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# Analysis of a Stagnation Point Flow with Hybrid Nanoparticles over a Porous Medium

1
Department of Mathematics, Davangere University, Shivagangothri, Davangere, India

2
Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

* Corresponding Authors: M. Hatami. Email: ;

*Fluid Dynamics & Materials Processing* **2023**, *19*(2), 541-567. https://doi.org/10.32604/fdmp.2022.022002

**Received** 16 February 2022; **Accepted** 25 April 2022; **Issue published** 29 August 2022

## Abstract

The unsteady stagnation-point ﬂow of a hybrid nanoﬂuid over a stretching/shrinking sheet embedded in a porous medium with mass transpiration and chemical reactions is considered. The momentum and mass transfer problems are combined to form a system of partial differential equations, which is converted into a set of ordinary differential equations via similarity transformation. These ordinary differential equations are solved analytically to obtain the solution for velocity and concentration proﬁles in exponential and hypergeometric forms, respectively. The concentration proﬁle is obtained for four different cases namely constant wall concentration, uniform mass ﬂux, general power law wall con-centration and general power law mass ﬂux. The effect of different physical parameters such as Darcy number (*Da*), mass transpiration parameter (

^{-1}*V*), stretching/shrinking parameter (

_{C}*d*), chemical reaction parameter (

*β*) and Schmidt number (

*S*) velocity and concentration proﬁle is examined. Results show that, the axial velocity will decreases as the shrinking sheet parameter increases, regardless of whether the suction or injection case is examined. The concentration decreases with an increase in the shrinking sheet parameter and the chemical reaction rate parameter.

_{C}## Keywords

Nomenclature

| Constants |

| Concentration field |

| Molecular diffusivity |

| Inverse Darcy number |

| Chemical reaction parameter |

| Permeability of porous medium |

| Pressure |

| Schmidt number |

| Suction/injection parameter |

| Mass transfer velocity |

| Velocities along |

| Cartesian coordinates |

Greek symbols | |

| Chemical reaction parameter |

| Similarity variable |

| Dynamic viscosity |

| Kinematic viscosity |

| Density |

Subscripts | |

| Hybrid nanofluid parameter |

| Wall condition |

| Ambient condition |

Abbreviations | |

HNF | Hybrid nanofluid |

MHD | Magneto hydrodynamics |

ODEs | Ordinary differential equations |

PDEs | Partial differential equations |

PST | Prescribed surface temperature |

PHF | Prescribed heat flux |

Highlights

• This work investigates the unsteady stagnation point flow and mass transfer with chemical reaction.

• The system of partial differential equations is converted into system of ordinary differential equations via similarity transformations.

• The concentration profile is obtained for cases such as constant wall concentration, uniform mass flux, general power law wall concentration and general power law mass flux.

• The axial velocity decreases as the shrinking sheet parameter increases.

The mass transfer and momentum boundary layer flow have practical interest in the field of polymer process and electrochemistry. Also, the hybrid nanofluid (HNF) flow is a significant field in industry and become an interest field for researchers due to its wide applications. There are many significances of heat transfer over stretching sheet, due to its advantages mentioned by many researchers [1,2]. Aly et al. [3,4] made a comparison between the significances of HNF over NF for the magneto-hydrodynamic (MHD) flow and heat transfer by considering the effect of partial slip. Turkyilmazoglu [5] found the multiple solutions for MHD slip flow of viscoelastic fluid and Anusha et al. [6,7] investigated the unsteady inclined MHD flow for Casson fluid with hybrid nanoparticles in a porous media. Also, Mahabaleshwar et al. [8] investigated the MHD flow behaviour and mass transfer due to porous media. Fang et al. [9] examined the unsteady stagnation point flow and heat transfer to obtain the closed form solutions for prescribed wall temperature and wall heat flux. Mahabaleshwar et al. [10,11] made a research on the MHD flow with carbon nanotubes by considering the effect of mass transpiration and radiation on it. Suresh et al. [12,13] investigated the effect of hybrid nanofluid on heat transfer characteristics and observed that a hybrid nanofluid of (Al2O3-Cu/H2O) had significant heat transfer. Momin [14] investigated the laminar flow in an elevated funnel using mixed convection with a (Al2O3-Cu/H2O) hybrid nanofluid. Recently, the mixed convective flow with radiation is studied by Patil et al. [15] considering the couple stress fluid flow for first order chemical reaction. Furthermore, Mahabaleshwar et al. [16] examined the MHD non-Newtonian fluid flow and heat transfer due to porous surface with heat source/sink by different solution methods. Mahabaleshwar et al. [17] investigated the steady flow with HNF with mass suction, mathematically, and found the solution in algebraic decaying form. Nakhchi et al. [18,19] studied the effect of CuO-water nanofluid on the improvement of entropy production for a double pipe heat exchanger and a double V-cut twisted tapes, respectively. Recently many works are done on HNF flow by researchers such as Zainal et al. [20] on MHD flow due to quadratic stretching/shrinking sheet, Umair et al. [21] on radiative mixed convective flow, more recent developments and applications of HNF are investigated by Sarkar et al. [22], Vishalakshi et al. [23] studied the effect of slips and mass transpiration on the flow over porous sheet and Sneha et al. [24] on dusty HNF. The effect of nanoparticles on the flow is also investigated by Jalali et al. [25] and Bhandari [26].

