# Electro-Magnetohydrodynamic Fractional-Order Fluid Flow with New Similarity Transformations

Authors: Fuad S. Al-Duais; Saeed Ahmed Asiri; Anwar Saeed; Jihen Majdoubi; Mohammad Mahtab Alam; Taza Gul
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

The fractional-order differential equations that exist in the field of science and engineering have been studied in this paper. The 2D fluid flow problems are recommended for the classical- and fractional-order analysis. It has been found that the nonlinear fractional-order problems are more realistic than the classical models to describe the proposed flow problems since the behavior of the stress of the model problem is not linear. The similarity variable in this study has been used in fractional form to transform the modeled governing equations from partial PDEs. The acquired equations are the nonlinear ordinary differential equations (ODEs) in fractional form. The electromagnetic field has also been imposed into the fluid motion to calculate the embedded constraints in the case of classical and noninteger orders. The nonlinear problems are attempted using the FDE-12 technique. The important physical phenomena including Nusselt number and skin friction are also calculated in case of the integer- and noninteger-order problems. The electric and magnetic fields are examined and discussed in the case of fractional form and classical form.


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## Body

## 1. Introduction

One of the demanding and competitive fields among researchers is the modeling of noninteger-order problems related to science and technology. The fractional models are more realistic than classical models to describe the nonlinear phenomena because the classical models do not completely represent all the requirements of the nonlinear problems. Moreover, the fractional models are more appropriate to calculate the actual influence of the parameters in the limited domain because the parameter impact in the physical problems at small intervals is very necessary for the parameter range and limitations. In 1967, the concept of the noninteger-order derivative was introduced by Caputo [1] to discuss problems involving a fractional differential equation with initial conditions. That idea was further extended [2–4] to implement the noninteger-order derivative concept to the problems happening in the field of engineering, biomedical, and industry. The various noninteger-order operators were introduced and used by the scientists [5–14] to handle the more realistic physical problems. These problems include higher-order problems that occur in the field of science, wave equations and fraction hybrid differential operators, Mumps virus with optimal control, q-integrodifferential equations, solution of the fractional Burgers equation, and solution of the fractional Allen–Cahn equations. However, the initial idea of the noninteger-order derivative was limited to the time-fractional derivative. Later, the idea of fractional-order derivative was further refined [15, 16] and the space variables have been introduced to handle the noninteger-order boundary value problems independent of time. The handling of nonlinear problems in the form of PDEs is not an easy job so the similarity transforms were implemented to convert the model PDEs into the nonlinear classical-order ODEs [17–21]. The noninteger-order similarity variables are used by El Rasouli et al. [22] to alter the classical PDEs into the noninteger-order ODEs for the gas flow model. Mohammadein et al. [23] used a similar idea for the boundary layer flow past an extending sheet.The above idea is further improved in the recent study by using the same variable idea to alter the PDEs in the noninteger-order ODEs avoiding the separating of variable concepts. Considering the common parameters, the recent idea matches with the existing literature [17, 18] for the integer-order analysis. The acquired outcomes are also contrasted with the classical model. The results are obtained using the FDE-12 technique [24, 25].Since in most of the problems the stress is not linear, that is why the fractional-order derivative approach is very essential to handle these kinds of problems.The recent work is the generalized form of the 2D model, and one can easily get the existence model by puttingn=1 as an integer case. Also, the electrical and magnetic fields are jointly used to improve the novelty.


