Thermosoluted Marangoni convective flow towards a permeable Riga surface

Anum Shafiq; Ghulam Rasool; Lehlohonolo Phali; Chaudry Masood Khalique

doi:10.1515/phys-2020-0167

Open Access Published by De Gruyter Open Access September 7, 2020

Thermosoluted Marangoni convective flow towards a permeable Riga surface

Anum Shafiq , Ghulam Rasool , Lehlohonolo Phali and Chaudry Masood Khalique

From the journal Open Physics

https://doi.org/10.1515/phys-2020-0167

Abstract

This study reveals the characteristics of chemical reaction on Marangoni mixed convective stream towards a penetrable Riga surface. The heat and mass phenomena are analysed within the sight of Dufour and Soret impacts. The administering partial differential equations system is converted into three nonlinear ordinary differential equations utilizing appropriately adjusted transformations. The resultant system of highly nonlinear equations is analytically solved by invoking the homotopy analysis method. Thereafter, the convergence of series solutions is discussed. The impact of appropriate parameters on various flow fields is thoroughly explained with the help of graphs and tables. The wall drag coefficient and relevant flux rates are arranged and discussed for dimensionless parameters. The outcomes show that the stronger Dufour effect of liquid causes a notable incremental variation in heat and mass flux, whereas an opposite trend is noted in the heat flux rate for the Soret effect. However, the mass flux is still found increasing for the stronger Soret effect.

Keywords: Marangoni convection; diffusion-thermo and thermal-diffusion effects; chemical reaction phenomena; Riga surface

1 Introduction

It is known that various surface tension gradients prompt Marangoni convection. Such type of convection has applications in the fields of welding and precious stone development. Likewise, it is generally utilized as a part of the semiconductor preparation. Marangoni impact is used during the manufacture of integrated circuits. Marangoni stream instigated by surface pressure varieties along the fluid–fluid or fluid–gas interfaces appears to have been investigated first [1,2]. Researchers’ enthusiasm in Marangoni convection emerges in material processing of space crafts, in which gravity drive is little in correlation with the thermo slender one. Thus, Marangoni convection has extraordinary significance because of its implementation in fields of space processing and microgravity sciences [3,4]. Boiling tests have demonstrated that the impact of Marangoni stream is critical in microgravity and might be essential in earth gravity [5,6]. Various investigations of Marangoni stream in different geometries have been carried out in refs. [7,8]. The comparable arrangement of Marangoni streams for the situation with an outer weight inclination was examined by Golia and Viviani [9]. A similitude investigation of simply speed profile for Marangoni stream that surface tension variety is linearly identified with the surface position was carried out by Arafune and Hirata [10]. Similarity solutions of surface pressure fluctuated as a quadratic capacity of temperature was computed by Slavtchev and Miladinova [11]. Sastry [12] examined the impact of magnetohydrodynamic (MHD) Marangoni convective flow in a nanoliquid along the stretching surface and also considered the radiation and first-order chemical reaction phenomena. Characteristics of the Cattaneo–Christov heat flux model of Marangoni flow in a copper–water nanoliquid were studied by Xu and Chen [13]. Chen et al. [14] analysed the impact of solid matrix in Marangoni flow of power law nanoliquid in a porous medium. The significance of Marangoni mixed convection flow Casson fluid was investigated in ref. [15], in which the Joule heating and nonlinear radiation phenomena were additionally considered. In ref. [16], the authors investigated the impacts of particle shape on Marangoni convective flow of a nanoliquid. Hayat et al. [17] analysed the significance of Marangoni convective flow of carbon–water nanoliquid towards a radiative surface. Biswas and Manna [18] studied the MHD Marangoni flow in an open square cavity. Some very interesting and related articles can be seen in refs. [19,20,21,22,23,24].