Motivated by these investigations, the current work investigates the unsteady stagnation point flow of

The incompressible unsteady stagnation point flow over stretching/shrinking sheet is embedded in porous media with mass transpiration and mass transfer with chemical reaction as shown in Fig. 1. The

and the boundary conditions (B.Cs) are,

The following similarity transformation is introduced (see Fang et al. [9]) to transform the system of PDEs (1) to (4) into system of ODEs in order to find out the analytical solution,

And therefore the velocities along x- and y-axes are obtained as,

and wall mass transpiration velocity is,

On applying x-momentum with

Use Eqs. (7), (8) and (10) in Eqs. (2) and (4) to get,

The B.Cs associated with this Eqs. (5) and (6) also reduced as

here,

Also,

Since

and

where,

On differentiate Eq. (16) w.r.t x to obtain equation as

The pressure

where,

Consider a special case with

Defining a new transformation

And the associated B.Cs will reduces as,

Assume the solution of Eq. (22) is in the form,

On applying B.Cs defined in Eq. (23),

And on using Eqs. (24) and (25) in Eq. (22) will give,

Its roots will obtained as,

here, to obtain physically feasible solution Eq. (27) must satisfy

The solution will become,

And axial velocity is given by

2.3 Solution for Concentration

For the special case

Use Eq. (29) in above,

Then use Eq. (28) in above,

On introducing the new variable

And the B.Cs in terms of new variable are,

Then the general solution of Eq. (33) in terms of

And Eq. (35) in terms of

To obtain value of constant

where,

then wall mass flux is given by,

Define the transformation for Eq. (4) as,

Use Eq. (40) in Eq. (4) for

With B.Cs,

The Eq. (41) is similar to Eq. (31b) upto their general solution,

On using B.Cs as in Eq. (42), Eq. (43) will imply.

2.5 General Power Law Wall Concentration

In the current section we examined the general condition by assuming the wall concentration with a power law dependence on both time and distance as

by applying this new definition in Eq. (4) to obtain the following ODE for

With the following B.Cs,

For

By introducing new variable

where,

And the B.Cs will reduce to,

Now define

Here

Eq. (51) is the standard form of hypergeometric differential equation and its solution will be in the form of,

It gives the solution of Eq. (49) in the form,

And the wall mass flux is,

2.6 General Power-Law Mass Flux

In this section, the general condition is assumed with the wall concentration by a power law dependence on both time and distance

Use Eq. (56) in Eq. (4) for

With B.Cs as,

Eq. (57) is the same as Eq. (46), therefore their solutions are similar. We can obtain the solution of Eq. (57) as,

On applying B.C as in Eq. (58) to Eq. (59), we obtain the solution of Eq. (57) as follows:

The current work with nanoparticles dispersed in the base fluid is well agreement with the investigation of Fang et al. [9] in the absence of nanoparticles, porous media, stagnation point parameter and chemical reaction parameter.

The examination of unsteady stagnation point flow over porous media with mass transpiration and mass transfer with chemical reaction for

Fig. 2 demonstrates the solution domain

Fig. 3 depicts the transverse velocity due to stretching sheet and different values of

Fig. 4a is the plot for axial velocity due to shrinking sheet

The concentration profiles for different values of shrinking sheet parameter are demonstrated in Fig. 7 for suction and non-permeability cases in Figs. 7a and 7b, respectively. It can be seen that

The concentration profile for different values of

The concentration profile for different values of mass transpiration parameter due to shrinking sheet

The concentration profile of mass flux case for different values of shrinking sheet parameter is demonstrated in Fig. 10 for suction, non-permeability and injection cases, respectively in Figs. 10a–10c. It can be seen that

The concentration profile of mass flux case for different values of

The concentration profile of mass flux case for different values of mass transpiration parameter due to shrinking sheet

The concentration profile of mass flux case for different values of

The concentration profile of mass flux case for different values of

Fig. 15 shows the general power law wall concentration profile for different values of shrinking sheet parameter for suction, non-permeability and injection cases in Figs. 15a–15c, respectively. It can be seen that

Furthermore, Fig. 16 demonstrates the general power law wall concentration profile for different values of

The general power law wall concentration profile for different values of mass transpiration parameter due to shrinking sheet

The general power law wall concentration profile for different values of

Finally, Figs. 19 and 20 demonstrate the stream line graphs for upper branch solution and lower branch solution, respectively for some values of time t = 0.1, 0.5, 1. Stretching velocity of the wall is away from the origin since stream velocity is towards the origin. It can be seen that as the time increases, the stagnation point is moving away from the origin and towards the positive y direction.

The current work examined the unsteady stagnation point HNF flow over porous sheet with mass transpiration and mass transfer with chemical reaction. The momentum and mass transfer problem are solved analytically to obtain the solution for velocity and concentration profiles in exponential form and hypergeometric form respectively. The concentration profile is obtained for four different cases such as constant wall concentration, uniform mass flux, general power law wall concentration and general power law mass flux. The effect of different physical parameters like Darcy number

• The solution domain will expand as mass transpiration decreases for shrinking sheet case.

• Transverse velocity increases with increase in

• Axial velocity decreases with increase in mass transpiration and it decreases with increase in

• The axial velocity will decrease as the shrinking sheet parameter increases both in suction and injection cases.

• Concentration profile

• The concentration profile of mass flux case

• Injection case had more mass transfer than suction case.

• The general power law wall concentration profile

Acknowledgement: The author T. Anusha is thankful to Council of Scientific and Industrial Research (CSIR), New Delhi, India for financial support in the form of Junior Research Fellowship: File No. 09/1207(0003)/2020-EMR-I.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

**APA Style**

*Fluid Dynamics & Materials Processing*,

*19*

*(2)*, 541-567. https://doi.org/10.32604/fdmp.2022.022002

**Vancouver Style**

**IEEE Style**

*Fluid Dyn. Mater. Proc.*, vol. 19, no. 2, pp. 541-567. 2023. https://doi.org/10.32604/fdmp.2022.022002