Buoyancy Effect on a Micropolar Fluid Flow Past a Vertical Riga Surface Comprising Water-Based SWCNT–MWCNT Hybrid Nanofluid Subject to Partially Slipped and Thermal Stratification: Cattaneo–Christov Model

Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21521, Saudi Arabia Department of Electricals and Electronic Engineering, Ardahan University, Muhendislik Fakultesi, Ardahan 75000, Turkey Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematical Sciences, Universiti Kebangsaan Malaysia, UKM Bangi, Selangor 43600, Malaysia Department of Mathematics and Social Sciences, Sukkur IBA University, Sukkur 65200, Sindh, Pakistan Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah 21589, Saudi Arabia


Introduction
Because of heat transfer enhancement application, nanofluids are still very interesting to study. Many previous studies have shown that the performance of nanofluid heat transfer is higher than usual fluids. So, it is better to discuss the nanofluid instead of regular fluid. Nanoliquids have a significantly higher thermal conductivity than other liquids, which is one of their most important properties. In industries such as nuclear reactors, transportation, food, electronics, and biomedicine, nanofluids play a significant role. Nanoparticles are very small (1 nm-100 nm) particles that increase the conductivity of normal fluids when applied to them. e shape of nanoparticles is made of metal oxide, carbon tubes (SWCNT and MWCNT), carbide, silicon, and nitride, etc. Choi [1] initially presented nanofluids and used a large range of nanofluids in production processes in the industries. Nadeem et al. [2] evaluated the flow towards a moving wedge with three different nanoparticles and induced magnetic field. Dianchen et al. [3] deliberated the mass and heat transport in the occurrence of carbon nanotubes with the influence of the heat generation coefficient. e convection boundary layer that flows across a vertical cone with carbon nanotubes in the existence of magnetohydrodynamics is scrutinized by Rahmat et al. [4]. Shafiq et al. [5] addressed the significance of SWCNTs and MWCNTs through a static wedge under the magnetohydrodynamics (MHD) impact. In the involvement of a uniform magnetic field, Sheikholeslami and Bhatti [6] explored the nanofluid forced convective heat transfer in a porous semiannulus. Saleem et al. [7] numerically investigated the heat transfer enhancement utilizing nanoparticles past a flat surface. Recent studies of nanofluids are scrutinized through attempts [8][9][10][11][12][13][14][15][16][17][18][19][20].
Hybrid nanofluids are new nanofluid categories that produced a small particle of metal. Hybrid nanofluids are of remarkable use in modern areas including applied science, engineering, biology, and agriculture. Heat and cooling storage performance can be improved with hybrid nanofluids at a low cost. As a result, nanofluids are more advantageous because they can be used in hybrid fuel turbines, diesel engine oil, and chillers enhancement. MWCNT nanoparticles affect engine oil preparation, as they increase CNT to oil ratios and thermal conductivity. In [21][22][23][24], some recent research on hybrid nanoliquid has been noted. Mehryan et al. [25] investigated the effects of the combined convection Cu-Al2O3/water hybrid nanoliquid and Al2O3/ water nanoliquid within a square cavity induced by a warm cylinder oscillation. Sundar et al. [26] tested the MWCNT-Fe2O4/water hybrid nanofluid thermal conductivity within the temperature range of 30°C and 60°C. Baghbanzadeh et al. [27] found out the viscosity and thermal conductivity of the MWCNT/SiO2 hybrid nanofluids. Mackolil and Mahanthesh [28] scrutinized the radiated Casson and nanofluids' flow with numerical computations of exact and statistical under mass and heat flux boundary conditions. Hussain and Muhammad [29] discussed the convective carbon/water flow of wall and hall characteristics in peristaltic with the influence of Soret and Dufour.
Nadeem et al. [30] investigated the interaction of MWNCT and SWNCT across the oscillatory state with the heat transfer. Mahanthesh et al. [31] studied heat transport of hybrid MoS 2-Ag nanoliquid flowing over an isothermal wedge with the importance of viscous dissipation and Joule heating. Temperature and volume concentration on dynamics viscosity of hybrid nanofluids in the presence of ethylene glycol and MWCNT were studied by Afshari et al. [32]. Mackolil and Mahanthesh [33] discussed the analysis of TiO 2-EG nanoliquid in Marangoni convection including temperature-dependent surface tension and nanoparticle aggregation.
Impacts of EMHD (Electro-magnetohydrodynamic) by fluid flows perform an elementary role in the production of momentum and the importance found in thermal reactor, microcoolers, liquid chromatography, and managing the flow in a network of fluidic. According to early research, Gailitis [34] invented the Riga plate flow control method.
is system is an electromagnetic actuator, and it retains magnets permanently; a continuous separation of the boundary layer helps decline the pressure drag. Zaib et al. [35] considered the entropy generation effects on the combined convection micropolar nanofluid flow past a vertical Riga surface. Abbas et al. [36] investigated the entropy production on viscous nanoliquid towards a Riga sheet. Ahmad et al. [37] examined the combined convective flow including nanofluid out of a Riga plate and obtained the analytical solution by a perturbation method. e Riga plate was considered by Magyari and Pantokratoras [38] to examine the influence of Lorentz force on a Blasius electrically conducting fluid flow. e micropolar liquid is a unique form of non-Newtonian liquid and the most well-known. Liquids with microstructure are called micropolar liquids. Non-Newtonian liquids in our everyday lives include oils, fruit, juice, mud, emulsions, toothpaste, butter, and ointments. Anandha Kumar et al. [39][40][41] studied the micropolar fluid under different effects. is exploration inspired to examine the influence of mixed convection on the micropolar SWCNT-MWCNT\water hybrid nanofluid via a vertical Riga sheet with Cattaneo-Christov heat flux, variable viscosity, viscous dissipation, and thermal stratification. To the best of the reviewer's awareness, no work has been conducted so far on hybrid nanofluids with combined buoyancy effect across a Riga sheet in the existence of micropolar fluid, thermal stratification, and variable viscosity. e transformed equations are explained numerically by applied the MAT-LAB well-known bvp4c technique. e effect of distinct characteristics on heat and velocity profile is shown graphically and displayed in the table.

