Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 223 https://internationalpubls.com Flow of Hybrid Dust Micropolar Nanofluids Along the Symmetric Riga Surface with Heat Generation and Application of Cattaneo- Christov Theory 1 Dr. E. Rama, 2 M Annapoorna 1Associate Professor, Department of Mathematics, Osmania University, Hyderabad, India 2Degree Lecturer, Department of Mathematics, TSWRDCW, Telangana, India Article History: Received: 01-02-2024 Revised: 15-04-2024 Accepted: 30-04-2024 Abstract: This study investigates the mechanical characteristics of a hybrid nanofluid containing dust and micropolar particles, flowing under mixed convection past a symmetric Riga surface using the Cattaneo-Christov (C-C) heat flux theory with consideration for heat generation effects. The analysis focuses on dust micropolar flow within a porous medium with a combination of hybrid nanoparticles CNTs - F𝑒3𝑂4 with blood. The mathematical model describing this system involves a set of partial differential equations (PDEs), which are transformed into dimensionless ordinary differential equations (ODEs) and eventually into a form that can be solved using the MATLAB program. Graphical representations are employed to examine and discuss the impact of various flow parameters on the velocity and temperature profiles of the dust micropolar hybrid nanofluid. Several key findings emerge from this investigation. It is anticipated that an increase in the interaction between fluid particle parameters will result in a decrease in the temperature of the fluid phase and an increase in the temperature of the dust phase. Moreover, the study reveals that the Nusselt number is influenced by the mixed convection effects. Additionally, larger values of the heat generation parameter lead to higher temperatures in both the fluid phase and the dust phase. Conclusion: In summary, this study comprehensively investigates the dynamics of nanofluid flows over a Riga surface, considering parameters like 𝑅𝑑, Pr, and Ξ²v. The findings highlight their significant impact on temperature distribution, fluid flow patterns, and friction forces. The results offer insights crucial for optimizing engineering systems like cooling technologies and heat exchangers. This research advances our understanding of nanofluid dynamics and provides valuable guidance for future studies and practical applications. Keywords: Micropolar Dust phase, Porous medium, a symmetric Riga Surface, Cattaneo-Christov Theory, Heat generation, Thermal radiation. 1. Abstract This study investigates the mechanical characteristics of a hybrid nanofluid containing dust and micropolar particles, flowing under mixed convection past a symmetric Riga surface using the Cattaneo-Christov (C-C) heat flux theory with consideration for heat generation effects. The analysis focuses on dust micropolar flow within a porous medium with a Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 224 https://internationalpubls.com combination of hybrid nanoparticles CNTs - F𝑒3𝑂4 with blood. The mathematical model describing this system involves a set of partial differential equations (PDEs), which are transformed into dimensionless ordinary differential equations (ODEs) and eventually into a form that can be