# Thermal Radiation and Viscous Dissipation Impacts of Water and Kerosene-Based Carbon Nanotubes over a Heated Riga Sheet

Authors: R. Prabakaran; S. Eswaramoorthi; K. Loganathan; Sonam Gyeltshen
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

This research communication explains the flow of water/kerosene-based CNTs (carbon nanotubes) over a Riga sheet. The energy analysis is debated with the availability of radiation, viscous dissipation, and convective heating conditions. Two divisions of CNTs, like, SWCNTs (single-wall carbon nanotubes) and MWCNTs (multiwall carbon nanotubes), are considered. The pertinent variables are applied to overhaul the governing mathematical models into ODE expressions. The overhaul expressions are analytically solved by HAM (homotopy analysis method) and numerically solved by MATLAB bvp4c scheme. The repercussions of relevant parameters on nanoliquid velocity, nanoliquid temperature, skin friction coefficient, and local Nusselt number are inspected via graphs, tables, and charts. Our computational results are compared with the previous researcher results and have the most acceptable agreement. Our outcomes are used to understand the flow attributes, behavior, and how to predict it for those working in the design of thermal equipment in thermal industry. It is noticed that the nanoliquid velocity decayed in counter to the unsteady parameter, and it enhances for larger values of Hartmann number. The nanoliquid temperature enriches when raising the Biot number and radiation parameter. The change in unsteady parameter decays the surface shear stress, while reverse results are found in the local Nusselt number.