The stream of electrically leading fluids can be controlled with the help of electromagnetic body powers. The movement of liquids of high electrical conductivity r (e.g. fluid metals, semiconductor dissolves) would be affected altogether by external magnetic fields of moderate qualities of 1T. This is traditional MHD stream control. In liquids (e.g. seawater) aside, the streams instigated by an outer attractive field alone are too tiny, and an outside electric field must be connected to accomplish a productive stream control (electro-MHD stream control). A crossed electric and attractive field was used to change the structure of a pressure gradient driven boundary layer and to balance out its movement by backing off its development. Mehmood et al. [25] investigated the significance of thermal-diffusion/diffusion-thermo on oblique stagnation point stream of couple stress casson liquid over an expanding horizontal Riga surface using a higher order chemical reaction. Nayak et al. [26] analysed the partial slip and viscous dissipation effects on radiative tangent hyperbolic nanoliquid stream over a vertical permeable Riga surface. Shafiq et al. [27] examined the significance of stagnant flow of Walters’ B liquid along a radiative Riga surface. Characteristics of homogenous–heterogeneous reactives on radiative NaCl–CNP nanoliquid flow past a convectively heated vertical Riga surface were studied in ref. [28]. Naseem et al. [29] studied the analytical solution of third-grade nanofluidic stream towards the Riga surface and also considered the Cattaneo–Christov heat flux model. In refs. [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45], some relevant and fruitful research articles are listed.

In the majority of investigations identified with heat/mass processes, Soret/Dufour impacts are usually ignored because of the undeniable fact that these two factors are of smaller importance than impacts illustrated due to Fourier and Ficks. If density difference exists in the flow regime, such impacts are noteworthy. For instance, when species are presented on the surface in liquid domain with various densities compared to the surrounding liquid, both impacts are second-order phenomena and have significant importance in areas such as petrology, hydrology, geosciences and so on. The Soret impact is used in the mixture of lighter molecular weight ( H 2 , He) and isotopical separation as well as in medium-low molecular weight ( N 2 , air ) gases [46,47,48,49,50,51].

This investigation is conducted to analyse the Marangoni mixed convective flow towards a Riga surface. Soret as well as Dufour effects are considered to examine the heat flux rate and the mass flux rate for their industrial importance. First-order homogeneous chemical reaction term is also taken into account. Numerical and graphical results of the considered problems are solved via a homotopy analysis technique [52,53,54,55,56,57,58,59,60,61,62]. Wall drag coefficient and relevant flux rates are arranged and talked about for dimensionless rising parameters.

2 Coordinate system and problem formulation

Consider the steady Marangoni convective flow towards a Riga surface with surface tension because of temperature and concentration gradients (Figure 1). Isotopes have been separated by utilizing the Soret impact and the same is utilized in the blend of gases having low atomic weight (H 2 , He) . Dufour impact was observed to be extensive for medium atomic weight ( N 2 , air), so it cannot be ignored. The liquid properties are supposed to be constant in restricted temperature extend. Not at all like Boussinesq impact in buoyancy induced flow, Marangoni impact goes about as a boundary condition on the governing equations for flow. Chemical reaction phenomena are assumed to exist. Taking the aforementioned assumptions into consideration, we have the following problems:

(1) ∂ v ∂ y + ∂ u ∂ x = 0 ,

(2) u ∂ u ∂ x + v ∂ u ∂ y = μ nf ρ nf ∂ 2 u ∂ y 2 + π j 0 M 0 exp − π b y 8 ρ nf ,

(3) u ∂ T ∂ x + v ∂ T ∂ y = α ∂ 2 T ∂ y 2 + D m k T C s C p ∂ 2 C ∂ y 2 ,

(4) u ∂ C ∂ x + v ∂ C ∂ y = D m k T T m ∂ 2 T ∂ y 2 + D m ∂ 2 C ∂ y 2 − k 1 ∗ ( C − C ∞ ) ,

with

(5) μ ∂ u ∂ y y = 0 = − σ C ∂ C ∂ x y = 0 , − σ T ∂ T ∂ x y = 0 v ( x , 0 ) = v ω , C ( x , 0 ) = C ∞ + A ∗ x 2 , T ( x , 0 ) = T ∞ + A x 2 , u ( x , ∞ ) → 0 , C ( x , ∞ ) → C ∞ , T ( x , ∞ ) → T ∞ ,