solved using the MATLAB program. Graphical representations are employed to examine and discuss the impact of various flow parameters on the velocity and temperature profiles of the dust micropolar hybrid nanofluid. Several key findings emerge from this investigation. It is anticipated that an increase in the interaction between fluid particle parameters will result in a decrease in the temperature of the fluid phase and an increase in the temperature of the dust phase. Moreover, the study reveals that the Nusselt number is influenced by the mixed convection effects. Additionally, larger values of the heat generation parameter lead to higher temperatures in both the fluid phase and the dust phase. 2. Introduction Recently, nanotechnology has been widely used in therapeutic methods such as the use of polymeric nanoparticles, silica nanoparticles, gold nanomaterials, magnetic nanoparticles, and nanotubes (CNTs). Graphite tubes with diameters commonly measured in nano-meters are known as (CNTs). There are two kinds of carbon nanotubes: first kind Multi-wall carbon nanotubes (MWCNTs) have more than one grapheme layer and are seamless cylinders of carbon allotropes with a diameter of 5.0–20 nm., which were discovered by researchers Radushkevich and Lukyanovich [1]. Iijima's seminal work on helical microtubules of graphitic carbon in 1991 expanded the understanding of carbon nanostructures, setting the stage for their diverse applications [2]. Since then, research has extensively explored the synthesis, properties, and applications of CNTs, including their use in nanofluids to improve thermal conductivity, as demonstrated by Murshd et al. [3] and Choi et al. [7]. Moreover, the development of hybrid nanofluids, combining different nanoparticles to achieve synergistic effects, has opened new avenues for enhancing heat transfer properties. Studies by Rahman et al. [14] and Khan et al. [23] have investigated the thermal characteristics of hybrid nanofluids, revealing their potential for various applications. Additionally, the incorporation of magnetic nanoparticles, such as Fe3O4, in nanofluids has enabled precise control over fluid flow and heat transfer, as demonstrated by Manaa et al. [17] and Mehryan et al. [18]. In parallel, advancements in fluid dynamics have led to the exploration of complex flow phenomena over surfaces with specific geometries, such as the Riga surface. This surface, introduced by Gailitis in 1961 [37], offers unique flow characteristics that influence heat transfer processes. Recent studies by Abbas et al. [41] and Islam et al. [43] have investigated the flow of micropolar fluids over Riga surfaces, shedding light on the underlying mechanisms governing these flows. The primary objective of this study is to investigate two-dimensional, incompressible dust micropolar flow within a porous medium, focusing on the flow over a symmetric Riga plate. Blood serves as the base liquid, with hybrid nanoparticles consisting of carbon nanotubes (CNTs) and Fe3O4 incorporated into the fluid. The analysis considers various factors, including heat generation, thermal radiation, and mixed convection effects. Notably, the Cattaneo-Christov (C-C) heat flux theory is employed to formulate the energy and mass Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 225 https://internationalpubls.com equations governing the flow. Through the use of tables, charts, and figures, the study explores the influence of key parameters on velocity, temperature, hybrid nanofluid concentration, local skin friction, and local Nusselt number, providing insights into the complex interactions within the system. 