Mathematical modeling of fractional derivatives for magnetohydrodynamic fluid flow between two parallel plates by the radial basis function method

Saman Hosseinzadeh , Seyed Mahdi Emadi , Seyed Mostafa Mousavi, Davood Domairry Ganji

Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran

Keywords: Magnetohydrodynamic fluid Fractional equation Radial basis function method Numerical method

ABSTRACT Investigations into the magnetohydrodynamics of viscous fluids have become more important in recent years, owing to their practical significance and numerous applications in astro-physical and geo-physical phenomena. In this paper, the radial base function was utilized to answer fractional equation associated with fluid flow passing through two parallel flat plates with a magnetic field. The magnetohydrodynam- ics coupled stress fluid flows between two parallel plates, with the bottom plate being stationary and the top plate moving at a persistent velocity. We compared the radial basis function approach to the numer- ical method (fourth-order Range-Kutta) in order to verify its validity. The findings demonstrated that the discrepancy between these two techniques is quite negligible, indicating that this method is very reliable. The impact of the magnetic field parameter and Reynolds number on the velocity distribution perpendic- ular to the fluid flow direction is illustrated. Eventually, the velocity parameter is compared for diverse conditions α, Reynolds and position (y), the maximum of which occurs at α= 0.4. Also, the maximum velocity values occur in α= 0.4 and Re = 10 0 0 and the concavity of the graph is less for α= 0.8.

The characteristics of various complicated and nonlocal systems could be described using fractional differential equations. They can be found in mathematics, geology, biochemistry, biomechanics, economics, management theory and other scientific and technical fields [ 1 , 2 ]. Fractional derivatives could be used to analyze nonlin- ear systems that describe a diversity of occurrences. Due to the obvious importance of the topic, a numerous of scholars have fo- cused on fractional equations in recent years [3–6] . The Brinkmann type fluid fraction (BTF) model of a hybrid nanoparticle retainer was studied by Ikram et al. [7] . The method of the Laplace trans- form has been solved. The constant proportional Caputo fraction operator (CPC) was found to have a superior memory effect to the Caputo-Fabrizio fraction operator (CF).

The effects of a uniform transverse magnetic field on tran- sient free convective currents of a nanofluid with generalized heat transfer between two vertical parallel plates were investigated by Ahmed et al. [8] . The velocity field was solved semi-analytically using the Laplace transform and numerical algorithms for the in- verse Laplace transform as given by Stehfest and Tzou. Graphical representations of the impacts of derivative fraction sequence and physical factors on nanofluid flow and heat transmission are pre- sented.

In addition, Khan et al. [9] have investigated a new idea of frac- tal fraction operators with the power law core for heat transfer in a fluid flow problem, and found that it increased the velocity and temperature fraction parameter when compared to the fractal operator for the effect of temperature and velocity characteristics. Moreover, Sin et al. [10] studied the unstable flow of a viscoelastic fluid between two parallel plates caused by the accelerated motion of the lower plate momentum, and derived the Barry Bernstein, Elliot Kearsley, and Louis Zapas (K-BKZ) fraction equation from the Maxwell fraction model based on the data. One-way flows between two plates were simulated and compared using the Maxwell frac- tional model and the K-BKZ fractional model, and the influence of flow velocity on different parameters of the K-BKZ fractional model was visually studied.

It is also worth mentioning that heat transmission through For- sheimer’s medium in a fluid flow type of unstable magnetic hy- drodynamic (MHD) differential statistically investigated by Shoaib Anwar and Rasheed [11] . Finite element discretization for spatial variables and fractional time derivative discretization for fractional time derivatives have been evaluated. Liu et al. [12] developed a model to characterize the Oldroyd-Bfluid unstable MHD flux be- tween parallel heat transfer plates, introducing a fraction derivative in the construct equation. To solve the problem numerically, they suggested a spectral localization approach with rapid exponential sum-based algorithms (SOE). Abor and Anwar [13] investigated ac- curate solutions for temperature and velocity field distributions based on Caputo-Fabrizoi fractional derivatives, noting that the set of partial differential equations governing the new Caputo-Fabrizoi fractional derivatives has been renamed conventional derivatives.

