Next Article in Journal / Special Issue
Diversity of Iron Oxides: Mechanisms of Formation, Physical Properties and Applications
Previous Article in Journal
Ferroelectric Polarization in an h-BN-Encapsulated 30°-Twisted Bilayer–Graphene Heterostructure
Previous Article in Special Issue
Investigation of the Prospects for the Use of Iron-Containing Nanocomposites Doped with Rare Earth Elements as Catalysts for the Purification of Aqueous Media
Comment published on 11 April 2024, see Magnetochemistry 2024, 10(4), 26.
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:

MHD Hybrid Nanofluid Flow over a Stretching/Shrinking Sheet with Skin Friction: Effects of Radiation and Mass Transpiration

Angadi Basettappa Vishalakshi
Rudraiah Mahesh
Ulavathi Shettar Mahabaleshwar
Alaka Krishna Rao
Laura M. Pérez
3,* and
David Laroze
Department of Mathematics, Shivagangotri, Davangere University, Davangere 577007, India
Department of Mathematics, Government Degree College, Chodavaram, Andhrapradesh 531036, India
Departamento de Física, FACI, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile
Instituto de Alta Investigación, CEDENNA, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile
Author to whom correspondence should be addressed.
Magnetochemistry 2023, 9(5), 118;
Submission received: 27 February 2023 / Revised: 19 April 2023 / Accepted: 21 April 2023 / Published: 27 April 2023


The study of inclined magnetohydrodynamics (MHD) mixed convective incompressible flow of a fluid with hybrid nanoparticles containing a colloidal combination of nanofluids and base fluid is presented in the current research. Al2O3-Cu/H2O hybrid nanofluid is utilized in the current analysis to enhance the heat transfer analysis. The impact of radiation is also placed at energy equation. The main research methodology includes that the problem provided equations are first transformed into non-dimensional form, and then they are obtained in ordinary differential equations (ODEs) form. Then using the solutions of momentum and transfers equations to solve the given ODEs to get the root of the equation. The main purpose includes the resulting equations are then analytically resolved with the aid of suitable boundary conditions. The results can be discussed with various physical parameters viz., stretched/shrinked-Rayleigh number, stretching/shrinking parameter, Prandtl number, etc. Besides, skin friction and heat transfer coefficient can be examined with suitable similarity transformations. The main significance of the present work is to explain the mixed convective fluid flow on the basis of analytical method. Main findings at the end we found that the transverse and tangential velocities are more for more values of stretched/shrinked-Rayleigh number and mass transpiration for both suction and injection cases. This is the special method it includes stretched/shrinked-Rayleigh number, it contributes major role in this analysis. The purpose of finding the present work is to understand the analytical solution on the basis of mixed convective method.

