Effects of Homogeneous and Heterogeneous Chemical Features on Oldroyd-B Fluid Flow between Stretching Disks with Velocity and Temperature Boundary Assumptions

Khan, Nargis; Hashmi, Muhammad Sadiq; Khan, Sami Ullah; Chaudhry, Faryal; Tlili, Iskander; Shadloo, Mostafa Safdari

doi:https://doi.org/10.1155/2020/5284906

Mathematical Problems in Engineering

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Recent Trends in Special Functions and Analysis of Differential Equations

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Volume 2020 | Article ID 5284906 | https://doi.org/10.1155/2020/5284906

Effects of Homogeneous and Heterogeneous Chemical Features on Oldroyd-B Fluid Flow between Stretching Disks with Velocity and Temperature Boundary Assumptions

Nargis Khan,¹Muhammad Sadiq Hashmi,²Sami Ullah Khan,³Faryal Chaudhry,⁴Iskander Tlili,^5,6and Mostafa Safdari Shadloo⁷

Guest Editor: Praveen Agarwal

Received21 Dec 2019

Revised10 Mar 2020

Accepted16 Mar 2020

Published30 Apr 2020

Abstract

This research endeavors the rheological features of Oldroyd-B fluid configured by infinite stretching disks in presence of velocity and thermal slip features. Additionally, the effects of homogeneous and heterogeneous chemical features are also considered. The transmuted flow equations are analytically solved with help of the homotopy analysis method (HAM). It is observed that the homogeneous chemical reaction parameter enhances the concentration distribution, while the heterogeneous reaction reduces the concentration profile. With implementations of temperature jump conditions, the heat transfer from the surfaces of both disks can be effectively controlled. The impacts of various dimensionless parameters are elaborated through graphs and tables.

1. Introduction

The fluid flow between stretching disks is the main motivation of investigators in recent years due to its leading applications in turbine engines, compression, mechanical components transient loading, semiconductor manufacturing, rotating wafers, injection modeling, power transmission, viscometer, lubrications, radial diffusers, geophysics, biomechanics, geothermal, oceanography, thrust bearings, etc. The usage of microdevices has many practical applications in different scientific areas such as surgery, biotechnology, electronic cooling, microchannels, heat pipes, and pumps. The heat and fluid flow characteristics are different for both microdevices and macroscale counterparts. This difference is constituted by velocity slip and temperature jump. The velocity slip is an important feature to analyze the behavior of microflows because no-slip boundary conditions are not applicable to the fluid flow in microelectro-mechanical-systems (MEMS). Also, no-slip boundary conditions show the impractical behavior for the cases such as corner flow, spreading of liquid on a solid substrate, and extrusion of polymer melts from a capillary tube. Therefore, no-slip boundary condition is replaced by slip boundary condition. Further, in the slip flow regime, temperature jump is significantly used to determine the heat transfer. Because of such applications of slip flow, many interesting contributions have been made by investigators in recent years. For instance, Zheng et al. [1] investigated the stretched flow of viscous fluid in presence of velocity as well as thermal slip features. The peristaltic transport of Carreau fluid through a channel with various flow features with application of velocity slip, temperature, and concentration jump has been inspected by Vajravelu et al. [2]. Khan et al. [3] discussed the double diffusion slip flow of viscous fluid over a vertical plate. Xiao et al. [4] presented a mathematical model for fully developed slip flow in a microtube gas problem. This interesting continuation contains the velocity slip of order two and the assumptions of temperature jump constraints. They claimed an effective change up to 15% in the local Nusselt number at room temperature. Similar slip effects have been performed by Rooholghdos and Roohi [5] for a nanoscale flat plate and a microscale cylinder. Another useful contribution regarding the gas flow associated with thermal slip conditions was examined by Le and Roohi [6]. The peristaltic transport of viscous fluid in an asymmetric channel in presence of velocity and temperature boundary conditions has been discussed by Sinha et al. [7]. El-Aziz and Afify [8] examined the heat transfer characteristics for slip flow of Casson fluid subjected to the induced magnetic field. Khan et al. [9] determined the analytical solution based on the Galerkin technique for an upper convected flow of Maxwell fluid in presence of slip features. Muhammad et al. [10] examined the entropy generation aspects in the flow of nanofluid under the action of the second-order slip. The investigation for fractional Maxwell fluid in presence of slip effects and porous medium was performed by Aman et al. [11].