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## Body

## 1. Introduction

In recent decades, enhancing the rate of liquid thermal transport in classical base liquids (engine oil, water, polymer-based solutions) has been a crucial challenge for modern researchers and technologists. One approach to enrich the liquid thermal conductivity is to mix nanometer size metallic particles (carbides, fullerene, oxides, nitrides, metals, and carbon nanotubes) with the classical base liquids. Carbon nanotubes are cylindrical format tubes with precious attributes such as robust thermal conductivity and enormous energy. These CNTs are segregated into SWCNTs (single-wall carbon nanotubes) and MWCNTs (multiwall carbon nanotubes) and are highly used in many industrial processes, like, drug delivery, nanotube transistors, microwave amplifiers, pancreatic cancer test, and optics. Haq et al. [1] noted that the higher surface shear stress attains in engine oil-based CNTs compared to ethylene glycol-based CNTs in their MHD flow of CNTs over a stretching surface. The consequences of gasoline oil-based hybrid CNTs past a curved stretched surface by Muhammad et al. [2], and noticed that the nanoliquid velocity enhances for enriching both CNTs volume fractions. Gholinia et al. [3] proved that the nanoliquid temperature for SWCNTs is larger than the MWCNTs for the problem of the 3D flow of hybrid nanoliquids on a cylinder. They utilize ethylene glycol, water, and engine oil which are the base fluids. The 3D flow of carbon nanotubes past a variable ticked stretching surface was employed by Imtiaz et al. [4]. They concluded that the nanoliquid flow improves when increasing the nanoparticle shape factor. Anuar et al. [5] proved that the 5% quantity of volume fraction on water-based CNTs increases the heat transfer gradient up to 26% for the case of MHD flow of water-based CNTs past a nonuniform stretching surface. The time-dependent flow of Maxwell nanoliquid suspended with CNTs with four different base fluids was numerically investigated through Laplace transform technique by Aman et al. [6]. Anuar et al. [7] concluded that the strengthening of nanoparticle volume fraction is noted to enrich the heat transfer attributes on the surface in a flow of CNTs in an exponentially shrinking/stretching sheet. Heat transfer evaluation for Blasius flow of water/ethylene glycol-based CNTs over a plane plate was disclosed by Ferdows et al. [8]. They detected that the MWCNTs have less heat transfer than SWCNTs.Flow through porous space plays a notable role in many areas of biomedical engineering, pharmaceuticals, environmental engineering, crude oil production, spacecraft, etc. Many researchers are used extensively to model the flow through Darcy’s law. But this law is applicable only for low velocity and low porosity cases. Also, this law is ineffectual when there is a larger flow speed and inconsistent porosity. Forchheimer [9] overcome this restriction to add a square velocity factor in the Darcian model. Later, this new model was called the Darcy-Forchheimer flow model (DFFM). Shafiq et al. [10] noticed that the Forchheimer number upshots the nanoliquid flow in 3D Darcy-Forchheimer flow of nanoliquid in a rotating frame. The Darcy-Forchheimer flow of 3D nanoliquid with bioconvection and the activation energy was demonstrated by Tayyab et al. [11]. They acknowledged that the larger quantity of porosity parameter deteriorates the nanoliquid velocity and it improves the nanoliquid temperature. Alzahrani and Khan [12] proved that the Forchheimer number leads to the decay of the speed of working fluid in a mixed convective flow of nanoliquid with the Darcy-Forchheimer law. The Darcy-Forchheimer flow of hybrid nanoliquid in a rotating disk with first order velocity slip condition was analyzed by Haider et al. [13]. They detected that the porosity parameter leads to improving the skin friction coefficient. Alzahrani [14] is used as the optimal homotopy analysis method for solving the problem of 3D Darcy-Forchheimer flow of water-based CNTs subject to homogeneous- heterogeneous chemical reactions. The forced convective and stagnation-point flow of viscous liquid in a Forchheimer porous medium past a shrinking sheet was illustrated by Bakar et al. [15].Nowadays, many researchers have been interested in analyzing the viscous dissipation impact because of its numerous industrial usages, like, as the design of electric ovens, paper creation, and crystal production. Pal et al. [16] proved that the heat transfer gradient declined for the presence of Eckert number to study the MHD flow of nanoliquid in a heated sheet with viscous dissipation. The MHD and Ohmic dissipative flow of Maxwell liquid on a stretching sheet with the available of thermal radiation was numerically addressed by Hsiao [17]. Their study proved that the mass transfer gradient grows when the Eckert number rises. Butt et al. [18] studied the impact of viscous dissipation and MHD flow of viscous liquid with entropy generation past a cylinder inserted in a porous medium. They detected that the Eckert number leads to improve the thickness of the thermal boundary layer. Alshehri and Shah [19] numerically analyzed the problem of the viscous dissipative flow of a hybrid nanoliquid with radiation. They detected that the nanoliquid thermal field improves when enhancing the Eckert number. Tang et al. [20] proved that the Eckert number leads to an upswing in the liquid temperate in their study of Sisko nanoliquid suspended with Au nanoparticles. The entropy optimization of Powell-Eyring nanoliquid with viscous dissipation impact was scrutinized by Rooman et al. [21], and they also see that the liquid warmness escalates when improving the Eckert number values. Further researches include, the viscous dissipative flow of hybrid nanoliquid by Muhammad et al. [22], nanoliquid and Casson liquid by Gopal et al. [23, 24], Powell-Eyring hybrid nanoliquid by Aziz et al. [25], blood-gold Carreau nanoliquid and dusty liquid by Koriko et al. [26], and dusty hybrid nanoliquid by Mallikarjuna et al. [27].The outcomes of heat source/sink acquire an essential role in many industrial processes, like, foodstuffs storage, packed bed reactors, rubber sheets production, disposal of radioactive waste material, and thermal cooling systems. Sharma and Gandhi [28] reported that the larger temperature-dependent heat source/sink parameter improves the nanoliquid temperature in their time-dependent MHD viscous liquid flow with nonuniform heat source/sink and Joule heating. The flow of dusty liquid past a stretching sheet under the influence of a nonuniform heat sink/source was scrutinized by Gireesha et al. [29]. They revealed that the temperature-dependent heat sink/source parameter leads to strengthening the wall temperature gradient. Elgazery [30] inspected the time-dependent 2D flow of nanoliquid past a stretching sheet with a nonuniform heat source/sink and inclined magnetic field. They obtained that the heat transfer gradient aggravates for a higher magnitude of the heat sink parameter, whereas it subsides when improving the heat source parameter. The impact of inconsistent heat source/sink of the 3D flow of Fe3O4/Al2O3 nanoparticles past a Riga plate was studied by Ragupathi et al. [31]. They have seen that the heat source parameter develops the thermal boundary layer thickness. Sandhyarani et al. [32] noted that the heat sink/source parameter declines the heat transfer gradient for the problem of stagnation and MHD flow of Williamson liquid past a sheet with an inconsistent heat sink/source. The heat transport analysis of the nonlinear radiative flow of Carreau and Casson liquid on a heated stretching surface with a nonuniform heat source/sink and magnetic dipole was investigated by Mahanthesh et al. [33].The innovation of this analysis is estimating the unsteady and Darcy-Forchheimer flow of water/kerosene-based CNTs over a Riga plate subjected to nonlinear heat sink/source and thermal radiation impacts. In many industries depends on the heating/cooling of the surfaces, the existence of convective heat interchange between surrounding liquid and plate surface cannot be spurned because this type of flow model is a crucial role in many industrial processes, like, cooling of the nuclear reactor, aerodynamic process, material drying, and design of thermal equipment. The following are the key objectives of the ongoing research:(1)
Extend the existing mathematical model to water/kerosene-based CNTs with viscous dissipation and nonlinear heat sink/source(2)
What are the impacts of physical parameters on nanoliquid flow distributions?(3)
How does the hybrid nanoliquid affect the heat transfer rates?