1. Introduction

Researchers show interest in stretching sheet problems with hybrid nanofluid due to its numerous significances in many fields, viz., production of paper, the creation of rubber sheets, numerous technological uses, etc. The model for a nanofluid motion induced by a stretching (shrinking) sheet was considered in [1]. The hybrid nanofluid, which is made up of various nanoparticles dispersed in the base fluid, has recently been introduced as an idea for enhancing nanofluids. It is anticipated that it will provide more substantial thermophysical and rheological characteristics, as well as improving heat transfer properties. Further, the magnetohydrodynamics (MHD) focuses on how an electrically conducting fluid moves in a magnetic field, which may be used to control how the system transfers heat. Theoretically, magnetic fields might cause the Lorentz force, a drag force that slows the flow of a fluid and raises its temperature and concentration in a flowing medium. The rate of thermal efficiency will be increased by hybrid nanofluid. When there is a problem with the sheet stretching, heat transfer can be used to determine a product’s quality. The problem of stretching sheet was initiated by Sakiadis 1961 (a,b) [2,3] and obtained the solution of flow problems involving continuously moving surfaces. Crane [4] expanded this for the stretching surfaces for the viscous flow with the distance linearly varying from the slit. MHD Casson flow past a stretching sheet is investigated computationally by Kumar et al. [5]. In the presence of a stretching sheet, Anderson et al. [6,7] examined the MHD power law fluid and viscoelastic fluid. Later, a great deal of study was done on fluid flow in the presence of nanofluid and hybrid nanofluid with different features such as radiation and mass transpiration. Aly [8] and Hassan [9] investigated the nanofluid flow over a porous stretching/shrinking sheet with the effect of MHD, radiation, partial slip and also suction/injection effects. Anusha et al. [10,11] explains the flow velocity due to the presences of nanoparticles in the base fluid by considering MHD effect, mass transpiration, Brinkman effect over a porous stretching/shrinking surface. The research on the most current development of hybrid nanofluid carried out by Sarkar et al. [12] used three different types of base fluid; he draws the conclusion that the right hybridization process is most beneficial for the hybrid nanofluids thermal efficiency. Devi and Devi [13] carried out an investigation with Cu-Al2O3/H2O hybrid nanofluid in the presence of stretching sheet. Mishra et al. [14] worked on MHD nanofluid flow along with Soret and Dufour effects in the presence of stenosed artery with variable viscosity. Sharma et al. [15] studied the EMHD Jeffrey nanofluid flow with entropy generation and thermal radiation. Then, Gandhi et al. [16] discussed the applications of Koo–Kleinstreuer–Li Correlations by using computer simulations of EMHD Casson nanofluid. In this work, the authors deal with a non-Newtonian nanofluid blood flow incorporating CuO and Al2O3 nanoparticles and perform an electromagnetohydrodynamic analysis under external fields, finding that it can be helpful in the diagnosis of hemodynamic irregularities.
In the recent developments, as a development of nanofluid, hybrid nanofluid is introduced; it is made by mixing two different types of nanoparticles with the base fluids, and this is expected to provide an extra characteristic namely thermophysical and rheological characteristics. Hybrid nanofluid is widely used in many heat transfer fields, as it is shown in Ref. [17].
With careful observations of above articles, the present work examines the dual nature of a MHD heat transfer of a fluid with radiation and inclined angle. The main objective includes that the given partial differential equations (PDEs) are first converted into non-dimensional form and then transformed into ordinary differential equations (ODEs). Exact analytical solutions are then obtained using suitable manipulation and this study has been achieved with the help of various physical parameters. In addition, skin friction and Nusselt number can be verified analytically with suitable parameters. The present work contains numerous industrial applications, such as blowing of glass and extrusion of growing crystal, in the field of polymer sheets as well as cooling of metallic plates. The manuscript is organized as follows: in Section 2, the problem is presented and the analytical solution is given. In Section 3, the results are commented on and discussed. In particular, we show how dimensionless transverse and tangential velocity spatially behave, as well as the effect of the Prandtl number on Skin friction and the impact of the Rayleigh number on the Nusselt number for several values of the parameters. Finally, the conclusions are exposed in Section 4.