3. Objectives This paper aims to address several objectives. Firstly, it investigates the behavior of two- dimensional, incompressible dust micropolar flow adjacent to a symmetric Riga plate. Secondly, it explores the implications of using blood as the base liquid and incorporating hybrid nanoparticles, specifically carbon nanotubes (CNTs) and Fe3O4, into the fluid medium. Thirdly, the study delves into the effects of various factors such as heat generation, thermal radiation, and mixed convection on the flow dynamics. Additionally, the paper employs the Cattaneo-Christov (C-C) heat flux theory to formulate energy and mass equations governing the flow, offering insights into the thermal characteristics of the system. Lastly, through the utilization of tables, charts, and figures, the research aims to elucidate the influence of key parameters like velocity, temperature, hybrid nanofluid concentration, local skin friction, and local Nusselt number, facilitating a comprehensive understanding of the flow behavior and its underlying mechanisms. 4. Problem Formulation Consider the problem of two-phase, 2D steady laminar flow and heat transfer in a micropolar dusty fluid embedded in a porous vertical Riga surface. Figure 2 shows a theoretical and numerical examination of shaped hybrid nanofluid flow with attendance of linear thermal radiation and heat generation effects using the C-C heat flux model. Figure 1: Riga plate geometry. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 226 https://internationalpubls.com Figure 2: Flow configuration and coordinate system. We have considered velocity and temperature in The x-y coordinate system in Cartesian coordinates, are the denote velocity is π‘’βˆž= cx and temperature is 𝑇𝑀(x)=π‘‡βˆž+bx with 𝑇𝑀(x) > π‘‡βˆž , respectively, In this context, the coordinate system considers the direction of x along the Riga's surface, while y represents the direction normal to it., π‘‡βˆž,b and c represent the ambient temperature and positive constants, respectively. The vector representations of the governing equations for a motion micropolar flow and dust flow with spherical properties are as follows: 4.1 Micropolar fluid βˆ‡. V = 0, (1) πœŒβ„Žπ‘›π‘“(V. βˆ‡)V = (πœ‡β„Žπ‘›π‘“ + k)βˆ‡ 2V + k(βˆ‡ Γ— 𝑁) + π‘˜βˆ—π‘ (𝑒𝑝 βˆ’ u) βˆ’ πœ‡β„Žπ‘›π‘“ 𝐾𝑝 𝑒 + π‘”π›½β„Žπ‘›π‘“(𝑇 βˆ’ π‘‡βˆž) + 𝑓, (2) π‘—πœŒβ„Žπ‘›π‘“(V. βˆ‡)𝑁 = j (πœ‡β„Žπ‘›π‘“ + k 2 ) βˆ‡2N + π‘˜(βˆ‡ Γ— V) βˆ’ 2kN, (3) (πœŒπ‘π‘)β„Žπ‘›π‘“(V. βˆ‡)𝑇 + βˆ‡π‘ž π‘Ÿ βˆ’ [𝑄0(𝑇 βˆ’ π‘‡βˆž) + πœŒπ‘(𝑐𝑝)𝑓 πœπ‘‡ (𝑇𝑝 βˆ’ T) + πœŒπ‘ πœπ‘£ (𝑒𝑝 βˆ’ u) 2 ] = βˆ’βˆ‡. q̌. (4) Where q̌ is heat flux. The Cattaneo-Christov expression (C-C) for heat flux is given as q̌ + 𝛾𝑑(V. βˆ‡q̌ βˆ’ q̌. βˆ‡V) = βˆ’π‘˜β„Žπ‘›π‘“βˆ‡T where 𝛾𝑑 represents thermal relaxation time, V signifies velocity, 𝑓 the Lorentz force. 