## 2. Mathematical Formulation

This analysis directs the 2D time-dependent Darcy-Forchheimer flow of CNTs over a Riga sheet. Here, water and kerosene as a base liquid and the heat transference phenomenon are conducted through thermal radiation. Furthermore, the viscous dissipation and nonlinear heat sink/source are also adapted into the energy expression. Let thex− axis be kept along the plate and the y− axis perpendicular to the plate. In our analysis, two types of CNTs, SWCNTs (single-wall carbon nanotubes) and MWCNTs (multiwall carbon nanotubes), are considered. The plate surface is warmed through the hot liquid with liquid temperature Tf, creating a heat transference coefficient hc. Let Tw is the temperature of the plate while T∞ is the ambient temperature such that Tw>T∞. The physical configuration of the flow model and flow chart are presented in Figures 1 and 2. The governing nanoliquid model under the above presumption can be expressed as follows, see Pal et al. [16], Madhukesh et al. [34], and Chaudhary and Kanika [35]:
(1)Vx+Wy=0.(2)Vt+VVx+WVy=νnfVyy−νnfk1∗V−cbk1∗V2+πJ0M8ρnfExp−πa1V.(3)Tt+VTx+WTy=αnfTyy+16σ∗3k∗ρcpnfT∞3Tyy+μnfρcpnfVy2+1ρcpnfknfVwxνnfAATf−T∞f′+T−T∞BB.Figure 1
Physical configuration of the flow model.Figure 2
Flow chart.The corresponding boundary conditions are, see Chakraborty et al. [36] and Nandeppanavar and Shakunthala [37]:
(4)V=Vw=ax1−ξt;W=−W01−ξt;−knfTy=hcTf−Taty=0,V⟶0;T⟶T∞asy⟶∞.All the flow variables are provided in the nomenclature part.The thermophysical properties of CNTs are defined as:(5)μnfμf=11−ϕ2.5;ρnfρf=1−ϕ+ϕρCNTρf;ρcpnfρcpf=1−ϕ+ϕρcpCNTρcpf,αnf=knfρcpnf;knfkf=1−ϕ+2ϕkCNT/kCNT−kflnkCNT+kf/2kf1−ϕ+2ϕkf/kCNT−kflnkCNT+kf/2kf.Define, see Haq et al. [1],
(6)H=aνf1−ξty;V=a1−ξtxf′H;W=−aνf1−ξtfH;θ=T−T∞Tf−T∞.Substituting Equation (6) in Equations ((2)–(3)), we have:
(7)A1A2f′″H+fHf′H−f′H2−Af′H+H2f″H−A1A2λf′H−Frf′H2+A2HaExp−βRH=0.(8)A3A41Prθ″H+1PrA443Rdθ″H+fHθ′H−f′HθH+A1A4Ecf″H2−AθH+H2θ′H+A3A1A2A41PrAAf′H+BBθH.The reduced boundary conditions are:(9)f0=fw;f′0=1;f′∞=0;θ′0=−BiA31−θ0;θ∞=0.All the flow parameter details are provided in the nomenclature part.Here:(10)A1=11−ϕ2.5;A2=11−ϕ+ϕρCNT/ρf;A3=1−ϕ+2ϕkCNT/kCNT−kflnkCNT+kf/2kf1−ϕ+2ϕkf/kCNT−kflnkCNT+kf/2kf;A4=1−ϕ+ϕρcpCNTρcpf.The skin friction coefficients and local Nusselt number are expressed as follows:(11)CfRe=A1f″0;NuRe=−A3+43Rdθ′0.


