 0.1, ϕ2 = 0.2. MWCNT HAM Results MWCNT Numerical Results φa∗ Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.1 2.9679 -1.1936 0.4782 -0.6832 2.9679 -1.1936 0.4782 -0.6832 0.2 3.0006 -1.1910 0.4782 -0.6832 3.0006 -1.1910 0.4782 -0.6832 0.3 3.0546 -1.1869 0.4782 -0.6832 3.0546 -1.1869 0.4782 -0.6832 0.4 3.1286 -1.1812 0.4782 -0.6832 3.1286 -1.1812 0.4782 -0.6832 Table 15: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of ϕ1 on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, δ = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, N = 0.1, ϕ2 = 0.2, φa∗ = 0.1. MWCNT HAM Results MWCNT Numerical Results ϕ1 Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.1 2.9679 -1.1936 0.4782 -0.6832 2.9679 -1.1936 0.4782 -0.6832 0.2 2.9716 -0.8503 0.4782 -0.6832 2.9716 -0.8503 0.4782 -0.6832 0.3 2.9775 -0.4353 0.4782 -0.6832 2.9775 -0.4353 0.4782 -0.6832 0.4 2.9871 -0.0187 0.4782 -0.6832 2.9871 -0.0187 0.4782 -0.6832 Table 16: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of ϕ2 on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, δ = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, Γ2 = −1.3, N = 0.1, ϕ1 = 0.1, φa∗ = 0.1. MWCNT HAM Results MWCNT Numerical Results ϕ2 Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.1 2.9727 -1.4207 0.4782 -0.6832 2.9727 -1.4207 0.4782 -0.6832 0.2 2.9679 -1.1936 0.4782 -0.6832 2.9679 -1.1936 0.4782 -0.6832 0.3 2.9675 -0.8044 0.4782 -0.6832 2.9675 -0.8044 0.4782 -0.6832 0.4 2.9690 -0.2957 0.4782 -0.6832 2.9690 -0.2957 0.4782 -0.6832 N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 20 of 34 Table 17: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of Γ2 on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, δ = 0.1, β = 0.7, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5, N = 0.1, ϕ1 = 0.1, ϕ2 = 0.2, φa∗ = 0.1. MWCNT HAM Results MWCNT Numerical Results Γ2 Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) −1.3 2.9690 -0.2957 0.4782 -0.6832 2.9690 -0.2957 0.4782 -0.6832 0.1 2.9690 -0.2957 -0.0368 0.0526 2.9690 -0.2957 -0.0368 0.0526 0.3 2.9690 -0.2957 -0.1104 0.1577 2.9690 -0.2957 -0.1104 0.1577 0.5 2.9690 -0.2957 -0.1839 0.2628 2.9690 -0.2957 -0.1839 0.2628 Table 18: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of β on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, M = 0.5, δ = 0.1, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5 Γ2 = −1.3, N = 0.1, ϕ1 = 0.1, ϕ2 = 0.2, φa∗ = 0.1. MWCNT HAM Results MWCNT Numerical Results β Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.1 2.9690 -0.2957 -0.1839 1.8394 2.9690 -0.2957 -0.1839 1.8394 0.3 2.9690 -0.2957 -0.1839 0.6131 2.9690 -0.2957 -0.1839 0.6131 0.5 2.9690 -0.2957 -0.1839 0.3679 2.9690 -0.2957 -0.1839 0.3679 0.7 2.9690 -0.2957 -0.1839 0.2628 2.9690 -0.2957 -0.1839 0.2628 Table 19: Computational of multi wall carbon nanotubes nanoparticales (Alumina+Ethylene-glycol with water) for different values of M on Φ′′(0), −ϑ′(0), −φ′(0) and −′(0) with Sr = −0.2, P ∗ r = 0.1, δ = 0.1, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5 Γ2 = −1.3, N = 0.1, β = 0.7, ϕ1 = 0.1, ϕ2 = 0.2, φa∗ = 0.1. MWCNT HAM Results MWCNT Numerical Results M Φ′′(0) −ϑ′(0) −φ′(0) −′(0) Φ′(0) −ϑ′(0) −φ′(0) −′′(0) 0.5 2.9690 -0.2957 -0.1839 0.2628 2.9690 -0.2957 -0.1839 0.2628 0.7 2.9690 -0.2956 -0.1839 0.2628 2.9690 -0.2956 -0.1839 0.2628 1.5 2.9690 -0.2955 -0.1839 0.2628 2.9690 -0.2955 -0.1839 0.2628 2.6 2.9690 -0.2953 -0.1839 0.2628 2.9690 -0.2953 -0.1839 0.2628 Sr = 0.1 Sr = 0.3 Sr = 0.5 Sr = 0.7 Sr = 0.2 Sr = 0.4 Sr = 0.6 Sr = 0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 Υ Φ (Υ ) Sr = 0.1 Sr = 0.3 Sr = 0.5 Sr = 0.7 Sr = 0.2 Sr = 0.4 Sr = 0.6 Sr = 0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Υ ϑ (Υ ) Sr = -0.1 Sr = -0.3 Sr = -0.5 Sr = -0.7 Sr = -0.2 Sr = -0.4 Sr = -0.6 Sr = -0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 -3 -2 -1 0 1 Υ ϕ (Υ ) Sr = 0.1 Sr = 0.3 Sr = 0.5 Sr = 0.7 Sr = -0.1 Sr = -0.2 Sr = -0.3 Sr = -0.4 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 Υ  (Υ ) Figure 4: Impact of Sr on single wall carbon nanotubes and multi wall carbon nanotubes with E1 = 0.1, E2 = 0.3, ϕ1 = 0.1, ϕ2 = 0.2, N = 1, M = 2, β = 1, P ∗ r = 2, S∗c = 2, h = 0.3, δ = 0.1, Γ1 = 1.1, Γ2 = 1.5, φa∗ = 1. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 21 of 34 Plotting Φ(Υ), ϑ(Υ), φ(Υ), and (Υ) for both the single wall carbon nanotubes and multi wall carbon nanotubes cases in Figure 4 illustrates the impact of squeeze number Sr on the velocity and temperature distribution for both heating and cooling surfaces. The top plate is moving away from the lower stationary plate in this instance because the squeeze number values have been configured to be greater than zero, but the lower plate is moving away from the upper plate due to the reverse inequality of the squeeze number. A sketch of Sr’s impact on Φ(Υ) is presented for both the single wall carbon nanotubes and multi wall carbon nanotubes scenarios. Actually, when Sr increases, the top plate descends, applying more stress to the nanoparticles and raising the velocity components Φ(Υ). A graph shows how Sr affects θ(η). For greater Sr, the top plate travels downward and the number of nanoparticle collisions rises, which raises the temperature and causes the concentration profiles of φ(Υ) to fall. The impact of the parameter Sr on the concentration profiles φ(Υ) and (Υ) is evident in Figure 4 for both single wall carbon nanotubes and multi wall carbon nanotubes. Due to a rise in Sr, it is noticed that the concentration profiles, φ(Υ) fall and (Υ) increases. The result indicates a decrease in viscosity due to an increase in the homogenous chemical process. On the other hand, Figure 4 shows that the φ(Υ) is the inverse of the (Υ). S*c = 0.1 S*c = 0.2 S*c = 0.3 S*c = 0.4 S*c = -0.1 S*c = -0.3 S*c = -0.5 S*c = -0.7 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 -1 0 1 2 3 4 Υ ϕ (Υ ) S*c = 1 S*c = 3 S*c = 5 S*c = 7 S*c = -1 S*c = -3 S*c = -5 S*c = -7 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Υ  (Υ ) Figure 5: Impact of S∗ c on single wall carbon nanotubes and multi wall carbon nanotubes with E1 = 0.1, E2 = 0.3, ϕ1 = 0.1, ϕ2 = 0.2, N = 1, M = 2, β = 1, P ∗ r = 2,Sr = 2, h = 0.3, δ = 0.1, Γ1 = 1.1, Γ2 = 1.5, φa∗ = 1. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 22 of 34 β = -0.1 β = -0.2 β = -0.3 β = -0.4 β = 0.1 β = 0.2 β = 0.3 β = 0.4 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 1.2 1.4 1.6 Υ ϕ (Υ ) β = 0.1 β = 0.3 β = 0.5 β = 0.7 β = 0.2 β = 0.4 β = 0.6 β = 0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Υ  (Υ ) Figure 6: Impact of β on single wall carbon nanotubes and multi wall carbon nanotubes with E1 = 0.1, E2 = 0.3, ϕ1 = 0.1, ϕ2 = 0.2, N = 1, M = 2, S∗c = 2, P ∗ r = 2,Sr = 1, h = 0.3, δ = 0.1, Γ1 = 1.1, Γ2 = 1.5, φa∗ = 1. Figure 5 demonstrates that for single wall carbon nanotubes and multi wall carbon nanotubes, the concentration boundary layer width decreases as the concentration pro- file φ(Υ) grows with increasing values of Schmidt’s number S∗c. It is shown that when the diffusion coefficient of species falls, the reactant concentration increases more quickly; larger values of S∗c correspond to a quicker increase in the flow field concentration. Con- versely, when the Schmidt number S∗c increases, the concentration profiles φ(Υ) and (Υ) drop.The effects of β = DA DB = 1, the ratio of the diffusion coefficients, are displayed for both the single wall carbon nanotubes and multi wall carbon nanotubes cases on φ(Υ) and (Υ) in Figure 6. Figure 6 illustrates how the concentration profiles φ(Υ) and (Υ) grow with higher values of β. This diagram shows that φ(Υ) and (Υ) are reversed. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 23 of 34 N = 1 N = 2 N = 3 N = 4 N = -1 N = -2 N = -3 N = -4 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Υ ϑ (Υ ) δ = -0.2 δ = -0.4 δ = -0.6 δ = -0.8 δ = 0.2 δ = 0.4 δ = 0.6 δ = 0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Υ ϑ (Υ ) P*r = 1 P*r = 2 P*r = 3 P*r = 4 P*r = -1 P*r = -2 P*r = -3 P*r = -4 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Υ ϑ (Υ ) M = 1 M = 3 M = 5 M = 7 M = -2 M = -4 M = -6 M = -8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 Υ ϑ (Υ ) Figure 7: Impact of N,M,P ∗ r , δ on single wall carbon nanotubes and multi wall carbon nanotubes with E1 = 0.1, E2 = 0.3, ϕ1 = 0.1, ϕ2 = 0.2, S∗c = 2, Sr = 1, h = 0.3, Γ1 = 1.1, Γ2 = 1.5, φa∗ = 1. Figure 7 shows how the temperature profile is affected by the Prandtl number P ∗ r , magnetic parameter, porosity parameter, and small parameter. The analysis reveals that in both single wall carbon nanotubes and multi wall carbon nanotubes cases, Φ(Υ) and ϑ(Υ) exhibit an inclining performance for high values of P ∗ r . Since P ∗ r is the ratio of momentum diffusivity to thermal diffusivity, greater values of P ∗ r essentially increase the thickness of the boundary layer, which increases the cooling impact of the nanoparticle. Due to the closer packing of the nanoparticles in single wall carbon nanotubes compared to multi wall carbon nanotubes. Figure 7 shows that with a regular increase in the parameter M , the temperature distribution rises in the single wall carbon nanotubes and falls in the multi wall carbon nanotubes. In general, the increased values of M N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 24 of 34 in the multi wall carbon nanotubes lead to an enhancement of the fluid’s temperature behaviour. Regular reductions in M will cause the fluid’s thermal boundary layer thickness to decrease. The escorting hybrid ferrofluid physically generates the Lorentz force, which is a form of resistive force. In addition, the flow produces a certain quantity of thermal energy. For the porosity parameter N and tiny parameter δ, comparable outcomes are seen. φa* = 15 φa* = 20 φa* = 25 φa* = 30 φa* = 4 φa* = 8 φa* = 12 φa* = 16 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 Υ Φ (Υ ) φa* = 0.1 φa* = 0.2 φa* = 0.3 φa* = 0.4 φa* = -0.2 φa* = -0.4 φa* = -0.6 φa* = -0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Υ ϑ (Υ ) φa* = 0.1 φa* = 0.3 φa* = 0.5 φa* = 0.7 φa* = 0.1 φa* = 0.3 φa* = 0.5 φa* = 0.7 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 1.0 Υ ϕ (Υ ) φa* = 0.1 φa* = 0.2 φa* = 0.3 φa* = 0.4 φa* = -0.2 φa* = -0.4 φa* = -0.6 φa* = -0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 Υ  (Υ ) Figure 8: Impact of φa∗ = 1 on single wall carbon nanotubes and multi wall carbon nanotubes with E1 = 0.1, E2 = 0.3, ϕ1 = 0.1, ϕ2 = 0.2, N = 1, M = 2, S∗c = 2, P ∗ r = 2,Sr = 1, h = 0.3, δ = 0.1, Γ1 = 1.1, Γ2 = 1.5, β = 1. For the phenomena of both the single wall carbon nanotubes and multi wall carbon nanotubes, Figure 8 illustrates the effect of the Hartmann number φa∗ on Φ(Υ), ϑ(Υ), φ(Υ), and (Υ) against the similarity variable Υ.In single wall carbon nanotubes and multi N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 25 of 34 wall carbon nanotubes flow, the fluid velocity increases as we move along the horizontal axis, and the velocity is enhanced in the single wall carbon nanotubes and multi wall carbon nanotubes