where dependence of surface tension can be defined as

(6) σ = σ 0 − ( T − T ∞ ) σ T − ( C − C ∞ ) σ C ,

with

(7) σ T = − ∂ σ ∂ T C , σ C = − ∂ σ ∂ C T ,

where thermal diffusivity, kinematic viscosity, liquid density, temperature and ambient temperature, concentration and ambient concentration, diffusion coefficient, thermal diffusion and the specific heat at constant pressure and concentration susceptibility are denoted by α , ν , ρ , ( T , T ∞ ) , ( C , C ∞ ) , D m , k T and ( C s , C p ) . k 1 ∗ denotes the chemical reaction parameter, j 0 denotes the applied intensity of the current in electrodes, M 0 denotes the magnetic strength offered by magnets, b denotes the width of electrodes and magnets and T m denotes the mean liquid temperature.

(8) η = C 1 y , f ( η ) = C 2 x − 1 ψ ( x , y ) , C ( η ) = C ( x , y ) − C ∞ A ∗ x − 2 , θ ( η ) = T ( x , y ) − T ∞ A x − 2 ,

where

(9) A = Δ T L 2 , A ∗ = Δ C L 2 , C 1 = ρ A d σ d T c μ 2 , 3 C 2 = ρ 2 μ A d σ d T c 3 ,

using the above equation, the transformed governing systems of equations are as follows:

(10) f ′ ″ − ( f ′ ) 2 + Q e − β η + f f ″ = 0 ,

(11) θ ″ − 2 Pr f ′ θ + Pr D u ϕ ″ + Pr f θ ′ = 0 ,

(12) ϕ ″ + S c f ϕ ′ − 2 S c f ′ ϕ + S r S c θ ″ − S c k 1 ϕ = 0 ,

along with

(13) − 2 ( 1 + γ ) = f ″ ( 0 ) , f ( 0 ) = − C 2 V ω , θ ( 0 ) = 1 , ϕ ( 0 ) = 1 , 0 = f ′ ( ∞ ) , θ ( ∞ ) = 0, ϕ ( ∞ ) = 0 .

The dimensionless quantities

(14) Q = π M 0 j 0 x 8 ρ nf u e 2 , β = l 0 π b x ( r − 2 ) / 3 , Pr = υ α , D u = D m K T C P C V ν Δ C Δ T , S c = ν D m , S r = D m k T T m ν Δ T Δ C , k 1 = k 1 ∗ D m ,

respectively, indicate modified Hartman number ( Q ) , Dufour number ( D u ) , dimensionless constant ( β ) , Prandtl number ( Pr ) , Schmidt number ( S c ) , Soret number ( S r ) and chemical reaction ( k 1 ) .

Figure 1

Physical diagram.

Local Nusselt and Sherwood numbers are as follows:

(15) Nu x = x q w k ˆ ( T − T ∞ ) = − θ ′ ( 0 ) , Sh x = x q m k ˆ ( C − C ∞ ) = − ϕ ′ ( 0 ) ,

where

(16) q w = − k ∂ T ∂ y y = 0 , q m = − k ∂ C ∂ y y = 0 .

3 Homotopy analysis method (HAM)

We now find a series of solutions for problems (10)–(13). Thus,

(17) f 0 ( η ) = − e − η ( 2 − 2 e η + f 0 e η + 2 r − 2 r e η ) , θ 0 ( η ) = e − η , ϕ 0 ( η ) = e − η ,

(18) ℒ 1 ∗ ( f ) = − d f d η + d 3 f d η 3 , ℒ 2 ∗ ( θ ) = − θ + d 2 θ d η 2 , ℒ 3 ∗ ( ϕ ) = − ϕ + d 2 ϕ d η 2 ,

where ( f 0 , θ 0 , ϕ 0 , ℒ 1 ∗ , ℒ 2 ∗ , ℒ 3 ∗ ) are initial solutions and auxiliary linear operators, respectively.

