Abstract
This paper focuses on the comparative study of natural convection flow of fractional Maxwell fluid having uniform heat flux and radiation. The well-known Maxwell fluid equation with an integer-order derivative has been extended to a non-integer-order derivative, i.e., fractional derivative. The explicit expression for the temperature and velocity is acquired by utilizing the Laplace transform (LT) technique. The two fractional derivative concepts are used (Caputo and Caputo–Fabrizio derivatives) in the formulation of the problem. Utilizing the Mathcad programming, the effect of certain embedded factors and fractional parameters on temperature and velocity profile is graphically presented.
1. Introduction
To specify the performance of non-Newtonian fluids, numerous models have been applied. The Maxwell fluid is the first viscoelastic rate type fluid, which is also extensively utilized. The differential form and rate type models have gotten a lot of attention among them. In recent years, this model has shown some achievements in portraying the reactions of some polymeric liquids. In industry and engineering, viscoelastic fluids pass through many processes, such as synthetic propellants and so on. Because of the simplicity of the Maxwell fluid, many investigators are paying particular attention to it [1–4]. Khan et al. [5] researched on heat transfer of Maxwell fluid through an infinite vertical plate. In this study, they obtained the analytical solutions for temperature and velocity via LT. Such a model was studied by Khan et al. [6] using fractional CF derivative.
The subject of fractional calculus is as old as standard calculus. Fractional calculus began when L’ Hospital wrote to Leibnitz about the significance of when . Leibnitz replied in 1695 saying that it might be an apparent paradox from which one day useful repercussions would be drawn. Between the 17th century and the early 20th century, the subject of fractional calculus stayed more or less dormant. The subject matter has been found in applications in all kinds of problems in various fields over the last few years, like fluid flow, reaction, diffusion, relaxation, rheology, reaction-diffusion, oscillation, anomalous diffusion, physics, electrical network, chemistry, dynamical problems, and so on. Nowadays, many researchers are using the concept of fractional derivative because fractional-order differential equation solutions define real-life situations effectively than the solution obtained through the corresponding integer-order differential equations. Aman et al. [7] discussed about heat, velocity, and shear stress of fractional Maxwell model in a flexible medium using numerical LT. The semianalytical solutions for Maxwell fluid with fractional derivative were discussed in [8, 9]. The solutions for generalized Maxwell fluid using Fourier and Laplace transform can be determined by Fetecau et al. [10]. Maxwell nanofluids were examined by Aman et al. [11] using four distinct molecular liquids. The impact of second-order fractional Maxwell (MHD) fluid was investigated by Liu and Guo [12].The flow of Maxwell fluid with MHD effects flowing over a stretching sheet was investigated numerically by Shateyi and Marewo [13]. Mohi [14] discussed the closed-form solution of fractional Maxwell of MHD effects using Laplace and Fourier transform. The comparative research was performed in [15] on MHD Maxwell fluid with Newtonian heating on a boundary layer. Abro and Shaikh [16] investigated the analytical solutions for Maxwell fluid over a vibratory plane. Asjad et al. [17] presented the comparisons between Caputo and Caputo–Fabrizio fractional derivatives on second-grade fluid over Newtonian heating. Raza and Ullah [18] used the fractional Maxwell fluid to compare the fractional derivatives of C and CF using the Laplace transformation. Maxwell fluid’s natural convection between two parallel plates was discussed by Wang et al. [19]. The exact solution for Maxwell MHD fluid in a perforated medium was obtained by Khan et al. [20]. Zheng et al. [21] investigated extended Maxwell flow due to a vibratory and uniform moving plate. Fetecau et al. [22] used the LT to solve the second problem of Stokes for Maxwell fluids. Farooq et al. [23] presented the MHD Maxwell flow through the infinitely stretched surface of nanomaterials. Many interesting and very useful results related to the cylindrical Maxwell model can be found in [24–27]. The analytical and semianalytical solutions for Maxwell fluid between two moving plates were obtained by Hisham et al. [28]. Free convection flow has many applications in science and engineering like determining heat losses or heat load for heating, ventilating, air conditioning, and so on. Azhar et al. [29] considered the mixed convection flow of fractional nanofluids with uniform heat flux and heat source. Toki [30] examined the natural convection flow of unsteady MHD fluid and found the exact solutions of flow parameters. Fetecau et al. [31] studied the influence of radiation and permeability on MHD flow moving via an oscillating vertical plate with uniform heat flux.
The above literature motivates us to work on the natural convection flow of the fractional Maxwell model. The Caputo and CF fractional derivative approaches are used to determine the solutions. The numerical Laplace transform is utilized for the solutions of velocity and temperature. We see the comparisons between standard-order derivative and fractional-order derivative. Finally, we observe the graphical representation of various embedded parameters like Maxwell fluid factor, fractional parameter, and Grashof and Prandtl numbers.