flows for the continuous positive change in the values of φa∗, while for the simultaneous cases, the same behaviour is shown in Figure 8. Furthermore, the liquid velocity is examined to ensure that the single wall carbon nanotubes and multi wall carbon nanotubes achieve maximum performance. In addition, ϑ(Υ) exhibits the inverse behaviour. Figure 8 illustrates the influence of the Hartmann number on the thermal profile. Reduced thermal profile is the result of a rise in the Hartman number. Hartman number’s effect on homogeneous and heterogeneous profiles is seen in Fig. 8, which displays results that are opposite to one another. E1 = 0.2 E1 = 0.4 E1 = 0.6 E1 = 0.8 E1 = -0.2 E1 = -0.4 E1 = -0.6 E1 = -0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Υ ϑ (Υ ) E2 = 0.2 E2 = 0.4 E2 = 0.6 E2 = 0.8 E2 = -0.2 E2 = -0.4 E2 = -0.6 E2 = -0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Υ ϑ (Υ ) Figure 9: Impact of E1, E2 on single wall carbon nanotubes and multi wall carbon nanotubes with φa∗ = 1, ϕ1 = 0.1, ϕ2 = 0.2, N = 1, M = 2, S∗c = 2, P ∗ r = 2,Sr = 1, h = 0.3, δ = 0.1, Γ1 = 1.1, Γ2 = 1.5, β = 1. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 26 of 34 ϕ2 = 0.1 ϕ2 = 0.3 ϕ2 = 0.5 ϕ2 = 0.7 ϕ2 = 0.2 ϕ2 = 0.4 ϕ2 = 0.6 ϕ2 = 0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 Υ Φ (Υ ) ϕ2 = 0.2 ϕ2 = 0.3 ϕ2 = 0.4 ϕ2 = 0.5 ϕ2 = 0.2 ϕ2 = 0.4 ϕ2 = 0.6 ϕ2 = 0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Υ ϑ (Υ ) Figure 10: Impact of ϕ2 on single wall carbon nanotubes and multi wall carbon nanotubes with φa∗ = 1, E1 = 1.1, E2 = 1.5, ϕ1 = 0.1, Γ1 = 1.1, Γ2 = 1.5, N = 1, M = 2, S∗c = 2, P ∗ r = 2,Sr = 1, h = 0.3, δ = 0.1, β = 1. Figure 9 illustrates the concentration profile ϑ(Υ) and the temperature-increasing be- haviour in single wall carbon nanotubes and multi wall carbon nanotubes for higher local Eckert and Eckert numbers. Increasing the Eckert number, which measures frictional heat and viscous dissipation, leads to a rise in temperature. The thickness of the ther- mal boundary layer increases as the Eckert number increases, but the gradient of hybrid nanofluid temperature at the plate’s surface, which is calculated using the local Nusselt number, decreases. This phenomenon may be explained by an increase in viscous dissi- pation, which raises the flow’s thermal conductivity and strengthens the momentum and thermal boundary layers. Figures (10–11) illustrate the effects of the volume fraction ϕ1 and ϕ2 of nanoparticles on the profiles of Φ(Υ), ϑ(Υ), φ(Υ), and (Υ) for (single wall carbon nanotubes) and (multi wall carbon nanotubes). However, it is clear that when the volume fraction parameters ϕ1 and ϕ2 are increased, the (single wall carbon nanotubes) and (multi wall carbon nanotubes) velocities are enhanced in the same manner. As a physical phenomenon, the rise in velocity is caused by the inverse relationship between the nanofluid’s volume and dynamic viscosity. As a result, as the volume fraction increases, the normal fluid’s viscosity falls, boosting fluid flow. Physically speaking, the higher volume fraction produces more energy transferred through the fluid flow associated with the uneven development of the ultra-fine materials, which leads to a notable increase in the friction factor and the rate of heat transfer. In both flows, the heat transfer rate and the friction factor are increased by the squeezed parameter. For volume concentration parameters, the inverse pattern is seen in both the homogeneous and heterogeneous profiles. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 27 of 34 ϕ1 = 0.1 ϕ1 = 0.3 ϕ1 = 0.5 ϕ1 = 0.7 ϕ1 = 0.2 ϕ1 = 0.4 ϕ1 = 0.6 ϕ1 = 0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Υ Φ (Υ ) ϕ1 = 0.2 ϕ1 = 0.3 ϕ1 = 0.4 ϕ1 = 0.5 ϕ1 = 0.2 ϕ1 = 0.4 ϕ1 = 0.6 ϕ1 = 0.8 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Υ ϑ (Υ ) ϕ1 = 0.1 ϕ1 = 0.2 ϕ1 = 0.3 ϕ1 = 0.4 ϕ1 = -0.1 ϕ1 = -0.2 ϕ1 = -0.3 ϕ1 = -0.4 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Υ ϕ (Υ ) ϕ1 = 0.1 ϕ1 = 0.3 ϕ1 = 0.5 ϕ1 = 0.7 ϕ1 = -0.2 ϕ1 = -0.3 ϕ1 = -0.4 ϕ1 = -0.5 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Υ  (Υ ) Figure 11: Impact of ϕ1 on single wall carbon nanotubes and multi wall carbon nanotubes with φa∗ = 1, E1 = 1.1, E2 = 1.5, ϕ2 = 0.2, Γ1 = 1.1, Γ2 = 1.5, N = 1, M = 2, S∗c = 2, P ∗ r = 2,Sr = 1, h = 0.3, δ = 0.1, β = 1. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 28 of 34 Γ1 = 2 Γ1 = 4 Γ1 = 6 Γ1 = 8 Γ1 = -1 Γ1 = -3 Γ1 = -5 Γ1 = -7 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 Υ ϕ (Υ ) Γ2 = 0.1 Γ2 = 0.2 Γ2 = 0.3 Γ2 = 0.4 Γ2 = -0.1 Γ2 = -0.2 Γ2 = -0.3 Γ2 = -0.4 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.8 1.0 1.2 1.4 1.6 1.8 Υ ϕ (Υ ) Γ1 = 2 Γ1 = 4 Γ1 = 6 Γ1 = 8 Γ1 = -1 Γ1 = -3 Γ1 = -5 Γ1 = -7 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Υ  (Υ ) Γ2 = 2 Γ2 = 3 Γ2 = 5 Γ2 = 7 Γ2 = -2 Γ2 = -3 Γ2 = -4 Γ2 = -5 Solid Lines : MWCNT Dotted Lines : SWCNT 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 Υ  (Υ ) Figure 12: Impact of Γ1,Γ2 on single wall carbon nanotubes and multi wall carbon nanotubes with φa∗ = 1, E1 = 1.1, E2 = 1.5, ϕ1 = 0.1, ϕ2 = 0.2, N = 1, M = 2, S∗c = 2, P ∗ r = 2,Sr = 1, h = 0.3, δ = 0.1, β = 1. Figure 12 shows the effects of the equations describing the homogeneous strength parameter Γ1 and the heterogeneous strength parameter Γ2 on the concentration profiles φ(Υ) and (Υ). An increase in φ(Υ) and (Υ) concentration profiles. It is shown that when Γ1 increases, the homogeneous chemical reaction parameter also increases, and viscosity reduces as a result. At decreasing strengths of the homogeneous reaction parameter, the concentration profile rises while the thickness of the border layer falls. It is observed that the concentration distribution decreases as the heterogeneous reaction intensity Γ2 increases. Increased concentration fields are caused by highly dispersed particles, which are strongly correlated with higher values of Γ2. Nevertheless, Figure 12 illustrates how the impact on concentration profiles, φ(Υ), shows that a homogeneous strength parameter, N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 29 of 34 Γ1, and a heterogeneous strength parameter, Γ2, are opposite of one another. Figure 13: Impact on Φ(Υ), ϑ(Υ), φ(Υ) and (Υ) for multi wall carbon nanotubes with Sr = −0.2, P ∗ r = 0.1, δ = 0.1, δ = 0.1, E1 = 0.1, E2 = 0.2, S∗c = 0.1, Γ1 = 0.5 Γ2 = −1.3, N = 0.1, M=0.5, β = 0.7, ϕ1 = 0.1, ϕ2 = 0.2, φa∗ = 0.1. N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 30 of 34 Figure 13 illustrates the effect of the similarity variable Υ on Φ(Υ), ϑ(Υ), φ(Υ), and (Υ) for the carbon nanotubes phenomenon in the case of three dimensions 7. Conclusion This work examines heat transmission, two-dimensional fluid flow between two squeez- ing plates, Joule heating, viscous dissipation, and homogeneous and heterogeneous chem- ical reactions in the presence of carbon nanotubes nanoparticles. There hasn’t been a study that looks at the combined effects of carbon nanotube nanoparticles on unstable flow between two squeezing plates. Subsequently, we used all the data to look at how a uniform and heterogeneous chemical reaction affected the flow between two compressing plates with carbon nanotube nanoparticles attached. New insights into the interplay of homogeneous-heterogeneous chemical processes, viscous dissipation, and Joule heating on mass and heat transfer behaviour are