0th order deformations

(19) ℏ 1 N 1 ∗ [ f ˆ ( η ; p ˆ ) ] = 1 − p ˆ p ˆ ℒ 1 ∗ [ f ˆ ( η ; p ˆ ) − f 0 ( η ) ] ,

(20) f ˆ ( η ; p ˆ ) η = 0 = − f 0 , ∂ 2 f ˆ ( η ; p ˆ ) ∂ η 2 η = 0 = − 2 ( 1 + γ ) , ∂ f ˆ ( η ; p ˆ ) ∂ η η → ∞ = 0 ,

(21) ℏ 2 N 2 ∗ [ f ˆ ( η ; p ˆ ) , θ ˆ ( η ; p ˆ ) ] = 1 − p ˆ p ˆ ℒ 2 ∗ [ θ ˆ ( η ; p ˆ ) − θ 0 ( η ) ] ,

(22) θ ˆ ( η ; p ˆ ) η = 0 = 1 , θ ˆ ( η ; p ˆ ) η → ∞ = 0 ,

(23) ℏ 3 N 3 ∗ [ f ˆ ( η ; p ˆ ) , θ ˆ ( η ; p ˆ ) , ϕ ˆ ( η ; p ˆ ) ] = 1 − p ˆ p ˆ ℒ 3 ∗ [ ϕ ˆ ( η ; p ˆ ) − ϕ 0 ( η ) ] ,

(24) ϕ ˆ ( η ; p ˆ ) η = 0 = 1 , ϕ ˆ ( η ; p ˆ ) η → ∞ = 0 ,

with

(25) N 1 ∗ [ f ˆ ( η ; p ˆ ) ] = ∂ 3 f ˆ ∂ η 3 + f ˆ ∂ 2 f ˆ ∂ η 2 − ∂ f ˆ ∂ η 2 + Q e − β η ,

(26) N 2 ∗ [ f ˆ ( η ; p ˆ ) , θ ˆ ( η ; p ˆ ) , ϕ ˆ ( η ; p ˆ ) ] = ∂ 2 θ ˆ ∂ η 2 + Pr f ˆ ∂ θ ˆ ∂ η − 2 Pr ∂ f ˆ ∂ η θ ˆ + Pr D u ∂ 2 ϕ ˆ ∂ η 2 ,

(27) N 3 ∗ [ f ˆ ( η ; p ˆ ) , θ ˆ ( η ; p ˆ ) , ϕ ˆ ( η ; p ˆ ) ] = ∂ 2 ϕ ˆ ∂ η 2 − 2 S c ∂ f ˆ ∂ η ϕ ˆ + S c f ˆ ∂ ϕ ˆ ∂ η + S r S c ∂ 2 θ ˆ ∂ η 2 − S c k 1 ϕ ˆ .

Here, embedding parameter p ˆ ∈ [ 0 , 1 ] and ℏ 1 , ℏ 2 and ℏ 3 are nonzero auxiliary parameters.

mth-order problems are as follows:

(28) ℒ 1 ∗ [ f ˆ m ( η ) − χ m f ˆ m − 1 ( η ) ] = ℏ 1 ℛ m f ˆ ( η ) ,

(29) f ˆ m ( η ; p ˆ ) η = 0 = 0 , ∂ 2 f ˆ m ( η ; p ˆ ) ∂ η 2 η = 0 = 0 , ∂ f ˆ m ( η ; p ˆ ) ∂ η η → ∞ = 0 ,

(30) ℒ 2 ∗ [ θ ˆ m ( η ) − χ m θ ˆ m − 1 ( η ) ] = ℏ 2 ℛ m θ ˆ ( η ) ,

(31) θ ˆ m ( η ; p ˆ ) η = 0 = 0 , θ ˆ m ( η ; p ˆ ) η → ∞ = 0 ,

(32) ℒ 3 ∗ [ ϕ ˆ m ( η ) − χ m ϕ ˆ m − 1 ( η ) ] = ℏ 3 ℛ m ϕ ˆ ( η ) ,

(33) ϕ ˆ m ( η ; p ˆ ) η = 0 = 0 , ϕ ˆ m ( η ; p ˆ ) η → ∞ = 0 ,

(34) ℛ m f ( η ) = f m − 1 ' ' ' ( η ) + ∑ k = 0 m − 1 f k ″ f m − k − 1 − ∑ k = 0 m − 1 f m − k − 1 ′ f k ′ + Q e − γ η ,

(35) ℛ m θ ( η ) = θ m − 1 ″ ( η ) + Pr ∑ k = 0 m − 1 f m − k − 1 θ k ′ − 2 Pr ∑ k = 0 m − 1 f m − k − 1 ′ θ k + D u Pr ϕ m − 1 ″ ,

(36) ℛ m ϕ ( η ) = ϕ m − 1 ″ ( η ) − 2 S c ∑ k = 0 m − 1 f m − k − 1 ′ ϕ k + S c ∑ k = 0 m − 1 f m − k − 1 ϕ k ′ + S c S r θ m − 1 ″ − S c k 1 ϕ m − 1 ,

(37) χ ˜ m = 0 , m ≤ 1 1 , m > 1 .

