# Approximate Analytical Study of Time-Dependent MHD Casson Hybrid Nanofluid over a Stretching Sheet and Considering Thermal Radiation

Authors: Ali Rehman; Zabidin Salleh; Abd Allah A. Mousa; Ebenezer Bonyah; Waris Khan
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The heat transfer ratio plays an important role in the production sector, and hybrid nanofluid has more heat transfer as compared to the base fluid. Two sorts of hybrid nanofluids have been used for heat enhancement applications. The present research paper is aimed at investigating an approximate analytical study of time-dependent MHD Casson hybrid nanofluid on an extending surface along with thermal radiation. The novelty of present research is that the first time-dependent Casson MHD flow of hybrid is addressed analytically in the form of a series solution along with flexible properties on an extending surface. Transforming the nonlinear partial differential equation to nonlinear ordinary differential equation, we used the defined similarity transformation. The governing nonlinear equations are solved with help of the approximate analytical method presented by Liao. The impact of different parameters like Casson parameter, unsteady parameter, magnetic field parameter, porosity parameter, Prandtl number, Eckert number, radiation parameter, and Grashof number is presumed in the form figures for velocity and temperature profile. The current research article has a good comparison with the previously published work.


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## Body

## 1. Introduction

In engineering processes, magneto hydrodynamics (MHD) has important applications, for example, design of a cooling system, liquid system, cooling of the nuclear reactor, thermal insulators, geothermal system, polymer, petroleum, and blood-pumping machines. The researcher takes more interest in the problem involved in chemical reaction on unsteady MHD-free convection flow, due to some important application in numerous industrial methods. Mekheimer [1] studies convection flow with the induced magnetic field by using peristaltic flow. Ellahi and Riaz [2] investigate MHD flow with the help of third grad fluid. Mukhopadhyay and Gorla [3] study time-dependent magneto hydrodnamics boundary layer flow by using moveable surface. The study of non-Newtonian fluids has more important as compared to Newtonian fluid, due to the inclusive variety application in organic field such blood at little sheer rate suspension, muds, chime, apple sauce, honey, and shampoos. These are the many applications of non-Newtonian fluids, the mathematical problem for Casson fluid is presented to investigate the mechanism of pseudoplastic yield stress liquids. Nadeem et al. [4] used linearly extending sheet to discuss 3D Casson fluid. Benazir et al. [5] used vertical cone and flat with nonuniform heat source to study unsteady magneto hydynamics Casson fluid flow in this study, and they show that growth of the Casson fluid parameter decreases liquid flow and increases shear stress in the flow system. Nowadays, for energy resources, nanofluids are the effective agents. These fluids are used for improvement temperature spread devices used in several industries and engineering fields. It is not sufficient to acquire energy but to control absorptions of energy so can only solve this problem one to agree advance heat transfer fluids to rheostat the expenses of energy and the development the maximum heat transfer which is the requirement of the manufacturing and other related scientific arenas. The study of nanofluid is more important due to some key application in different parts of industries because of high heat transfer ratio. Parmar et al. [6] investigate the Casson flow and blood flow. Nadeem et al. [7] investigate the consequence of radioactivity for Casson fluid. Ullah et al. [8] used porous sheet to study the non-Newtonian fluid flow. Kamran et al. [9] investigate Casson nanofluid flow numerical along with joule heating and slip effect. Archana et al. [10] study the consequence of radiation on the energy expression of Casson fluid. Gireesha