4.2 Dust fluid βˆ‡. V𝑝 = 0, (5) πœŒπ‘(V𝑝. βˆ‡)V𝑝 = βˆ’π‘˜ βˆ—π‘ (𝑒𝑝 βˆ’ u), (6) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 227 https://internationalpubls.com πœπ‘‡π‘π‘š(V𝑝. βˆ‡)𝑇𝑝 = βˆ’(𝑐𝑝)𝑓(𝑇𝑝 βˆ’ T). (7) The governing equations for both the fluid and dust phases are developed using the assumptions stated above [25-28]. 4.3 Fluid flow and heat transfer in micropolar systems πœ•π‘₯𝑒 + πœ•π‘¦π‘£ = 0 (8) π‘’πœ•π‘₯𝑒 + π‘£πœ•π‘¦π‘’ = ( πœ‡β„Žπ‘›π‘“+k πœŒβ„Žπ‘›π‘“ )πœ•π‘¦π‘¦u + π‘˜βˆ—π‘  πœŒβ„Žπ‘›π‘“ (𝑒𝑝 βˆ’ u) + k πœŒβ„Žπ‘›π‘“ πœ•π‘¦π‘ βˆ’ πœ‡β„Žπ‘›π‘“ πœŒβ„Žπ‘›π‘“πΎπ‘ 𝑒 + πœ‹π‘—0𝑀0 8πœŒβ„Žπ‘›π‘“ 𝑒(βˆ’ πœ‹ π‘Ž 𝑦) + π‘”π›½β„Žπ‘›π‘“(𝑇 βˆ’ π‘‡βˆž) (9) π‘’πœ•π‘₯𝑁 + π‘£πœ•π‘¦π‘ = ( πœ‡β„Žπ‘›π‘“+ k 2 πœŒβ„Žπ‘›π‘“ )πœ•π‘¦π‘¦N βˆ’ π‘˜ π‘—πœŒβ„Žπ‘›π‘“ (2N + πœ•π‘¦π‘’) (10) π‘’πœ•π‘₯𝑇 + π‘£πœ•π‘¦π‘‡ + 𝛾𝑑 (2π‘’π‘£πœ•π‘₯𝑦𝑇 + 𝑒 2πœ•π‘₯π‘₯𝑇 + 𝑣 2πœ•π‘¦π‘¦π‘‡ + (π‘’πœ•π‘₯𝑒 + π‘£πœ•π‘¦π‘’)πœ•π‘₯𝑇 + (π‘’πœ•π‘₯𝑣 + π‘£πœ•π‘¦π‘£)πœ•π‘¦π‘‡ βˆ’ 𝑄0 (πœŒπ‘π‘)β„Žπ‘›π‘“ (π‘’πœ•π‘₯𝑇 + π‘£πœ•π‘¦π‘‡)) = π‘˜β„Žπ‘›π‘“ (πœŒπ‘π‘)β„Žπ‘›π‘“ πœ•π‘¦π‘¦π‘‡ βˆ’ πœ•π‘¦π‘ž π‘Ÿ + πœŒπ‘(𝑐𝑝)𝑓 πœπ‘‡(πœŒπ‘π‘)β„Žπ‘›π‘“ (𝑇𝑝 βˆ’ 𝑇) + πœŒπ‘ πœπ‘£(πœŒπ‘π‘)β„Žπ‘›π‘“ (𝑒𝑝 βˆ’ 𝑒) 2 + 𝑄0 (πœŒπ‘π‘)β„Žπ‘›π‘“ (𝑇 βˆ’ π‘‡βˆž) (11) 4.4 Heat transfer and dust particle flow πœ•π‘₯𝑒𝑝 + πœ•π‘¦π‘£π‘ = 0 (12) π‘’π‘πœ•π‘₯𝑒𝑝 + π‘£π‘πœ•π‘¦π‘’π‘ = π‘˜βˆ—π‘  πœŒπ‘ (u βˆ’ 𝑒𝑝) (13) π‘’π‘πœ•π‘₯𝑇𝑝 + π‘£π‘πœ•π‘¦π‘‡π‘ = βˆ’ (𝑐𝑝)𝑓 πœπ‘‡π‘π‘š (𝑇𝑝 βˆ’ 𝑇) (14) The corresponding boundary conditions for given problem is 𝑒 = 𝑒w = cx , v = 0 , 𝑇 = 𝑇𝑀(π‘₯) = π‘‡βˆž + 𝑏π‘₯, 𝑁 = βˆ’π‘›πœ•π‘¦π‘’ π‘Žπ‘‘ 𝑦 = 0 (15) 𝑒 β†’ 0 , 𝑒𝑝 β†’ 0 , 𝑣𝑝 β†’ v, 𝑇 β†’ π‘‡βˆž , 𝑇𝑝 β†’ π‘‡βˆž , 𝑁 β†’ 0 π‘Žπ‘  𝑦 β†’ ∞ (16) Here (𝑒, 𝑣) and (𝑒𝑝, 𝑣𝑝) these denote the velocity components along the x and y axes, respectively , Οβ„Žπ‘›π‘“ , (ρ𝑐𝑝)β„Žπ‘›π‘“ ,πœ‡β„Žπ‘›π‘“ , π‘˜hnf and πœŽβ„Žπ‘›π‘“ represent the density, volumetric heat capacity, viscosity, and thermal conductivity of the hybrid nanofluid, respectively. 𝐾𝑝 permeability of porous medium , π‘˜βˆ— = 6πœ‹π‘πœ‡π‘“ is the Stokes drag constant, S is the density of the dust particles, the length is 𝑗 = πœˆβ„Žπ‘›π‘“ 𝑐 and 𝐾 = π‘˜ πœ‡π‘“ is the parameter of material T, the velocity of angular is N, πœπ‘£ and πœπ‘‡ are the particle phase relaxation and energy equilibrium times, and 𝑄0 is the heat generation. The term π‘žπ‘Ÿ is refers to the Rosseland radiative heat flux, which is defined as π‘žπ‘Ÿ = βˆ’ 4πœŽβ‹† 3π‘˜βˆ—β‹† πœ•π‘¦π‘‡ 4 = βˆ’ 16πœŽβ‹† 3π‘˜βˆ—β‹† π‘‡βˆž 3πœ•π‘¦π‘‡. Table 1 Utilized models for the thermophysical properties of the hybrid nanofluid [6,10]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 228 https://internationalpubls.com Properties Hybrid-nanofluid Nanofluid Dynamic viscosity πœ‡hnf = πœ‡bf(1 βˆ’ πœ‘2) βˆ’2.5 πœ‡bf = πœ‡f(1 βˆ’ πœ‘1) βˆ’2.5 Density 𝜌hnf = (1 βˆ’ πœ‘2)𝜌bf + πœ‘2𝜌2, 𝜌bf = (1 βˆ’ πœ‘1)𝜌f + πœ‘1𝜌1 Heat capacity (πœŒπ‘π‘)hnf = (1 βˆ’ πœ‘2)(πœŒπ‘π‘)bf + πœ‘2(πœŒπ‘π‘)2, (πœŒπ‘π‘)bf = (1 βˆ’ πœ‘1)(πœŒπ‘π‘)f + πœ‘1(πœŒπ‘π‘)1 Thermal conduc. π‘˜hnf π‘˜bf = (π‘˜2+(οΏ½Μ‚οΏ½βˆ’1)π‘˜bf)βˆ’(οΏ½Μ‚οΏ½βˆ’1)πœ‘2(π‘˜bfβˆ’π‘˜2) (π‘˜2+(οΏ½Μ‚οΏ½βˆ’1)π‘˜bf)+πœ‘2(π‘˜bfβˆ’π‘˜2) , π‘˜bf π‘˜f = (π‘˜1+(οΏ½Μ‚οΏ½βˆ’1)π‘˜f)βˆ’(οΏ½Μ‚οΏ½βˆ’1)πœ‘1(π‘˜fβˆ’π‘˜1) (π‘˜1+(οΏ½Μ‚οΏ½βˆ’1)π‘˜f)+πœ‘1(π‘˜fβˆ’π‘˜1) Electrical conduc. 𝜎hnf 𝜎bf = (1 + 3( 𝜎2 𝜎bf βˆ’1)πœ‘2 ( 𝜎2 𝜎bf +2)βˆ’( 𝜎2 𝜎bf βˆ’1)πœ‘2 ), 𝜎bf 𝜎f = (1 + 3( 𝜎1 𝜎f βˆ’1)πœ‘1 ( 𝜎1 𝜎f +2)βˆ’( 𝜎1 𝜎f βˆ’1)πœ‘1 ) As a base fluid, blood was employed. 