Influence of Convective Boundary Condition on heat and mass transfer in a Walters B fluid over a ve

Abstract

Keywords: Thermal Radiation; Similarity Variables; local Weissenberg number; thermal-diffusion; Homotopy Analysis Method

Introduction

The interaction of heat and mass transfer by natural convection in laminar boundary layer flows has received significant attention in the years past and extensively studied in the literature for both steady and unsteady phenomenon of Newtonian fluid due to its numerous applications in science and engineering field. Among the early

investigation revealed by Ali et’al. [1] shows that thermal radiation interaction enhances the wall shear stress as well as the surface heat transfer rate while investigating the natural convection-­radiation interaction in boundary layer flow over the horizontal surface. Arpaci [2] studied the effect of thermal radiation on the laminar free convection from a

*Corresponding author. *E-mail address: bjakinbo@fce-abeokuta.edu.ng This paper was recommended for publication in revised form by Regional Editor Omid Mahian Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

heated vertical plate. Recently, other researchers also made their contribution to the literature. Afify and El-Aziz [3] revealed that the heat transfer rates for both pseudoplastic and dilatant nano fluids are insensitive to change in viscosity for lower values of Biot number and declined by relatively strong convective heating with higher values of Biot number. Sarafraz et’al. [4] reported that the only influence of sub-cooling temperature is found to decrease the corresponding heat flux related to the onset of nucleate boiling. El-Aziz and Nabil [5] justified among others that the velocity slip leads to a faster rate of cooling of the stretching sheet only in the case of free convection flow regime. El-Aziz [6] investigated thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer by hydromagnetic three-dimensional free convection over a permeable stretching surface with radiation. The result shows among others that the maximum effect of the thermal-­diffusion and diffusion-thermo on the velocity occurs in the absence of the magnetic field when the plate is impermeable. El-Aziz [7] examined the radiation effect on the flow and heat transfer over an unsteady stretching sheet where it was pointed out that the effect of radiation parameter on the temperature distribution of a steady flow is more pronounced than that of unsteady flow. Abo-Eldahab and El Aziz [8] investigated the effect of blowing/suction on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption. Oahimire and Olajuwon [9] e­ xamined the effect of radiation absorption and thermo-diffusion on MHD heat and mass transfer flow of a micropolar fluid in the presence of heat source. The result shows among others that in the presence of the uniform magnetic field, an increase in the strength of the applied magnetic field decelerates the fluid motion along the wall of the plate inside the boundary layer, whereas the micro-rotational velocity of the fluid along the wall of the plate increase. Makinde [10] reported the effect of heat and mass transfer on MHD over a moving vertical plate with a convective boundary condition. However, the law of Newtonian fluid has been proved to be in good agreement with Newton’s second law of motion which work very well for air, water and other fluid delineated with Navier-Stokes and conservation of energy equation but failed while dealing with more complex fluid, especially with the emergence of viscoelastic fluid or polymeric liquid. The deficiency encountered in the theory of Newtonian fluid and recent development in Science and Technology with its numerous biological and industrial applications, such as polymer solution, paint ink, and cake butter e.t.c had made its studies interesting to all and recently studied in the literature. El-Aziz [11] found that a viscoelastic fluid is more sensitive to the variable fluid properties effect than a Newtonian fluid. Labropulu et’al. [12] examined the stagnation-point flow of the Walters’ B fluid with slip where the effect of condition and the viscoelasticity were to increase the velocity near the wall. Shivakumara et’al. [13] reported that the