## 3. Solution Methodology

### 3.1. Analytical Solution

The achieved ODE models, Equations ((7)–(8)) with the associated Condition (9) are solved analytically by exploiting HAM scheme. Liao created this scheme, see Liao [38]. This method is a potent mathematical technique for solving highly nonlinear problems, see Mabood et al. [39] and Dawar et al. [40].For this purpose, we fix the following initial approximation, linear operator and linear properties aref0=fw+1−e−H;θ0=Bie−H/A3+Bi; £f=f′″−f′ and £θ=θ″−θ, and £fß1+ß2eH+ß31/eH=0=£θß4eH+ß51/eH, respectively, where ßjj=1−5 are constants. The zeroth-order deformation problems:
(12)1−q£ffH,q−f0H=qhfM1fH,q1−q£θθH,q−θ0H=qhθM2fH,q,θH,qwhere q∈0,1 is an embedding parameter and M1 and M2 are nonlinear operators.After handling thenth order HAM, we acquire the following expressions:
(13)fnH=fn⊗H+ß1+ß2eH+ß31eH.θnH=θn⊗H+ß4eH+ß51eH.where fn⊗H and θn⊗H are the particular solutions.The HAM parameters (hf and hθ) are the authoritative for solution convergent, see Prabakaran et al. [41] and Eswaramoorthi et al. [42]. The range values of hf are [-1.2, -0.2] (W-SWCNTs), [-1.2,-0.2] (W-MWCNTs), [-1.2,-0.15] (K-SWCNTs), and [-1.2,-0.15] (K-MWCNTs) and the range values of hθ are [-2, -0.3] (W-SWCNTs), [-2,-0.3] (W-MWCNTs), [-2,-0.29] (K-SWCNTs), and [-2,-0.29] (K-MWCNTs), collected in Figures 3(a) and 3(b) and fix hf=−0.7 and hθ=−1.2 for better convergent.Figure 3
h-curves.
(a)(b)


### 3.2. Numerical Solution

The remodeled ODE Equations ((7)–(8)) with the corresponding boundary Condition (9) are solved numerically by applying MATLAB bvp4c procedure. In this regard, we convert the higher term into first-order equations, see Ali et al. [43]. Let f=D1,f′=D2,f′′=D3,θ=D4,θ′=D5.The system of equations are:(14)D1′=D2.D2′=D3.D3′=D22−D1D2+AD2+H/2D3+A1A2λD2+FrD22−A2HaExp−βRHA1A2.D4′=D5.D5′=D2D4−D1D5+AD4+H/2D5−A1/A4EcD32−A3/A1A2A41/PrAAD2+BBD41/A4PrA3+4/3Rd.With the corresponding conditions:(15)D10=fw;D20=1;D2∞=0;D50=−BiA31−D40;D4∞=0.To solve the above problem by applying MATLAB bvp4c procedure with error is10−5 and step size is 0.05.


## 3.1. Analytical Solution

The achieved ODE models, Equations ((7)–(8)) with the associated Condition (9) are solved analytically by exploiting HAM scheme. Liao created this scheme, see Liao [38]. This method is a potent mathematical technique for solving highly nonlinear problems, see Mabood et al. [39] and Dawar et al. [40].For this purpose, we fix the following initial approximation, linear operator and linear properties aref0=fw+1−e−H;θ0=Bie−H/A3+Bi; £f=f′″−f′ and £θ=θ″−θ, and £fß1+ß2eH+ß31/eH=0=£θß4eH+ß51/eH, respectively, where ßjj=1−5 are constants. The zeroth-order deformation problems:
(12)1−q£ffH,q−f0H=qhfM1fH,q1−q£θθH,q−θ0H=qhθM2fH,q,θH,qwhere q∈0,1 is an embedding parameter and M1 and M2 are nonlinear operators.After handling thenth order HAM, we acquire the following expressions:
(13)fnH=fn⊗H+ß1+ß2eH+ß31eH.θnH=θn⊗H+ß4eH+ß51eH.where fn⊗H and θn⊗H are the particular solutions.The HAM parameters (hf and hθ) are the authoritative for solution convergent, see Prabakaran et al. [41] and Eswaramoorthi et al. [42]. The range values of hf are [-1.2, -0.2] (W-SWCNTs), [-1.2,-0.2] (W-MWCNTs), [-1.2,-0.15] (K-SWCNTs), and [-1.2,-0.15] (K-MWCNTs) and the range values of hθ are [-2, -0.3] (W-SWCNTs), [-2,-0.3] (W-MWCNTs), [-2,-0.29] (K-SWCNTs), and [-2,-0.29] (K-MWCNTs), collected in Figures 3(a) and 3(b) and fix hf=−0.7 and hθ=−1.2 for better convergent.Figure 3
h-curves.
(a)(b)


