# 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

Authors: Ahmed Mohammed Alshehri; Hasan Huseyin Coban; Shafiq Ahmad; Umair Khan; Wajdi Mohamad Alghamdi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper provides a comprehensive analysis of the mixed convective flow that comprises SWCNT-MWCNT/water hybrid nanofluid containing micropolar fluid through a partially slipped vertical Riga surface. A Cattaneo–Christov heat flux model is used to examine the heat transport rate. The energy equation is gaining more significance with the effect of viscous dissipation and thermal stratification. The flow model is transformed by convenient transformation into nondimensionless form. The numerical results of nonlinear complex equations are collected using the bvp4c built-in function from MATLAB which is based on the finite difference method. The graphical results are obtained for both hybrid nanofluid and simple nanofluid. The temperature distribution for hybrid nanofluid is higher than that for simple nanofluid when the solid volume fraction increases. The axial friction factor increases with solid volume fraction, porosity parameter, and mixed convection parameter. The velocity graph varies inversely with nanofluid volume fraction and micropolar parameter.


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## Body

## 1. 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. The 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. The 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 [8–20].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 [21–24], 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. This 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. The Riga plate was considered by Magyari and Pantokratoras [38] to examine the influence of Lorentz force on a Blasius electrically conducting fluid flow.The 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. [39–41] studied the micropolar fluid under different effects. This 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. The transformed equations are explained numerically by applied the MATLAB well-known bvp4c technique. The effect of distinct characteristics on heat and velocity profile is shown graphically and displayed in the table.


## 2. 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. The physical justification of the considered model is explained in Figure1 where v¯ and u¯ are velocity components in the directions y and x, respectively.Figure 1
Physical description of the problem.Mathematically, viscosity in the variable form is determined by Shafiq and Nadeem [13]:(1)μfT¯=μ∞δT¯−δT¯∞+1.Applying the boundary layer approximation, the basic governing equations take place the form as [35, 42, 44, 45](2)∂v¯∂y+∂u¯∂x=0,(3)u¯∂u¯∂x+v¯∂u¯∂y=u¯∞du¯∞dx+1ρhnf∂∂yμhnfT¯+κ∂u¯∂y+ρβhnfρhnfgT¯−T¯∞+πJ0M08ρhnfe−π/dy−1ρhnfμhnfT¯K∗u¯−u¯∞+κρhnf∂N¯∂y,(4)v¯∂N¯1∂y+u¯∂N¯1∂x=γhnfjρhnf∂2N¯1∂y2−κjρhnf2N¯1+∂u¯∂y,(5)∂T¯∂xu¯+∂T¯∂yv¯+λ1u¯2∂2T¯∂x2+2u¯v¯∂2T¯∂x∂y+v¯2∂2T¯∂y2+u¯∂v¯∂x∂T¯∂y+v¯∂u¯∂y∂T¯∂x+v¯∂v¯∂y∂T¯∂y+u¯∂u¯∂x∂T¯∂x−μhnfT¯ρCphnf2u¯∂u¯∂y∂2u¯∂x∂y+2v¯∂u¯∂y∂2u¯∂y2=αhnf∂2T¯∂y2+μhnfT¯ρCphnf∂u¯∂y2.The subjected boundary conditions are(6)u¯|y=0=lμhnfT¯∂u¯∂y|y=0,v¯|y=0=0,T¯|y=0=T¯wx=T¯0+Bx,N¯1|y=0=−n∂u¯∂y|y=0,u¯|y⟶∞⟶u¯∞x,T¯|y⟶∞⟶T¯∞=T¯0+B1x,N¯1|y⟶∞⟶0.The 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. The limiting case condition n=0 is imposed for the strong concentration where the intense particles near the shrinking/stretching surface do not swivel. The strong concentration and the no-slip condition N=0 both are similarly matched at the given limiting case. The 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:(7)αnf=knfρCpnf,μnf=μf1−ϕ−2.5,ρCpnf=ρCpSWCNTϕ+ρCpf1−ϕ,ρnf=ϕρSWCNT+1−ϕρf,ρβnf=ρβSWCNTϕ+ρβf1−ϕ,knfkf=1−ϕ+2ϕkSWCNT/kSWCNT−kflnkSWCNT+kf/kf1−ϕ+2ϕkf/kSWCNT−kflnkSWCNT+kf/kf.Hybrid nanofluid:(8)αhnf=khnfρCphnf,μhnf=μf1−ϕ1−2.51−ϕ2−2.51+θθr,ρCphnf=ϕ2ρCpSWCNT+1−ϕ21−ϕ1ρCpf+ϕ1ρCpMWCNT,ρhnf=ϕ2ρSWCNT+1−ϕ21−ϕ1ρf+ϕ1ρMWCNT,ρβhnf=ρβSWCNTϕ2+1−ϕ21−ϕ1ρβf+ϕ1ρβMWCNT,khnfkbf=1−ϕ2+2ϕ2kSWCNT/kSWCNT−kbflnkSWCNT+kbf/kbf1−ϕ2+2ϕ2kbf/kSWCNT−kflnkSWCNT+kf/kf,where(9)kbfkf=1−ϕ1+2ϕ1kMWCNT/kMWCNT−kflnkMWCNT+kf/kf1−ϕ1+2ϕ1kf/kMWCNT−kflnkMWCNT+kf/kf.Here the solid volume fraction of MWCNT and SWCNT are, respectively, exemplified byϕ1, and ϕ2 and kf identify the water-based thermal conductivity of the fluid; specific heat is symbolized by Cp. The deviation amount for thermo-physical properties applied in this examination is categorized in Table 1.Table 1
Thermo-physical properties of water and CNTs (SWCNT and MWCNT) [42].
Physical characteristicsWaterMWCNTsSWCNTsCpJkg−1K−14179796.00425.00ρkgm−3997.116.002600.00KWm−1K−10.6133000.006600.00Pr6.2——