2. Mathematical Analysis

The current study takes into account a Newtonian fluid flow with inclined angled MHD on the surface of Al2O3-Cu-H2O hybrid nanofluid particles. The flow is mixed convective and the radiation is also present in the heat transfer equation. Here, the initial equations are in the form of mixed convection, and these equations are solved analytically. The Cartesian coordinate systems (x,y) are introduced with its origin schematically indicated in Figure 1. The hybrid nanofluid physical properties are explained in Table 1. These suppositions allow for the following modifications to be made to N-S equations: [18,19,20]:
u x + v y = 0
u u x + v u y = ν h n f 2 u y 2 + g β T T σ h n f B 0 2 ρ h n f S i n 2 τ u
u T x + v T y = χ h n f 2 T y 2 1 ρ C p h n f q r y
The B. Cs supported to motion equations are [21,22]
u = d α x , v = V w , T = T + γ T w T x at y = 0
u = 0 , T T as   y
here, Equation (4a,b,c) indicate the linear stretching in x direction, compatible condition, stretching of the sheet and it excludes dynamics at the distance away from the field, respectively. It also (4d,e) indicates temperature along the plate along with x axis, and distance for away from the plate, respectively.
According to Rosseland, the approximation thermal radiation can be simplified as (See Refs. [23,24,25,26,27])
q r = 4 σ * 3 k * T 4 y
here, expand T 4 on the basis of Taylor’s series and ignored some terms to get the equation as
T 4 = 4 T 3 T 3 T 4
Upon inserting Equation (6) into Equation (5), the following outcome is obtained.
q r y = 16 T 3 σ * 3 k * T 2 y 2
Next, we use the following dimensionless variables for further calculation [28,29]
X , Y = α ν x , y , U , V = u , v α ν θ = T T T w T ,
With the help of the above Equation (8) to change Equation (1) to (3) as given below [30]
U X + V Y = 0
U U X + V U Y = ε 1 ε 2 2 U Y 2 + R a s Pr θ ε 3 ε 2 M S i n 2 τ U = 0  
U θ X + V θ Y = 1 Pr ε 1 ε 2 + R ε 4 2 θ Y 2  
The parameters used in Equation (6)–(8) can be defined as
λ = γ α ν , stretching/shrinking boundary,
R a s = β g T w T ν χ ν α 3 / 2 , stretched/shrinked-Rayleigh number
Pr = ν χ , Prandtl number.
M = σ f B 0 2 ρ f α indicates magnetic parameter.
R = 16 σ * T 3 3 k * κ f indicates radiation parameter.
The constant parameters of Equations (9)–(11) can be defined as follows
ε 1 = μ h n f μ f = 1 1 ϕ 1 2.5 1 ϕ 2 2.5 ,
ε 2 = 1 ϕ 2 1 ϕ 1 + ϕ 1 ρ S 1 ρ f + ϕ 2 ρ S 2 ρ f ,
ε 3 =   σ h n f σ f = σ b f σ s 2 1 + 2 ϕ 2 + 2 σ b f 1 ϕ 2 σ s 2 1 ϕ 2 + σ b f 2 + ϕ 2 where , σ b f = σ f σ s 1 1 + 2 ϕ 1 + 2 σ f 1 ϕ 1 σ s 1 1 ϕ 1 + σ f 2 + ϕ 1
ε 4 = ρ C p h n f ρ C p f = 1 ϕ 2 1 ϕ 1 + ϕ 1 ρ C p S 1 ρ C p f + ϕ 2 ρ C p S 2 ρ C p f
Table 1. Heat-transfer characteristics of hybrid nanofluids (see Refs. [30,31,32,33,34,35]).
Table 1. Heat-transfer characteristics of hybrid nanofluids (see Refs. [30,31,32,33,34,35]).
SR. No.Thermophysical PropertiesLiquid Phase (Water)CopperAlumina
1 C P (J/kgK) 4179385765
2 ρ (kg/m3) 997.189333970
3 κ (W/mK) 0.61340040
4 σ (Sm−1) 0.055.97 × 10735 × 106
Introduce a stream function ψ x , y , and the velocities u & v in terms of ψ x , y , as follows
ψ X , Y , U = ψ Y , V = ψ X
Inserting Equation (12) in Equations (10) and (1), to yield the following equations [36]
ε 1 ε 2 3 ψ Y 3 + ψ , ψ Y X , Y + R a s Pr θ ε 3 ε 2 M S i n 2 τ ψ Y = 0
ε 1 ε 2 + R ε 4 2 θ Y 2 + Pr ψ , θ X , Y = 0  
Reduces B. Cs are as follows [37]
ψ Y = d X , ψ X = V C , θ = λ X at Y = 0 ψ Y = 0 , θ 0 as Y
The solution of Equations (13) and (14) can be assumed as
ψ = X f Y , θ = λ X f 1 Y
On substituting Equation (16) into Equations (13) and (14) to obtain the following results
ε 1 ε 2 d 3 f 1 d Y 3 + f d 2 f 1 d Y 2 d f d Y 2 + R a s Pr λ f 1 ε 3 ε 2 M S i n 2 τ d f d Y = 0
ε 1 ε 2 + R ε 4 d 2 f 1 d Y 2 + Pr f d f 1 d Y d f d Y f 1 = 0  
B.Cs related to f and f 1 can be yielded by substituting Equation (16) in Equation (15a,b)
f 0 = V C , d f d Y η = 0 = d , d f d Y η 0 f 1 0 = 1 , f 1 0
Assume Equation (17) has a solution that takes the form
f Y = V C + d 1 e δ Y γ , f 1 Y = e δ Y
Using the above assumed solution into Equation (17) to yield the result given below.
δ = V C ± V C 2 + 4 ε 1 d ( ε 2 d 2 R a s Pr λ ε 2 + ε 3 M S i n 2 τ d ) 2 ε 1 d
To increase the results of this paper, we introduce the global quantities Skin friction ( C f ) and Nusselt number ( N u ) as follows
C f = 1 X 2 ψ Y 2 = f Y Y 0 = δ d
N u = 1 X θ Y = λ f 1 0 = λ δ
Using these results, we discuss the characteristics of some special parameters.