## 3.2. Numerical Solution

The remodeled ODE Equations ((7)–(8)) with the corresponding boundary Condition (9) are solved numerically by applying MATLAB bvp4c procedure. In this regard, we convert the higher term into first-order equations, see Ali et al. [43]. Let f=D1,f′=D2,f′′=D3,θ=D4,θ′=D5.The system of equations are:(14)D1′=D2.D2′=D3.D3′=D22−D1D2+AD2+H/2D3+A1A2λD2+FrD22−A2HaExp−βRHA1A2.D4′=D5.D5′=D2D4−D1D5+AD4+H/2D5−A1/A4EcD32−A3/A1A2A41/PrAAD2+BBD41/A4PrA3+4/3Rd.With the corresponding conditions:(15)D10=fw;D20=1;D2∞=0;D50=−BiA31−D40;D4∞=0.To solve the above problem by applying MATLAB bvp4c procedure with error is10−5 and step size is 0.05.


## 4. Results and Discussion

The key role of this section is to examine the consequences of various pertinent parameters on nanoliquid velocity, nanoliquid temperature, skin-friction coefficient, and local Nusselt number for water/kerosene-based carbon nanotubes over a Riga plate. Table1 provides the thermo-physical properties of SWCNTs, MWCNTs, kerosene, and water. Table 2 gives the comparison of −f″0 to Ibrahim and Shankar [44] for distinct values of fw with A=λ=ϕ=Fr=Ha=0 and found that our numerical results exactly matched with Ibrahim and Shankar [44]. The HAM order of approximations is illustrated in Table 3, and it is found that 12th order is sufficient for all computations. The fluctuations of SFC of water and kerosene-based SWCNTs and MWCNTs for distinct values of A, Ha, Fr, λ, and fw are given in Tables 4 and 5. It is unveiled that the surface shear stress downturns when escalating the size of A, Fr, λ, and fw. However, it enhances for a larger quantity of Ha. In addition, the larger SFC attained in MWCNTs compared to the SWCNTs for both base liquids. Tables 6 and 7 were taken for the overview of the LNN of water and kerosene-based SWCNTs and MWCNTs for distinct values of R, Ec,AA,BB, and Bi. It is acknowledged that the higher amount of R and Bi leads to develop the heat transfer gradient and Ec, AA, and BB deduce the heat transfer gradient.Table 1
Thermo-physical properties.
Physical characteristicsSWCNTsMWCNTsKeroseneWaterk660030000.1450.613ρ26001600783997.1cp42579620904179Table 2
Comparison of−f′′0 for distinct values of fw with A=λ=ϕ=Fr=Ha=0 to Ibrahim and Shankar [44].
fwPresent studyRef. [44]01.0000001.00000.51.2807761.2808Table 3
Order of approximations.
OrderWaterKeroseneSWCNTsMWCNTsSWCNTsMWCNTs−f″0−θ′0−f″0−θ′0−f″0−θ′0−f″0−θ′011.197410.114161.153460.122441.22430.105091.136880.1167751.230270.112551.164760.123191.275620.101761.142430.11830101.230260.112421.164740.123001.275680.101651.142410.11809121.230260.112421.164740.123001.275680.101661.142410.11809151.230260.112421.164740.123001.275680.101661.142410.11809201.230260.112421.164740.123001.275680.101661.142410.11809251.230260.112421.164740.123001.275680.101661.142410.11809301.230260.112421.164740.123001.275680.101661.142410.11809351.230260.112421.164740.123001.275680.101661.142410.11809Table 4
The fluctuations of SFC of water-based SWCNTs and MWCNTs for distinct values ofA, Ha, Fr, λ, and fw.
AHaFrλfwSWCNTsMWCNTsHAMNumericalHAMNumerical00.50.40.20.4−1.48806−1.48806−1.40807−1.408060.3−1.60099−1.60099−1.51573−1.515740.7−1.74581−1.74581−1.65380−1.653801−1.84908−1.84908−1.75233−1.752330.300.40.20.4−1.84060−1.84060−1.75824−1.758240.3−1.69589−1.69589−1.61180−1.611800.7−1.50723−1.50723−1.42078−1.420781−1.36849−1.36849−1.28025−1.280250.30.500.20.4−1.46607−1.46607−1.38680−1.386800.5−1.63313−1.63313−1.54645−1.546451−1.78576−1.78576−1.69234−1.692341.5−1.92701−1.92695−1.82738−1.827320.30.50.400.4−1.48098−1.48098−1.39054−1.390540.2−1.60099−1.60099−1.51573−1.515740.4−1.71237−1.71237−1.63130−1.631300.6−1.81677−1.81677−1.73919−1.739190.30.50.40.2−0.6−1.12169−1.12169−1.07830−1.07830−0.3−1.24519−1.24519−1.19186−1.191860−1.38562−1.38562−1.32019−1.320190.3−1.54406−1.54406−1.46417−1.464170.6−1.72107−1.72107−1.62427−1.62427Table 5
