Fractional Modeling of Non-Newtonian Casson Fluid between Two Parallel Plates

Qayyum, Mubashir; Afzal, Sidra; Ahmad, Efaza

doi:https://doi.org/10.1155/2023/5517617

Journal of Mathematics

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5517617 | https://doi.org/10.1155/2023/5517617

Fractional Modeling of Non-Newtonian Casson Fluid between Two Parallel Plates

Mubashir Qayyum,¹Sidra Afzal,¹and Efaza Ahmad¹

Academic Editor: Serkan Araci

Received26 Nov 2022

Revised10 Feb 2023

Accepted21 Feb 2023

Published08 Mar 2023

Abstract

In this manuscript, fractional modeling of non-Newtonian Casson fluid squeezed between two parallel plates is performed under the influence of magneto-hydro-dynamic and Darcian effects. The Casson fluid model is fractionally transformed through mixed similarity transformations. As a result, partial differential equations (PDEs) are transformed to a fractional ordinary differential equation (FODE). In the current modeling, the continuity equation is satisfied while the momentum equation of the integral order Casson fluid is recovered when the fractional parameter is taken as . A modified homotopy perturbation algorithm is used for the solution and analysis of highly nonlinear and fully fractional ordinary differential equations. Obtained solutions and errors are compared with existing integral order results from the literature. Graphical analysis is also performed at normal and radial velocity components for different fluid and fractional parameters. Analysis reveals that a few parameters are showing different behavior in a fractional environment as compared to existing integer-order cases from the literature. These findings affirm the importance of fractional calculus in terms of more generalized analysis of physical phenomena.

1. Introduction

Scientists and researchers around the globe are recently more focused on capturing fractional phenomena instead of integer-order problems. A major reason for considering fractional environment is to consider a more general problem with memory effect that is helpful in many engineering and physical issues. In this regard, different types of fractional derivatives are considered in which various kernels are involved. Chu et al. [1] analyzed MHD flow of a Newtonian fluid containing hybrid nanoparticles through an ABC fractional model by utilizing the Laplace transform method. They studied temperature and velocity profiles for water, kerosene, and engine oil. Kang et al. investigated [2] a time-variant traffic flow model in fractional order by using the modeling principle of the Bass model. The results of this fractional model were compared with six other traffic models to check efficiency. Yavuz et al. [3] studied fractional second-grade fluid model by using the classical Caputo fractional operator. Exact solutions to the problem were computed through the Laplace transform method. Ferromagnetic fluid was analyzed by Abro [4] through the fractal-Laplace transform. A fractal-fractional differential operator was used in modeling that was principally focused on power law. Chen et al. [5] analyzed service of data latency through the fractional fluid model use network systems with multi-hop. Rate-type fluid models are presented by Abro and Atangana [6] using nonlocal differentiation. Reyaz et al. [7] studied fractional Casson fluid flow under impact of thermal radiation passing over an oscillating plate. Arif et al. [8] investigated engine oil characteristics at ramped wall temperature using a fractional Casson fluid model. Prabhakar’s fractional approach on Casson fluid with generalized Fourier law was studied by Sarwar et al. [9]. Ur Rehman et al. [10] fractionally analyzed Casson fluid with Fourier and Fick’s laws. Electro-osmotic flow of a fractal-fractional modeled Casson fluid was examined by Murtaza et al. [11] by using a finite difference approach for solution purposes.