Setting p ˆ = 0 and 1, we have

(38) f 0 ( η ) = f ˆ ( η ; 0 ) , θ 0 ( η ) = θ ˆ ( η ; 0 ) , ϕ 0 ( η ) = ϕ ˆ ( η ; 0 ) ,

(39) f ( η ) = f ˆ ( η ; 1 ) , θ ( η ) = θ ˆ ( η ; 1 ) , ϕ ( η ) = ϕ ˆ ( η ; 1 ) .

As p ˆ varies from 0 to 1, the functions f ˆ ( η ; p ˆ ) , θ ˆ ( η ; p ˆ ) and ϕ ˆ ( η ; p ˆ ) change from initial ( f 0 , θ 0 , ϕ 0 ) to final ( f , θ , ϕ ) solutions. Employing the Taylor theorem leads to

(40) f ˆ ( η ; p ˆ ) = f 0 ( η ) + ∑ m = 1 ∞ p ˆ m f ˆ m ( η ) , f ˆ m ( η ) = 1 m ! ∂ m f ˆ ∂ p ˆ m p ˆ = 0 ,

(41) θ ˆ ( η ; p ˆ ) = θ 0 ( η ) + ∑ m = 1 ∞ p ˆ m θ ˆ m ( η ) , θ ˆ m ( η ) = 1 m ! ∂ m θ ˆ ∂ p ˆ m p ˆ = 0 ,

(42) ϕ ˆ ( η ; p ˆ ) = ϕ 0 ( η ) + ∑ m = 1 ∞ p ˆ m ϕ ˆ m ( η ) , ϕ ˆ m ( η ) = 1 m ! ∂ m ϕ ˆ ∂ p ˆ m p ˆ = 0 .

In order for the above series to be convergent for p ˆ = 1 , we choose suitable auxiliary parameters and thus obtain

(43) f ( η ) = f ˆ 0 ( η ) + ∑ m = 1 ∞ f ˆ m ( η ) ,

(44) θ ( η ) = θ ˆ 0 ( η ) + ∑ m = 1 ∞ θ ˆ m ( η ) ,

(45) ϕ ( η ) = ϕ ˆ 0 ( η ) + ∑ m = 1 ∞ ϕ ˆ m ( η ) .

Special solutions of equations (31)–(36) are as follows:

(46) f ˆ m ( η ) = f m ∗ ( η ) + G 1 ∗ + G 2 ∗ e η + G 3 ∗ e − η ,

(47) θ ˆ m ( η ) = θ m ∗ ( η ) + G 4 ∗ e η + G 5 ∗ e − η ,

(48) ϕ ˆ m ( η ) = ϕ m ∗ ( η ) + G 6 ∗ e η + G 7 ∗ e − η ,

in which special solutions are f m ∗ , θ m ∗ and ϕ m ∗ .

4 Analysis of the solutions

We now want to make sure that the series solution we obtained is convergent. It should be noted that the auxiliary estimator ℏ plays a pivotal role in managing and modifying the region of convergence for the final series solution. Thus, we sketch the HAM curves in Figures 2 and 3 and conclude that the admissible values are ℏ 1 ∈ [ − 0.35 , − 0.1 ] , ℏ 2 ∈ [ − 0.37 , − 0.2 ] and ℏ 3 ∈ [ − 0.39 , − 0.15 ] at the 12th order approximation.

$Figure 2 h-Curve of f ′ ″ (0) f^{\prime} ^{\prime\prime} \mathrm{(0)} and θ ′ (0) \theta ^{\prime} \mathrm{(0)} at the 12th order approximation.$

Figure 2

h-Curve of f ′ ″ (0) and θ ′ (0) at the 12th order approximation.