The fluctuations of SFC of kerosene-based SWCNTs and MWCNTs for distinct values ofA, Ha, Fr, λ, and fw.
AHaFrλfwSWCNTsMWCNTsHAMNumericalHAMNumerical00.50.40.20.4−1.54358−1.54358−1.38085−1.380850.3−1.66010−1.66010−1.48668−1.486680.7−1.80950−1.80950−1.62241−1.622411−1.91599−1.91599−1.71930−1.719300.300.40.20.4−1.89783−1.89783−1.73023−1.730230.3−1.75423−1.75423−1.58317−1.583170.7−1.56710−1.56710−1.39130−1.391301−1.42953−1.42953−1.25012−1.250120.30.500.20.4−1.52112−1.52112−1.35982−1.359820.5−1.69320−1.69320−1.51690−1.516911−1.85040−1.85040−1.66046−1.660461.5−1.99587−1.99582−1.79335−1.793290.30.50.400.4−1.54340−1.54340−1.35960−1.359600.2−1.66010−1.66010−1.48668−1.486680.4−1.76877−1.76877−1.60376−1.603760.6−1.87089−1.87090−1.71288−1.712880.30.50.40.2−0.6−1.15102−1.15102−1.06321−1.06321−0.3−1.28152−1.28152−1.17343−1.173430−1.43053−1.43053−1.29771−1.297710.3−1.59932−1.59932−1.43689−1.436890.6−1.78848−1.78848−1.59140−1.59140Table 6
The fluctuations of LNN of water-based SWCNTs and MWCNTs for distinct values ofR, Ec, AA, BB, and Bi.
REcAABBBiSWCNTsMWCNTsHAMNumericalHAMNumerical00.50.10.10.50.325970.325970.335590.3355910.477970.477970.500450.5004520.627900.627900.662650.6626530.775390.775390.821860.821860.400.10.10.50.529810.529810.535930.535930.30.444110.444110.455460.455460.60.358400.358400.374990.374990.90.272700.272700.294520.294520.40.500.10.50.398780.398780.412170.412170.20.375160.375160.391450.391450.40.351530.351530.370740.370740.60.327900.327900.350020.350020.40.50.100.391220.391220.405180.405180.20.382440.382430.398260.398250.40.372320.372310.390510.390500.60.360330.359990.381720.381660.40.50.10.1−1−1.03704−1.03715−1.08198−1.08222−0.5−0.46577−0.46577−0.48501−0.48501000000.50.386970.386970.401810.4018110.713580.713570.740110.74011Table 7
The fluctuations of LNN of kerosene-based SWCNTs and MWCNTs for distinct values ofR, Ec, AA, BB, and Bi.
REcAABBBiSWCNTsMWCNTsHAMNumericalHAMNumerical00.50.10.10.50.326980.326980.342150.3421510.465530.465530.500740.5007420.602880.602880.657260.6572630.738630.738630.811390.811390.400.10.10.50.526770.526770.535710.535710.30.440210.440210.457760.457760.60.353650.353650.379810.379810.90.267090.267090.301850.301850.40.500.10.50.394570.394570.415490.415490.20.370440.370430.396090.396090.40.346300.346300.376700.376700.60.322160.322160.357300.357300.40.50.100.50.386850.386850.408770.408770.20.377850.377850.402670.402670.40.367430.367410.395930.395920.60.354940.354370.388380.388350.40.50.10.1−1−0.98734−0.98737−1.06140−1.06149−0.5−0.45008−0.45008−0.48134−0.48134000000.50.382500.382500.405790.4057910.711590.711590.752520.75253Figures4(a)–4(d) are drawn to show the upshot of fw, Ha, Fr, and A on nanoliquid velocity profile. It is evident from the outcomes that nanoliquid flow speed boosted up for larger Ha values while reverse measurements are found for positive changes of fw, Fr, and A. Physically, a bigger modified Hartmann number produces a greater electrical field, resulting in a larger wall-parallel Lorentz force. As a result, the fluid velocity increases. To scrutiny, the significant’s of AA, A, Bi, and Rd on nanoliquid temperature profile are drawn in Figures 5(a)–5(d), and it is found that nanoliquid temperature upturns when escalating the values of AA, Bi, and Rd, and it fallout when raising the quantity of A. Physically, the Biot number is related to the heat transfer coefficient. If the heat transfer coefficient is enhanced, the fluid temperature will rise. This is why enhancing the fluid temperature leads to a higher Biot number. Intensifying the thermal radiation parameter accelerates the transfer of energy to the fluid. That is why the fluid’s temperature is rising. Figures 6(a) and 6(b) are taken to inspect the affects of BB and ϕ on nanoliquid temperature profile. The results show that the nanoliquid becomes warmer when enriching the availability of BB and ϕ.Figure 4