effect of thermal modulation disappears at large frequencies in all the cases of thermal modulation while investigating the effect of thermal modulation on the onset of convection in Walters B viscoelastic fluid-saturated porous medium. Rana et’al. [14] and (Aggarwal and Verma [15]) reported that Walters’ (model B') visco-elastic fluid behaves like an ordinary Newtonian fluid due to the vanishing of the viscoelastic parameter. Pandey et’al. [16] investigated the characteristic of Walter’s B visco-elastic Nanofluid layer heated from below. It is reported among other that the Kinematic visco-elasticity parameter destabilizes the oscillatory convection and no has effects on stationary convection. Other authors like Thirumurugan and Vasanthakumari [17], Sharma et’ al. [18], Kango et’al. [19], Rana [20–21] also contributed to the literature about Walters’ B fluid. Going by the previous effort of other researchers in the literature, much attention has not been given to the impact of the Boit number on Walters’ B fluid. On that note, this work is set to examine the influence of Convective Boundary Condition on heat and mass transfer in a Walters’ B fluid over a vertical stretching surface with thermal-diffusion effect, having considered unaddressed to the best of our knowledge in the literature. The boundary layer equations governing the system are then solved via the Homotopy Analysis Method being a modern method for solving both linear and nonlinear differential equations.

Mathematical Formulation

In this article, the steady flow of heat and mass transfer over a vertical surface of Walters’ B viscoelastic fluid is considered. We assumed that the plate experienced heat by convection at a temperature Tf which provides heat transfer coefficient hf. A magnetic field B0 of uniform strength is executed in y-direction while the induced magnetic field is not taken into account due to the magnetic Reynolds number that is really small in the most fluid used in industries and the joule heating is neglected as it really small to affect the motion of free convection. The x-axis is taken along the direction of the main flow and y-axis normal to it. The temperature and concentration of the fluid is respectively considered as T and C, Cw is the plate surface concentration while T∞ and C∞ respectively denote the ambient temperature and concentration. The heat and mass transfer characteristics areconsidered in the presence of non-uniform heat generation/absorption and thermal-diffusion effect. (See Fig. 1). The stretching sheet is moving with a velocity uw(x) = ax and a > 0. On the account of elastic properties of the fluid which are important in extensional behaviors of polymer, the stress tensor S* for Walters’ B fluid is expressed as (See Nadeem et’al. [22]); S * = 2η0 − 2k0

= hf[Tf–T(x,0)], C(x,0) = Cw U(x,∞) = 0, T(x,∞) = T∞, C(x,∞) = C∞(6) The velocity components acting in x and y directions are respectively denoted as u and v, qr is the radiation heat flux, Cp is the specific heat at constant pressure, ѵ is the kinematic viscosity, βc is the concentration expansion coefficient, Dm is the mass diffusivity, α is the thermal diffusivity, g is the acceleration due to gravity, ρ is the density, Tm is the mean fluid temperature, σ is the fluid electrical conductivity, KT is

Figure 1. Flow configuration and coordinate system. δe is the convected δt derivative of a tensor quantity in relation to the material δ e ∂e motion, expressed as = + v.∇e – e∇V – (∇V)T.e, η0 = δ t ∂t ∞ ∫0 N(τ)dτ denotes the limiting viscosity at small shear rates, ∞ k0 = ∫0 τN(τ)dτ represents the short memory coefficient, while N(τ) is the distribution function with relaxation time τ. Keeping in mind the short memory, the term involving ∞ k0 = ∫0 τnN(τ)dτ (at n ≥ 2) is neglected. On the account of the assumption stated above and under usual Boussinesq’s approximation, the governing equation for Walters’ B fluid, in agreement with Mihra et’ al. [23] is given by

where k* is the means of absorption coefficient and σ* is the Sterfan-Boltzmann constant. We assumed that the temperature variation within the flow is such that the term T4 may be simplified as a linear function of temperature by expanding T4 in a Taylor series about T∞ and neglecting higher-order terms, gives

invoking (7) and (8) in equation (3), gives a modified equation of the form

∂u ∂u ∂2 u u +v =v 2 − ∂x ∂y ∂y  ∂3u ∂ 3 u ∂u ∂ 2 u ∂u ∂ 2 u  k0 u + v + − (3)  2 ∂y 3 ∂x ∂y 2 ∂y ∂x ∂y   ∂x ∂y σ B2u − 0 + g βT (T − T∞ ) + g βc (C − C∞ ) ρ