3. Results and Discussion

The present paper demonstrates the Newtonian flow of a fluid with radiation and inclined MHD. Al2O3-Cu-H2O hybrid nanofluid is also considered to obtain new results. The flow is caused due to the linearly stretching vertical plate and the flow is mixed convective in nature. The given PDEs of the form are transformed into ODEs with the help of similarity transformations. The resulting ODEs equations are solved analytically to obtain the analytical solution. A unique class of heat transfer fluid called hybrid nanofluid was developed through technical innovation to improve the efficiency of heat transmission for numerous industrial and engineering applications. In comparison to base fluid and regular nanofluid, hybrid nanofluid may provide a higher thermal efficiency. In many technological applications, including automobile cooling systems, power generation, microelectronics, heat exchangers, and air conditioning, thermal system efficiency optimization is essential. The impact of different parameters can be observed by using graphical arrangements. The results of C f and N u values are taken into account. Upper branch solution (ubs) and lower branch solution (lbs) are examined. Today, improving heat transmission is a major issue in engineering and industrial applications. While cooling liquids with low thermal conductivity, such as water, ethylene glycol, and oil, are frequently utilized as pure fluids in industrial applications, the improvement in heat transfer is constrained.
Figure 2a,b portray the impact of f Y as a function of Y for varying the R a s values for suction and injection cases, respectively. Here, the plots related to f Y raises for raising R a s values for both V C > 0 and V C < 0 cases. Figure 3a,b represents the effect of f Y verses Y for varying the ϕ 1 values. Here, the plots related to f Y raises for raising the ϕ 1 values for both V C > 0 and V C < 0 cases. That is the boundary layer thickness is increased due to the increased ϕ 1 values for both V C > 0 and V C < 0 .
Figure 4a,b represent the impact of f Y Y as a function of Y for varying the R a s values for both V C > 0 and V C < 0 cases, respectively. Here, the axial velocity is incresed for increasing R a s values for both V C > 0 and V C < 0 cases.
The C f verses Pr for varying the λ values is indicated in Figure 5, dual nature is observed in Figure 5, one is ubs and another one is lbs. Here, it is observed that λ value decreases with increasing of C f for lbs and λ value increases with increases of C f for ubs. Further, it is seen that Pr value raises for larger λ values.
The impact of N u on R a s for varying the values of λ is indicated in Figure 6, dual nature is observed in Figure 6, one is ubs and another one is lbs; it is observed that λ value decreases with increases of N u for lbs and λ value increases with increases of N u for ubs. In addition, it is seen that R a s value decreases with increases of λ .

