Physical Aspects of Homogeneous-Heterogeneous Reactions on MHD Williamson Fluid Flow across a Nonlinear Stretching Curved Surface Together with Convective Boundary Conditions

Ahmed, Kamran; Akbar, Tanvir; Muhammad, Taseer

doi:https://doi.org/10.1155/2021/7016961

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2021 | Article ID 7016961 | https://doi.org/10.1155/2021/7016961

Physical Aspects of Homogeneous-Heterogeneous Reactions on MHD Williamson Fluid Flow across a Nonlinear Stretching Curved Surface Together with Convective Boundary Conditions

Kamran Ahmed,¹Tanvir Akbar,¹and Taseer Muhammad²

Academic Editor: Amer Rasheed

Received08 Sept 2021

Revised20 Oct 2021

Accepted22 Oct 2021

Published15 Nov 2021

Abstract

This article is concerned with the fluid mechanics of MHD steady 2D flow of Williamson fluid over a nonlinear stretching curved surface in conjunction with homogeneous-heterogeneous reactions with convective boundary conditions. An effective similarity transformation is considered that switches the nonlinear partial differential equations riveted to ordinary differential equations. The governing nonlinear coupled differential equations are solved by using MATLAB bvp4c code. The physical features of nondimensional Williamson fluid parameter , power-law stretching index , curvature parameter , Schmidt number , magnetic field parameter , Prandtl number , homogeneous reaction strength , heterogeneous reaction strength , and Biot number are presented through the graphs. The tabulated form of results is obtained for the skin friction coefficient. It is noted that both the homogeneous and heterogeneous reaction strengths reduced the concentration profile.

1. Introduction

The fluid is subdivided into two main categories: non-Newtonian fluid and Newtonian. One of its types of non-Newtonian fluid is a shear-thinning (pseudoplastic) fluid [1]. Pseudoplastic takes attention due to its large commercial applicability. Polymer solutions as well as molten polymers, complex fluids, and suspensions like nail polish, whipped cream, blood, ketchup, and paint are the industrial and everyday applications of pseudoplastic fluids. Gogarty [2] considered the porous media to study the rheological properties of pseudoplastic fluids. Researchers use different models to investigate the behaviour of non-Newtonian fluid like the Ellis model, Williamson model, cross model, Carreaus model, and the power-law model, but the Williamson model for fluid flow takes more attention for the study of pseudoplastic fluid. Williamson [3] provides an experimentally verified model for the analysis of pseudoplastic fluids. In the last decade, several researchers [4–9] investigated the behaviour of pseudoplastic fluid by using the Williamson fluid model. Hayat et al. [10, 11] used the Homotopy analytical method to examine the impact of joule heating, thermal radiation, and Ohmic dissipation in the two-dimensional flow of Williamson fluid over a stretching surface.

From the last two decades, several investigators have focused on non-Newtonian fluid across nonlinear and linear stretching of a plate, flat surface, cylinder, or disk [12–16]. Flow across a curved surface is firstly introduced by Sajid et al. [17]. Later on, Abbas et al. [18] analyzed the heat transfer flow of MHD fluid across stretching curved surface. Ahmad et al. [19] examined the boundary layer flow across a curved surface embedded in a porous medium. Sanni et al. [20] investigated the flow of viscous fluid due to a nonlinear stretching curved surface. The effect of mass and heat transfer across a curve-shaped surface is numerically examined by Ramana et al. [21]. Saleh et al. [22] investigated the flow of unsteady micropolar fluid flow over a permeable curved stretching/shrinking surface. The transfer of heat and mass of an electrically conducting micropolar fluid with MHD effect across a curved stretching sheet is presented by Yasmin et al. [23]. Recently Kamran et al. [24] numerically investigated the flow of Williamson fluid over an exponential curved stretching surface. Gowda et al. [25] used Li (KKL) correlation, Koo–Kleinstreuer, and modified Fourier heat flux model to investigate the nanofluid flow over a curved stretching sheet.

Chemical reactions involve h-h (homogeneous and heterogeneous) reactions in many chemically reacting processes. In homogeneous reactions, reactants and products are in the same phase, whereas the reactants involve two or more phases in heterogeneous reactions. Williamson fluid flow across a stretching cylinder with h-h reactions is studied numerically by Malik et al. [26]. The effect of h-h reactions on boundary layer flow across a nonlinear stretching curved surface with convective boundary conditions is studied by Saif et al. [27]. Khan et al. [28] utilized the concept of h-h reactions on Oldroyd-B fluid flow between stretching disks. Ahmed et al. [29] explored the stagnation flow of h-h reactions in single-walled carbon nanotubes nanofluid towards a plane surface. Ali et al. [30] analyzed the effect of mass and heat transport on cross fluid with h-h reactions to investigate the behaviour of a chemical process. Javed et al. [31] examined the melting heat transfer with radiative effects and h-h reaction in thermally stratified stagnation flow embedded in a porous medium. Sreedevi et al. [32] implemented the FEM to examine the impact of h-h reactions on Maxwell nanofluid with heat and mass transfer flow over a horizontal stretched cylinder.