the thermal diffusion ratio, Q0 is the non-uniform heat genku (x ) eration/absorption coefficient defined by Q0 = w [A(Tw ρCρ xv – T∞) f '+ (T – T∞)B], where A connotes thespace-­dependent and B stands for the temperature-­dependent heat generation/absorption (see Hayat et’ al. [24]) while βT is the thermal expansion coefficient. Keeping in mind that the boundary layer is optically thick, therefore, the Rosseland approximation for heat transfer is considered (See Uddin et’al. [25]), hence, the radiative heat flux is modeled as

The appropriate boundary conditions for the problem are expressed as u(x,0) = uw(x) = ax, v(x,0) = 0, –k

∂T ∂T ∂2T 16σ * T∞3 ∂2T +v =α 2 + + Q0 ∂x ∂y ∂y 3k * ρCρ ∂y 2

The continuity equation (2) is automatically satisfied by the application of the stream function ψ defined by u=

In accordance with Almakki et’al. [26], the similarity solution for momentum, energy and concentration equations are obtained by the application of the appropriate transformation method defined as a , ψ = x av f (η), v T − T∞ C − C∞ θ (η) = , φ(η) = T f − T∞ CW − C∞

Here, η denotes independent similarity variable, θ(η) and φ(η) are dimensionless temperature and concentration respectively. Introducing (10) and (11) in (3), (9) and (5), result in f '''(η) + f(η) f ''(η) – (f '(η))2 + β[(f ''(η))2 – 2f '(η) f '''(η) + f(η) f (iv)(η)] –Mnf '(η) + λTθ(η) + λMφ(η) = 0

f(0) = 0, f '(0) = 1, θ'(0) = Bi[θ(0) – 1], φ (0) = 1 (15) (16)

where the prime symbol denotes the derivate with respect σ B02 ak to η, Mn = is the magnetic field, β = 0 is the local ρa v Grx Weissenberg Number, λT = is the thermal buoyancy (Re x )2 parameter, λM =

g βT (TW − T∞ )x 3 is the thermal Grash of Number, Gcx = v2 hf v g βC (CW − C∞ )x 3 is the solutal Grashof Number, Bi = 2 k a v

u(x ) is the Relyold Number, Pr = v 4σ * T∞3 vρCp is the prandtl number, Ra = is the radiation kk * k K T (T f − T∞ ) v parameter, Sr = is the Soret number, Sc = Tm (Cw − C∞ ) Dm is the Biot number, Rex =

is the Schmidtl number. Keeping in mind the engineering application of the study, the local skin friction coefficient, the local Nusselt number, and the Local Sherwood number are respectively presented in the form 2τ w xqw , Nu = , 2 ρuw k(Tw − T∞ ) xqm Sh = Dm (Cw − C∞ )

 4σ ∂T 4   ∂T  qw =  − k  ,  = −  ∂ y  y = 0  3K * ∂ y  y = 0

 ∂C  qm =  − Dm  ∂y  y = 0  By the introduction of (18) in (17) with the above transformation technique, an expression for local Skin-friction, the local Nusselt number, and the local Sherwood number formulated and given as 1

 ∂u  ∂2 u ∂2 u ∂u ∂v   +v 2 +2 τ w =  µ − k0  u  , ∂y ∂x ∂y   y = 0  ∂x ∂y  ∂y

− 4 Re x2 C f = (1 − β ) f ′′(0), Re x 2 Nu = −  1 + Ra θ ′(0),  3  (19) −

Re x 2 Sh = φ ′(0) where Rex = uw(x)/ѵ represents the Reynold number, τw is the shear stress along with the plate, qw is the surface heat and qm is the surface mass.