Problem Formation
Consider the two-dimensional micropolar hybrid nanofluid flow with variable viscosity embedded in the porous vertical Riga surface with viscous dissipation. In the existence of thermal stratification and the Cattaneo-Christov heat flux model, the heat equation is further performed. e physical justification of the considered model is explained in Figure 1 where v and u are velocity components in the directions y and x, respectively.
Mathematically, viscosity in the variable form is determined by Shafiq and Nadeem [13]:   Mathematical Problems in Engineering (1) Applying the boundary layer approximation, the basic governing equations take place the form as [35,42,44,45] zv zy e subjected boundary conditions are e mathematical symbols in the governing equations are, namely, defined in the nomenclature. From the micropolar model, n is steady with the given closed interval [0, 1]. e limiting case condition n � 0 is imposed for the strong concentration where the intense particles near the shrinking/stretching surface do not swivel. e strong concentration and the no-slip condition N � 0 both are similarly matched at the given limiting case. e microstructure particles have a very small influence near the shrinking surface of the sheet, and this behavior is noted for the condition n ≠ 0. Particularly in the condition, n � 0.5 indicates weak concentration. Also, the turbulent boundary layer flows are produced due to the pertinent condition, n � 1.
Further, the thermal conductivity, variable viscosity, specific heat, and density for SWCNT-MWCNT/water (hybrid nanofluid) and SWCNT/water (nanofluid) are defined as follows [42]: Simple nanofluid: Mathematical Problems in Engineering Hybrid nanofluid: where Here the solid volume fraction of MWCNT and SWCNT are, respectively, exemplified by ϕ 1 , and ϕ 2 and k f identify the water-based thermal conductivity of the fluid; specific heat is symbolized by C p . e deviation amount for thermophysical properties applied in this examination is categorized in Table 1.   4 Mathematical Problems in Engineering