As a further matter, Abdulhameed et al. [14] used the Laplace transform methodology, the Riemann sum approximation method, and the Stehfest algorithm to solve the derived fractional momen- tum equation that contains the body’s electromagnetic force from the two-layer electric field (EDL). Imran [15] studied the use of the recently introduced fractal fraction operators with the power law core in fluid dynamics, formulated the governing equation result- ing from the problem with the fractal fraction derivative operator with the power law core, and solved the fractal fraction model us- ing the Laplace transform. Akhtar [16] offered a comparison of un- stable flows between two parallel planes of a pair stress fluid with two different Caputo (single core derivative) and Caputo-Fabrizio time fraction derivatives (fractional-time derivative). The Laplace transform for the time variable and the finite Fourier transform were used to solve the problem.

Besides, Abro et al. [17] used a laparoscope to investigate the Caputo-Fabrizio and Atangana-Baleanu partial derivatives on the nanofluid hydrodynamic magnetic flux in a porous medium, and developed mathematical modeling of the governing equations us- ing modern fractional derivatives and general solutions for the ve- locity field and temperature distribution. Bhatti et al. [18] used a differential conversion (DTM) approach to solve nonlinear cou- pled differential equations and compared the answers to physical factors, as well as employing a table to assess the Nusselt num- ber. The DTM method for solving nonlinear differential equations is adaptive and stable, they concluded.

In another work, Abro et al. [19] studied Maxwell’s magne- tohydrodynamic fluid (MHD) heat transfer problem on a vertical plate embedded in a porous medium using the Atangana-Baleano fraction derivative, as well as analytical solutions for temperature and velocity field distributions using the Laplace transform tech- nique. Both sine and cosine oscillations were obtained. Arif et al. [20] used a new fractional-fractional derivative concept to model the coupled stress fluid (CSF) with the combined effect of heat transfer and mass, as well as a non-dimensional fractal-fractional model of coupled stress fluid in the Riemann-Level concept with numerical power. They solved using the implicit finite difference approach and discovered that fractal-fraction solutions are more general in the channel than classical and CSF motion fraction solu- tions.

The employment of a fractional operator, such as the Mittag- Leラer function, has a significant impact on modeling for many physical processes, as studied by Tassaddiq et al. [21] . The effects of Newtonian heating on generalized cascade fluid flow, as well as the MHD and porous magnetic effects for such fluids, were studied using this modern fractional actuator.

Khan et al. [22] investigated heat transfer in a generalized caisson fluid with an unstable MHD flux on a vertical plane, as well as the SA-NaAlg fluid fraction model utilizing the Atangana- Baleanu fraction (ABFD) non-local nucleus fraction derivative. Shoaib et al. [23] to analyze numerical simulations for compress- ing two-dimensional MHD nanofluid currents between two paral- lel plates and BN data sets, researchers used artificial intelligence- based post-diffusion neural networks with the Lnberg-Marquardt (BNN-LMA) method. By varying the number of presses, Hartmann number, and heat source parameter, LMA created an Adam numer- ical solution for various MHD-SNFM scenarios. The combined ef- fects of heat transfer and magnetic field on the free convective flow of two-phase magnetohydrodynamics of electrically conduc- tive caustic soda fluid between parallel plates were studied by Ali et al. [24] .

In two-dimensional fractional integral equations, Safinejad and Moghaddam [25] proposed a network-free local radial basis func- tion (RBF) method to solve the fractional differential integral equa- tion and the two-dimensional Voltra fractional integral equation, demonstrating that the local RBF method is much more efficient than the global RBF method and concluding that the local RBF method is suitable for large dimensions. Liu and Li [26] devised a differential squaring method based on the Hermit radial function (HRBF-DQ), and they utilized a quadruple type of radial basis func- tions for the computations, and they used the local form without the RBF-DQ network for the general condition of irregular geome- try. The thin-spline radial base function was used to approximate the answer in regular and irregular domains in Mohebbi and Saf- farian [27] study of the numerical solution of a two-dimensional variable order fraction cable issue.