Figure 4 delineates variations of f ′ ( η ) in response to a change for values of γ . Here, f ′ ( η ) decreases for larger γ . Figure 5 provides the impact of modified Hartman number Q on f ′ ( η ) . Enhancement in Q corresponds to rise in f ′ ( η ) and thickness of momentum boundary layer. As a matter of fact, higher values of Q correspond to rise in the external electric field and as a result enhances the velocity field. The characteristics of various values of ( f 0 < 0 ) and ( f 0 > 0 ) , i.e. suction and injection, on velocity distribution are given by Figure 6. It is found that the first parameter reduces the velocity profile, whereas the second parameter augments the profile. Figure 7 illustrates the significance behaviour of dimensionless number β on f ′ ( η ) . Velocity and associated boundary layer width rise by enhancement in dimensionless estimator β .

$Figure 3 h-Curve of ϕ ′ (0) \phi ^{\prime} \mathrm{(0)} at the 12th order approximation.$

Figure 3

h-Curve of ϕ ′ (0) at the 12th order approximation.

$Figure 4 γ \gamma versus f ′ ( η ) f^{\prime} (\eta ) .$

Figure 4

γ versus f ′ ( η ) .

$Figure 5 Q versus f ′ ( η ) f^{\prime} (\eta ) .$

Figure 5

Q versus f ′ ( η ) .

$Figure 6 f 0 {f}_{0} versus f ′ ( η ) f^{\prime} (\eta ) .$

Figure 6

f 0 versus f ′ ( η ) .

Behaviour of γ on θ ( η ) is demonstrated in Figure 8. Increment in γ shows the decrease in temperature profile. Figure 9 illustrates the impact of Q on temperature field. It is depicted that θ ( η ) reduces for higher estimates of Q. However, associated boundary layer width also reduces for higher estimates of Q. The behaviour of Prandtl parameter on θ ( η ) is depicted in Figure 10. We note that temperature field diminishes when values increase. Due to the definition of Prandtl number, enhancement of Pr means higher momentum diffusivity or lesser thermal diffusivity, which causes the reduction in the thermal boundary layer thickness. Figure 11 discloses variations of D u on temperature field. Both θ ( η ) and its layer thickness are enhanced via larger D u .

$Figure 7 β \beta versus f ′ ( η ) f^{\prime} (\eta ) .$

Figure 7

β versus f ′ ( η ) .

$Figure 8 γ \gamma versus θ ( η ) \theta (\eta ) .$

Figure 8

γ versus θ ( η ) .

$Figure 9 Q versus θ ( η ) \theta (\eta ) .$

Figure 9

Q versus θ ( η ) .

$Figure 10 P r versus θ ( η ) \theta (\eta ) .$

Figure 10

P _r versus θ ( η ) .

In Figure 12, the behaviour of the various values of S r on ϕ ( η ) is displayed. Here, the concentration field is a clearly mounting function of S r . The impact on concentration profile due to change in reaction rate parameter k is shown in Figure 13, where concentration reduces with rise in k. This is due to the fact that the higher reaction rate causes the thickness of concentration boundary layer. Figure 14 shows that the increase in Schmidt number causes a reduction in concentration and associated boundary layer concentration thickness.

$Figure 11 D u versus θ ( η ) \theta (\eta ) .$

Figure 11

D _u versus θ ( η ) .

$Figure 12 S r versus ϕ ( η ) \phi (\eta ) .$

Figure 12

S _r versus ϕ ( η ) .

$Figure 13 k versus ϕ ( η ) \phi (\eta ) .$

Figure 13

k versus ϕ ( η ) .

$Figure 14 S c {S}_{\text{c}} versus ϕ ( η ) \phi (\eta ) .$

Figure 14

S c versus ϕ ( η ) .

Table 1 represents convergence of our problem. It is noted that momentum, energy and concentration equations converge at 20th and 25th order of approximation for both energy and concentration, respectively. Table 2 gives a short comparison of Prandtl with the previous literature. An agreement is noticed. In Table 3, we present numerical results/data of relevant flux rates for numerous parameters. We note that the heat flux rate augments for higher r , Pr , Q and D u , whereas it reduces through higher values of f 0 , β , S c , k and S r . On the other side, the local mass flux rate shows anti-augmented behaviour by increasing f 0 , β , Pr and k, whereas it increases for r , Q , D u , S c and S r .