𝐹𝑒3𝑂4 nanoparticles and carbon nanotubes (CNTs) were utilized in a ratio of 50/50, respectively. In two steps, samples were generated using carbon nanotubes and 𝐹𝑒3𝑂4 nanoparticles. The thermophysical characteristics of blood and both types of 𝐹𝑒3𝑂4and CNTs are shown in Table 2. Table 2. Thermal properties of the base fluid (blood), 𝐹𝑒3𝑂4, and CNTs [23-24]. The equations are converted into a dimensionless format by introducing the following non- dimensional parameters: πœ‚ = y ( 𝑐 πœˆπ‘“ ) 1 2⁄ , 𝑒(u𝑝)(πœ‚) = cx𝐹′(𝐹𝑝 β€²)(πœ‚), 𝑣 = βˆ’(π‘πœˆπ‘“) 1 2⁄ 𝐹(𝐹𝑝 β€²)(πœ‚), N = cx ( 𝑐 πœˆπ‘“ ) 1 2⁄ H(Ξ·), ΞΈ(θ𝑝)(πœ‚) = 𝑇(𝑇𝑝)βˆ’π‘‡βˆž π‘‡π‘€βˆ’π‘‡βˆž . } (17) Equations (8) and (12) are satisfied automatically and Eqs. (9) - (16) yield ( πœ‡β„Žπ‘›π‘“ πœ‡π‘“ + 𝐴)Fβ€²β€²β€² βˆ’ πœŒβ„Žπ‘›π‘“ πœŒπ‘“ (Fβ€² 2 βˆ’ FFβ€²β€² βˆ’ π›½β„Žπ‘›π‘“ 𝛽𝑓 Ξ»πœƒ) + L𝛽𝑣(𝐹𝑝 β€² βˆ’ Fβ€²) + AHβ€² βˆ’ πœ‡β„Žπ‘›π‘“ πœ‡π‘“ πœ–Fβ€² + πœ’π‘’(βˆ’π΅πœ‚) = 0 (18) (πœŒπ‘π‘)𝑓 (πœŒπ‘π‘)β„Žπ‘›π‘“ ( πΎβ„Žπ‘›π‘“ 𝐾𝑓 + 𝑅𝑑) πœƒ β€²β€² + Pr [Lβ𝑇(πœƒπ‘ βˆ’ πœƒ) + L𝛽𝑣𝐸𝑐(𝐹𝑝 β€² βˆ’ Fβ€²) 2 βˆ’ Ξ΄(Fβ€² 2 πœƒ βˆ’ πΉπΉβ€²β€²πœƒ βˆ’ πΉπΉβ€²πœƒβ€² + 𝐹2πœƒβ€²β€²) βˆ’ (πœŒπ‘π‘)𝑓 (πœŒπ‘π‘)β„Žπ‘›π‘“ Q(πΉβ€²πœƒ βˆ’ πΉπœƒβ€²)) βˆ’ πΉβ€²πœƒ + πΉπœƒβ€²] + (πœŒπ‘π‘)𝑓 (πœŒπ‘π‘)β„Žπ‘›π‘“ QΞΈ = 0 (19) πœŒπ‘“ πœŒβ„Žπ‘›π‘“ (( πœ‡β„Žπ‘›π‘“ πœ‡π‘“ + 𝐴 2 )𝐻′′ βˆ’ ABβˆ—( 2𝐻 +𝐹′′)) βˆ’ 𝐹′𝐻 + 𝐹𝐻′ = 0 (20) Feature 𝝆 (kg/m πŸ‘) 𝒄𝒑(J/kg.K) π’Œ (W/m.K) πœ·π’™πŸπŸŽπŸ“(πΎβˆ’1) Human blood 1053 3594 0.492 0.18 SWCNTs 2600 425 6600 27 MWCNTs 1600 765 3000 44 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 229 https://internationalpubls.com F𝑝𝐹𝑝 β€²β€² βˆ’ 𝐹𝑝 β€²2 + 𝛽𝑣(F β€² βˆ’ 𝐹𝑝 β€²) = 0 (21) 𝐹𝑝 β€²πœƒπ‘βˆ’Fπ‘πœƒπ‘ β€² + γ𝛽𝑇(πœƒπ‘ βˆ’ πœƒ) = 0 (22) with 𝐹(0) = 0, 𝐹′(0) = 1, πœƒ(0) = 1,𝐻(0) = βˆ’n𝐹′′(0) (23) 𝐹′(𝐹𝑝 β€²)(∞) β†’ 0, 𝐹𝑝 (∞) β†’ 𝐹(∞), πœƒ(πœƒπ‘)(∞) β†’ 0, 𝐻(∞) β†’ 0 (24) where, 25 10 25 10 1 (1 ) (1 ) hnf f CNTs FO D   οͺ οͺ = = βˆ’ βˆ’ . 1 (1 ) (1 ) CNTs FOhnf FO CNTs CNTs FO f f f D   οͺ οͺ οͺ οͺ      οƒΆ  οƒΆ = = βˆ’ βˆ’ + +  οƒ·   οƒ·  οƒΈ   οƒΈ . 2 ( ) ( )( ) (1 ) (1 ) ( ) ( ) ( ) CNTs FOhnf FO CNTs CNTs FO f f f D   οͺ οͺ οͺ οͺ      οƒΆ  οƒΆ = = βˆ’ βˆ’ + +  οƒ·   οƒ·  οƒΈ   οƒΈ . 3 ( ) ( ) ( ) (1 ) (1 ) ( ) ( ) ( ) p hnf p CNTs p FO FO CNTs CNTs FO p f p f p f c c c D c c c    οͺ οͺ οͺ οͺ      οƒΆ  οƒΆ = = βˆ’ βˆ’ + +  οƒ·   οƒ·   οƒ·  οƒΈ   οƒΈ . ( ) ( ) ( ) ( ) 4 ( 1) ( 1) ( ) ( 1) ( ) ( 1) ( 1) ( ) . ( 1) ( ) CNTs f CNTs f CNTshnf f CNTs f CNTs f CNTs FO f FO f FO FO f FO f FO K m K m K KK D K K m K K K K m K m K K K m K K K οͺ οͺ οͺ οͺ + βˆ’ βˆ’ βˆ’ βˆ’ = = ο‚΄ + βˆ’ + βˆ’ + βˆ’ βˆ’ βˆ’ βˆ’ + βˆ’ + βˆ’ The parameters and non-dimensional numbers are specified as Pr = ( πœ‡π‘π‘ π‘˜ )𝑓 , πœ† = πΊπ‘Ÿπ‘₯ 𝑅𝑒2 , πΊπ‘Ÿπ‘₯ = 𝑔𝛽𝑓(π‘‡π‘€βˆ’π‘‡βˆž)π‘₯ 3 πœˆπ‘“ 2 , 𝑅𝑒 = π‘₯𝑒𝑀 πœˆπ‘“ , πœ– = πœˆπ‘“ 𝑐𝐾𝑝 , 𝐸𝑐 = (𝑐π‘₯)2 (𝑐𝑝)𝑓 (π‘‡π‘€βˆ’π‘‡βˆž) , 𝑅𝑑 = 16πœŽβˆ—π‘‡βˆž 3 3π‘˜βˆ—βˆ—πΎπ‘“ , 𝑄 = 𝑄0 𝑐(πœŒπ‘π‘)𝑓 , Ξ΄ = c𝛾𝑇, 𝐴 = π‘˜ πœ‡π‘“ , πœ’ = πœ‹π‘—0𝑀0 8𝑐2π‘₯πœŒπ‘“ , 𝛽𝑣 = 1 cπœπ‘£ , 𝛽𝑇 = 1 cπœπ‘‡ ,πœπ‘£ = π‘š π‘˜βˆ— , π‘š = πœŒπ‘ 𝑠 , Ξ³ = (𝑐𝑝)𝑓 π‘π‘š , B = πœ‹ π‘Ž √ πœˆπ‘“ 𝑐 , 𝜌p = 𝐿𝜌f. The significant physical quantities in engineering include the skin friction coefficient denoted as 𝐢𝑓 and the local Nusselt number represented by 𝑁𝑒.These parameters are defined as: 𝐢𝑓 = πœπ‘€ πœŒπ‘“π‘’w 2 , 𝑁𝑒 = π‘₯π‘žπ‘€ π‘˜π‘“ (π‘‡π‘€βˆ’π‘‡βˆž) (25) where πœπ‘€ = πœ‡β„Žπ‘›π‘“(πœ•π‘¦π‘’)𝑦=0 , π‘žπ‘€ = βˆ’π‘˜β„Žπ‘›π‘“(πœ•π‘¦π‘‡)𝑦=0 + π‘ž π‘Ÿ. The dimensionless form of Eq. (16) takes the form. 