### 2.1. Similarity Transformation Conversion

The following similarity transformation is used [42]:(10)η=cνfy,θη=T¯−T¯∞T¯w−T¯0,N¯1=cxcνfgη,u¯=cxf′η,v¯=−cνffη,where νf represents the kinematic viscosity of the carrier-based liquid.Applying equation (10), equations (2)–(6) transmuted to the following dimensional form of ODEs as follows:(11)1A21+θrθ+Kf″′+Hae−Λη+A1ff″+1−f′2−θrθ′f″A21+θrθ2+Kg′−Pm1+θrθA2f′−1+λA4θ=0,(12)1+θrθ−1A2+K2g″−K2g+Kf″−A1f′g−fg′=0,(13)A6θ″+PrA5fθ′−f′θ−S1f′−γθf′2+S1f′2−ff″θ′−S1ff″−ff′θ′+f2θ″+EcPrf″21+θθrA2−2γEcPr1+θθrA2f″′ff″−f″2f′=0,while the transform appropriate conditions are(14)f′0=sf″0A21+θθr,g0=−nf″0,f0=0,andθ0=1−S1,f′η=1,θη=0andgη=0,atη⟶∞.The dimensionless parameter involved in equations (11)–(14) is identified as follows:(15)A1=1−ϕ21−ϕ1+ϕ1ρMWCNTρf+ϕ2ρSWCNTρf,Λ=πυfdc,γ=cλ1,Ha=πJ0M08c3/2Rex1/2ρfυf,A4=ρSWCNTρfβSWCNTβfϕ2+1−ϕ21−ϕ1+ϕ1ρMWCNTρfβMWCNTβf,θr=δT¯w−T¯∞,A2=1−ϕ12.51−ϕ22.5,A3=1−ϕ21−ϕ1+ϕ1ρCpMWCNTρCpf+ϕ2ρCpSWCNTρCpf,j=υfc,A6=khnfkf,Pm=υfcK∗,K=κμf,A5=1−ϕ21−ϕ1+ϕ1ρCpMWCNTρCpf+ϕ2ρCpSWCNTρCpf,λ=gβfbc2=GrxRex2,Pr=νfαf,s=lμfcυf.


### 2.2. Engineering Physical Quantities of Interest

The local skin friction is stated as(16)Cf=τwρfu¯∞2,where τw=κ+μhnfT¯∂u¯/∂y+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(17)Rex1/2Cfx=1A1K1−n+11+θrθ0A2f″0,whereas the local Reynolds number is indicated as Rex=xu¯∞/νf.