## 2. Formulation of the Problem

The fluid flow is assumed in two-dimensional space considering the steady motion of the fluid towards the stretched surface. The electromagnetic field is imposed on the flow pattern in the vertical direction. All the assumptions are similar to the published work [17]. The elementary equations are displayed as(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=υ∂2u∂y2−1ρσB02u−E0B0,(3)u∂T∂x+v∂T∂y=kρcp∂T2∂y2.The boundary conditions are(4)u=xb,v=0,T=Twaty=0,(5)u⟶0,T⟶T∞,aty⟶∞.Here,υ,ρ,cp,E0,k,σ,B0, and T stand for kinematic viscosity, density, specific heat, electric conductivity, thermal conductivity, magnetic field, and temperature distribution, respectively. The similarity variable in fractional form with appropriate transformations is defined as [26, 27](6)ψ=xbυf1/2fη,(7)u=∂ψ∂y,(8)v=−∂ψ∂x,(9)Θη=T−T∞Tw−T∞,(10)ηα=yαΓα+1bυf⇒η=ybυf.Upon using Equation (6) in Equations ((1)), ((2)), ((3)), ((4)), the noninteger-order ODEs are obtained as(11)α2η3α−3fα+2+α2α−12η2α−3+η2α−2fα+1+αα−12ηα−2fα+αα−1ηα−1fαf+α2η2α−2fα+1f−α2η2α−2fα2+EM−αηα−1fαM=0,(12)α2η2α−2Θα+1+α−1ηα−2Θα+Prηα−2fΘα=0.Reduced conditions are(13)f0=0,(14)fα0=1,(15)θ0=1,(16)f∞=0,(17)θ∞=0.α=1 reduce the above equations into the classical order as(18)f‴+ff″−f′2+ME−Mf′=0,θ″+Prfθ′=0.In Equations (11) and (12), α,M,Pr, and E stand for the order exponent, magnetic field parameter, Prandtl number, and electric field as(19)M=σB02bρf,E=E0B0uw,Pr=νfαf.


### 2.1. Physical Quantities of Interest

The physical number is stated as(20)Cfx=τw1/2ρuw2,(21)Nux=xqwkTw−T∞.The simplified form of Equation (20) using the similarity variable is(22)CfxRex0.5=α+1f″0,NuxRex−0.5=−θ′0.


## 2.1. Physical Quantities of Interest

The physical number is stated as(20)Cfx=τw1/2ρuw2,(21)Nux=xqwkTw−T∞.The simplified form of Equation (20) using the similarity variable is(22)CfxRex0.5=α+1f″0,NuxRex−0.5=−θ′0.


## 3. Caputo Fractional Derivatives

The brief and basic theory of the Caputo is displayed as follows.


### 3.1. Definition 1

Letb,α,t∈R,b>0, and t>b. According to the Caputo fraction derivative, using α as the fractional order from the function g∈Cn is derived as(23)bCDtαgt=1Γn−α∫btgnςt−ςα+1−ndς,n−1<α<n∈N.


### 3.2. Property

Letgt,ht:a,b⟶R be such that bCDtαgt and bCDtαht exist almost everywhere, and let e1,e2∈R. Then, bCDtαe1gt+e2ht exists almost everywhere and(24)bCDtαe1gt+e2ht=e1bCDtαgt+e2bCDtαht.


## 3.1. Definition 1

Letb,α,t∈R,b>0, and t>b. According to the Caputo fraction derivative, using α as the fractional order from the function g∈Cn is derived as(23)bCDtαgt=1Γn−α∫btgnςt−ςα+1−ndς,n−1<α<n∈N.


## 3.2. Property

Letgt,ht:a,b⟶R be such that bCDtαgt and bCDtαht exist almost everywhere, and let e1,e2∈R. Then, bCDtαe1gt+e2ht exists almost everywhere and(24)bCDtαe1gt+e2ht=e1bCDtαgt+e2bCDtαht.


## 4. Solution Methodology

Equations (11), (12), (13) are selected as(25)y1=f,y2=f′,y3=f″,y4=θ,y5=θ′,y1=0,y2−1=0,y3,y4−1=0,y5.The above-selected functions are solved using the FDE-12 technique as mentioned in [24–27].