Homotopy Analysis Method

The solution of the differential equation has been the utmost priority of every researcher in mathematical modeling. They are solved by different methods among which are Shooting Techniques with Runge-Kutta method, Variation iteration method and Weighted Residual Method e.t.c. Homotopy Analysis Method is chosen and used over other methods being a modern Method and very efficient in solving both bounded and unbounded domain of differential equations. Subject to the rule of the solution and boundary conditions (15) – (16), we choose the initial guess (See Hayat et’al. [24], and Liao [27]) f 0 (η) = 1 − exp(−η), θ 0 (η) =

φ0 (η) = exp(−η) as the initial linear approximations of f(η), θ(η) and φ(η). The auxiliary linear operations Lf , Lθ, and Lφ are; L f [ f (η; r )] =

Lf[C1 + C2 exp(η) + C3 exp(–η)] = 0, Lθ[C4 + C5 exp(–η)] = 0, Lφ[C6 + C7 exp(–η)] = 0 (22) where C1, C2, ..., C7 are constants. ZERO ORDER DEFORMATION PROBLEM. (1 – r)Lf[f(η;r) – f0(η)] = rℏfHf(η)Nf[f(η;r), θ(η;r), φ(η;r)]

where r∈[0,1] is the same as the embedding parameter defined above. The increase in embedding parameter r from Zero to One corresponds to a variation of the function f(η;r), θ(η;r) and φ(η;r) from initial guess f0(η), θ0(η) and φ0 (η) to the solutions f(η), θ(η) and φ(η). Using Taylor series with respect to r, we have ∞

(1 – r)Lφ[f(η;r) – φ0(η)] = rℏφHφ(η)Nφ[f(η;r), θ(η;r), φ(η;r)]

∂θ (η; r ) |η = 0 = Bi[θ (η = 0; r ) − 1], ∂η φ(η = 0; r ) = 1

where L and N are called Linear and non-linear function respectively (for Algebra Equation) or Linear and Nonlinear operators (for differential Equations) and r∈[0,1] is the embedding parameter, with the following boundary conditions (See Akinbo and Olajuwon [28]). f (η = 0; r ) = 0,

1 ∂m f (η; r ) 1 ∂mθ (η; r ) , θ m (η) = , m m ! ∂η m ! ∂θ m 1 ∂mφ(η; r ) m ! ∂φ m

Obviously, the convergence of the series (33) is subject to the auxiliary parameter ℏ. Assuming ℏ is chosen such that the series (33) converge at r = 1, we have ∞

∂f (η; r ) |η →0 = 0, θ (η → ∞; r ) = 0 = φ(η → ∞; r ) (27) ∂η

∂2 f (η; r )  ∂f (η; r )  ∂3 f (η; r ) − + η − f ( ; r )  ∂η  ∂η 2 ∂η 3  ∂f (η; r ) ∂2 f (η; r ) β 2 − ∂η ∂η 2  2 ∂ 4 f (η; r )  ∂2 f (η; r )   − f (η; r )  ∂η2   ∂η 4 

Mth-ORDER DEFORMATION PROBLEM Following Hayat et’al. [29], the mth-order deformation are considered as follow (28)

2 1 + 4 Ra  ∂ θ (η; r ) + Pr ∂θ (η; r ) f (η; r ) +  3  ∂η2 ∂η (29) ∂f (η; r ) + Bθ (η; r ) = 0 A ∂η

∂2φ(η; r ) ∂φ(η; r ) ∂2θ (η; r ) + η + = 0 (30) Scf ( ; r ) Sr ∂η 2 ∂η ∂η 2

Lf[fm(η) – χmfm–1(η)] = ℏRfm(η), Lθ[θm(η) – χmθm–1(η)] = ℏRθm(η) Lφ[φm(η) – χmφm–1(η)] = ℏRφm(η)(35) having the following boundary conditions. ∂f m (η = 0; 0) ∂θ m (η = 0; 0) = 0, = ∂η ∂η (36) Bi[θ m (η = 0; 0)], φm (η = 0; 0) = 0