Furthermore, Li et al. [28] looked into the local RBF technique, which uses Laplacera conversion to estimate the solution of Klein- Gordon linear time fraction equations. The recommended local RBF strategy, when combined with the Laplace transform, is an ex- cellent way for approximating the Klein-Gordon linear time frac- tion equations. Qiao et al. [29] examined a local compact in- tegrated radial basis function (CIRBF) approach for solving the fraction’s convective-diffusion-temporal response equations, find- ing that with lattice refinement, the proposed CIRBF scheme was accurate and converged quickly. To connect physical space with RBF weight space, the second node derivative and node function values were used. To substitute time discretization, the second- order modified Grunwald approach was adopted.

Several scholars have dealt with magnetohydrodynamic fluids and their associated fractional equations, according to the litera- ture. Many common methods are inefficient due to the fractional and partial terms. The rest of the methods, if successful, have a significant error rate. As a result, there is a considerable demand for an efficient solution with high response accuracy for this sort of problem. The radial basis function approach is a method that can solve complicated fraction problems successfully. However, no study has been conducted on employing the radial base function method to solve the analytical and comprehensive problem of a flow of magnetohydrodynamic passing through two flat plates. The primary aim of this investigation is to propose a new and pre- cise technique for solving fractional equations regarding the flow of magnetohydrodynamic fluid across two parallel flat plates.

Two infinite horizontal parallel stiffplates with a distancedapart are considered. Assume the space between the plates is filled with a non-compressible pair stress fluid. Assume that the plates and fluid are both at rest. As indicated in Fig. 1 , we use a Cartesian coordinate system with the bottom plate’s origin and theη?axis perpendicular to the plates. The two plates are identified by the lettersη= 0 andη=d.

Fig. 1. Schematic of magnetohydrodynamic fluid flow.

Fig. 2. The computational approach in this paper.

and the beginning and boundary conditions that go with it pre- senting the dimensionless elements that go with it

into Eq. (1) , we get

whereM=σ/μis the magnetic field parameter andRe=ρV0d/μistheReynoldnumber.

We present some fundamental concepts and properties of frac- tional calculus theory, which will be applied throughout this arti- cle.

A real functionf(x),x>0 , is said to be in the spaceCμ,μ∈R, if there exists a real numberξ(>μ), such thatwheref1(x)∈C[ 0,∞), and it is said to be in the spaceiff(m)∈Cμ,m∈N.

The Riemann-Liouville fractional integral operator of orderα≥0 , of a functionf∈Cμ,μ≥?1,is defined as:

Properties of the operatorJαcan be found in Podlubny [30] , Samko et al. [31] and Oldham and Spanier [32] , we mention only the following. Forf∈Cμ,μ≥?1,α,β≥0 andγ>?1 :

The Riemann-Liouville derivative has certain limitations when used to describe real-world phenomena using fractional differen- tial equations. As a result, we’ll use Caputo’s modified fractional differential operatorDα, which he proposed in his paper on vis- coelasticity theory [33] .

The fractional derivativef(x)inthe Caputo senseisdefined as: Form?1 〈α≤m,m∈N,x〉 0,f∈

Also, two of its fundamental features are required here.

Ifm?1〈α≤m,m∈N,x〉 0,fthenDαJαf(x)=f(x)and

The Caputo fractional derivatives are used because they allow for the inclusion of traditional beginning and boundary conditions in the issue formulation. The fractional partial differential equa- tions that arise in fluid mechanics are considered in this work, and the fractional derivatives are taken in the Caputo sense as follows.

The Caputo time-fractional derivative operator of orderα>0 is defined as for m to be the smallest integer that exceedsα:

The references given can be used to learn more about the math- ematical properties of fractional derivatives and integrals.

The radial basis function technique in multivariate approxima- tion is one of the most realistic methods in modern theory, and it is utilized in the geometry of the dimensional independence prob- lem for high precision and flexibility, as well as its ease of imple- mentation (particularly when the goal is to internalize).Fis re- ferred to as radial when:

If this characteristic holds,F(x) values are solely determined by ‖x‖ . As a result of this

Some of the most widely used radial basis functions are in- cluded in Table 2 .

Table 1 Initial and boundary conditions in this study.

Table 2 Radial basic function categories.

As illustrated in Table 2 , the radial basis functions are divided into two groups.

? Radial base functions that are infinitely stable

These functions can be differentiated indefinitely and are highly influenced by the state parameter.

? Radial base functions that are finitely stable

Basic functions in this category are not infinitely differentiable. These functions do not have any free parameters and are less pre- cise than the basic functions in the first group.

The nature of the problem and the amount of precision re- quired should determine the best option. The following is the pro- cedure for solving a mathematical issue using RBF. We get the fol- lowing result ifZis an ODE andz(ε)is the function to be calculated [ 5 , 6 ].

With the following boundary conditions:

We also consider the solution of a related ODE to be a sequence with n known constant coefficients (λ) and n specified radial basis functions (n is determined by the needed precision and conver- gence):

From the series forz(ε)and its derivatives inε= 0 , we may derive the following:

And whenx=L:

Equations (18) and (19) can be replaced in Eq. (15) , yielding:

To find out the values in Eq. (17) , we primell needn+ 1 equa- tions andn+ 1 unknowns. Because the available terms are insuffi- cient, we must repeat Eq. (15) until we obtain the following vari- ables:

When we apply the boundary conditions to the derivatives of the differential equationZ, we obtain more terms:

Eventually,n+ 1 equations and unknowns can be solved, and the result is written in sequence form asz(ε).

The fractional derivatives in Equation are used.

Equation (24) could be solved by selecting an appropriate radial basis function from Table 2 and applying the boundary conditions of Eq. (4) . Because the equation has no harmonious answer and the response is a sequence of that function, the Gaussian function is used as the radial basis function.

Table 3 Comparison of RBF solution with numerical solution for α= 0 . 4

Table 4 Comparison of RBF solution with numerical solution for α= 0 . 6

Table 5 Comparison of RBF solution with numerical solution for α= 0 . 8

Table 6 Comparison of RBF solution with numerical solution for α= 1 . 0

As a result, the time and position components of velocity are as follows.

In the previous equation,nis two, andSistands for radial basis function centers.

Gis also the same as the adjustment coefficient, which is 0.2 (shape parameter). As a result, the velocity function will look somewhat like this:

by substituting Eq. (28) in Eq. (24) :

The following equations are obtained by applying the boundaryand initial conditions.

By solving Eqs. (30) –(35) while considering(Re= 10 0 0,M= 0.5), we will have:

Figure 2 depicts the procedure of doing the calculations in the current study, which is the primary framework of this work.

In this study, we applied a novel and effective methodology of radial base function to calculate fractional equations associated with the flow of magnetohydrodynamic fluid passing through two parallel horizontal planes. The figures depict the effect of Reynolds number, magnetic field parameter, and coefficientαon the veloc- ity component perpendicular to the flow direction. To validate the radial basis function technique, Tables 3–6 were utilized to demon- strate its efficiency and correctness. This approach was compared to the fourth-order Range-Kutta numerical method in these ta- bles. Figure 3 shows a three-dimensional image of the simultane- ous effect of two parameters y and t on the velocity distribution, based on Eq. (24) . In this figure, various values ofαare investi- gated, which for Fig. 3 d are 0.4, 0.6, 0.8 and 1, respectively. ac- cording to Fig. 3 , the two parameters y and t have a direct effect on the numerical value of the velocity distribution. The output of Eq. (24) based on different values of position and time is shown in Fig. 4 . These graphs are calculated based on the values of param- etersα,ReandM, respectively 0.4, 1200 and 0.5. The variations of the velocity distribution by location for various times are shown in Fig. 5 , and this diagram is examined forα= 0.4 ,Re= 1200 andM= 0.4 , As the time increases, the speed value for all y increases and Also in graphv(y,t)?y, the velocity values for times 0.2, 0.4 and 0.6 are higher than times 0, 0.8 and 1. Figure 6 demonstrates the impact of time on the velocity distribution in disparate y the maximum value for different incremental values of position 0, 0.2, 0.4, 0.6, 0.8 and 1 occurred at time 0.5, The speed values are 0, 1.19352, 2.38703, 3.58055, 4.77406 and 5.96758 respectively. The diagram of velocity distribution according to t parameter for dif- ferent values ofαis shown in Fig. 7 . In this graph, the values ofα,ReandM, are 1, 1200 and 0.5, respectively, with increasing time to 0.5 forα, 0.4 and 0.6 and increasing the time up to 0.6 forα0.8 and 1 is the ascending velocity distribution, while with increas- ing time more than the corresponding value, the velocity distribu- tion will decrease. Figure 8 examines the effect of various values ofαon the velocity distribution at diverse locations as the loca- tion (y) increases, the velocity value for allαvalues increases and, in this diagram, the parameterαis an important criterion in ve- locity changes so that by increasing the value ofαfrom 0.4 to 1, the slope of the diagram decreases. Thus, it can be concluded that in graphv(y,t)?ythe parameterαis a determining criterion. The velocity distribution diagram in terms of time parameter for vari- ous values ofα0.4 and 0.8 in different Reynolds 10 0 0, 150 0, 20 0 0 is shown in Fig. 9 . As can be seen from the corresponding figure, the maximum velocity forα= 0.4 in different Reynolds is higher thanα= 0.8 Also, the maximum velocity values occur inα= 0.4 andRe= 10 0 0 and the concavity of the graph is less forα= 0.8 . Figure 10 shows the behavior of disparate values of magneto (M) on velocity. In this diagram, the coefficientsα0.3, 0.6 and 0.9 are considered. Which are compared inα0.4 and 0.8 fory= 1 andRe= 1200 . So that the maximum velocity forM= 0.9 inα= 0.4 is 58.35% higher thanM= 0.9 andα= 0.8 and 28% higher thanM= 0.6 andα= 0.4 and also the concavity of the graphs is also increasing.