The impact offw (a), Ha (b), Fr (c), and A (d) on f′H for water-SWCNTs (black), water-MWCNTs (blue), glycerin-SWCNTs (red), and glycerin-MWCNTs (purple).
(a)(b)(c)(d)Figure 5
The impact ofAA (a), A (b), Bi (c), and Rd (d) on θH for water-SWCNTs (black), water-MWCNTs (blue), glycerin-SWCNTs (red), and glycerin-MWCNTs (purple).
(a)(b)(c)(d)Figure 6
The impact ofBB (a) and ϕ (b) on θH for water-SWCNTs (black), water-MWCNTs (blue), glycerin-SWCNTs (red), and glycerin-MWCNTs (purple).
(a)(b)The modifications of SFC for different combinations ofHa, A, fw, λ, and Fr are draw in Figures 7(a)–7(d). It can be detected that plate surface shear stress improves when improving the modified Hartmann number while it suppresses for larger values of A, fw,λ, and Fr. Figures 8(a)–8(d) are presented to explain the impact of Ha, A, fw, λ, and Fr on local Nusselt number. It can be detected from these figures that the local Nusselt number escalates when heightening the quantity of Ha and A, and the opposite trend attains when rising the values of fw, λ, and Fr. The streamlines for both CNTs and both nanoliquids for the combinations of Rd and Bi and Rd and Ec are demonstrated in Figures 9(a)–9(d) and Figures 10(a)–10(d) for both cases.Figure 7
The modifications of SFC for different combinations ofHa, A, fw, λ, and Fr for water-SWCNTs (black), water-MWCNTs (blue), glycerin-SWCNTs (red), and glycerin-MWCNTs (purple).
(a)(b)(c)(d)Figure 8
The modifications of LNN for different combinations ofHa, A, fw, λ, and Fr for water-SWCNTs (black), water-MWCNT s(blue), glycerin-SWCNTs (red), and glycerin-MWCNTs (purple).
(a)(b)(c)(d)Figure 9
Stream line forRd and Bi on LNN for water-SWCNTs (a), water-MWCNTs (b), glycerin-SWCNTs (c), and glycerin-MWCNTs (d).
(a)(b)(c)(d)Figure 10
Stream line forRd and Ec on LNN for water-SWCNTs (a), water-MWCNTs (b), glycerin-SWCNTs (c), and glycerin-MWCNTs (d).
(a)(b)(c)(d)Figures11(a)–11(d) is drawn for to scrutinize the growing/lessening percent of SFC on A, Ha, λ, and fw for all cases. In the unsteady parameter case, the maximum lessening percentage 5.13% appeared in kerosene-based MWCNTs when changing A from 0 to 0.2, and the minimum lessening percentage 4.10% appeared in kerosene-based SWCNTs when changing A from 0.6 to 0.8. In the modified Hartmann case, the maximum growing percentage 6.59% appeared in kerosene-based MWCNTs when changing Ha from 0.6 to 0.8, and the minimum growing percentage 5.06% appeared in kerosene-based SWCNTs when changing Ha from 0 to 0.2. In the porosity parameter case, the maximum lessening percentage 9.35% appeared in kerosene-based MWCNTs when changing λ from 0 to 0.2, and the minimum lessening percentage 5.16% appeared in kerosene-based SWCNTs when changing λ from 0.6 to 0.8. In the suction/injection parameter case, the maximum lessening percentage 16.05% appeared in kerosene-based SWCNTs when changing fw from 0 to 0.4, and the minimum lessening percentage 13.86% appeared in kerosene-based MWCNTs when changing fw from −0.8 to −0.4.Figure 11
Growing/lessening percent of SFC onA (a), Ha (b), λ (c), and fw (d) for all cases.