et al. [11] study the impression of Casson nanofluid flow and chemical reaction. Numerous other investigations are related to the Casson fluid, Souayeh et al. [12], Ullah et al. [13], and Aziz and Afify [14]. In the view of the above study, recently, several revisions have been conducted around the nanofluid, but the researchers take more interest in the new type introduced nanofluid known as hybrid nanofluid. Hybrid nanofluid is defined as two or more types of nanofluid are suspended in the base fluid. The interest of the researchers in this subject is the high heat transfer ratio improvement and invention rate saving that can be attained by the using these nanofluids. The researchers take more interest in this new type of nanofluids to solve the problems of real world. Initially, the idea of hybrid nanofluid increases the advanced structures of usual nanofluid was investigated by Suresh et al. [15]. Devi and Devi [16, 17] used to move surface to investigate the problems of MHD hybrid nanofluid. Tayebi and Chamkha [18] used an annulus to study the heat transfer of hybrid nanofluid. Ghadikolaei et al. [19] study the features of TiO2-Cu/H2O a hybrid nanofluid in the presence of resistance force. Hayat et al. [20] used the rotating disc to study flow Ag-CuO/water hybrid nanofluid. Yousefi et al. [21] used stretching cylinder to study the stagnation point flow of aqueous titanic-copper hybrid. Subhani and Nadeem [22] used stretching surface to study the behavior of Cu-TiO2/H2O hybrid nanofluid. Li et al. [23] under seismic conditions examine modeling response of magnetorheological fluid dampers. The influence of MHD (magneto hydrodynamics) flow of Casson fluid is examined by Crane et al. [24]. Nadeem et al. [25] used stretching sheet to examine the 2D viscous fluid flow. The mass diffusion on MHD (magneto hydrodynamics) flow over a moveable surface was deliberated by Ali and Sandeep [26]. The unsteady Casson fluid over extending sheet was studied by Vyas and Ranjan [27]. Chamka and Takhar [28] studied the MHD flow on nonlinear extending surface by permeable surface. Sandhya et al. [29] discuss buoyancy forces and activation energy on the MHD radiative flow over an exponentially stretching sheet. Lund et al. [30] study double comparison of MHD flow of Casson fluid. Ibrahim and Anbessa [31] study 3D MHD flow of nanofluid along with ion slip effects. Varun Kumar et al. [32] used stretching surface to investigate modeling on Casson nanofluid flow along with the influence of magnetic field. Yusuf et al. [33] study magneto-bioconvection flow of Williamson nanofluid over an inclined plate with gyro tactic microorganisms and entropy generation. Kumar et al. [34] study the impact of thermophoretic element testimony heat transmission through the dynamics of Casson fluid flow on a stretching needle. Rehman and Salleh [35] used stretching surface to study effect on MHD flow and heat transmission on hybrid nanofluids. Rehman and Salleh [36] used stretching surface to study time-dependent boundary layer stagnation point flow of nanofluid. The purpose of the present research paper is to investigate approximate analytical study of time-dependent MHD Casson hybrid nanofluid on an extending surface along with thermal radiation. The novelty of present research paper is that the first time-dependent Casson MHD flow of hybrid is addressed along with variable properties over stretching surface and the approximate solution one for velocity and other for temperature profile is analyzed by analytical method. Two sorts of hybrid nanofluids TiO2+Ag+blood and TiO2+blood are heat enhancement applications. Transforming the nonlinear partial differential equation to nonlinear ordinary differential equation, we used the defined similarity transformation. The governing nonlinear ODE (ordinary differential equations) is solving with approximate analytical method [37]. With the help of graphs, the influence of dissimilar parameters is presented. Also the consequence of Cf and Nu are obtainable in the form of tables. The current research article has good comparison with the already published work.