## 5. Results and Discussion

The two-dimensional flow on an extending surface is reflected. The electromagnetic term is considered vertically to the flow field including the momentum and energy equations. The main purpose of the research is to alter the modeled PDEs equations into noninteger-order ODEs by using similarity transformations. The fractional operator is used asη to alter the governing equations in the noninteger-order nonlinear ODEs. The nonlinear ODEs are then solved with the help of the FDE-12 method. The upshot of the constraints is observed using the classical- and noninteger-order systems. The noninteger-order exponent α is exhibited in Figures 1 and 2 for the momentum and thermal boundary layers. The results obtained show that the noninteger-order results are compact in relation to conventional results. The electric field parameter improves the velocity profile in both classical- and fractional-order cases as shown in Figures 3 and 4. The result matches the existing literature in the case of the classical models. The noninteger-order improvement in the velocity field is comparatively compressed as displayed in Figure 4. The impact of the parameter M versus the fluid motion for its increasing value is shown in Figures 5 and 6. The fluid motion declines in both the classical shown in Figure 5 and fractional displayed in (Figure 6) for the rising values of M. The resistive forces existing in the magnetic field do not allow the fluid to flow freely, and the results are compacted via using the noninteger-order derivatives as shown in Figure 6. The resistance is accrued due to the existence of the Lorentz force. The rising credit of theMelevates the energy transferenceθηas shown in Figures 7 and 8. The impact of the parameter M is the same for both classical (Figure 7) and fractional (Figure 8). Again, the changes that occur in the temperature distribution are relatively compact in the noninteger case. Fractional-order impact versus drag force and Nusselt number are shown in Table 1. The parameter M drops the fluid velocity for its larger values, and the impact is the same as shown in Table 2 taking the values of αin both increasing and decreasing forms like α=0.95,0.90,0.85, and 0.80 or α=0.80,0.85,0.90,and0.95, while this retort is opposing in the case of Nusselt number. The electric field E declines the drag force in both classical and fractional cases as shown in Table 2. The decline rate is comparatively small using the noninteger form. The consequences of the parameters M and Pr versus energy transition are shown in Table 3. The larger magnitude of M enhances the heat transfer rate while the larger magnitude of Pr opposes the heat propagation rate. The variation of these parameters is also compared with the existing literature considering classical results. The comparison of the classical results authenticates the obtained results.Figure 1
Impact ofα versus f′η. When M=0.1,E=0.3, and Pr=6.2.Figure 2
Impact ofα versus θη. When M=0.1,E=0.3, and Pr=6.2.Figure 3
Impact ofE versus f′η. When α=1,M=0.3, and Pr=6.2.Figure 4
Impact ofE versus f′η. When α=1,M=0.3, and Pr=6.2.Figure 5
Impact ofM versus f′η. When α=1,E=0.3, and Pr=6.2.Figure 6
Impact ofM versus f′η. When E=0.3 and Pr=6.2.Figure 7
Impact ofM versus θη. When α=1,E=0.3, and Pr=6.5.Figure 8
Impact ofM versus θη. When α=1,E=0.3, and Pr=6.5.Table 1
α+1f″0 and −θ′0. When M=0.1,E=0.3, and Pr=6.2.
αα+1f″0−θ′010.7325471.745840.950.7101791.732940.900.6913561.718650.850.676081.702720.800.6643321.68486Table 2
α+1f″0 and −θ′0. When E=0.3 and Pr=6.2.
α,Mf″0−θ′0α,Mf″0−θ′0α=1,M=0.0,E=0.10.6546861.68489α=0.95,E=0.1,M=0.00.69661.73290M=0.10.7263031.70721α=0.90,M=0.10.6993561.71865M=0.50.7395221.71849α=0.85,M=0.50.7191291.70257M=10.828821.73258α=0.80,M=10.7493241.68455α=1M=0.1,E=0.10.7263031.70721α=0.90,M=0.10.6993561.71865E=0.30.71410211.70721α=0.85,E=0.30.6952311.71865E=0.50.70212021.70721α=0.80,E=0.50.6914521.71865Table 3
2f″0and−θ′0. When α=1..
MPrf″0 (present)f″0 [18]−θ′0 (present)−θ′0 [18]0.06.20.7175340.7179841.745881.745980.10.7325470.7329951.745841.745990.50.7916130.7918741.745671.7458910.8633060.8637171.745461.745870.16.40.8633060.8638281.734821.734986.60.8633060.8638971.725141.725686.80.8633060.8635641.715541.71567


## 6. Conclusion

The fluid flow model in the form of boundary layer flow is considered in the two-dimensional space. The basic governing equations are transformed from the PDEs into the noninteger-order form of ODEs. The transform variable used in this transformation is in the noninteger order, and as a result, the high nonlinear ODE system is achieved. The influence of the physical parameters is obtained and shown graphically. The results obtained show that the influence of parameters is compact in comparison with conventional results. Moreover, it has been observed that fractional-order models are highly nonlinear. In very limited models, the stress is linear and in most of the fluid flow problems, the stress is nonlinear. Therefore, the fractional-order derivative is more essential and appropriate to deal the nonlinear problems.The recent work is the generalized form of the 2D model, andn=1 becomes a special case for the classical model. Electric and magnetic field parameter results are obtained in both cases. The electric field improves the fluid motion while the magnetic field declines the fluid velocity.



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