4. Concluding Remarks

Al2O3-Cu/H2O hybrid nanofluid is used in the current analysis to improve the heat transfer analysis. The study of inclined magnetohydrodynamics (MHD) mixed convective incompressible flow of a fluid with hybrid nanoparticles contains a colloidal combination of nanofluids and base fluid. The energy equation also includes the impact of radiation. The primary research methodology is first transforming the equations provided by the problem into a non-dimensional form before obtaining them as ordinary differential equations (ODEs). The provided ODEs are then solved using the momentum and transfer equation solutions to yield the equation’s root. The main goal entails solving the resulting equations analytically with the aid of appropriate boundary conditions. The yielded ODEs solved exactly to get the proper results. Several physical parameters, such as the stretched/shrinked-Rayleigh number, the stretching/shrinking parameter, the Prandtl number, etc., can be used to discuss the results.
  • f Y Y is more for more R a s values for both V C > 0 and V C < 0 .
  • f Y is more for more values of R a s , and ϕ 1 for both V C > 0 and V C < 0 .
  • Pr value increases with increases of λ .
  • R a s value decreases with increases of λ .
  • Dual nature is observed.
  • The present study helps to motivate the future researchers to conduct the investigations on stretching sheet problems with the help of mixed convective flow with hybrid nanofluid.

Author Contributions

Conceptualization, U.S.M.; methodology, A.B.V. and R.M.; validation, A.B.V. and R.M.; formal analysis, U.S.M., A.K.R., L.M.P. and D.L.; investigation, A.B.V. and R.M.; data curation, writing—original draft preparation, U.S.M., A.K.R., L.M.P. and D.L.; writing—review and editing, A.K.R., L.M.P. and D.L.; visualization, A.B.V., R.M. and A.K.R.; supervision, U.S.M., A.K.R., L.M.P. and D.L. All authors have read and agreed to the published version of the manuscript.


L.M.P. acknowledges partial financial support from ANID through Convocatoria Nacional Subvención a Instalación en la Academia Convocatoria Año 2021, Grant SA77210040. D.L. acknowledges partial financial support from Centers of Excellence with BASAL/ANID financing, AFB220001, CEDENNA. Also, Dr. U. S. Mahabaleshwar is thankful to the Institute for Advanced Studies at University of Tarapacá (Arica, Chile) for the hospitality during his stay at the University of Tarapacá.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.


SymbolDescriptionS.I. Unit
d Coefficient of stretching/shrinking parameter
f Y Dimensionless transverse velocity
f Y Y Dimensionless tangential velocity
g Gravity(N)
k * Coefficient of mean absorption
K Permeability parameter(m2)
T Temperature(K)
V C Mass transpiration
Greek symbols
α Dimensional stretching/shrinking parameter
β Thermal expansion coefficient(K−1)
χ Thermal diffusivity(m2s−1)
λ Dimensional stretching/shrinking parameter
θ Temperature profile
ν Kinematic viscosity(m2s−1)
ρ Density(kgm−3)
μ Dynamic viscosity of nanofluid(kgm−1s−1)
σ * Stefan–Boltzmann constant(Wm−2s−4)
w Quantities at wall
Quantities at for stream
f Fluid
h n f Hybrid nanofluid
B. CsBoundary conditions
ODEOrdinary differential equations
PDEPartial differential equations