Fig. 3. The surfaces show solutions of Eq. (24) for (a) α= 0 . 4 , (b) α= 0 . 6 , (c) α= 0 . 8 , (d) α= 1

Fig. 4. The solution of Eq. (24) based on (a) position and (b) time when (α= 0 . 4 , Re = 1200 , M = 0 . 5)

Fig. 5. The velocity distribution by location at various times when (α= 0 . 4 , Re = 1200 , M = 0 . 5)

Fig. 6. The profile of velocity by time in various locations when (α= 0 . 4 , Re = 1200 , M = 0 . 5)

Fig. 7. Behavior of velocity vs. time for distinct value of α when ( y = 1 . 0 , Re = 1200 , M = 0 . 5 )

Fig. 8. Behavior of velocity vs. position for distinct value of α when ( t = 1 . 0 , Re = 1200 , M = 0 . 5 )

Fig. 9. Behavior of velocity vs. time for distinct value of Re and α when ( y = 1 . 0 , M = 0 . 5 )

Fig. 10. Behavior of velocity vs. time for distinct value of M and αwhen ( y = 1 . 0 , Re = 1200 )

In order to gain a better comprehension of the behavior of magneto-hydrodynamic fluids, we employed the radial basis func- tion methodology to answer fractional equations relevant to this kind of fluid. According to the physics of the problem, the fluid flow is positioned between two flat and parallel plates, with the top plate moving at a steady velocity and the bottom plate being absolutely stationary. Using the radial basis function methodology, the governing fraction equation was solved after determining the dimensionless parameters according to the boundary and initial conditions. The following are some of the most significant findings from this research.

? Different values of time and position are investigated for vari- ous parametersα,ReandM.

? The effect of location (y) on velocity changes is investigated that with increasing y, the velocity increases in time 0.5. So that the concavity of their graph increases.

? The velocity parameter is compared for diverse conditionsα, Reynolds and position (y), the maximum of which occurs atα= 0.4.

? In examining twoαstates, we find thatMis directly related to velocity.

Declaration of Competing Interest

The authors whose names are listed immediately below certify that they have no affiliations with or involvement in any organiza- tion or entity with any financial interest (such as honoraria; educa- tional grants; participation in speakers’ bureaus; membership, em- ployment, consultancies, stock ownership, or other equity interest; and expert testimony or patent licensing arrangements), or non- financial interest (such as personal or professional relationships, affiliation, knowledge or beliefs) in the subject matter or materi- als discussed in this manuscript. The manuscript is original and it does not submit in another journal.