## 2. Mathematical Formulation

The two-dimensional time-dependent incomparable laminar boundary layer MHD flow of hybrid nanofluid over a permeable inclined movable surface. The impact in density difference with temperature only arises on the body force term, and later, changed temperature induces the buoyancy forces. The velocity of the moving surface in the direction of applied force inx direction is uwx,t, and the mass transfer is vwt which is normal to the moving surface. The surface (wall) temperature is Twx,t, and uniform temperature away from the moving surface is T∞. Under these assumptions, the governing model equations are as follows:
(1)∂u∂x+∂v∂y=0,(2)∂u∂t+u∂u∂x+v∂u∂y=1+1β∂2u∂y2−σhnfρhnfB02tu+gT−T∞cosα,(3)∂T∂t+u∂T∂x+u∂T∂y=khnfρCphnf∂2T∂y2−1ρCphnf∂qr∂y+μhnfρCphnf∂u∂y2+σhnfρCphnfB02tu2,wheret represents time, u and v represent velocity in x and y directions, ρhnf is the density of the hybrid nanofluid, μhnf represents the coefficient of viscosity, Cp represents the specific heat, βT is the convection term, T is the temperature of the wall, and T∞ is temperature away from the surface; the kinematic viscosity of the hybrid nanofluid is denoted by νhnf, the electrical conductivity of the hybrid nanofluid is σhnf, the strength of the external magnetic field is denoted by B0, and qr is the radiation heat flux. The relevant boundary condition is given below.
(4)u=uwx,t=cx1−λt,v=vwt,T=Twx,t,y=0,u⟶o,T⟶T∞,y⟶∞.In Equation (4), c represents the early extending rate, λ represents constant, and the subscripts w and ∞ are attitude for surface and boundary layer edge. To convert the nonlinear partial differential equation (Equations (1) and (2)) to nonlinear ordinary differential equation, we used the following similarity transformation.
(5)η=cν∗1−λty,ψ=ν∗c1−λtxfη,θη=T−T∞Tw−T∞.Using Equation (5) in Equations (1)–(3), Equation (5) satisfied Equation (1) identically, and convert Equations (2) and (3) to the following form.
(6)1−ϕ1−ϕ2−2.51−ϕ1−ϕ2+ϕ1ρ1/ρf+ϕ2ρ2/ρf1+1βf‴−Sf′+η2f″−1−ϕ12.51−ϕ12.5φ+Mf′−f′2+ff″+Grθ=0,(7)1−ϕ1−ϕ2−2.51−ϕ1−ϕ2+ϕ1ρ1/ρf+ϕ2ρ2/ρfkskhnfθ″+β1θ′2+1−ϕ12.51−ϕ12.5PrSη/2θ′+2Sθ+f′θ−fθ′−Ecf″21+R+β1θ.The transform boundary conditions for the defined problem are as follows:(8)f0=0,f′0=1,θ0=1,f′∞=0,θ′∞=0,whereS, φ, M, Gr, Pr, R, Ec, and β are time-dependent parameter, porous medium parameter, magnetic parameter, Grashof number, Prandtl number, radiation parameter, Eckert number, and Casson parameter and defined in
(9)S=λc,φ=ν∗1−λtk∗c,M=σB021−λtρc,Gr=gβTxTw−T∞uw2cosα,Pr=να,Ec=uw2CpTw−T∞.The skin friction inx and y directions is as follows:
(10)Rex1/2Chnf=−1−ϕ1−ϕ2−2.51−ϕ1−ϕ2+ϕ1ρ1/ρf+ϕ2ρ2/ρff″0.Nusselt number and Sherwood number is given by(11)Rex−1/2Nux=−kskhnfθ′0,


### 2.1. OHAM Solution

The model governing Equations (6) and (7) is analyzed with the help of approximate analytical OHAM
(12)Ewη+Jwη+rη=0,Cwη=0.In Equation (12), E represents linear operator, η represents the independent variable, rη represents the unknown function, J represents the nonlinear operator, and Cw represents the boundary operator for the problem.The initial solution for the definedf′η and θη is
(13)f0η=η3+e−η,(14)θ0η=12e−η.These two initial guess are obtained with the help of linear operator.(15)Ef=f‴+ff″=0,Eθ=θ″=0.The residual error for Equations (6) and (7) is defined by Liao [37].
(16)εmf=1n1+1∑j1=1n1κf∑j1=1n1f1ηη=jδη,(17)εmθ=1n+1∑j=1nκθ∑j=1nfηη=jδη,∑j=1nθηη=jδη,(18)εmt=εmf+εmθ.