Similarity Transformation Conversion.
e following similarity transformation is used [42]: where ] f represents the kinematic viscosity of the carrierbased liquid. Applying equation (10), equations (2)-(6) transmuted to the following dimensional form of ODEs as follows: while the transform appropriate conditions are 14) e dimensionless parameter involved in equations (11)- (14) is identified as follows: e local skin friction is stated as where τ w � ((κ + μ hnf (T))(zu/zy) + κN 1 )| y�0 is the wall shear stress for the micropolar hybrid nanofluid. Using (10) into equation (16), the following reduced dimensionless form of the skin friction takes place as whereas the local Reynolds number is indicated as

Methodology of the Numerical Solution
e system of equations (11)- (13) is obtained in the form of dimensionless ODEs along with appropriate boundary stipulations (14), after using self-similarity transformations (10). e systems of equations are highly nonlinear and difficult to solve exactly. erefore, the solution to the current investigation is achieved numerically by the MATLAB bvp4c technique. It is a built-in function from MATLAB and is based on the finite difference scheme which is generally known as Lobatto IIIA formula. is method is working only when the dimensionless ODEs are in the firstorder ODEs. For the working process of the considered method, we need to transform our equations from the higher third and second order into a first order by implementing the new variables. Let the new variables are as follows: with the transform initial conditions 6 Mathematical Problems in Engineering To solve the aforementioned equations, the system required initial early guesses at the mesh point to accomplish our conditions (24). For clearer and better understanding of the current method bvp4c, the detail flow chart has also been added (see Figure 2). A convergence criterion 10 −8 is provided for the solution which has been obtained. We have set appropriate finite values η ⟶ ∞, that is, η � η ∞ � 2.5 to 4, based on the values of the suggested variables.

Results and Discussion
is section of the work demonstrates the consequences of the involved sundry parameter in the considered problems on velocity profiles, microrotation profiles, temperature profiles, and shear stress. e impacts of these parameters are shown through various different graphs (see . For the computation purposes, we have fixed the value of the constraint throughout the simulation as the following Pr � 6.2, s � 0.7, n � 0.5, Λ � 0.5, c � 0.3, Ha � 0.1, θ r � 1.0, λ � −1.1, P m � 0.7, and K � 0.25. e visual findings are addressed for both SWCNT-MWCNT/water hybrid nanofluid and SWCNT/ water nanofluid. e current issue demonstrates strong alignment with the recently published article that is explained in Table 2.

e Behavior of Different Characteristics on the Velocity
Field. e impact on velocity distribution of the s (slip parameter), Ha (modified Hartman number), P m (porosity parameter), θ r (variable viscosity parameter), ϕ 2 (solid volume fraction), and K (micropolar parameter) is depicted in Figures 3-8. Figure 3 is considered to capture the influence of s on velocity f ′ (η). On velocity distribution, it displays rising behavior. On both nanofluid and hybrid nanofluid, the thickness of the velocity boundary layer is often observed to be decreasing. Figure 4 explains that as Ha increased, the velocity field increases. Physically, the Ha values result in the enhancement of internal and external forces including electric forces and adhesive. In these forces, the momentum flow increases, as an effective fluid velocity rises. e influence of P m on f ′ (η) is sketched in Figure 5. With the increasing estimate, P m , the thickness of the momentum boundary layer declines. Figure 6 highlights that  f ′ (η) is affected by variation in θ r . Physically, due to the addition in θ r , momentum transfer exists by the small fluid viscosity which boosts the velocity. Figure 7 analyzes that the velocity field decreases as we increase the solid volume fraction. It is because of the larger thermal conductivity of CNTs with higher ϕ 2 which declines the fluid velocity. e velocity of fluid diminishes with the increase of K, while its corresponding momentum boundary layer thickness improves (see in Figure 8).