Table 1

Convergence of homotopy solutions when γ = 0.3 , f 0 = 0.5 , Q = 0.7 , β = 0.4 , S c = 0.5 , k = 0.3 , Pr = 0.4 , D u = 0.4 and S r = 0.6

Order of approximations	f ′ ″ (0)	− θ ′ (0)	− ϕ ′ (0)
1	3.199	1.195	0.8967
5	2.812	1.893	0.7127
10	2.783	2.959	0.7035
13	2.743	3.653	0.6542
17	2.724	4.472	0.6308
20	2.689	5.236	0.6190
25	2.689	5.794	0.6142
30	2.689	5.794	0.6142
50	2.689	5.794	0.6142

Table 2

Comparison with the previous literature

Pr	Nu_x present	Nu_x (Hayat et al. [63])
1.0	1.154	1.153
1.5	1.425	1.424
2.0	1.651	1.652

Table 3

Data for heat and mass flux rates

r	f ₀	Q	B	Pr	D _u	S _c	k	S _r	Re_xNu_x	Re_xSh_x
0.0	0.2	0.3	0.2	1.0	1.0	1.3	0.5	1.0	7.7787	1.6218
0.3									8.0360	1.6461
0.5									8.2525	1.6682
0.4	0.0	0.3	0.2	1.0	1.0	1.3	0.5	1.0	8.3977	1.8278
	0.2								8.1246	1.6544
	0.4								7.7862	1.4913
0.4	0.3	0.0	0.2	1.0	1.0	1.3	0.5	1.0	5.6733	1.0834
		0.2							6.9233	1.3843
		0.4							7.4585	1.5739
0.4	0.3	0.3	0.1	1.0	1.0	1.3	0.5	1.0	7.1137	1.4928
			0.4						6.8240	1.4117
			0.8						6.3151	1.3202
0.4	0.3	0.3	0.2	1.0	1.0	1.3	0.5	1.0	7.2526	1.4902
				1.3					7.3424	1.4336
				1.6					8.0656	1.4051
0.4	0.3	0.3	0.2	1.0	0.1	1.3	0.5	1.0	7.7030	1.4623
					0.4				7.8895	1.4952
					0.8				8.3994	1.5972
0.4	0.3	0.3	0.2	1.0	1.0	0.0	0.5	1.0	10.555	0.32503
						0.5			8.1359	0.86162
						1.0			8.1085	1.34311
0.4	0.3	0.3	0.2	1.0	1.0	1.3	0.0	1.0	5.6332	1.5234
							0.2		4.3562	1.3263
							0.4		3.2476	1.1452
0.4	0.3	0.3	0.2	1.0	1.0	1.3	0.5	0.0	8.5465	1.2563
								0.5	7.2365	1.4128
								1.0	6.5233	1.8563

5 Concluding remarks

This investigation deals with the impact of thermosoluted Marangoni convective flow towards a Riga surface. Chemical reaction of first order is considered. Main findings can be summarized as follows:

Behaviour of γ was the same on both velocity and temperature distributions.
The suction parameter caused a reduction in the velocity profile, whereas injection parameter increased the velocity profile.
Due to enhancement in Q , the dimensionless velocity distribution increased along the flow regime, whereas temperature distribution reduced.
Stronger effect of D u of liquid caused a noteworthy increment in local heat and mass flux rates.
Stronger effect of S r of fluid prompted a reduction in local heat transfer rate, but increased local Sherwood number along the flow.

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Received: 2019-12-07

Revised: 2020-05-30

Accepted: 2020-07-02

Published Online: 2020-09-07

This work is licensed under the Creative Commons Attribution 4.0 International License.

Thermosoluted Marangoni convective flow towards a permeable Riga surface

Abstract

1 Introduction

2 Coordinate system and problem formulation

3 Homotopy analysis method (HAM)

4 Analysis of the solutions

5 Concluding remarks

References

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