Non-Newtonian fluids have many applications in chemical engineering, mechanical, industrial, and biological fields due to their visco-elastic properties. Non-Newtonian fluids do not obey Newton’s law of viscosity and behave differently under varying physical conditions. Many researchers have analyzed different non-Newtonian fluid models in the literature. Gireesha et al. [12] studied the two-phase flow of a Maxwell fluid with a heat source and sink. Mahanthesh et al. [13] then analyzed the three-dimensional flow of Maxwell fluid flow impacted by the nonlinear thermal radiation effect. Krupalakshmi et al. [14] simulated two-dimensional steady flow of a non-Newtonian fluid over a heated stretching sheet. Das et al. [15] analyzed flow of an ethylene glycol-based hybrid nanofluid with special reference to entropy generation. The flow of a nanofluid influenced by quadratic radiative heat flux is simulated by Mahanthesh [16]. Casson fluids possess shear thinning properties and are hence much useful in manufacturing of paints, extraction of crude oil, and the food industry. Divya et al. [17] investigated peristaltic flow in the Casson fluid model influenced by a magnetic field in the radial direction passing through a nonuniform channel. Thermal properties were studied under convective boundary conditions. This model characterized blood flow through arterial walls. Khan and Ali [18] gave a theoretical analysis of the classical Graetz problem for the model of Casson fluid. A built-in technique called bvp4c was employed to solve the flow problem. El-Kabeir et al. [19] examined the mixed convective flow of a Casson nanofluid with partial slip at the boundary. Aneja et al. [20] studied the flow of Casson fluid through a partially heated porous square cavity with natural convection. The finite element method was utilized to solve the system of highly nonlinear coupled equations. Fluid flow and heat transfer of Casson fluid passing through a partially heated trapezoidal cavity were scrutinized by Hamid et al. [21]. For solution purpose, the finite element method was employed with the Galerkin optimizer. Rashidi et al. [22] analyzed blood flow through the Casson fluid model with special reference to heat and mass transfer. Das et al. [23, 24] analyzed Casson nanofluid with copper nanoparticles in a porous microchannel. Vs and Pai [25] studied Casson fluid flow between two plates with suction and injunction phenomena. The flow problem is solved by using the homotopy perturbation technique and compared with a finite difference scheme. Hamarsheh et al. [26] simulated the Casson nanofluid’s natural convective flow over a horizontal cylinder with methanol-base fluid. Alwawi et al. [27] numerically analyzed the heat transfer effects of Casson nanofluid over a solid sphere. Several other Casson fluid models are scrutinized by researchers in literature [28–30].

In order to solve governing differential equations of flow problems various analytical and semi-numerical approaches are employed. Due to the highly nonlinear and fractional nature of the current problem, we apply a hybrid of the Laplace transform with the homotopy perturbation algorithm. Many fractional problems in various fields of mathematics and engineering have recently been solved using this technique due to its efficiency in tackling fractional environment. Morales-Delgado et al. [31] used the Laplace homotopy perturbation technique to solve the coupled system of the time-fractional Keller–Segal chemotaxis model with nonsingular kernel. Qayyum et al. [32] studied soliton solutions of the Korteweg-de Vries system with the help of a Laplace transform and homotopy perturbation technique. Dispersive long-wave and Hirota–Satsuma KdV systems were solved, and better results were obtained through LHPM when compared with existing results in the literature. Coupled Whitham–Broer–Kaup equations in fractional order were investigated by Nonlaopon et al. [33] with Mittag–Leffler and exponential laws in Caputo sense of fractional derivatives. The system of partial differential equations was solved with the help of the Laplace transform coupled with the homotopy perturbation method. Sulaiman et al. [34] studied fractional viscous Burgers’ equations with kernels of Mittag-Leffler type utilizing Laplace transform with homotopy perturbation technique. The uniqueness and existence of the purposed model were validated and the effect of various parameters on displacement was studied graphically. Ahmad et al. [35] solved a third order fractional PDE undergoing exponential decay by utilizing the LHPM scheme to tackle a fractional derivative with Mittag–Leffler type kernel. Shokhanda et al. [36] analyzed a two-mode fractional Burgers’ equation with the LHPM technique. Yavuz and Ozdemir [37] used the inverse Laplace homotopy perturbation scheme to numerically investigate the fractional heat equations. Pandey et al. [38] computed approximate solutions for the space and time fractional diffusion equation in two dimensions using the Laplace transform with homotopy perturbation method. Fang et al. [39] used the same semi-analytical approach to find solutions for the fractionally ordered Camassa–Holm equation. Pandey et al. [40] applied the Laplace homotopy perturbation scheme to reaction diffusion equations in fractional order to evaluate the approximate series solution. Nonlinear time-fractional differential equations were evaluated by Zhang et al. [41] by employing the LHPM algorithm. This technique was also utilized by Jalili et al. [42] to work out the approximate results for a Newtonian fluid model at fractional order.

This study focuses on fractional modeling and simulation of an unsteady Casson fluid model squeezed between two plates under magneto-hydro-dynamic effect and porosity. A literature review reveals that the unsteady Casson fluid flow has not been modeled in a fractional environment and thoroughly analyzed along with error analysis and solution validation. In this regard, the unsteady Casson fluid model is fractionally transformed into a more general and comprehensive fluid model in fractional order. This complex model is solved by utilizing the modified homotopy perturbation technique. Residual errors are computed and compared with existing results in the literature. Graphical analysis is done to analyze the behavior of fluid flow with varying fluid parameters. This study is divided into the following sections. Section 1.1 provides basic definitions, Section 2 presents problem formulation and mathematical modeling, methodology used to solve the flow problem is given in Section 3, results are discussed in Section 4 and finally, conclusions are made in Section 5.