### 2.2. Analysis of OHAM

This approximate analytical method is commonly used for the solution of nonlinear boundary value problem. For justification, consider the following nonlinear boundary value problem.(19)EFη+rη+JFη=0CF,dFdη=0.In Equation (19), E represents linear operator, J represents the nonlinear operators, rη represents the known function, Fη represents the indefinite function, and C represents the boundary operative for the defined problem. The deformation equation for OHAM is given below.
(20)1−qEFη,q+rη=ZqEFη,q+rη+JFη,qCFη,q,dFη,qdη.In Equation (20), q represents an implanting parameter, and its range is from 0 to 1, and Zq represents a nonzero supplementary function with the Z0=1 for q=0 and q=1.We also haveFη,0=F0η and Fη,1=Fη respectively. Thus, as q increases, from 0 to 1, the solution Fη,q varies from F0η to Fη.
(21)L′F0η+Fη=0,BF0,dF0dη.The power series inq is
(22)Zq=D1q+D2q+.⋯In Equation (22), D1 and D2 represent the unknown constant which can be obtained. The required solution is presented as follows:
(23)Fη,q,D1⋯D2m,(24)Fη,q,D1⋯Dm=F0η+∑k≥1Fkη,q,D1⋯Dmqk.Putting Equation (21) in Equation (22) and comparing the coefficients of like power of q primes to the governing equation F0η,F1η up to Fkη,
(25)EF1η=D1J0F0η,CF1,dF1dη=0,(26)EFkη−Fk−1η=DkJ0F0η+∑i=2k−2DiEFk−1η+Jk−1F0η,F1η⋯Fk−1ηCFk,dFkdη=0,k=1,2,.⋯JmF0η,F1η⋯Fmη are the coefficients of pm, which are gained by expanding JFη,p,D1⋯Dm in the form of power series about the embedding parameter q. Likewise,
(27)JFη,q,D1⋯Dm=JF0η+∑k≥JkF0η,F1η⋯Fkηqk.JFη,q,D1⋯Dm is assumed in Equation (26), and the convergence of Equation (26) depends on the auxiliary constant Di,i=1,2,3⋯. Equation (27) converges when q=1, and we gain
(28)Fη,D1,D2,⋯Dm=F0η+∑k≥2Fkη,D1,D2⋯.Themth-order approximation for the problem is
(29)Fη,D1,D2,⋯Dm=F0η+∑k≥1mFmη,D1,D2⋯Dm.The residual error for the problem is(30)R′η,D1,D2⋯Dm=EFη,D1,D2⋯Dm+Fη+JFη,D1,D2⋯Dm.R′η,D1,D2⋯Dm=0 and Fη,D1,D2⋯Dm have an exact solution which in overall impossible particularly in nonlinear problems. For Di,i=1,2,⋯,m, we used the method of least square.
(31)∂J∂D1=∂J∂D2⋯∂J∂Dm=0,(32)JD1,D2,⋯Dm=∫abR′2η,D1,D2⋯Dmdη.


## 2.1. OHAM Solution

The model governing Equations (6) and (7) is analyzed with the help of approximate analytical OHAM
(12)Ewη+Jwη+rη=0,Cwη=0.In Equation (12), E represents linear operator, η represents the independent variable, rη represents the unknown function, J represents the nonlinear operator, and Cw represents the boundary operator for the problem.The initial solution for the definedf′η and θη is
(13)f0η=η3+e−η,(14)θ0η=12e−η.These two initial guess are obtained with the help of linear operator.(15)Ef=f‴+ff″=0,Eθ=θ″=0.The residual error for Equations (6) and (7) is defined by Liao [37].
(16)εmf=1n1+1∑j1=1n1κf∑j1=1n1f1ηη=jδη,(17)εmθ=1n+1∑j=1nκθ∑j=1nfηη=jδη,∑j=1nθηη=jδη,(18)εmt=εmf+εmθ.