d 3 f m −1 (η) m −1 d2 f (η) + ∑ fn (η) m −1−2n − 3 dη d η n=0

[30], we consider the rule of coefficient ergodicity and rule of solution existence and choose the auxiliary functions as

d 2 f (η) d 2 f m −1−n (η) − β∑ n 2 dη dη2 n=0 df (η) + λT θ m −1 (η) + λ M φm −1 (η) Mn m −1 dη m −1

d θ m −1 (η) 4 + Rmθ (η) = 1 + Ra   3  dη2 m −1 (η) dθ Pr ∑ fn (η) m −1−n + (39) dη n=0 df (η) A m −1 Qθ m −1 (η) dη 2

m −1 d 2φm −1 (η) dφ (η) + Sc ∑ fn (η) m −1−n + 2 dη d η n=0

Convergence Of The HAM Solution

The convergence solution of this present investigation is considered in line with Liao [31], Akinbo and Olajuwon [28] suggestions. The non-zero auxiliary parameters ℏf, ℏθ and ℏφ help to monitor and control the convergence of the series solution. By the application of the following parameters Bi = λT = Sr = λM = β = 0.1, Sc = 0.62, Pr = 0.72, Ra = 0.7, Mn = 1, A = B = 0.01. The admissible values of ℏf, ℏθ and ℏφ are presented at the range where ℏ – curve becomes parallel which in turns give –1.2 ≤ ℏf ≤ –0.3, –1.3 ≤ ℏθ ≤ –0.4 and –1.5 ≤ ℏφ ≤ –0.4 for ℏf, ℏθ and ℏφ respectively as demonstrated in figures (2–4) below The exact approximate solution for the convergence of the governing equations which corresponds to momentum, energy and concentration equations are presented in table 1. The dimensionless equations meet the far-field domains and the iteration series converges at 20th – order

and χm = 0 for m ≤ 1, χm = 1 for m > 1 having the following as a general solution fm(η) = f*m(η) + C1 + C2exp(–η) + C3exp(–η)(41) θm(η) = θ*m(η) + C4 + C5exp(η)

where f*m(η), θ*m(η) and φ*m(η) represent the particular solution of equation (35). In agreement with Olubode et’al.

for momentum and concentration equations while the energy equation converges at 22th – order of iterations while table 2 agreed with Hayat et’al.[24]. However, the results are validated using Galerkin Weighted Residual

Table 2. Validation with Hayat et’al.[24] at Mn = 0, λT = 0, λM = 0 Hayat et’al.[24] Present Result β

Table 3. Validation/Numerical result for local Skin-friction coefficient, Local Nusselt number, and Local Sherwood number Parameters β

Discussion Of Results

In this study, computation analysis is carried out via the Homotopy Analysis Method (HAM) at 20th – order to meet the far-field boundary conditions. This is done by holding Bi = λT = Sr = λM = β = 0.1, Sc = 0.62, Pr = 0.72, Ra = 0.7, Mn =1, A = B = 0.01 constant for each varying parameter. We observed that almost all the values of the local Skin1

Friction Re 2 C quantitatively displayed negative as shown x

temperature within the boundary layer which in turn reduces the momentum layer thickness. In that wise, smaller values of Pr enhance the thermal conductivity and enable the heat to diffuse quickly from the heated surface than higher values. Figures (10–11) reveal the influence of magnetic interaction (Mn) on velocity and temperature profiles. It is noticed from fig.10 that higher values of Mn pioneer resistive forces called Lorentz force that resist the motion of the fluid and reduces the velocity profile and its boundary layer thickness. However, a reverse phenomenon is observed in the temperature of the fluid. An increase in Mn causes frictional heating across the boundary layer which increases the layer temperature of which its aftermath increases the thermal boundary layer thickness.