1.1. Preliminaries

Definition 1. The Caputo fractional derivative of a function can be expressed as follows [43]:

Definition 2. The Laplace transform of Caputo’s fractional derivative is as follows [44]:

2. Mathematical Formulation of Fractional Casson Fluid

A detailed modeling and mathematical formulation of non-Newtonian fractional Casson fluid is given in the following steps.

2.1. Fluid Model

Consider an incompressible flow of a Casson fluid passing between two plates that are distance apart in the -direction. We define , where represents squeezing motion of both plates and is the initial gap among the plates. Squeezing motion of plates is characterized by whereas plates recede when . Major flow assumptions are as follows:(i)Fluid is at rest at time between two plates distance apart.(ii)Fluid motion is caused by squeezing/receding motion between two plates at .(iii)Magneto-hydro-dynamic effect acts perpendicular to the axis along with Darcian effects.(iv)The stress tensor of incompressible Casson fluid is given as follows [28–30].

The governing equations of flow characterized by the mentioned assumptions are as follows [45]:

This flow problem is subject to the following boundary conditions:

Now, we introduce a vorticity function as follows:

Final velocity equation is obtained by using equations (5) and (6) in equation (8), which is given as follows:

2.2. Similarity Transformations

The similarity transforms for unsteady squeezing flow of fractional Casson fluid are as follows [45, 46]:andwhere is the fractional parameter such that .

2.3. Nondimensionalization in Fractional Environment

2.3.1. Continuity Equation

By introducing equation (11) in equation (4), we check that fractional transform satisfies the continuity equation identically.

2.3.2. Momentum Equation

The momentum equation in equation (9) is fractionally nondimensionalized by using equation (11) as follows:where the dimensionless quantities are defined as follows:where squeeze number implies squeezing and corresponds to receding motion of the plates. Obtained momentum equation (12) is subject to following nondimensional boundary conditions after using equation (10) in equation (7)

2.4. Model Verification

To verify the fully fractional developed model, we put in equation (12) and recover the following momentum equation derived in [45, 47].

2.5. Skin Friction

The coefficient of skin friction is as follows:and after applying similarity transforms from equation (10) in above equation, we obtain dimensionless skin friction as follows:where is the local Reynolds number .

3. Proposed Methodology for Fractional-Order Problem

To demonstrate the basic concept of modified homotopy perturbation algorithm [32], we firstly develop a homotopy as follows:where and represent the linear and nonlinear parts of the function , respectively. is the fractional differential operator and () is some known function. In case of order boundary value problems, dummy initial conditions are used to initialize the solution process.

In the next step, we construct a homotopy of given equation after applying definition of Laplace transform as follows:where is the initial guess. Expansion of in terms of power series gives the following equation:

Substituting equation (20) in equation (19) and then comparing similar coefficients of leads us to different order problems.

Problem at i-th order is given as follows:

Operating Laplace transform inverse gives the following equation:

Hence, the approximate solution of the fractional, nonlinear differential equation is as follows:

In case BVPs, right side boundary conditions will be used to find optimal values of dummies constants of initial conditions. Residual errors of concerned fractional differential equation can be found by

4. Results and Discussion

In this section, initial numerical analysis at different values of fluid parameters is performed to check the efficiency and validity of the proposed approach in the case of unsteady squeezing of Casson fluid in fractional space. In next phase, the effects of different parameters on the normal and radial velocity are analyzed graphically. Our discussion of results is based on fractional and fluid parameters in the following subsections.

4.1. Error Analysis and Numerical Validation

Tables 1 and 2 show numerical validation and efficiency of the proposed methodology (modified homotopy perturbation algorithm) by comparing it with HPM and RK4 in the case of an unsteady squeezing flow of Casson fluid at . In Table 1, residual error through modified HPM method varying between . Similarly, Table 2 shows comparison of modified HPM results with HPM and RK4 at different values of non-Newtonian parameter . In this case, modified HPM errors are varying in range . Analysis of these tables shows that the current results obtained through the proposed methodology are better as compared to HPM and RK4. This analysis provides firm evidence for the validation and stability of the proposed methodology in the current study.