## 2.1. Similarity Transformation Conversion

The following similarity transformation is used [42]:(10)η=cνfy,θη=T¯−T¯∞T¯w−T¯0,N¯1=cxcνfgη,u¯=cxf′η,v¯=−cνffη,where νf represents the kinematic viscosity of the carrier-based liquid.Applying equation (10), equations (2)–(6) transmuted to the following dimensional form of ODEs as follows:(11)1A21+θrθ+Kf″′+Hae−Λη+A1ff″+1−f′2−θrθ′f″A21+θrθ2+Kg′−Pm1+θrθA2f′−1+λA4θ=0,(12)1+θrθ−1A2+K2g″−K2g+Kf″−A1f′g−fg′=0,(13)A6θ″+PrA5fθ′−f′θ−S1f′−γθf′2+S1f′2−ff″θ′−S1ff″−ff′θ′+f2θ″+EcPrf″21+θθrA2−2γEcPr1+θθrA2f″′ff″−f″2f′=0,while the transform appropriate conditions are(14)f′0=sf″0A21+θθr,g0=−nf″0,f0=0,andθ0=1−S1,f′η=1,θη=0andgη=0,atη⟶∞.The dimensionless parameter involved in equations (11)–(14) is identified as follows:(15)A1=1−ϕ21−ϕ1+ϕ1ρMWCNTρf+ϕ2ρSWCNTρf,Λ=πυfdc,γ=cλ1,Ha=πJ0M08c3/2Rex1/2ρfυf,A4=ρSWCNTρfβSWCNTβfϕ2+1−ϕ21−ϕ1+ϕ1ρMWCNTρfβMWCNTβf,θr=δT¯w−T¯∞,A2=1−ϕ12.51−ϕ22.5,A3=1−ϕ21−ϕ1+ϕ1ρCpMWCNTρCpf+ϕ2ρCpSWCNTρCpf,j=υfc,A6=khnfkf,Pm=υfcK∗,K=κμf,A5=1−ϕ21−ϕ1+ϕ1ρCpMWCNTρCpf+ϕ2ρCpSWCNTρCpf,λ=gβfbc2=GrxRex2,Pr=νfαf,s=lμfcυf.


## 2.2. Engineering Physical Quantities of Interest

The local skin friction is stated as(16)Cf=τwρfu¯∞2,where τw=κ+μhnfT¯∂u¯/∂y+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(17)Rex1/2Cfx=1A1K1−n+11+θrθ0A2f″0,whereas the local Reynolds number is indicated as Rex=xu¯∞/νf.


## 3. Methodology of the Numerical Solution

The 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). The systems of equations are highly nonlinear and difficult to solve exactly. Therefore, 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. This method is working only when the dimensionless ODEs are in the first-order 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:(18)f=y1,f′=y2,f″=y3,(19)yy1=11+θrθA2+K−1−A11+y1y3−y22+θry7y3A21+θrθ2−PmA21+θrθ1−y2−Hae−Λη−Ky5−λA4A2θ,(20)g=y4,g′=y5,(21)yy2=1A21+θrθ+K2−1A1y2y4−y1y5+K2y4+y3,(22)θ=y6,θ′=y7,(23)yy3=−1A6−γPrA5y12EcPry321+y6θrA2+PrA5y1y7−y2y6−y2S1−γS1y22+y22y6−S1y1y3−y1y3y7−y1y2y7+2γEcPr1+y4θrA2y2y32−y1y3yy1,with the transform initial conditions(24)y02−sA21+θθry03=0,y01=0,y04+ny03=0,y06−1+S1=0,yinf2−1=0,yinf4=0,yinf6=0.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∞=2.5to4, based on the values of the suggested variables.Figure 2
Flow chart of the considered numerical solution.


## 4. Results and Discussion

This 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. The impacts of these parameters are shown through various different graphs (see Figures3–18). 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, γ=0.3, Ha=0.1, θr=1.0, λ=−1.1, Pm=0.7, and K=0.25. The visual findings are addressed for both SWCNT-MWCNT/water hybrid nanofluid and SWCNT/water nanofluid. The current issue demonstrates strong alignment with the recently published article that is explained in Table 2.Figure 3
Influence ofs on f′η.Figure 4
Effect ofHa on f′η.Figure 5
Upshot ofPm on f′η.Figure 6
Effect ofθr on f′η.Figure 7
Result ofϕ2 on f′η.Figure 8
Influence ofK on f′η.Figure 9
Impression ofK on gη.Figure 10
Upshot ofϕ2 on gη.Figure 11
Result ofθr on gη.Figure 12
Impact ofϕ2 on θη.Figure 13
Results ofγ on θη.Figure 14
Influence ofλ on temperature θη.Figure 15
Result ofEc on temperature θη.Figure 16
Influence ofS1 on θη.Figure 17
Rex1/2Cfx versus ϕ2 for λ.Figure 18
Rex1/2Cfx versus ϕ2 for Pm.Table 2
Variation of the skin friction coefficient previously published papers whenλ=1 and Pm=K=θr=ϕi=0.
PrRex1/2CfxAman et al. [43]Zaib et al. [35]Present result0.71.70631.70631.70641.01.67541.67541.67537.01.51791.51791.5180101.49281.49281.4928201.44851.44851.4486401.41011.41011.4102501.39891.39891.3989601.39031.39031.3903801.37731.37761.3773


### 4.1. The Behavior of Different Characteristics on the Velocity Field

The impact on velocity distribution of thes (slip parameter), Ha (modified Hartman number), Pm (porosity parameter), θr (variable viscosity parameter), ϕ2 (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. The influence of Pm on f′η is sketched in Figure 5. With the increasing estimate, Pm, the thickness of the momentum boundary layer declines. Figure 6 highlights that f′η is affected by variation in θr. Physically, due to the addition in θr, 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 ϕ2 which declines the fluid velocity. The velocity of fluid diminishes with the increase of K, while its corresponding momentum boundary layer thickness improves (see in Figure 8).