## 2.2. Analysis of OHAM

This approximate analytical method is commonly used for the solution of nonlinear boundary value problem. For justification, consider the following nonlinear boundary value problem.(19)EFη+rη+JFη=0CF,dFdη=0.In Equation (19), E represents linear operator, J represents the nonlinear operators, rη represents the known function, Fη represents the indefinite function, and C represents the boundary operative for the defined problem. The deformation equation for OHAM is given below.
(20)1−qEFη,q+rη=ZqEFη,q+rη+JFη,qCFη,q,dFη,qdη.In Equation (20), q represents an implanting parameter, and its range is from 0 to 1, and Zq represents a nonzero supplementary function with the Z0=1 for q=0 and q=1.We also haveFη,0=F0η and Fη,1=Fη respectively. Thus, as q increases, from 0 to 1, the solution Fη,q varies from F0η to Fη.
(21)L′F0η+Fη=0,BF0,dF0dη.The power series inq is
(22)Zq=D1q+D2q+.⋯In Equation (22), D1 and D2 represent the unknown constant which can be obtained. The required solution is presented as follows:
(23)Fη,q,D1⋯D2m,(24)Fη,q,D1⋯Dm=F0η+∑k≥1Fkη,q,D1⋯Dmqk.Putting Equation (21) in Equation (22) and comparing the coefficients of like power of q primes to the governing equation F0η,F1η up to Fkη,
(25)EF1η=D1J0F0η,CF1,dF1dη=0,(26)EFkη−Fk−1η=DkJ0F0η+∑i=2k−2DiEFk−1η+Jk−1F0η,F1η⋯Fk−1ηCFk,dFkdη=0,k=1,2,.⋯JmF0η,F1η⋯Fmη are the coefficients of pm, which are gained by expanding JFη,p,D1⋯Dm in the form of power series about the embedding parameter q. Likewise,
(27)JFη,q,D1⋯Dm=JF0η+∑k≥JkF0η,F1η⋯Fkηqk.JFη,q,D1⋯Dm is assumed in Equation (26), and the convergence of Equation (26) depends on the auxiliary constant Di,i=1,2,3⋯. Equation (27) converges when q=1, and we gain
(28)Fη,D1,D2,⋯Dm=F0η+∑k≥2Fkη,D1,D2⋯.Themth-order approximation for the problem is
(29)Fη,D1,D2,⋯Dm=F0η+∑k≥1mFmη,D1,D2⋯Dm.The residual error for the problem is(30)R′η,D1,D2⋯Dm=EFη,D1,D2⋯Dm+Fη+JFη,D1,D2⋯Dm.R′η,D1,D2⋯Dm=0 and Fη,D1,D2⋯Dm have an exact solution which in overall impossible particularly in nonlinear problems. For Di,i=1,2,⋯,m, we used the method of least square.
(31)∂J∂D1=∂J∂D2⋯∂J∂Dm=0,(32)JD1,D2,⋯Dm=∫abR′2η,D1,D2⋯Dmdη.