Impact of Different Characteristics on Microrotation
Profiles. e characteristic of the micropolar parameter K on the microrotation field is exhibited in Figure 9. It is visible from the observation of certain figure that g(η) is reduced by growing K. It is further observed that near the surface, the microrotation profile enhances for both simple and hybrid nanofluids. Figures 10 and 11 are designed to see the variance of the microrotation field for various parameters of flow, including solid volume fraction ϕ 2 and variable viscosity θ r . It is seen from these figures that the microrotation profile g(η) enhances both ϕ 2 and θ r . Figures 12-16 justify that how c (thermal relaxation time), ϕ 2 (solid volume fraction), E c (Eckert number), S 1 (thermal stratification parameter), and λ (mixed convection parameter) affect θ(η). θ(η) is plotted in Figure 12 as ϕ 2 varies. Due to an increase in the solid volume fraction, the conductivity of thermal enhances which improves the temperature θ(η). In Figure 13, θ(η) reduces with improving values of c. Physically, it is because particles express a nonconductive attitude as we escalate c, i.e., it takes longer for particles to disperse heat to their neighboring particles, which is responsible for lowering the distribution of temperature. Figure 14 confirms that θ(η) decreases with λ. θ(η) improves with improving E c (see in Figure 15). e mechanical fluid's energy is converted to thermal energy for an increase of the Eckert number (E c ) from 0 to 1 due to internal molecular friction. As seen from Figure 16, θ(η) reduces S 1 . Physically, change S 1 decreases the difference in temperature between the surface and ambient fluid that contributes to the smaller temperature. Figures 17 and 18 illustrate the skin friction Re 1/2

Effect of Several Parameters on Skin Friction.
x C fx against ϕ 2 (solid volume fraction) with the impact of λ and P m . Figure 17 scrutinized the increasing behavior when λ and ϕ 2 boost. Figure 18 demonstrated the impact of solid volume fraction and porosity parameter on Re (1/2) x C fx . From the figure, it is explained that, for both values of P m and ϕ 2 , Re 1/2 x C fx boosts.

Concluding Remarks
is article concisely reports on the mixed convective micropolar fluid flow that comprises SWCNT-MWCNT/water hybrid nanofluid towards a partially slipped vertical Riga sheet. e present model is original and new. e problem gains more importance with the effect of thermal stratification, viscous dissipation, and Cattaneo-Christov heat flux. To resolve the transformed ordinary differential equation, the MATLAB bvp4c technique is used to solve the system numerically. e effect of various characteristics on velocity and heat transfer is inspected. Micropolar nanofluids may be used to increase the rate of cooling or heating in all electronic devices. Given below is an important description of such a problem: (i) Velocity f′(η) varies inversely with the micropolar parameter K and the nanofluid volume fraction, ϕ 2 (ii) Improving θ r and ϕ 2 strongly boosts the microrotation profile, but it is a contradictory behavior for K (iii) Axial friction factor enhances with mixed convection parameter, porosity parameter, and solid volume fraction (iv) θ(η) is an increasing function of ϕ 2 and E c , while decreasing function of S 1 , c, and λ (v) Velocity f ′ (η) enhances with s, Ha, θ r , and P m Nomenclature (x, y): Cartesian coordinates (u, v): Velocity components in x and y directions E c : Eckert number κ: Vortex viscosity d: Electrodes and magnets width K * : Porous medium permeability j: Microinertia density g: Gravitational acceleration J 0 : Density of applied current in the electrodes C p : Specific heat (J/kg K) M 0 : e permanent magnets' magnetization T: Temperature of the fluid T ∞ : Free-stream temperature (K) Ha: Modified Hartman number s: Slip parameter K: Micropolar parameter l: Slip length u ∞ (x): Free-stream velocity of the fluid N: Component of microrotation f(η): Dimensionless velocity component g(η): Dimensionless angular velocity component Pr: Prandtl number Gr x : Grashof number Re x : Local Reynolds number C f : Surface drag force.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.