4.2. Normal Velocity Profile

Figures 1(a)–1(f) depict behavior of normal fluid velocity for increasing values of the fluid parameter , the fractional parameter , the squeeze number , an unsteady magnetic , and porosity parameters. In Figure 1(a), with an increase in the fluid parameter normal velocity shows different behavior about point . For , normal velocity increases with increasing fluid parameter, while decrease in fluid velocity in normal direction is observed for . Fractional parameter is varied between (0, 1) in Figure 1(b). Normal fluid velocity increases with increasing fractional parameter when and it decreases otherwise. For squeezing motion of plates, i.e., in Figure 1(c) velocity in normal direction decreases for and increases when . Contrasting behavior of velocity is observed in Figure 1(d) for receding motion between the two plates. The magnetic parameter decreases normal fluid velocity for due to dominant Lorentz-like drag force on fluid flow. This drag force does not remain dominant after and hence velocity starts to increase in normal direction (see Figure 1(e)). Opposite behavior in case of higher porosity parameter is observed in Figure 1(f) about .

(a)

(b)

(c)

(d)

(e)

(f)

4.3. Radial Velocity Profile

Figures 2(a)–2(f) present simulations of radial velocity profile for fluid parameter , fractional parameter , squeeze number , unsteady magnetic and porosity parameters. The velocity profile in radial direction shows increasing behavior in contrast to the normal velocity of the fluid. Radial velocity gradually increases with elevated non-Newtonian parameter in Figure 2(a). The fractional parameter decreases fluid velocity in radial direction as seen in Figure 2(b). For squeeze number , i.e., when the plates are moving closer, velocity in radial direction decreases as observed in Figure 2(c). In contrast, radial velocity increases in the case of receding plates in Figure 2(d) as more space for fluid motion along the radius develops when plates are moving apart. An increase in the magnetic parameter decreases radial velocity (see Figure 2(e)) due to a dominant drag force throughout the domain. In Figure 2(f) increase in fluid velocity is observed for higher values of the porosity parameter . The velocity boundary layer is enhanced with higher values of due to which resistance among fluid layers decreases causing an increase in fluid velocity along radial direction.

(a)

(b)

(c)

(d)

(e)

(f)

4.4. Skin Friction

Skin friction at the wall is simulated numerically in Table 3 against the fractional parameter , fluid parameter , squeeze number , magnetic parameter , and porosity parameter . It is observed that increase in and increases the skin friction at the boundary. Moreover, an increase in , , and decreases the skin friction of fractional Casson fluid.

5. Conclusions

The objective of the current manuscript is fractional modeling and simulation of the unsteady squeezing flow of Casson fluid. The incompressible Casson fluid is considered to have MHD and Darcian effects in a fully fractional environment. The obtained model is validated by recovering the ODE of integer-order at . In current modeling, the obtained equations are fractionally transformed, and solved through modified homotopy perturbation algorithm. An error analysis is performed for checking the convergence and validity of the applied methodology. It is noted that the solutions are improved using modified HPM as compared to existing results in the literature. The key findings of this study are as follows:(i)Radial velocity increases throughout the domain in contrast to normal fluid velocity.(ii)Decrease in radial velocity is observed against increasing whereas radial velocity increases in case of higher .(iii)Velocity in normal direction shows similar behavior as radial velocity for and inverse behavior is observed when .(iv)Furthermore, the squeeze number and porosity parameter present opposite behavior in fractional enthronement when compared with the existing integer-order model given in [45].

This study provides the basis for fractional modeling of the unsteady squeezed flow of Casson fluid in particular and non-Newtonian fluid in general. The steps presented in current modeling can be extended to different non-Newtonian fluid models like Maxwell, Carreau, and Sutterby fluids under various boundary conditions.

Nomenclature

:	Spatial coordinates
:	Temporal coordinate
:	Velocities in and directions
:	Extra stress tensor
:	Rate of deformation
:	Initial gap between plates
:	Constant quantity
:	Dynamic viscosity
:	Yield stress
:	Density
:	Magnetic field strength
:	Casson fluid parameter
:	Electric conductivity
:	Kinematic viscosity
:	Porosity rate .

Data Availability

All the data are available within manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Mubashir Qayyum 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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