The fluid flow encountered the heat transportation process conveying a diverse engineering and industrial significance in the metal cooling, petroleum engineering, chemical processing, food industries, thermophysical systems, fiber spinning, manufacturing of metallic sheets, and various nuclear processes. Besides this, the thermal performance of disc-shaped bodies had engaged many scholars because of its practical applications in the era of aeronautical sciences. Many engineering and mechanical processes like thermal power generation and heat transfer to automatic control systems encountered the applications of these phenomena. Due to such recurrent applications, several researchers investigate the flow over or flow between two disks. The initial contribution on this topic was led by Kármán and Uber [12] by considering viscous fluid flow between two infinite disks. This study was further extended by many researchers with different flow features. Hayat et al. [13] studied the heat transfer characteristics based on the Fourier law of conduction in third-grade liquid configured by two porous disks. Turkyilmazoglu [14] simulated the numerical solution of hydromagnetic fluid flow near the stagnation point subject to disk rotation. Heat transfer analysis in the hydromagnetic fluid flow caused by a rotating shrinking disk was also performed numerically by Turkyilmazoglu [15]. Soid et al. [16] applied the numerical technique to observe heat transfer phenomenon in viscous fluid for a radially stretching disk. Yin et al. [17] examined the flow thermal characteristics of nanofluid flow due to a rotating disk. Turkyilmazoglu [18] numerically examined the flow of Newtonian fluid through a vertically moving disk. Hashmi et al. [19] analytically explored the mixed convection flow of Oldroyd-B fluid placed between isothermal stretching disks. The idea of flow over stretching surfaces is extremely useful and involved a large number of practical applications in manufacturing processes [20–23]. The spontaneous idea of flow due to a moving surface was originally advised by Sakiadis [24, 25] which encouraged the investigators to pay attention in this direction. The exact solution for a stretching flow problem was successfully provided by Wang [26]. Another investigation in this direction has been suggested by Fang [27] which conferred the viscous fluid flow induced due to a stretched disk. In another attempt, Fang and Zhang [28] derived an exact solution based on the mathematical formulation of Navier Stokes equations modeled in cylindrical coordinates. In fact, such type of flow between two infinite stretching disks arises due to accelerated stretching velocity. Gorder et al. [29] discussed the axisymmetric flow between two infinite stretching disks. Mohyud-Din and Khan [30] implemented effects of nonlinear thermal radiation in flow of Casson fluid concedes between two stretching disks. Slip flow in presence of thermo-diffusion effects in flow of viscous fluid between stretching disks was suggested by Rashidi and his coworkers [31]. Analytical solution based on the homotopy analysis method for flow of viscous fluid through a stretchable disk has been depicted by Khan et al. [32]. In another investigation, Khan et al. [33] examined the viscous dissipation and joule heating effects on the axisymmetric flow of viscous fluid between stretching disks. Khan et al. [34] studied the entropy generation effects on flow of carbon nanotubes between two rotating and stretching disks. The heat transfer analysis based on Cattaneo–Christov heat flux expressions for the flow of micropolar fluid induced by a nonlinear stretching disk was focused by Doh et al. [35]. Renuka et al. [36] computed an analysis solution for the flow of nanofluid, additionally featuring entropy generation features induced by a stretchable spinning disk.