𝑅𝑒 1 2⁄ 𝐢𝑓 = πœ‡β„Žπ‘›π‘“ πœ‡π‘“ Fβ€²β€²(0),𝑅𝑒 βˆ’1 2⁄ 𝑁𝑒 = βˆ’( π‘˜β„Žπ‘›π‘“ π‘˜π‘“ + 𝑅𝑑)πœƒ β€²(0). (26) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 230 https://internationalpubls.com 5. Methods Non-linear governing ODEs (18)-(22) and corresponding boundary conditions (23)- (24) are solved using MATLAB bvp4c. We converted the nonlinear governing ODE into a set of 1𝑓𝑑- order differential equations. The procedure involved considering the following factors: F = YΜ‚1, 𝐹 β€² = Y2Μ‚, F β€²β€² = YΜ‚3, πœƒ = YΜ‚4, πœƒ β€² = YΜ‚5, H = YΜ‚6, H β€² = YΜ‚7, 𝐹𝑝 = YΜ‚8, 𝐹𝑝 β€² = YΜ‚9, θ𝑝 = YΜ‚10. (27) The first-order differential equations are as follows: Fβ€²β€²β€² = [ 1 D+𝐴 ] (𝐷1(οΏ½Μ‚οΏ½2 2 βˆ’ YΜ‚1YΜ‚3 βˆ’ 𝐷2Ξ»YΜ‚4) βˆ’ L𝛽𝑣(YΜ‚9 βˆ’ YΜ‚2) βˆ’ AYΜ‚7 + Dπœ–YΜ‚2 βˆ’ πœ’π‘’ (βˆ’π΅πœ‚)), (28) πœƒβ€²β€² = [ 1 (1 𝐷3(𝐷4+𝑅𝑑)βˆ’Prδ⁄ οΏ½Μ‚οΏ½1 2) ] ((βˆ’Pr) [Lβ𝑇(YΜ‚10 βˆ’ YΜ‚4) + L𝛽𝑣𝐸𝑐(YΜ‚9 βˆ’ YΜ‚2) 2 βˆ’ Ξ΄(οΏ½Μ‚οΏ½2 2YΜ‚4 βˆ’ YΜ‚1YΜ‚3YΜ‚4 βˆ’ YΜ‚1YΜ‚2YΜ‚5) βˆ’ 1 𝐷3 Q(YΜ‚2YΜ‚4 βˆ’ YΜ‚1YΜ‚5)) βˆ’ YΜ‚2YΜ‚4 + YΜ‚1YΜ‚5] + 1 𝐷3 QYΜ‚4), (29) 𝐻′′ = [ 1 (1 𝐷1(D+𝐴 2⁄ )⁄ ) ] (( 1 𝐷1 )π΄π΅βˆ—(2YΜ‚6 + YΜ‚3) + YΜ‚2YΜ‚6 βˆ’ YΜ‚1YΜ‚7), (30) 𝐹𝑝 β€²β€² = 1 Y8 (οΏ½Μ‚οΏ½9 2 + 𝛽𝑣(YΜ‚2 βˆ’ YΜ‚9)), (31) πœƒπ‘ β€² = 1 Y8 (YΜ‚9YΜ‚10 + γ𝛽𝑇(YΜ‚10 βˆ’ YΜ‚4)). (32) The boundary conditions equations (23)-(24) are as follows: YΜ‚1 = 0, YΜ‚2 = 1, YΜ‚4 = 1, YΜ‚6 = 𝑛YΜ‚3 π‘Žπ‘‘πœ‚ = 0 (33) YΜ‚2 = 0, YΜ‚9 = 0, YΜ‚8 = YΜ‚1, YΜ‚4 = 0, , YΜ‚10 = 0, YΜ‚6 = 0 π‘Žπ‘  πœ‚ ⟢ ∞ (34) The Matlab bvp4c solver employs these first-order differential equations and boundary conditions to investigate the impact of variables on consecutive profiles. 6. Results and Discussion A mathematical exploration has been exhibited to examine the impacts of the dust micropolar hybrid flow of hybrid nanofluids over a Riga surface subjected to the influence of C-C with heat generation impact. The consequences of several considerable system parameters on the flow pattern have been thoroughly debated in the previous section. Some important observations are summarized below: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 231 https://internationalpubls.com Table 3: The specific numerical values of local Nusselt Number for various values of π‘ƒπ‘Ÿ when 𝐴 = πœ† = 𝐿 = 𝛽𝑣 = νœ€ = πœ’ = 𝐡 = 𝑅𝑑 = 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = 𝐡1 = 𝑛 = 𝛾 = 0. Pr Mahdy [47] Das et al.[48] Present Study 0.72 1.00 3.00 7.00 10.0 0.80868 1.00000 1.92368 3.07224 3.72067 0.80876 1.00000 1.92357 3.07314 3.72055 0.80863 1.00000 1.92368 3.07225 3.72067 (SWCNT-Fe3O4) (SWCNT-Fe3O4) --------- (MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-3: Impact of 𝑅𝑑 πœƒ(πœ‚) Figure-4: Impact of 𝑅𝑑on πœƒπ‘(πœ‚) when 𝐴 = πœ† = 𝐿 = 𝛽𝑣 = νœ€ = πœ’ = 𝐡 = πœ‚ = 𝑅𝑑 = 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = 𝐡1 = 𝑛 = 𝛾 = 0.1 . Figure 3-4: Depict the influence of 𝑅𝑑 on temperature profiles (ΞΈ) for SWCNT-Fe3O4 and MWCNT-Fe3O4 nanofluids. Increasing 𝑅𝑑 leads to notable changes in temperature distribution along the Riga surface (Ξ·), indicating alterations in heat transfer characteristics. These changes are attributed to enhanced thermal radiation effects, impacting heat transfer rates near the surface. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 232 https://internationalpubls.com (SWCNT-Fe3O4) (SWCNT-Fe3O4) --------- (MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-5: Impact of π‘ƒπ‘Ÿ πœƒ(πœ‚) Figure-6: Impact of π‘ƒπ‘Ÿ on πœƒπ‘(πœ‚) when 𝐴 = πœ† = 𝐿 = 𝛽𝑣 = νœ€ = πœ’ = 𝐡 = 𝑅𝑑 = 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = 𝐡1 = 𝑛 = 𝛾 = 0.1. These figures demonstrate how the Prandtl number (Pr) influences temperature profiles (πœƒπ‘) for different nanofluid compositions. They show how changing Pr affects the distribution of temperature within the nanofluid, providing insights into the thermal behavior of the system. (SWCNT-Fe3O4) (SWCNT-Fe3O4) --------- (MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-7: Impact of 𝛽𝑣 on 𝐹′(πœ‚) Figure-8: Impact of 𝛽𝑣 on 𝐹𝑝 β€²(πœ‚) when A = Ξ» = L = Ξ²v = Ξ΅ = Ο‡ = B = Rd = Ξ²T = Ec = Ξ΄ = Q = B1 = n = Ξ³ = 0.1 Figure 7-8: Here, the effect of the fluid-particle interaction parameter 𝛽 on velocity profiles (F and 𝐹𝑝 β€²) is depicted. These figures explore how variations in 𝛽 impact the flow velocity in both the fluid and particle phases, indicating changes in the flow characteristics. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 233 https://internationalpubls.com (SWCNT-Fe3O4) (SWCNT-Fe3O4) -------- (MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-9: Impact of 𝛽𝑇 on πœƒ(πœ‚) Figure-10: Impact of 𝛽𝑇 on πœƒπ‘(πœ‚) when A = Ξ» = L = Ξ²v = Ξ΅ = Ο‡ = B = Rd = Ξ²T = Ec = Ξ΄ = Q = B1 = n = Ξ³ = 0.1 Figure 9-10: These figures illustrate the influence of the fluid particle interaction parameter for temperature (𝛽𝑇) on temperature profiles (ΞΈ and πœƒπ‘) for different nanofluid compositions. They show how altering Ξ²T affects the temperature distribution within the nanofluid, providing insights into the heat transfer behavior of the system. (SWCNT-Fe3O4) SWCNT-Fe3O4) --------- (MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-11: Impact of 𝐸𝑐 on πœƒ(πœ‚) Figure-12: Impact of 𝐸𝑐 on πœƒπ‘(πœ‚) when A = Ξ» = L = Ξ²v = Ξ΅ = Ο‡ = B = Ξ· = Rd = Ξ²T = Ec = Ξ΄ = Q = B1 = n = Ξ³ = 0.1 Figure 11-12: The impact of Eckert number (Ec) on temperature profiles (ΞΈ and ΞΈp) for various nanofluid compositions is demonstrated here. These figures highlight how changing Ec influences the temperature distribution within the nanofluid, indicating alterations in the heat transfer characteristics. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 234 https://internationalpubls.com (SWCNT-Fe3O4) (SWCNT-Fe3O4) ---------( MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-13: Impact of 𝐴 on 𝐹′(πœ‚) Figure-14: Impact of 𝐴 on 𝐹𝑝 β€²(πœ‚) when A = Ξ» = L = Ξ²v = Ξ΅ = Ο‡ = B = Rd = Ξ²T = Ec = Ξ΄ = Q = B1 = n = Ξ³ = 0. Figure 13-14: Here, the effect of the micropolar parameter (A) on velocity profiles (Fβ€² and 𝐹𝑝 β€²) for different nanofluid compositions is illustrated. These figures explore how variations in A impact the flow velocity in both the fluid and particle phases, indicating changes in the flow behavior. (SWCNT-Fe3O4) (SWCNT-Fe3O4) --------- (MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-15: Impact of 𝐴 on 𝐻 (πœ‚), when Figure-16: Impact of 𝑄 on πœƒ(πœ‚) when π΅βˆ— = 𝛽𝑣 = 0.1, νœ€ = πœ’ = 𝐡 = πœ‚ = 𝑅𝑑 𝐴 = πœ† = 𝐿 = π΅βˆ— = 𝛽𝑣 =, νœ€ = πœ’ = 𝐡 = 𝐴 = πœ† = 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = π‘ƒπ‘Ÿ = 𝑛 = 𝛾 = 0. 