Figures (12–13) present the variation influence of local Weissenberg number (β) on velocity and temperature profiles. It is noticed from the fig. 12 that the fluid velocity decline for the large values of β. This result agreed with the expectation as higher values of β are to improve the viscoelasticity through the tensile stress which opposes the fluid velocity and reduces its layer thickness. The effect of improving viscoelasticity generates more heat within the thermal boundary layer which in turn improves the thermal layer thickness. The effect of Schmidtl number for most encountered chemical in the application is varied in Figure 14 on concentration field. At higher values of Sc, the diffusion properties of the fluid experienced downfall which in turn falls the concentration profile near the boundary layer as well as

Figure 14. Concentration profile for different values of Sc.

Figure 16. Concentration profile for different values of Sr.

concentration boundary layer thickness (See Akinbo and Olajuwon [32]). Figures (15–16) present the effect of Soret Number (Sr) on temperature and concentration profiles. Physically, we observed from figure 15 that an increase in Sr contributes to the falling of the temperature which ultimately reduces the thermal boundary layer thickness. However, the opposite phenomenon is observed in the concentration field which ultimately boosts concentration boundary layer thickness (see Fig. 16). Figure 17 addresses the influence of the radiation parameter (Ra) on the temperature profile. It is noticed that the temperature of the fluid improves due to the increasing

values of Ra. This is true as higher values of Ra magnify the conduction of heat transfer to thermal heat transfer of which its aftermath effect strengthens the thermal boundary layer thickness. Figures (18–19) presents the influence of the internal heat generation/absorption parameter (A,B) > 0 on the temperature profile. As expected, an increase in (A,B) corresponds to the enhancement of more heat within the layer, which in turns improves the temperature of the fluid and strength the thermal boundary layer thickness Figure (20) presents the influence of Boit Number (Bi) on the temperature profile. An increase in Bi constitutes strong convective heating within the boundary which

Figure 19. Temperature profile for different values of B. ultimately maximizes the thermal boundary layer thickness. However, this enhancement paves way for the penetration of thermal effect to the quiescent fluid.

Conclusion

In this work, Homotopy Analysis Method is employed to solve the three dimensionless equations corresponding to momentum, energy, and concentration which describe the influence of Convective Boundary Condition on heat and mass transfer in Walters’ B fluid over a vertical plate with thermal-diffusion effect. The results of various embedded parameters are analyzed through graphs and tables. The following conclusions were drawn from the results obtained

Figure 20. Temperature profile for different values of Bi. • Setting β = 0, Walters’ B model behaves like an ordinary Newtonian fluid • The skin-friction quantitatively displayed negative, indicating that the drag forces are exerted on the fluid by the plate which in turns impede the flow • Higher values of (A,B) enhance the temperature which in turn pioneer the lightening of the surface and enable the fluid to flow faster. • The tensile stress effect is magnified for large values of Weissenberg number which ultimately declines the velocity boundary layer thickness. • The temperature distribution and surface heat transfer are boosted when Radiation intensity improves. • Large values of Biot number magnifies thermal effect and allow its penetration to the quiescent

fluid. The strength of the Biot number contributes to the drying of the materials component in the Engineering field. Compliance with ethical standards Conflict of interest: The authors declare that they no conflict of interest.

Nomenclature

Mn Magnetic field β local Weissenberg Number λT Thermal buoyancy parameter λM Mass buoyancy parameter Bi Biot number Pr Prandtl number, Ra Radiation parameter Sc Schmidtl number A space-dependent heat generation/absorption B Temperature-dependent heat generation/Absorption Q0 non-uniform heat generation/absorption Coefficient Dm mass diffusivity Greek symbols α thermal diffusivity βc concentration expansion coefficient η Similarity variable θ dimensionless temperature ѵ kinematic viscosity ψ Stream Function ρ density qr Radiation heat flux Cp specific heat at constant pressure βT temperature expansion coefficient g acceleration due to gravity σ fluid electrical conductivity hf heat transfer coefficients

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of thisarticle.

Ethics

There are no ethical issues with the publication of this manuscript.

References

1. The article references can be accessed from the .pdf file.

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