  1. Baranovskii, E.S. Flows of a polymer fluid in domain with impermeable boundaries. Comput. Math. Math. Phys. 2014, 54, 1589–1596. [Google Scholar] [CrossRef]
  2. Sakiadis, B.C. Boundary-layer behaviour on continuous solid surface. AICHE J. 1961, 7, 26–28. [Google Scholar] [CrossRef]
  3. Sakiadis, B.C. Boundary-layer behavior on continuous solid surfaces. II. The boundary layer on a continuous flat surface. AIChE J. 1961, 7, 221–225. [Google Scholar] [CrossRef]
  4. Crane, L.J. Flow past a stretching plate. Z. Angew. Math. Phys. 1970, 21, 645–647. [Google Scholar] [CrossRef]
  5. Kumaran, G.; Sandeep, N.; Ali, M.E. Computational analysis of magnetohydrodynamic Casson and Maxwell flows over a stretching sheet with cross diffusion. Results Phys. 2017, 7, 147–155. [Google Scholar] [CrossRef]
  6. Andersson, H.I.; Bech, K.H.; Dandapat, B.S. Magnetohydrodynamic flow of a power-law fluid over a stretching sheet. Int. J. Non-Linear Mech. 1992, 27, 929–936. [Google Scholar] [CrossRef]
  7. Andersson, H.I. MHD flow of a viscoelastic fluid past a stretching surface. Acta Mech. 1992, 95, 227–230. [Google Scholar] [CrossRef]
  8. Aly, E.H. Existence of the multiple exact solutions for nanofluids flow over a stretching/shrinking sheet embedded in a porous medium at the presence of magnetic field with electrical conductivity and thermal radiation effects. Powder Tech. 2016, 301, 760–781. [Google Scholar] [CrossRef]
  9. Aly, E.H.; Hassan, M.A. suction and injection analysis of MHD nano boundary-layer over a stretching surface through a porous medium with partial slip boundary condition. J. Comput. Theor. Nanosci. 2014, 11, 827–839. [Google Scholar] [CrossRef]
  10. Anusha, T.; Huang, H.-N.; Mahabaleshwar, U.S. Two dimensional unsteady stagnation point flow of Casson hybrid nanofluid over a permeable flat surface and heat transfer analysis with radiation. J. Taiwan Inst. Chem. Eng. 2021, 127, 79–91. [Google Scholar] [CrossRef]
  11. Anusha, T.; Mahabaleshwar, U.S.; Sheikhnejad, Y. An MHD of nanofluid flow over a porous stretching/shrinking plate with mass transpiration and Brinkman ratio. Trans. Porous Media 2021, 142, 333–352. [Google Scholar] [CrossRef]
  12. Sarkar, J.; Ghosh, P.; Adil, A. A review on hybrid nanofluids: Recent research, development and applications. Renew. Sust. Energ. Rev. 2015, 43, 164–177. [Google Scholar] [CrossRef]
  13. Devi, S.S.; Devi, S.P. Heat transfer enhancement of Cu-Al2O3/water hybrid nanofluid flow over a stretching sheet. J. Niegerian Math. Soc. 2017, 36, 419–433. [Google Scholar]
  14. Mishra, N.K.; Sharma, M.; Sharma, B.K.; Khanduri, U. Soret and Dufour effects on MHD nanofluid flow of blood through a stenosed artery with variable viscosity. Int. J. Mod. Phys. B 2023, 2350266. [Google Scholar] [CrossRef]
  15. Sharma, B.K.; Kumar, A.; Gandhi, R.; Bhatti, M.M.; Mishra, N.K. Entropy generation and thermal radiation analysis of EMHD Jeffrey nanofluid flow: Applications in solar energy. Nanomaterials 2023, 13, 544. [Google Scholar] [CrossRef]
  16. Gandhi, R.; Sharma, B.K.; Mishra, N.K.; Al-Mdallal, Q.M. Computer simulations of EMHD Casson nanofluid flow of blood through an irregular Stenotic Permeable Artery: Applications of Koo-Kleinstreuer-Li Correlations. Nanomaterials 2023, 13, 652. [Google Scholar] [CrossRef]
  17. Jana, S.; Salehi-Khojin, A.; Zhong, W.H. Enhancement of fluid thermal conductivity by the addition of single and hybrid nano-additives. Thermochim. Acta 2007, 462, 45–55. [Google Scholar] [CrossRef]
