Abstract
This theoretical study explores the impact of heat generation/absorption with ramp wall velocity and ramp wall temperature on the magnetohydrodynamic (MHD) time-dependent Oldroyd-B fluid over an unbounded plate embedded in a porous surface. The mathematical analysis of fractional governing partial differential equations has been established using systematic and powerful techniques of Laplace transform with its numerical inversion algorithms. The fractionalized solutions have been traced out separately through all fractional differential operators. Nondimensional parameters along with Laplace transformation are used to find the solution of temperature and velocity profiles. Fractional time derivatives are used to analyze the impact of fractional parameters (memory effect) on the dynamics of the fluid. While making a comparison, it is observed that the fractional-order model is the best to explain the memory effect as compared to classical models. The obtained solutions are plotted graphically for different values of physical parameters. Our results suggest that the velocity profile decreases by increasing the effective Prandtl number. Furthermore, the existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.
1. Introduction
The interest in studying problems involving non-Newtonian fluid flows has considerably grown for their wide range of applications: from drilling oil and gas wells and well completion operations to industrial processes involving waste fluids, synthetic fibre foodstuffs, and the extrusion of molten plastics. The attributes of fluid flow trace the diversity of the physical structure for non-Newtonian fluid flow. In such a fluid, stress and rate of strain have a nonlinear relationship. Oldroyd-B fluids have become a significant model of rate-type fluid. The procedure for the flow of rate-type fluids was discussed by Oldroyd [1]. Viscoelasticity has important implications due to the characterization of viscoelastic parameters (relaxation and retardation phenomenon), elastic shearing strain, thermal relaxation, and other rheological properties [2, 3]. In this regard, the thermodynamical analysis for the constitutive model of thermoplastic, viscoelastic, and viscoplastic was observed by Krairi and Doghri [4] through the Cattaneo–Christov heat model. The temperature distribution and relaxation time of the heat flux were emphasized by the temperature equation.
The technique of fractional calculus has been used to formulate mathematical modeling in various technological developments, engineering applications, and industrial sciences. Different valuable work has been discussed for modeling fluid dynamics, signal processing, viscoelasticity, electrochemistry, and biological structure through fractional time derivatives. This fractional differential operator found useful conclusions for experts to treat cancer cells with a suitable amount of heat source and has compared the results to see the memory effect of the temperature function. As compared to classical models, the memory effect is much stronger in fractional derivatives. From the past to the present, modeling of different processes is handled through various types of fractional derivatives and fractal-fractional differential operators, such as Caputo (power law), Atangana–Baleanu (Mittag–Leffler law), Caputo–Fabrizio (exponential law), Riemann–Liouville, and modified Riemann–Liouville (power law with boundaries) [5–16]. Ramped wall velocity and temperature with MHD fluid flow are gaining attention of many researchers. Physically, the implementation of ramped wall velocity with temperature in real-life problems has a significant role, but mathematically, it is difficult to handle such conditions. Ramped heating is used to control and increase the temperature with adiabatic conditions in an effective way. Firstly, Ahmed and Dutta [17] discussed the simultaneous use of ramped velocity and ramped temperature. Seth et al. [18–20] investigated heat and mass transfer phenomena with ramp temperature conditions. Recently, Tiwana et al. [21] and Anwar et al. [22] analyzed the MHD Oldroyd-B fluid under the effect of ramped temperature and velocity.
In context with fractional differential operators, convective flow with ramped wall temperature for nonsingular kernel was analyzed by Riaz et al. [23]. Moreover, Riaz et al. [24] investigated the study of heat and mass transfer in the MHD Oldroyd-B fluid with ramped wall temperature using local and nonlocal differential operators. Additionally, the recent studies on modern fractional differential operators and viscoelastic fluids can be traced out in [25–38]. For this problem, the noninteger differentiable operator is chosen for the fractional MHD Oldroyd-B model which is developed under thermal radiation, ramp velocity, and ramp temperature associated with physical initial and boundary conditions. The model is solved via the Laplace transform technique and inversion algorithm. The required results are displayed in graphs with physical arguments.
