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.
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, -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 putting 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
The boundary conditions are
Here, , and 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]
Upon using Equation (6) in Equations ((1)), ((2)), ((3)), ((4)), the noninteger-order ODEs are obtained as
Reduced conditions are
reduce the above equations into the classical order as
In Equations (11) and (12), , and stand for the order exponent, magnetic field parameter, Prandtl number, and electric field as
2.1. Physical Quantities of Interest
The physical number is stated as
The simplified form of Equation (20) using the similarity variable is
3. Caputo Fractional Derivatives
The brief and basic theory of the Caputo is displayed as follows.
3.1. Definition 1
Let and . According to the Caputo fraction derivative, using as the fractional order from the function is derived as
3.2. Property
Let be such that and exist almost everywhere, and let Then, exists almost everywhere and
4. Solution Methodology
Equations (11), (12), (13) are selected as
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 as 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 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 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 . 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 theelevates the energy transferenceas shown in Figures 7 and 8. The impact of the parameter 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 drops the fluid velocity for its larger values, and the impact is the same as shown in Table 2 taking the values of in both increasing and decreasing forms like and or , while this retort is opposing in the case of Nusselt number. The electric field 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 and versus energy transition are shown in Table 3. The larger magnitude of enhances the heat transfer rate while the larger magnitude of 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.
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, and 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.
Data Availability
The relevant data exist in the article.
Conflicts of Interest
No such interests exist.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Group Research Project under grant number RGP. 2/160/43.