In the recent decade, the study of combined heat and mass transportation has inspired the scientists to examine various aspects of the simultaneous phenomenon due to its arising applications in the real-world problems like reacting systems, cooling towers, marine engineering, distillation columns, hydrometallurgical industry, crop damage via freezing, and copse of trees. The collaboration amongst homogeneous and heterogeneous responses happening on some catalytic surfaces is correlated with the production and employment of chemical species at diverse rates within the fluid and on the catalytic surfaces. Merkin [37] developed a very useful mathematical model to explore the relationship between a surface-based reaction and homogeneous and heterogeneous reactions. Another useful contribution is from Kameswaran et al. [38] where flow of nanoparticles is immersed in a porous medium with additional features of binary chemical reactions. Rashidi et al. [31] address the effects of homogeneous/heterogeneous on a peristaltic transport in a channel. Hayat et al. [39] implemented the effects of second-order velocity slip to examine the flow of chemical reactive viscous nanofluid induced by a permeable stretching surface.

In this present analysis, our focus is to evaluate the driven transport of Oldroyd-B fluid considered within two infinite stretching disks in presence of homogeneous and heterogeneous reactions. Unlike typical studies, here the idea of second-order velocity slip and temperature jump boundary conditions has been implemented. According to the literature survey, no attempt has been made by researchers for such analysis and is presented for the first time. The present flow problem is utilized in presence of applied magnetic field effects which are useful in the industry of metal-working, chemical reactors, plasma materials, modern metallurgical, oil exploration, and extraction of geothermal energy. The analytical solutions of such transmuted flow equations are determined by employing the homotopy analysis method [40–45]. The accuracy of this method is successfully obtained and expressed in a tabular form. Finally, the important feature effective parameters are graphically underlined and discussed for some velocity, temperature, and concentration profiles with technical relevance.

2. Mathematical Modeling

We consider a two-dimensional flow of Oldroyd-B due to infinite stretching disks. Let flow be axisymmetric and considered fluid be incompressible. The velocity slip and temperature jump are also considered at the walls of stretchable disks. A magnetic field with strength B₀ is imposed in z-direction. The effects of electric and induced magnetic fields are neglected. It is assumed that both lower and upper disks are maintained at temperature T₁ and T₂, respectively. Following Merkin and Chaudhary [46], the mathematical expressions repressing the homogeneous-heterogeneous reactions are expressed as

The isothermal, first-order reaction associated with a catalyst surface is represented aswhere α and β stand for concentrations of chemical species and A, B, k_c, and k_s denote the rate constants. In the present analysis, both reactions are treated as processes which are isothermal. The analysis is performed by opting a cylindrical coordinate (r, θ, z). All the involved expressions are independent of θ due to axisymmetry. The constitutive partial differential equations for Oldroyd-B fluid in presence of chemical reactions are expressed aswhere and are the radial and axial components of velocities, respectively, is the pressure, is the fluid density, stands for dynamic viscosity of fluid, represents the kinematic viscosity, and are the stretching constants, is the constant of relaxation, is the retardation time, T is the temperature, is the thermal diffusivity, and are the diffusion species coefficient of and

2.1. Slip Boundary Conditions

As it has been mentioned earlier that the present flow problem is assisted with slip boundary conditions. For the velocity profile, the derivation of second-order velocity slip is based on the expansion of Taylor series from the first-order Maxwell conditions which are generally expressed as

For the present analysis, we propose the following second-order boundary conditions:where and represent the stretching rates, is the tangential momentum accommodation coefficient, and denotes the molecular mean-free path. It is a well-established fact that the molecular mean-free path is assumed positive, i.e., and.