## 3. Discussion

In this section, we discuss the effect of different parameters such asβ, M, S, φ, Pr, R, Gr, and Ec (Casson parameter, magnetic field parameter, time-dependent parameter, porous medium parameter, Prandtl number, radiation parameter, Grashof number, and Eckert number) on f′η and θη distribution. The similarity transformation is used to convert the given nonlinear partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE). Two sorts of hybrid nanofluids TiO2+Ag+blood and TiO2+blood are used for the enhancement heat ratio. In the present research paper, TiO2+blood represents the base fluid and TiO2+Ag+blood represents hybrid nanofluid. The flow investigation is discussing on a moveable surface along with magnetic field and couple stress along with nonlinear convection. The approximate analytical technique is used to obtain analytical solution of particular problem. The impacts of dissimilar parameter on velocity f′η and θη are accessible in figures [1–7, 40], the effects on velocity profile are presented in figures [1–4], and the effects of different parameter on temperature profile are presented in figures [5–7, 40]. Tables 1 and 2 represent the impact of different parameters on Cf and Nu for both hybrid nanofluid and base nanofluid. The table shows that Cf (skin friction) coefficient is increasing in both cases for both hybrid nanofluid and base nanofluid. For the increasing values of magnetic field parameter M and Casson parameter β, physically increasing these parameters, viscous force is increasing so as a result, Cf (skin friction) is increasing. Table 2 presents the effect of Ec and R on Nu (Nusselt number) for both hybrid nanofluid and base nanofluid. Nusselt number coefficient is increasing in on both hybrid nanofluid and base nanofluid for growing amount of Ec and R. The convergence of the hybrid nanofluid and base fluid is gained for 25th iteration for both hybrid nanofluid and base nanofluid in Tables 3 and 4. From Tables 3 and 4, we see that growing the iteration decreases the order of residual error and good convergence was achieved. The assessment of the current work and with the published research work is obtainable in Tables 5 and 6. Table 7 shows the thermophysical properties of silver titanium oxide. The thermophysical properties of the nanofluid and hybrid nanofluid are presented in Table 8. The variation in Casson parameter β on velocity profile is captured in Figure 1 for both hybrid nanofluid and base nanofluid on f′η, and from Figure 1, we see that f′η is the declining function of the Casson parameter β. Or there is inverse relation between Casson parameter and f′η distribution, that is, the growing magnitude of Casson parameter decreases the velocity distribution. This effect is due to construction of resistive forces known as viscous force. The power of this force improves with the increasing power of Casson parameter β which responds the motion of fluid within boundary layer and drops the thickness of boundary layer. The deviation in M (magnetic field parameter) on f′η profile is captured in Figure 2 for both hybrid nanofluid and base nanofluid on f′η, and from Figure 2, we see that f′η is the declining function of M. Or there is inverse relation between magnetic field parameter and f′η distribution, that is, the growing amount of M declarations in f′η distribution. This effect is due to the production of resistive type force known as Lorentz force. These forces improve with the growing asset of magnetic field parameter M. Due to this, the motion of fluid particles inside boundary layer drops the thickness of boundary layer. The variation in time-dependent parameter S on velocity profile is captured in Figure 3 and for both hybrid nanofluid and base nanofluid, and from Figure 3, we see that velocity is the decreasing function of time-dependent parameter S. The variation in porous medium parameter φ on f′η profile is captured in Figure 4 for both hybrid nanofluid and base nanofluid on f′η, and from Figure 4, we see that f′η is the declining function of the porous medium parameter φ or there is inverse relation between porous medium parameter and velocity distribution, that is, the growing magnitude of porous medium parameter decreases the velocity distribution. Such state happens due to the injection and suction of fluid particles. The strength of such force enhances with the rising strength of porous medium parameter φ, which counteracts the motion of fluid within boundary layer and drops the thickness of boundary layer. Figure 5 is schemed for Pr for both hybrid nanofluid and base nanofluid. From Figure 5, we see that θη is the declining function of Pr. We can say that the Prandtl number works as a cooling agent, and this time, the fact is true in the case of hybrid nanofluid other than