  18. Mahabaleshwar, U.S.; Sarris, I.E.; Hill, A.A.; Lorenzini, G.; Pop, I. An MHD couple stress fluid due to a perforated sheet undergoing linear stretching with heat transfer. Int. J. Heat Mass Transf. 2017, 105, 157–167. [Google Scholar] [CrossRef]
  19. Xenos, M.; Petropoulou, E.; Siokis, A.; Mahabaleshwar, U.S. Solving the Nonlinear Boundary Layer Flow Equations with Pressure Gradient and Radiation. Symmetry 2020, 12, 710. [Google Scholar] [CrossRef]
  20. Reddy, G.B.; Goud, B.S.; Shekar, M.N.R. Numerical solution of MHD mixed convective boundary layer flow of a nanofluid through a porous medium due to an exponentially stretching sheet with Magnetic effect. Int. J. Appl. Eng. Res. 2019, 14, 2074–2083. [Google Scholar]
  21. Jia, Q.; Muhammad, M.B.; Munawwar, A.A.; Mohammad, M.R.; El-Sayed Ali, M. Entropy Generation on MHD Casson Nanofluid Flow over a Porous Stretching/Shrinking Surface. Entropy 2016, 4, 123. [Google Scholar]
  22. Umair, K.; Aurang, Z.; Sakhinah, A.B.; Ishak, A. Stagnatiom-point flow of a hybrid nanoliquid over a non-isothermal stretching/shrinking sheet with charecteristics of inertial and microstructure. Case Stud. Therm. Eng. 2021, 26, 101150. [Google Scholar]
  23. Nandy, S.K.; Pop, I. Effects of magnetic field and thermal radiation on stagnation flow and heat transfer of nanofluid over a shrinking surface, Int. Commun. Heat Mass Tranf. 2014, 53, 50–55. [Google Scholar] [CrossRef]
  24. Cortell, R. Radiation effects for the Blasius and sakiadis flows with a convective surface boundary condition. Appl. Math. Comput. 2008, 206, 832–840. [Google Scholar]
  25. Nayak, M.K.; Akbar, N.S.; Pandey, V.S.; Khan, Z.H.; Tripathi, D. 3D free convective MHD flow of nanofluid over permeable linear stretching sheet with thermal radiation. Powder Tech. 2017, 315, 205–215. [Google Scholar] [CrossRef]
  26. Sreedevi, P.; Reddy, P.S.; Chamkha, A.J. Heat and Mass transfer analysis of nanofluid over linear and non-linear stretching surfaces with thermal radiation and chemical reaction. Powder Tech. 2017, 315, 194–204. [Google Scholar] [CrossRef]
  27. Umair, K.; Aurang, Z.; Ishak, A. Magnetic field effect on sisko fluid flow containing gold nanoparticles through a porous covered surface in the presence of radiation and partial slip. Mathematics 2021, 9, 921. [Google Scholar]
  28. Ashraf, M.B.; Hayat, T.; Alsaedi, A. Mixed convection flow of Casson fluid over a stretching sheet with convective boundary conditions and Hall effect. Bound. Value Probl. 2017, 137, 1–17. [Google Scholar]
  29. Patil, P.M.; Roy, S.; Chamkha, A.J. Mixed convection flow over a vertical power-law stretching sheet. Int. J. Num. Meth. Heat Fluid Flow. 2010, 20, 445–458. [Google Scholar] [CrossRef]
  30. Sharma, B.K.; Khanduri, U.; Mishra, N.K.; Mekheimer, K.S. Combined effect of thermophoresis and Brownian motion on MHD mixed convective flow over an inclined stretching surface with radiation and chemical reaction. Res. Pap. 2023, 37, 2350095. [Google Scholar] [CrossRef]
  31. Aly, E.H.; Pop, I. MHD flow and heat transfer near stagnation point over a stretching/shrinking surface with partial slip and viscous dissipation: Hybrid nanofluid versus nanofluid. Powder Technol. 2020, 367, 192–205. [Google Scholar] [CrossRef]