### 4.2. Impact of Different Characteristics on Microrotation Profiles

The characteristic of the micropolar parameterK on the microrotation field is exhibited in Figure 9. It is visible from the observation of certain figure that gη 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 ϕ2 and variable viscosity θr. It is seen from these figures that the microrotation profile gη enhances both ϕ2 and θr.


### 4.3. Effect of Various Parameters on Fluid Temperature

Figures12–16 justify that how γ (thermal relaxation time), ϕ2 (solid volume fraction), Ec (Eckert number), S1 (thermal stratification parameter), and λ (mixed convection parameter) affect θη. θη is plotted in Figure 12 as ϕ2 varies. Due to an increase in the solid volume fraction, the conductivity of thermal enhances which improves the temperature θη. In Figure 13, θη reduces with improving values of γ. Physically, it is because particles express a nonconductive attitude as we escalate γ, 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 θη decreases with λ. θη improves with improving Ec (see in Figure 15). The mechanical fluid’s energy is converted to thermal energy for an increase of the Eckert number (Ec) from 0 to 1 due to internal molecular friction. As seen from Figure 16, θη reduces S1. Physically, change S1 decreases the difference in temperature between the surface and ambient fluid that contributes to the smaller temperature.


### 4.4. Effect of Several Parameters on Skin Friction

Figures17 and 18 illustrate the skin friction Rex1/2Cfx against ϕ2 (solid volume fraction) with the impact of λ and Pm. Figure 17 scrutinized the increasing behavior when λ and ϕ2 boost. Figure 18 demonstrated the impact of solid volume fraction and porosity parameter on Rex1/2Cfx. From the figure, it is explained that, for both values of Pm and ϕ2, Rex1/2Cfx boosts.


## 4.1. The Behavior of Different Characteristics on the Velocity Field

The impact on velocity distribution of thes (slip parameter), Ha (modified Hartman number), Pm (porosity parameter), θr (variable viscosity parameter), ϕ2 (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. The influence of Pm on f′η is sketched in Figure 5. With the increasing estimate, Pm, the thickness of the momentum boundary layer declines. Figure 6 highlights that f′η is affected by variation in θr. Physically, due to the addition in θr, 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 ϕ2 which declines the fluid velocity. The velocity of fluid diminishes with the increase of K, while its corresponding momentum boundary layer thickness improves (see in Figure 8).


## 4.2. Impact of Different Characteristics on Microrotation Profiles

The characteristic of the micropolar parameterK on the microrotation field is exhibited in Figure 9. It is visible from the observation of certain figure that gη 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 ϕ2 and variable viscosity θr. It is seen from these figures that the microrotation profile gη enhances both ϕ2 and θr.


## 4.3. Effect of Various Parameters on Fluid Temperature

Figures12–16 justify that how γ (thermal relaxation time), ϕ2 (solid volume fraction), Ec (Eckert number), S1 (thermal stratification parameter), and λ (mixed convection parameter) affect θη. θη is plotted in Figure 12 as ϕ2 varies. Due to an increase in the solid volume fraction, the conductivity of thermal enhances which improves the temperature θη. In Figure 13, θη reduces with improving values of γ. Physically, it is because particles express a nonconductive attitude as we escalate γ, 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 θη decreases with λ. θη improves with improving Ec (see in Figure 15). The mechanical fluid’s energy is converted to thermal energy for an increase of the Eckert number (Ec) from 0 to 1 due to internal molecular friction. As seen from Figure 16, θη reduces S1. Physically, change S1 decreases the difference in temperature between the surface and ambient fluid that contributes to the smaller temperature.


## 4.4. Effect of Several Parameters on Skin Friction

Figures17 and 18 illustrate the skin friction Rex1/2Cfx against ϕ2 (solid volume fraction) with the impact of λ and Pm. Figure 17 scrutinized the increasing behavior when λ and ϕ2 boost. Figure 18 demonstrated the impact of solid volume fraction and porosity parameter on Rex1/2Cfx. From the figure, it is explained that, for both values of Pm and ϕ2, Rex1/2Cfx boosts.


## 5. Concluding Remarks

This article concisely reports on the mixed convective micropolar fluid flow that comprises SWCNT-MWCNT/water hybrid nanofluid towards a partially slipped vertical Riga sheet. The present model is original and new. The 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. The 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)
Velocityf′η varies inversely with the micropolar parameter K and the nanofluid volume fraction, ϕ2(ii)
Improvingθr and ϕ2 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)
θη is an increasing function of ϕ2 and Ec, while decreasing function of S1,γ, and λ(v)
Velocityf′η enhances with s,Ha,θr, and Pm



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