2.2. Temperature Jump Boundary Conditions

By using Taylor series second-order expansion for from the first order, Smoluchowski jump condition second-order jump conditions are proposed in [6] as follows:

The second-order temperature jump boundary conditions associated with the governing equations arewhere is the thermal accommodation coefficient and is the specific heat ratio. The other boundary conditions for the flow problem are prescribed by

Introducing the similarity variables,

In view of the above similarity variables, equations (4)–(10) yieldwhere is the wall stretching parameter, stands for the Reynolds number, is the Prandtl number, is the first-order velocity slip parameter, is the Hartmann number, is the second-order velocity slip parameter, is the first-order temperature jump parameter, stands for temperature jump parameter of the second order, represents the Schmidt number, is the ratio of the diffusion coefficient, denotes the Knudsen number, is the strength of the homogeneous reaction, and is the strength of the heterogeneous reaction and is defined as

The constant has been eliminated from equation (15) as the following procedure:in which is the Deborah number for relaxation time and for the retardation time. It is pointed out here that the diffusion coefficients of chemical species and are not equal in general. So, we remarked that constants and are of comparable size as a special case and subsequently and are equal, i.e., . Equations (16) and (17) lead to the following relation:

Following mathematical expressions are suggested for the wall skin friction coefficient, local Nusselt number, and local Sherwood number at both surfaces of disks:

3. Solution Methodology

To start our simulations, first we introduce the following initial guesses for velocity, temperature, and concentration profiles:with auxiliary linear operators:

The mathematical expressions associated with the zeroth-order deformation problem are defined aswhere , , and denote the auxiliary parameters and represents the embedding parameter. And,

The equations for the m-th deformations of the problem are

The series solution is computed iteratively for using MATHEMATICA software.

4. Convergence of Solution

In order to obtain the comfortable accuracy of the homotopic solution, the significance of auxiliary parameters cannot be denied. This task has been completed by preparing three -curves, organized for velocity, temperature, and concentration profiles for some dignified values of emerging parameters. The admissible values of such parameter guaranteed the convergence of the solution. The convergence of the derived series solution is controlled by auxiliary parameters , , and Therefore, we have sketched the -curves in Figure 1 to determine the admissible values of , , and These figures reveal that the convergence region lies within the domain , and

(a)

(b)

(c)

In Table 1, the computations have been performed to illustrate the convergence of the obtained solution for , and at various approximations. Close observations to the table suggest that accuracy of the solution has been obtained at the 15^th order of approximations.

5. Physical Interpretations of Results

In this section, the effects of various arising parameters on radial and vertical velocity components, pressure, temperature, and concentration fields are discussed with relevant physical significances.

5.1. Dimensionless Velocity and Pressure Profiles

Figure 2(a) shows the impact of the Hartmann number on the velocity vertical component by keeping other parameters fixed. The interface of stronger magnetic force is more valuable to decay the motion of fluid particles. A small increment in velocity was observed first which decreases up to a certain height. Physically, as increases, the Lorentz force boosts up which resists the flow of liquid due to which velocity decay occurs. Therefore, the presence of magnetic field combats the transport phenomena and subsequently diminishes the vertical velocity. The effects of wall stretching parameter on the velocity profile are shown in Figure 2(b). The vertical velocity component rises up with a variation of However, a change in the radial component is not similar to vertical components. Here, velocity increases at a specific range and then gradually decreases. Figure 2(c) delineates the significance of the Deborah number in terms of relaxation time on vertical and radial component of velocities. A rise in the vertical component of velocity is observed for larger values of the Deborah number; however, the radial component of velocity decreases smoothly after a small increment. The variation of material parameter on both vertical and radial velocity components is illustrated in Figure 2(d). The reverse trend is observed as compared to for both components. We observe from Figures 2(e) and 2(f) that when we increase of first- and second-order velocity slip constants the vertical velocity component also increases. Physically, with increase of velocity slip parameters, the stretching velocity affects the movement of fluid so velocity profiles get maximum values. Moreover, the amplitude of radial velocity increases up to a specific range due to the difference of the stretching rate. Figures 2(g) and 2(h) show that the skin friction coefficient increases with increase of both slip parameters. It is scrutinized from Figure 2(i) that pressure decreases in the whole domain by increasing values of the Hartmann number . It is found from Figure 2(j) that decay in pressure is observed by increasing the velocity slip parameter.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 2

(a)–(f) Graphs of vertical and radial components of velocity, (g)-(h) graphs of for skin friction, and (i)-(j) graphs of pressure for different values of . (a) Effects of the Hartmann number, (b) effects of the stretching parameter, (c) effects of the Deborah number of relaxation, (d) effects of the Deborah number of retardation, (e) effects of the first-order velocity slip parameter, (f) effects of the second-order velocity slip parameter, (g) influence of the first-order velocity slip parameter on the skin friction coefficient, (h) influence of the second-order velocity slip parameter on the skin friction coefficient, (i) Influence of Hartmann number on pressure (j) Influence of first-order velocity slip parameter on pressure.

5.2. Dimensionless Temperature Profile

In Figures 3(a) and 3(b), the dimensionless temperature is plotted to study the impact of the velocity slip parameter. The temperature decreases by increasing both velocity slip parameters. It is elucidated from Figures 3(c) and 3(d) that the distribution of temperature boosts up due to alteration of the first- and second-order temperature jump parameters. Physically, due to slip effect, more flow penetrates through the thermal boundary with an increase in temperature jump parameters. Figure 3(e) accomplishes the significance of the Prandtl number on the temperature profile. The impression declined the temperature of the fluid effectively. The dimensionless number depends upon thermal diffusivity which decreases by increasing Therefore, a decline in the temperature field is observed. Thus, higher values of correspond to lower thermal diffusivity and subsequently declining temperature distribution. Figure 3(f) exhibits the dominant effect of the Hartmann number on the temperature profile. As expected, the temperature of fluid increases by increasing . Physically, the applied magnetic field produces the Lorentz force, which creates a drag force which has a tendency to enhance the temperature of the fluid between both disks.

(a)

(b)

(c)

(d)

(e)

(f)

5.3. Dimensionless Concentration Profile

Taking into account of the concentration profile , the effects for various parameters are encountered. First, we consider the variation of the homogeneous reaction on An increase in results in diminishing of the concentration profile (Figure 4(a)). Figure 4(b) shows the consequence of heterogeneous reaction parameter on the concentration profile. The rate of mass transfer is enhanced by increasing Figure 4(c) shows that the rate of mass transfer solely decreases by varying Schmidt number . has an inverse relation with molecular diffusivity which decreases by increasing . The variation of different values of the strengths of the homogeneous parameter and heterogeneous reaction parameter on wall concentration on both disks is shown in Figures 5 and 6, respectively. These figures indicate that values of and increase by increasing while contradictory behavior is noted for .

(a)

(b)

(c)

(a)

(b)

(a)

(b)

5.4. Local Nusselt Number

Table 2 aims to elaborate the iterative numerical variation in the local Nusselt number against involved fluid parameters. We found that with the increase in the velocity slip parameter, the temperature profile at the lower disk increases. The heat transfer rate decreases by increasing the Hartmann number at the lower disk. However, opposite values for M are observed for the upper disk. Such observations are made as both disks are stretched with different velocities.

6. Conclusions

In this work, a chemically reactive flow of Oldroyd-B fluid subject to stretchable disks is considered in presence of homogeneous and heterogeneous chemical reactions. The homogeneous-heterogeneous reactions are considered in the concentration equation. The physical features are visualized for various involved parameters graphically. The important observations are summarized as follows:(i)The velocity distribution increases with variation of slip parameters while it decreases with the Deborah number for retardation time.(ii)The concentration distribution declines with increment of the Schmidt number and the homogeneous reaction while effects of the heterogeneous reaction parameter are quite reverse.(iii)The temperature distribution increases by increasing the Hartmann number while lower temperature distribution is observed for larger values of the Prandtl number.(iv)The presence of first- and second-order velocity slip results in an increment in the wall shear stress.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

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Copyright © 2020 Nargis Khan 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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