In Williamson fluid flow problem, researchers [33–39] used different similarity transformations to change governing nonlinear partial differential equations into ordinary differential equations. The literature survey shows that the Williamson fluid model is more effective than other models for studying pseudoplastic fluid, but the researcher studied locally similar Williamson fluid parameter. The MHD Williamson fluid flow across a curved surface with convective boundary condition and h-h reactions is not studied until now.

The objective of this study is to examine the effect of h-h reactions with convective boundary conditions on MHD Williamson fluid over a curved surface. This problem has many applications in engineering processes such as wire drawing, continuous casting, metal extrusion, paper production, glass fibre production, hot rolling, and crystal growing. Anappropriate similarity transformation [20, 27] is used to convert governing PDEs to ODEs.By fixing the value of nonlinear index parameterm = 1/3,which provides an entirely similar Williamson fluid parameter. The impact of different parameters, i.e., curvature parameter , nondimensional Williamson fluid parameter , Biot number , magnetic field parameter , Schmidt number , Prandtl number , homogeneous reaction strength , heterogeneous reaction strength , power-law stretching index on velocity, pressure, temperature, and concentration profiles, is presented through the graphs, whereas the results of skin friction coefficient and Nusselt number are depicted in a tabulated form.

2. Formulation of Problem

We examined the 2-dimensional steady, incompressible, fully developed magnetohydrodynamic Williamson fluid flow across a nonlinear stretching curved surface having radius . We consider coordinate system. The -axis represents the flow direction, whereas the radial direction is taken along . The stretching of the curved surface is along the -axis with velocity where is the initial stretching rate. The variable magnetic field is applied in the radial direction. For h-h reactions, we consider two chemical species A and C, respectively. Figure 1 shows the geometry of flow. In the case of Williamson fluid flow, Cauchy stress tensor is defined as in which represents extra stress tensor and defined as where ,, , and are the first Rivlin–Erickson tensor; positive time constant; limiting viscosities at zero and infinite shear stress rates, respectively; and is defined as , whereas . Here we consider the case in which and . The homogeneous reaction for cubic autocatalysis with two chemical species and is represented by the equation below:

For cubic autocatalysis, the heterogeneous reaction on the catalyst surface is mathematically represented as follows:where and are the rate constants and a and c represent the concentrations for chemical species A and C. Under these conditions, the governing boundary layer equations [9, 17–24, 26–32] are

The accompanying boundary conditions are [20, 27]where symbolizes the velocity component in the s direction and describes the velocity component in the r direction. Furthermore, represents the density, and are the kinematic viscosity and the electrical conductivity, respectively.

We introduce the following dimensionless variable transformations:where shows the dimensionless velocity, shows the similarity variable, and is the pressure.

Using equation (10) in equations (3)–(9), equation (3) satisfies identically, we getwhere represents the parameter for Williamson fluid; magnetic field parameter is expressed as , is the Schmidt number, is the Prandtl number, is the homogeneous reaction strength, is the Biot number, is the proportion of diffusion coefficients, and is the heterogeneous reaction strength.

Abolishing from the equations (11) and (12) produces the following equation:

When , then , we have

Thus equations (14) and (15) take the form

Along with boundary conditions

The local skin friction coefficient and Nusselt number are defined as

Shear stress and heat flux near to the surface are mathematically represented as

Using equation (10) in equation (21), equation (20) takes the formwhere .

3. Solution Procedure

The coupled nonlinear system of ODEs, (11), (13), (17), and (18), along with boundary conditions (16) and (19), are solved by using built-in MATLAB code bvp4c. In order to find out the solution of coupled ODEs, first of all, we rewrite equations (11) and (13), (17) and (18), with boundary conditions (16), (19) as an equivalent system of first-order differential equations by using the substitutions , . In the next step, we code these systems of first-order ODEs and the boundary conditions with function names “ex1ode” and “ex1bc” in MATLAB. Furthermore, we choose the interval of integration from 0 to 4 and divided it into 30 mesh points which formed the initial guess structure. Mesh selection and error control are built on the residual of the continuous solution. The relative error tolerance considered in this study is . Finally, we call “bvp4c” function

The “deval” built-in MATLAB function is used to evaluate the solution at a specific point. The Comparison of skin friction coefficient from the present study with the work of Kumar et al. [38] and Sajid et al. [17] by fixing and shows the acceptable result in Table 1. We assumed , , , , , = 0.5, , and in all discussions.

4. Results and Discussion

The physical interpretation of the results obtained in the above section is presented here. The results of different physical parameters are analyzed graphically and presented in the tabular form. The skin friction and Nusselt number are presented through the table, and the impact of power-law stretching index magnetic field parameter , the radius of curvature , Williamson fluid parameter , Schmidt number , homogeneous reaction strength , heterogeneous reaction strength , Prandtl number , Biot number on velocity , temperature , pressure , and concentration profiles is presented through the graphs.