(a)(b)(c)(d)The growing/lessening percent of LNN onA, Ha, λ, and fw for all cases are deliberated in Figures 12(a)–12(d). In the unsteady parameter case, the maximum growing percentage 1.64% appeared in water-based SWCNTs when changing A from 0 to 0.2, and the minimum growing percentage 0.83% appeared in kerosene based MWCNTs when changing A from 0.6 to 0.8. In the modified Hartmann number case, the maximum growing percentage 4.75% appeared in water-based SWCNTs when changing Ha from 0 to 0.2, and the minimum growing percentage 2.79% appeared in kerosene based MWCNTs when changing Ha from 0.6 to 0.8. In the porosity parameter case, the maximum lessening percentage 3.98% appeared in water-based SWCNTs when changing λ from 0.6 to 0.8, and the minimum lessening percentage 3.79% appeared in kerosene-based SWCNTs when changing λ from 0.2 to 0.4. In the suction/injection parameter case, the maximum growing percentage 2.23% appeared in water-based MWCNTs when changing fw from 0 to 0.4, and the minimum growing percentage 0.27% appeared in kerosene-based SWCNTs when changing fw from −0.8 to −0.4.Figure 12
Growing/lessening percent of LNN onA (a), Ha (b), λ (c), and fw (d) for all cases.
(a)(b)(c)(d)Figures13(a)–13(d) is provide for to discuss the growing/lessening percent of LNN on Rd, Ec, AA, and BB for all cases. In the radiation parameter case, the maximum growing percentage 24.65% appeared in water-based MWCNTs when changing Rd from 0 to 0.5, and the minimum growing percentage 12.82% appeared in kerosene-based SWCNTs when changing Rd from 1.5 to 2. In the Eckert number case, the maximum lessening percentage 16.32% appeared in kerosene-based SWCNTs when changing Ec from 0.6 to 0.8, and the minimum lessening percentage 9.70% appeared in kerosene-based MWCNTs when changing Ec from 0 to 0.2. In the AA parameter case, the maximum lessening percentage 7.49% appeared in kerosene-based SWCNTs when changing AA from 0.6 to 0.8, and the minimum lessening percentage 4.67% appeared in kerosene-based MWCNTs when changing Å from 0 to 0.2. In the BB parameter case, the maximum lessening percentage 4.49% appeared in kerosene-based SWCNTs when changing BB from 0.6 to 0.8, and the minimum lessening percentage 1.49% appeared in kerosene-based MWCNTs when changing BB from 0 to 0.2.Figure 13
Growing/lessening percent of LNN onRd (a), Ec (b), AA (c), and BB (d) for all cases.
(a)(b)(c)(d)


## 5. Conclusions

The present communication provides the time-dependent 2D flow of water/kerosene-based carbon nanotubes over a Riga plate with thermal radiation, viscous dissipation, and convective heating conditions. The suitable variables are implemented to restructure the governing mathematical models into an ODE model. These models are analytically solved by the HAM (homotopy analysis method) and numerically solved by the MATLAB bvp4c scheme. The main observations are as follows:(i)
The larger modified Hartmann number corresponds to more nanoliquid velocity and the opposite reaction attained for larger values of the unsteady parameter(ii)
The nanoliquid velocity decays for the higher suction/injection parameter and Forchheimer number(iii)
The nanoliquid temperature develops because of enhancement of radiation parameter, Biot number, space-dependent, and temperature-dependent heat source/sink parameters(iv)
The plate surface shear stress depresses because of the Forchheimer number, unsteady parameter, and porosity parameter(v)
The heat transfer gradient enriches due to the more presence of suction/injection and Forchheimer number(vi)
In the future, we extend this work to nonFourier heat flux theory and slip effect



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