nanofluids. Figure 6 is schemed for Grashof number Gr for both hybrid nanofluid and base nanofluid. It is noticeable from Figure 6 that the relation between θη and Grashof number Gr is inverse relation, that is, the large value of Grashof number Gr is declining in temperature distribution as shown in Figure 6. Figure 7 shows the relation between Eckert number Ec and θη is direct relation or θη is the increasing function of Eckert number Ec. Therefore, enhancing the of Eckert number Ec heat is reserved in nanofluid because of the drug or frictional force, and hence, θη enhances. Figure 8 shows the relation between radiation parameter R and temperature profile is direct relation or temperature profile is the increasing function of radiation parameter R Therefore, enhancing the of radiation parameter R heat is reserved in nanofluid because of the drug or frictional force, and hence, θη enhances.Table 1
The inspiration ofM and β on skin friction coefficient.
βMTiO2+Ag+bloodTiO2+blood0.110.52230.31370.220.55730.34180.330.59110.37560.440.63170.44340.550.67940.47540.660.75210.53160.770.79650.5641Table 2
Inspiration ofR and Ec on Nusselt number.
REcTiO2+Ag+bloodTiO2+blood0.510.21230.6071120.29340.68251.530.37450.7037240.39560.75632.550.43760.7912360.49590.80253.570.55020.8341Table 3
Convergence control parameter for particular method.
mεmfTiO2+Ag+bloodεmθTiO2+Ag+blood50.6438×10−10.6775×10−3100.4094×10−20.1873×10−5150.9443×10−30.9729×10−7200.7298×10−50.5413×10−8250.5787×10−70.1442×10−9Table 4
Convergence control parameter for particular method.
mεmfTiO2+bloodεmθTiO2+blood50.7991×10−10.8574×10−1100.5266×10−30.5759×10−2150.1138×10−50.8759×10−5200.9616×10−60.3721×10−7250.5133×10−90.6632×10−9Table 5
Comparison forf′η. And publish work.
mNumericalAnalyticalError111021.52….1.50….0.02…31.31….1.29….0.02…41.46….1.44….0.02…50.57….0.53….0.04…60.73….0.71….0.02…70.79….0.77….0.02…80.89….0.88….0.01…90.32….0.27….0.05…100.52….0.50….0.02…Table 6
Comparison forθη. And publish work.
ηNumericalAnalyticalError111021.21….1.19….0.02…31.32….1.30….0.02…41.40….1.35….0.05…51.54….1.51….0.03…61.64….1.60….0.04…71.75….1.70….0.05…81.50….1.40….0.1…91.24….1.20….0.04…101.15….1.10….0.05…Table 7
Riahi et al.’s [38] thermophysical properties of silver titanium oxide.
ρkCpβSilver (Ag)10.54292351.89×10−5Titanium (TiO2)42508.956860.9×10−5Table 8
Alghamdi et al. [39] show the thermophysical properties.
PropertiesNanofluidHybrid nanofluidDensityρhnf=1−ϕρf+ϕρsρhnf=1−ϕ21−ϕ1+ϕ1s1+ϕ2ρ2Heat capacityρCphnf=1−ϕρCpf+ϕρCpsρCphnf1−ϕ21−ϕ1ρCpf+ϕ1ρCps1+ϕ2ρρCps2Viscosityμhnf=μf/1−ϕ2.5μhnf=μf/1−ϕ12.51−ϕ22.5Thermal conductivitykhnf/knf=ks+n−1kf−n−1ϕkf−ks/ks+n−1kf+ϕkf−ks,khnf/knf=ks2+n−1kbf−n−1ϕ2kbf−ks2/ks2+n−1kbf+ϕ2kbf−ks2Figure 1
Inspiration of Casson parameterβ on velocity profile.Figure 2
Inspiration of magnetic field parameterM on velocity profile.Figure 3
Inspiration of unsteady parameterS on velocity profile.Figure 4
Inspiration of porosity parameterφ on velocity profile.Figure 5
Inspiration of Prandtl number onPr temperature profile.Figure 6
Inspiration of Grashof numberGr on temperature profile.Figure 7
Inspiration of Eckert numberEc on temperature profile.Figure 8
Inspiration of Reedition parameterR on temperature profile.


## 4. Conclusion

This paper investigates approximate analytical study of time-dependent MHD Casson hybrid nanofluid over a stretching sheet along with thermal radiation and on extending surface. Novelty of current research is that the first time-dependent Casson MHD flow of hybrid is addressed for hybrid nanofluid along with variable properties over stretching surface. Transforming nonlinear PDE (partial differential equation) to nonlinear ODE (ordinary differential equation), we used the defined similarity transformation. The governing nonlinear equations are solving with the help of approximate analytical method. The impression of different parameters like couple stress, magnetic field parameter, porous medium parameter, time-dependent parameter, Prandtl number, Grashof number, Eckert number, and radiation parameter is interpreted through graphs for velocity and temperature profile. The key finding of the present research article is as follows:(1)
Growing the value of Casson parameter declarations in velocity(2)
Growing the value of magnetic field parameter declarations in velocity(3)
Growing the value of time-dependent parameter declarations in velocity(4)
Growing the value of porous medium parameter declarations in velocity(5)
Growing the value of Prandtl number declarations in the temperature profile(6)
Growing the value of Grashof number declarations in the temperature profile(7)
Growing the value of Eckert number rise temperature profile(8)
Growing the value of radiation parameter rise temperature profile



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