Table 2 depicts the effect of , , , , , and , on and . As we increase the value of , there is an increase in and . The graph shows that increment in Williamson fluid parameter reduces the value of skin friction coefficient and Nusselt number because the collision of the fluid particles slows down. By the increase in the value of K, and both decrease. The higher the value of , the higher the increase in the value of and the decrease in the value of . By raising the value of and , is rising for both parameters, but remains unchanged because the velocity profile is independent of these parameters.

The effect of on is observed in Figure 2(a). The graph exhibits the obvious results that, by an increase in the value of , the radius of the surface enhances and hence boosts the velocity of the fluid. Figures 2(b)–2(d) display the changes of , , and profiles, for increasing radius of curvature . It is seen that there is a decrease in , , and for larger values of . This is because rising curvature makes the curved surface flat. The influence of on , , , and is shown in Figures 3(a)–3(d). It is noted that fluid velocity, pressure, and concentration for higher values of reduced. However, the temperature profile increases for greater values of M. Practically, this effect is shown when the magnetic field is applied perpendicular to the flow direction, which creates resistance for the fluid flow. Figures 4(a)–4(d) illustrate the effect of nonlinearity parameter on , , , and profiles. It is easy to discern that the velocity, temperature, and pressure decrease, whereas concentration profile increases by increasing m which is as expected practically.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

Figure 5 portrays the impact of on velocity and temperature profiles for different values of . We see that by raising the values of λ, the fluid velocity declines, as depicted in Figure 5(a). Figure 5(b) presents the effect of on the temperature profile. It is easy to detect an increase in the temperature of the fluid particles for a better value of . The Williamson fluid parameter is defined as the proportion of relaxation time to retardation time. The increasing values of Williamson fluid parameter enhance the relaxation time; as a result of this, the fluid particles need additional time to reinstate their previous path.

(a)

(b)

Figure 6(a) shows the effect of on . For better values of γ, the increment in the temperature profile is observed. Consequently, Biot number gives a simple index of the ratio of heat transfer resistance inside and at the surface of geometry. The nature of temperature for different values of is well portrayed in Figure 6(b). As thermal conductivity is inversely proportional to , a decrement is observed for higher values of . Hence for conducting fluids, is responsible for enhancing the cooling rate.

(a)

(b)

Figure 7(a) shows the effect of homogeneous parameter on concentration distribution . From this figure, we observe that the concentration profile decreases as there is a rise in . Hence it is concluded that reaction rate dominates diffusion coefficients. Figure 7(b) shows the strength of a heterogeneous reactionon. The plot shows that the concentration of the fluid and associated boundary layer thickness is reduced for higher values of . Figure 8 illustrates the effect of on . is the ratio of momentum to mass diffusivity. An increase in Sc means momentum diffusivity is dominated,resulting an increment in the concentration profile.

(a)

(b)

5. Conclusions

In this article, we have modelled the MHD flow of Williamson fluid across a nonlinear stretching curved surface with h-h reactions and convective boundary conditions. The governing partial differential equations are adapted into ordinary differential equations by using suitable similarity transformation. The impact of involving parameters on pressure, velocity, concentration, and temperature profiles are examined. Some examples of this problem in biochemical science and engineering processes are blood flow, plasma flow, lubrication flow, wire drawing, continuous casting, metal extrusion, paper production, glass fibre production, hot rolling, crystal growing, etc. The main consequences are noted below:(i)Williamson fluid parameter becomes globally similar byfixing .(ii)Skin friction coefficient rises for , , whereas it decreases for and K.(iii)For better values of m, M, , and , increases for and decreases for M, λ, and K.(iv)As we increase the values of , , and , decreases for , , and but increases for .(v)As we increase , , λ, γ, and , θ (η) decreases for, and , whereas it increases for , λ, and .(vi)As we increase the values m, M, , and , the concentration profile settles at lower values, whereas it settled as higher values for and .(vii)Pressure decreases for higher values of m, , and .

Nomenclature

:	Chemical species
:	Velocity component
:	Wall shear stress
:	Dynamic viscosity
:	Kinematic viscosity
:	Power-law stretching index
:	Williamson fluid parameter
:	Rate constant
:	Specific heat capacity
:	Velocity at surface
:	Similarity variable
:	Radius
:	Dimensionless temperature
:	Local Nusselt number
:	Curvature parameter
:	Prandtl number
:	Homogeneous reaction parameter
:	Skin friction coefficient
:	Dimensionless velocity
:	Coordinate axes
:	Concentrations
:	Fluid density
:	Dimensional pressure
:	Heat flux at the wall
:	Ambient fluid temperature
:	Convective coefficient
:	Convective surface temperature
:	Positive constant
:	Diffusion coefficient
:	Heterogeneous reaction parameter
:	The ratio of the diffusion coefficient
:	Schmidt number
:	Dimensionless concentration
:	Biot number
:	Local Reynolds number
h-h:	Homogeneous and heterogeneous
:	Thermal conductivity
T:	Temperature.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Kamran Ahmed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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