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = π‘ƒπ‘Ÿ = 𝑛 = 𝛾 = 0.1. Figure 15-16: These figures show the impact of heat generation parameter (Q) on temperature profiles (ΞΈ and πœƒπ‘) for different conditions. They demonstrate how altering Q affects the temperature distribution within the system, providing insights into the heat generation effects. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 235 https://internationalpubls.com (SWCNT-Fe3O4) (SWCNT-Fe3O4) --------- (MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-17 Impact of 𝑄 on πœƒπ‘(πœ‚) Figure-18: Impact of 𝐴 and πœ† on 𝐢𝑓. when A = πœ† = L = Ξ²v = Ξ΅ = Ο‡ = Bβˆ— = Ξ· = Rd = L = Ξ²v = Ξ΅ = Ο‡ = Bβˆ— = Ξ· = Rd = 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = π‘ƒπ‘Ÿ = 𝑛 = 𝐡 = 𝛾 = 0.1. 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = π‘ƒπ‘Ÿ = 𝑛 = 𝐡 = 𝛾 = 0.1. Figure 17-18: The influence of parameters A and Ξ» on skin friction coefficient (𝐢𝑓) for different conditions is depicted in these figures. They explore how changes in A and Ξ» affect the frictional forces experienced by the fluid, providing insights into the flow behavior over the surface. (SWCNT-Fe3O4) (SWCNT-Fe3O4) --------- (MWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure-19: Impact of 𝐴 and πœ† on 𝑁𝑒, Figure-20: Impact of π‘ƒπ‘Ÿ and 𝑅𝑑 on 𝐢𝑓 , When 𝐿 = 𝛽𝑣 = νœ€ = πœ’ = π΅βˆ— = πœ‚ = 𝑅𝑑 = π΅βˆ— = when 𝐴 = 𝐿 = 𝛽𝑣 = νœ€ = πœ’ = 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = π‘ƒπ‘Ÿ = 𝑛 = 𝐡 = 𝛾 = 0.1. 𝛽𝑇 = 𝐸𝑐 = 𝛿 = 𝑄 = πœ† = 𝑛 = 𝐡 = 𝛾 = 0.1. Figure 19-20: Here, the effect of Prandtl number (Pr) and non-linear thermal radiation parameter (𝑅𝑑 ) on skin friction coefficient (𝐢𝑓) for different conditions is illustrated. These Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 236 https://internationalpubls.com figures explore how variations in Pr and 𝑅𝑑 impact the frictional forces experienced by the fluid, providing insights into the heat transfer characteristics. (SWCNT-Fe3O4) --------- (MWCNT-Fe3O4) Figure 21: Impact of π‘ƒπ‘Ÿ and 𝑅𝑑 on 𝑁𝑒 when 𝐴 = πœ† = 𝐿 = 𝛽𝑣 = νœ€ = πœ’ = 𝐡 = πœ‚ = π΅βˆ— = Ξ²T = Ec = Ξ΄ = Q = n = Ξ³ = 0.1 Figure 21: This figure demonstrates the impact of Prandtl number (Pr) and non-linear thermal radiation parameter (𝑅𝑑 ) on the local Nusselt number (Nu) for specified conditions. It explores how variations in Pr and 𝑅𝑑 influence the convective heat transfer rate at the surface, providing insights into the heat transfer behavior of the system. Overall, these figures offer comprehensive insights into the influence of various parameters on the thermal and flow characteristics of micropolar hybrid nanofluid flows over a Riga surface, aiding in the understanding and optimization of such systems in engineering applications. 7. Conclusion In summary, this study comprehensively investigates the dynamics of nanofluid flows over a Riga surface, considering parameters like 𝑅𝑑, Pr, and Ξ²v. 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