provided by their combined inclusion.Squeeze flow, which has not been thoroughly investigated in previous research, will be systematically in- vestigated utilising hybrid nanofluids (carbon nanotubes + alumina nanoparticles). After transforming PDEs into ODEs, BVP4c and Homotopy analysis method solved the simpli- fied governing equation. Graphs have been used to show how different factors affect the concentration, temperature, and velocity profiles, as well as the skin friction coefficient and local Nusselt number change caused by the impacts of interest parameters. The analysis reveals several important trends in the flow and heat transfer characteristics of carbon nanotubes-based nanofluids under squeezing conditions. The parameters Sr, ϕ1, ϕ2, and φa∗ are found to enhance the fluid velocity, whereas Sc, β,Γ1, and Γ2 act to suppress it. Similarly, the fluid temperature rises with increasing P ∗ r , δ, φa ∗, ϕ1, and ϕ2, but decreases with N,M,E1, and E2. The incorporation of carbon nanotubess leads to a reduction in volume fraction and an increase in the squeezed Reynolds number, which subsequently enhances skin friction. Moreover, the local Nusselt number decreases with decreasing squeeze numbers and increasing Prandtl numbers, volume fractions, and Eckert numbers. A thicker carbon nanotubes nanofluid layer intensifies the cooling effect and simultane- ously increases both skin friction and the Nusselt number by enhancing the resistance force driving the flow motion, with single wall carbon nanotubess exhibiting a stronger in- fluence compared to multi wall carbon nanotubess. The Eckert number further influences the thermal field by raising the nanofluid temperature through viscous dissipation effects, more prominently in the case of single wall carbon nanotubess. Overall, both types of carbon nanotubes significantly affect skin friction and the Nusselt number, as confirmed by the numerical results. The semi-analytical Homotopy analysis method solutions and the numerical bvp4c results exhibit excellent agreement, indicating the high accuracy and reliability of the adopted methodologies. 8. Limitations and Future Work This section explicitly acknowledges the simplifying assumptions adopted in the present model, as well as the potential sources of error and avenues for future improvement. The N. Khan et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6395 31 of 34 added discussion emphasises the following points: Modeling Assumptions: The analysis assumes idealised homogeneous–heterogeneous reaction mechanisms, simplified nanoparticle–fluid interactions, and the use of a slip boundary condition model. These assumptions neglect possible agglomeration, Brown- ian motion, and non-Newtonian effects that may occur in real nanofluid systems. Thermal and Flow Conditions: The boundary conditions are treated as uniform and steady, whereas in practical applications variations in wall temperature, surface rough- ness, or unsteady squeezing may alter the system response. 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Math, 18 (4) (2025), 6395 34 of 34 φu, φl Upper and lower plates temperatures DΩ, D∗ B Diffusion coefficients of the chemical species β Ratio of the diffusion coefficients Γ1 Homogeneous reaction strength Γ2 Heterogeneous reaction strength Sr Squeeze Reynolds number P ∗ r Prandtl number φ∗a Hartman number Cf Skin-friction coefficient Nu Local Nusselt number Sh Sherwood number Ω1, Ω2 Chemical species E1 Local Eckert number E2 Eckert numbers N Porosity parameter δ Small parameter S∗ c Schmidt number M Magnetic field parameter ϕ1, ϕ2 The volume fraction of the individual particles hnf Hybrid nanofluid snf Single nanofluid