  32. Taherialekouhi, R.; Rasouli, S.; Khosravi, A. An experimental study on stability and thermal conductivity of water-graphene oxide/aluminium oxide nanoparticles as a cooling hybrid nanofluid. Int. J. Heat Mass Transf. 2019, 145, 118751. [Google Scholar] [CrossRef]
  33. KSneha, N.; Mahabaleshwar, U.S.; Bennacer, R.; Ganaoui, E.L. Darcy Brinkman equations for hybrid dusty nanofluid flow with heat transfer and mass transpiration. Computation 2021, 9, 118. [Google Scholar]
  34. Fang, T.; Shanshan, Y.; Pop, I. Flow and heat transfer over a generalized stretching/shrinking wall problem-Exact solutions of the Navier-Stokes equations. Int. J. Non-Linear Mech. 2011, 46, 1116–1127. [Google Scholar] [CrossRef]
  35. Iskandar, W.; Ishak, A.; Pop, I. Mixed convection flow over an exponentially stretching/shrinking vertical surface in a hybrid nanofluid. Alex. Eng. J. 2020, 59, 1881–1891. [Google Scholar]
  36. Sharma, B.K.; Kumar, A.; Gandhi, R.; Bhatti, M.M. Exponential space and thermal-dependent heat source effects on electro-magneto-hydrodynamic Jeffrey fluid flow over a vertical stretching surface. Int. J. Mod. Phys. B 2022, 30, 2250220. [Google Scholar] [CrossRef]
  37. Khanduri, U.; Sharma, B.K. Entropy analysis for MHD flow subject to temperature-dependent viscosity and thermal conductivity. In Nonlinear Dynamics and Applications; Springer: Cham, Switzerland, 2022; pp. 457–471. [Google Scholar] [CrossRef]
Figure 1. Schematic diagram of problem setup.
Figure 1. Schematic diagram of problem setup.
Magnetochemistry 09 00118 g001
Figure 2. (a,b): f Y verses Y for different R a s values at (a) V C > 0 and (b) V C < 0 .
Figure 2. (a,b): f Y verses Y for different R a s values at (a) V C > 0 and (b) V C < 0 .
Magnetochemistry 09 00118 g002aMagnetochemistry 09 00118 g002b
Figure 3. (a,b): f Y verses Y for different ϕ 1 values at (a) V C > 0 and (b) V C < 0 .
Figure 3. (a,b): f Y verses Y for different ϕ 1 values at (a) V C > 0 and (b) V C < 0 .
Magnetochemistry 09 00118 g003aMagnetochemistry 09 00118 g003b
Figure 4. (a,b): f Y Y verses Y for different R a s values at (a) V C > 0 and (b) V C < 0 .
Figure 4. (a,b): f Y Y verses Y for different R a s values at (a) V C > 0 and (b) V C < 0 .
Magnetochemistry 09 00118 g004aMagnetochemistry 09 00118 g004b
Figure 5. The impact of C f on P r for different λ values.
Figure 5. The impact of C f on P r for different λ values.
Magnetochemistry 09 00118 g005
Figure 6. The impact of N u on R a s for different λ values.
Figure 6. The impact of N u on R a s for different λ values.
Magnetochemistry 09 00118 g006
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Vishalakshi, A.B.; Mahesh, R.; Mahabaleshwar, U.S.; Rao, A.K.; Pérez, L.M.; Laroze, D. MHD Hybrid Nanofluid Flow over a Stretching/Shrinking Sheet with Skin Friction: Effects of Radiation and Mass Transpiration. Magnetochemistry 2023, 9, 118.

AMA Style

Vishalakshi AB, Mahesh R, Mahabaleshwar US, Rao AK, Pérez LM, Laroze D. MHD Hybrid Nanofluid Flow over a Stretching/Shrinking Sheet with Skin Friction: Effects of Radiation and Mass Transpiration. Magnetochemistry. 2023; 9(5):118.

Chicago/Turabian Style

Vishalakshi, Angadi Basettappa, Rudraiah Mahesh, Ulavathi Shettar Mahabaleshwar, Alaka Krishna Rao, Laura M. Pérez, and David Laroze. 2023. "MHD Hybrid Nanofluid Flow over a Stretching/Shrinking Sheet with Skin Friction: Effects of Radiation and Mass Transpiration" Magnetochemistry 9, no. 5: 118.

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop