Vieta-Lucas Collocation Technique for Examination of the Flow of Casson Fluid over a Slippery Stretching Sheet Which Is Impacted by Thermal Slip, Ohmic Dissipation, and Variable Thermal Conductivity

Eid, A.; Khader, M. M.; Megahed, Ahmed M.

doi:https://doi.org/10.1155/2023/8723343

Journal of Mathematics

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 8723343 | https://doi.org/10.1155/2023/8723343

Vieta-Lucas Collocation Technique for Examination of the Flow of Casson Fluid over a Slippery Stretching Sheet Which Is Impacted by Thermal Slip, Ohmic Dissipation, and Variable Thermal Conductivity

A. Eid,^1,2M. M. Khader,^3,4and Ahmed M. Megahed⁴

Academic Editor: M. M. Bhatti

Received01 Dec 2022

Revised15 Jan 2023

Accepted05 Apr 2023

Published25 Apr 2023

Abstract

This work aimed to present the influence of the magnetic field and Ohmic dissipation on the non-Newtonian Casson fluid on a vertical stretched sheet to numerically solve the problem. Here, the variable thermal conductivity is taken as a linear function of temperature. Electric fields, thermal slip, and viscous dissipation effects are taken into consideration. A collection of physical conditions on the sheet’s enclosing wall and the momentum and heat transport processes are expressed as partial differential equations (PDEs). Some of the similarity transformations are used to convert the collection of PDE into a system of ordinary differential equations. This system is numerically treated by implementing the Vieta-Lucas spectral collocation method. Some observations are made for the investigation of method convergence. The effect of some different parameters on the velocity and temperature profiles is graphically represented. Additionally, this area of study has significant practical applications in a variety of industries, including paper production, thermal power generation, nuclear reactors, cooling of metallic sheets, glass fiber, and lubrication.

1. Introduction

The flow of non-Newtonian fluids (n-NFs) has gained a great deal of interest in recent decades because it has a wide range of important technological applications. This type of fluid has many uses in industrial applications, including the widespread use of molten plastics, blood, slurries, and paints. Non-Newtonian fluids respond to stress in different ways: some grow more solid and some become more fluid. Depending on the amount of tension applied or the length of time the stress is applied, the properties of non-Newtonian fluids may alter. These n-NFs are divided into distinct classes based on their different properties. As a result, the expanding kinds of n-NFs have recently aroused the curiosity of researchers. The n-NFs models of many distinct types have been studied, including the power-law model [1], the viscoelastic model [2], the Carreau model [3], the Maxwell model [4], the Williamson model [5], the micropolar model [6, 7], the Powell-Eyring model [8], and the Sisko model [9].

The Casson model class, which combines shear-thinning with shear-thickening, is the most popular of the aforementioned non-Newtonian models. If the fluid’s shear stress does not exceed its yield stress, the Casson fluid model behaves similar to a solid. Researchers became particularly intrigued by Casson fluids because of their uses in the manufacture of biological fluids, pigments, and China clay [10]. There are a number of intriguing articles pertaining to the Casson fluid flow model [11–13]. Due to their many applications in fluid mechanics, biology, viscoelasticity, engineering, and physics, ordinary and PDEs have been the subject of many investigations. As a result, the solutions of ordinary differential equations (ODEs) of physical relevance have received a lot of attention. We used the Vieta–Lucas spectral collocation method (SCM) to solve the nonlinear ODEs that regulate the physical problem quantitatively.

Having carefully read the aforementioned works and to the best of our knowledge, for the non-Newtonian Casson fluid flow with viscous dissipation and slip velocity, no attempt has been made to obtain the numerical solutions by using the Vieta–Lucas spectral collocation method. Due to this, the goal of this research is to investigate the approximate solutions for the model under study by implementing the SCM based on the shifted Vieta–Lucas polynomials (VLPs) [14]. The most famous advantage of these methods is their capability to generate accurate outcomes with a very small degree error of freedom [14]. The orthogonality property of the shifted Vieta–Lucas polynomials is used to approximate functions on its domain. These polynomials have a main and important role in these methods for ODEs [15]. Many researchers used and implemented these polynomials to solve numerically many problems, such as in [16], they were used to solve the nonlinear generalized Benjamin–Bona–Mahony–Burgers equation, and in [17], they were used to solve the sinh–Gordon equation. The study’s novelty and purpose stem from the fact that it is the first of its type to implement the proposed numerical technique to solve the proposed model.

2. Mathematical Modeling of the Problem

We considered the flow of a non-Newtonian Casson fluid across the vertical rough stretching sheet that is electrically conducting, steady, and incompressible. In the energy equation relationship, the Ohmic dissipation, and viscous dissipation properties are taken into account. The motion of the Casson fluid is produced by velocity . The elastic sheet’s plane is the -axis selection, and the plane’s normal to it is the -axis as depicted in Figure 1. In this study, we assumed that the fluid flow is constrained by the presence of magnetic field of strength , whereas the fluid temperature is affected by the viscous dissipation phenomenon. Where and are the velocity components in the and directions, respectively. Likewise, the density of the Casson fluid is symbolized by . The controlling nonlinear expressions are as follows under the aforementioned assumptions [18, 19]:where are, respectively, the Casson parameter, fluid viscosity, thermal conductivity, electrical conductivity of fluid, magnetic field strength, temperature of fluid, specific heat, kinematic viscosity, gravitational acceleration, and the coefficient of thermal expansion. The subjected boundary constraints are as follows [20]:

Here, it is crucial to highlight that this study considers the slip velocity phenomenon, as shown by the first portion of equation (4), whereas the sheet is impermeable according to the second portion of the same equation, where denotes the thermal slip coefficient and denotes the slip velocity coefficient, both of which have the dimension . The dimensionless of velocity and temperature exists if we introduce the following transformations [21]:where is the dimensionless temperature and is the dimensionless velocity. Furthermore, the accurate estimation of the heat transfer mechanism can be achieved for the assumption of fluid variable thermal conductivity. Therefore, here, the thermal conductivity is assumed to obey the following relation [22]:where is the thermal conductivity parameter and is the ambient thermal conductivity. Evidently, equation (6) already meets the continuity equation (1). Utilizing the previously stated relations (6) and (7) leads to the following governing equations:with the following applicable boundary constraints:where the mixed convection parameter, slip velocity parameter, local electric parameter, magnetic parameter, thermal slip parameter, Prandtl number, Eckert number are defined, respectively, as follows:

The importance physical quantities in this study are the skin friction coefficient and the local Nusselt number which is given by the following equation:where is the local Reynolds number.

3. Procedure of Solution

3.1. Approximate the Solution

In this subsection, we give some definitions and properties of the shifted VLPs to solve the problem under study in . We used the transformation to generate a new orthogonal family of the VLPs on and so-called the shifted VLPs which is denoted by and may be given as follows [23, 24]:

The polynomials yield from the following recurrence relation:where, . It is easy to find that .

The function may be approximated by as a finite sum with the first -terms as follows:

Here, we used an approximate formula of of the approximated function defined in form (18), where the authors in [25] derived this formula in the following form:

For more details about these polynomials and the convergence analysis of approximations (18) and (19), see [25].

3.2. Procedure Solution Using SCM

We are going to apply the proposed method (SCM) to solve numerically systems (9) and (10). We approximated the unknowns and by and , respectively, in the following form:

By substituting from formula (16) and the formula (15) in equations (9) and (10), we can get the following equations:

We collocate the previous equations (21) and (22) with of nodes to obtain the following nonlinear system of algebraic equations:

We substituted from equation (20) in the boundary conditions (11) and (12) to expressed in the following equations:

Equations (23)–(26) construct a system of algebraic equations. Then, we used the Newton iteration method to solve this system for the unknowns , where , to obtain the approximate solution.

4. Code Verification

By completing the particular validations shown in Table 1, this section checks the technical correctness of the completed numerical code. For the case of Newtonian fluid , Table 1 compares the numerical values of that were reported by Hasnain et al. [26] for various values of when . The findings are seen to be in very good agreement with the published studies. Therefore, Table 1 ensures that the current numerical solutions are validated against earlier literature.

5. Main Results

The following figures and table describe many pertinent parameters related to dimensionless velocity, dimensionless temperature, local Nusselt number, and skin friction coefficient. Firstly, the range of the parameters that govern our model can be mentioned as and . Therefore, the fixed values for the same controlling parameters can be employed through the graphical illustrations: , and . The fluid velocity is considerably suppressed when subjected to an increased magnetic number, as seen in Figure 2(a). In actuality, a stronger magnetic field produces more drag force, which slows down the movement of the fluid. Therefore, it is projected that this drag force will have an enhanced effect on the thermal field at high magnetic number values as noted in Figure 2(b). Physically, adding a magnetic field increases magnetic irreversibility; however, Lorentz forces reduce fluid flow velocity and increase fluid temperature.

(a)

(b)

Figures 3(a) and 3(b) showed how the local electric parameter affected the profiles of velocity and temperature. The increasing value of the local parameter in these graphs demonstrates an increase in the momentum and temperature fields of the Casson fluid. As a result, at higher values of the local electric parameter, the boundary layer (BL) thickness increases. Physically, the presence of an electric field may provide an induction force for the fluid particles, causing the fluid flow motion within the boundary layer to increase.

(a)

(b)

The effects of the mixed convection parameter on the momentum field and the thermal field are presented in Figure 4 for different quantities of the parameter. Figure 4(a) shows that at higher values of , the velocity curves are of a rising nature through the BL region. Furthermore, the thermal field is slightly diminished by the same parameter as observed from Figure 4(b). Physically, there is an increase in fluid velocity and a minor reduction of the temperature distribution due to the strong mixed convection parameter, which works as a pressure gradient and dominates over the resistance.

(a)

(b)

Figure 5 shows, respectively, how and fields are affected by the Casson parameter . Physically, the Casson fluid behaves more similar to a Newtonian fluid when the values of are increased. Therefore, increasing results in a declining phenomenon in and .

(a)

(b)

Figure 6 shows the characteristics of and under the impact of the slip velocity parameter . Clearly, greater resistive forces via the fluid layers are produced by slip velocity parameter with bigger values, which reduce fluid velocity and marginally raise the temperature. Physically, it is obvious that as the slip velocity impact rises, the fluid flow becomes more impeded, limiting the thickness of the boundary layer and the temperature distribution.

(a)

(b)

Figures 7(a) and 7(b), respectively, show the thermal field’s behaviors for the ranges of the thermal conductivity parameter , and the Eckert number were taken into consideration. This analysis shows that increasing the values of and improves the temperature distribution. Additionally, the thickness of the thermal BL increases as the same parameters are gradually improved. Physically, an increase in the Eckert number increases the thermal system’s kinetic energy. As a result, the thermal field grew. Also, the fluid receives thermal energy from the thermal conductivity characteristic. More thermal energy will travel through the fluid when this value rises.

(a)

(b)

The curves in Figure 8 show how temperature profiles have changed over time about changes in the Prandtl and thermal slip parameters. Figure 8(a) shows temperature profiles for three different values of (,) at , while Figure 8(b) shows temperature profiles about the thermal slip parameter at . This graphical illustration shows that both parameters show a discernible diminishing in the sheet temperature and for the temperature profiles . Physically, these observations are made because the thermal properties between the heating fluid and the solid surface are greatly weakened by the advanced values of the Prandtl and thermal slip parameters.

(a)

(b)

To observe the behavior of parameters affecting the local Nusselt number and the skin friction coefficient , Table 2 is now generated. This table makes it abundantly clear that the and the have inverse relationships with the magnetic number. The local Nusselt number decreases as both the thermal slip parameter and the magnetic number rises, yet the upsurges. Additionally, the rises together with the value of the thermal conductivity parameter, whereas the declines. The skin friction coefficient also decreases when the Eckert number rises, as does the , whereas the reverse trend is noted for the Prandtl number. Additionally, the decreases, but the local Nusselt number increases when the local electric, mixed convection, Casson, and slip velocity parameters are improved.

6. Conclusions

Under the impacts of Ohmic dissipation and the slip velocity conditions, the mixed convection flow of non-Newtonian Casson fluid is studied numerically. To compute and outline the changing reactions of flow velocities and temperature when the physical parameters are altered in their proper ranges, SCM based on the shifted VLPs is introduced as a numeric technique. The important results are listed below:(1)The local electric parameter causes a rise in local temperature and the Nusselt number.(2)Increasing the values causes a decrease in the local Nusselt number and skin friction coefficient.(3)The heat transfer rate increases while the local skin friction reduces for high Casson and mixed convection parameter values.(4)By raising the value of the slip velocity parameter, the temperature distribution is enhanced with slight changes in thermal boundary layer thickness.(5)The presence of enhances the local skin friction coefficient, but it also limits fluid flow and raises the fluid temperature.(6)In contrast to and , the temperature marginally drops across the boundary layer.(7)Since the modified Darcy law is more appropriate for the non-Newtonian fluid flow within the porous medium, therefore, in the future, this work can be expanded by taking into consideration the impact of thermal slip and the variable heat flux on the flow behavior through a porous medium that is constrained by the modified Darcy law.

Nomenclature

:	Constant
:	Magnetic field strength
:	Specific heat
:	Skin friction coefficient
:	Electric field
:	Local electric parameter
:	Eckert number
:	Gravitational acceleration
:	Dimensionless stream function
:	Thermal slip coefficient
:	Magnetic parameter
:	Local Nusselt number
:	Prandtl number
:	Local reynolds number
:	Thermal slip parameter
:	Temperature of the fluid
:	Sheet temperature
:	Temperature away the sheet
:	and direction of the fluid velocity
:	Sheet velocity
:	Cartesian coordinates

Greek Symbols

:	Casson parameter
:	Fluid viscosity
:	Thermal conductivity
:	Ambient fluid thermal conductivity
:	Similarity variable
:	Nondimensional temperature
:	Kinematic viscosity
:	Thermal conductivity parameter
:	Coefficient of thermal expansion
:	Mixed convection parameter
:	Fluid density
:	Slip velocity parameter
:	Slip velocity factor
:	Electrical conductivity

Superscripts

:	Differentiation with respect to
:	Free stream condition
:	Wall condition.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-42.

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Copyright © 2023 A. Eid 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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