2. Problem Statement
We discuss unsteady magnetohydrodynamic (MHD) fractional convective Oldroyd-B fluid flow under Boussinesq approximations over an infinite plate. Figure 1 represents the flow geometry of the magnetized Oldroyd-B fluid. Under these presumptions, the governing equation for the Oldroyd-B fluid with appropriate conditions is defined as follows [22]:
The appropriate conditions are given as follows:
The dimensionless parameters in equations (1) to (3) are mentioned in the following:
Applying (4) into (1)–(3), we required a set of dimensionless governing equations in the form of PDE’s system presented as follows:where
The dimensionless corresponding conditions can be given as follows:
2.1. Governing Equations in terms of Singular and Nonsingular Kernels
We define Caputo time derivative with its Laplace transform defined in the following [39]:
The Caputo–Fabrizio fractional derivative and its Laplace transformation are defined as follows [40]:
The Atangana–Baleanu fractional derivative and its Laplace transformation are defined as follows [40]:
3. Temperature Profile via Fractional Operators
3.1. Temperature Profile via the Caputo Approach
Generating equation (6) for the fractional form, we imposed equation (8) on equation (6), and we have
We prefer to apply Laplace transform given in (8) on equation (11). The resultant form of the above expression is
The solution of equation (12) isand using equation (7), we find out the arbitrary parameter:
3.2. Temperature Profile via the Caputo–Fabrizio Approach
Generating equation (6) for the fractional form, we imposed equation (9) on equation (6), and we have
The solution of equation (16) isand using equation (7), we find out the arbitrary parameter:
3.3. Temperature Profile via the Atangana–Baleanu Approach
Generating equation (6) for the fractional form, we imposed equation (10) on equation (6), and we have
The solution of equation (20) isand using equation (7), we find out the arbitrary parameter:where and
4. Velocity Profile via Fractional Operators
4.1. Velocity Profile via the Caputo Approach
We utilize Laplace transformation for the solutions of the velocity profile given by equation (5):
We prefer to apply Laplace transform given in (8) on equation (23). The resultant form of the above expression is
The solution of the homogeneous part of (24) is
The required solution can be written as
4.2. Velocity Profile via the Caputo–Fabrizio Approach
We utilize Laplace transformation for the solutions of the velocity profile given by equation (5). We prefer to apply Laplace transform (9) on equation (5). The resultant form of the above expression is
The simplest form of the above equation is as follows:and using (7), we find out the arbitrary parameter:where , , , and
4.3. Velocity Profile via the Atangana–Baleanu Approach
We utilize Laplace transformation for the solutions of the velocity profile given by equation (5). We prefer to apply Laplace transform (10) on equation (5). The resultant form of the above expression isand using (7), we find out the arbitrary parameter:
As in the required velocity expressions (27), (30), and (32), we get the same result for the classical model as discussed in [22]. Furthermore, if we neglect and , then the results are identical which were obtained by Riaz et al. [35]. This shows the validation of our obtained results. We use classical computational and numerical techniques such as Stehfest’s [41] and Tzou’s algorithms [42] for the inverse of Laplace transform. Tzou’s calculation for our numerical inverse Laplace iswhere is the real part, represents the imaginary part, and is the natural number.
5. Results and Discussion
This section is dedicated to present physical interpretation of the obtained results via C, CF, and AB differential operators on the MHD fractional Oldroyd-B fluid over an infinite vertical plate on the porous medium. Analytical results are investigated via Laplace transformation with the inversion algorithm for velocity and energy profiles. The graphical representations are depicted for showing the influences of different physical parameters on velocity and temperature using the package of Mathcad-15. The time influence on all fractional derivative operators is analyzed in Figure 2. It clearly shows that, for the variation of time, the behavior of the velocity profile is the same. The resultant velocity of the ABC model is larger than the other fractional models. Effect of : the influence of fraction parameter on velocity can been seen through Figure 3. Clearly, fluid velocity reduces with the increase in the fractional parameter for small and large time. It is worth mentioning that profiles for these are best to explain the history (memory) of the fluids. While making comparison, velocity for the Atangana–Baleanu model is greatest because it has a nonlocal kernel. Velocity for CF is greater than C. This is because CF has a nonsingular kernel that imitates C with the singular kernel. Effect of : Figure 4 investigates the impact of the magnetic force on all fractional operators. This graphical representation indicates that, with an increase in the magnetic field, the velocity reduces due to Lorentz force. By increasing the parameter of , the Lorentz force also increases. Fluid flow on the boundary layer slows down due to this force. Effect of : Figure 5 shows the impact on for the velocity field versus time. It can be seen that the velocity field enhances by increasing . It is supported by the physical fact that is the fraction of buoyancy and viscous forces. An increase in means that the buoyancy force gets stronger near the plate such that it overcomes the viscous force and that the fluid gets accelerated. Effect of : Figure 6 shows the impact on the velocity field for . As increases, the thickness of the momentum boundary layer reduces which results in the deceleration of the fluid. As a relaxation time increment implies that the fluid will take extra time to calm, it readily justifies the decrease in velocity. It is quite a reverse behavior as compared to . Effect of : Figure 7 shows the behavior of velocity curves for . It is observed that velocity enhances with the increase in for all fractional models. The velocity behavior is also observed for the variation of time. Effect of : Figure 8 discusses the effect of using C, CF, and ABC models with the variation of time. Specific heat boundary thickness depends on . The thickness of the momentum and boundary layer is controlled by . It is seen from the graph that the decrease in the velocity is observed by the increase in the value of . Effect of : Figure 9 highlights the effect of the fractional parameter on the temperature profile for fractional models. With the increase in , the resultant temperature decreases. Temperature for CF and ABC is more as compared to C in all cases. Moreover, as , temperature curves for noninteger order approach integer order.
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6. Conclusion
This paper studies MHD Oldroyd-B fluid flow with ramped wall temperature and velocity under the influence of the thermal radiation in a porous medium. Fractional derivative operators with the inversion algorithm are used to acquire the solution of velocity and temperature. The significant remarks for this article are as follows:(1)Velocity curves show decreasing behavior for fractional parameters and . The velocity field decreases by increasing the value of (2)Velocity increases as increases for all fractional models.(3)Velocity profile is a decreasing function of for all fractional operators.(4)Velocity profile shows an opposite behavior for and for all fractional operators.(5)Temperature decreases by enhancing the value of the fractional parameter.(6)ABC fractional operator is more considerable as compared to all the other fractional operators.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly to writing this manuscript.
Acknowledgments
The authors are highly thankful and grateful for the support provided by the Deanship of Scientific Research at Majmaah University under Project no. 1439-41.