2. Mathematical Statement
Here, we will assume the unsteady oscillatory natural convection flow of Maxwell fluid under the effects of radiation and uniform heat flux. Initially, with the uniform temperature , the fluid and the plate are at rest. After some time, at , the plate begins to oscillate in x-direction, and its velocity is given by
Following are the governing equations for the flow model:
Appropriate initial-boundary conditions are
Using Rosseland approximations [22, 30–32] and accepting the small temperature variation among the temperature of the free stream and the fluid temperature utilizing the Taylor theorem on at , and ignoring the second- and higher-order terms, we obtainandwhere are, respectively, the Stefan Boltzman constant and the mean absorption coefficient.
Substituting (7) into (6) and then into (3), we obtain (see [30, 32]) the following form:
Now, we will make use of the following similarity transformations in order to reduce equations (2), (4), and (8) into the dimensionless form
Substituting the above dimensionless parameters into equations (2), (4), and (8) and removing from ¸ we get the following forms:
To obtain the fractional model, we replace the inner time derivative by the time fractional derivative and we acquire a set of fractional PDEs as follows:
By neglecting the sign and utilizing equation (9), we will get the following initial and boundary conditions:
3. Preliminaries
3.1. Caputo Fractional Derivative
The Caputo fractional derivative is given by
The LT of Caputo fractional derivative is given by
3.2. Caputo–Fabrizio Fractional Derivative
The Caputo–Fabrizio fractional derivative is given by
The LT of Caputo–Fabrizio fractional derivative is defined by the following formula:
Remark. If , then we obtain
4. Temperature Profile with Caputo Derivative
Taking the LT of equation (14) and utilizing the corresponding ICs and BCs, we obtain the following form:
The inverse LT of equation (22) is given by convolution product:wherewhere .
5. Temperature Profile with CF Derivative
Applying LT to equation (14) and introducing equations (19) and (20), we get
Inverse Laplace of equation (25) is obtained through the Faltung theorem, and we havewhere and
6. Nusselt Number
The Nusselt number measures the rate of heat transfer at the plate. The Nusselt number for both equations (22) and (25) is constant:
7. Temperature Profile in Ordinary Case
Taking LT on equation (11) and utilizing the related initial-boundary conditions, we get
The inverse LT of equation (29) isand such solution was obtained by Abro and Shaikh [16].
8. Velocity Profile with Caputo Derivative
Taking LT on equation (13) and related initial-boundary conditions and substituting equation (22) for , we find that
9. Velocity Profile with CF Derivative
Taking the LT on equation (13) and related initial-boundary conditions and substituting equation (25) for , we have
10. Velocity Profile in Ordinary Case
Now taking the Laplace transform on equation (10) and related initial and boundary conditions and also substituting equation (29) for , we obtain
On solving the ODE (33), then we acquire
We can write equation (34) into the following equivalent form:where .
Let
The Laplace inverse of (36) is
Now applying the inverse LT of equation (38), we get
Let
Applying the inverse LT of equation (40), we get
Taking the inverse LT on equation (35) and by the convolution theorem, we get
11. Shear Stress with Caputo Time Fractional Derivative
Taking Laplace transform on (15), we acquire
Differentiating (31) w. r. t , we get
Substituting (44) into (43), we obtain
12. Shear Stress with Caputo–Fabrizio Time Fractional Derivative
Taking LT of Caputo–Fabrizio fractional derivative on (15), we get
Differentiating (32) w. r. t , we have
Substituting (47) into (46), we obtain the following form:
13. Shear Stress in Ordinary Case
Taking Laplace transform on equation (12), we get
Differentiating equation (34) w. r. t we get
The above equation can be written as
Substituting equation (51) into equation (49), we havewhere
Taking the inverse LT on equation (52) and utilizing Faltung theorem, we get
Taking Laplace inverse transform on equation (53), we get
Due to the complex combination of Laplace transform in equations (32), (33), (47), and (50), analytical LP inversion is very difficult, so for the Laplace inversion, we use different numerical LP inversion methods like Stehfest’s numerical method, Tzou’s algorithms, and Zakian’s algorithms.
14. Numerical Discussion and Graphs
The aim of this research is to study the Maxwell fluid's natural convection flow with radiation and consistent heat flow. The differential model is developed into fractional order. There are two fractional derivative concepts that we used (Caputo and Caputo–Fabrizio derivatives). Solutions for temperature and velocity are extended to Caputo and Caputo–Fabrizio derivatives. Solutions are obtained through the Laplace transform method. The effect of various embedded factors on temperature and velocity is a key feature of the model. We are also interested in comparing the Caputo and Caputo–Fabrizio derivative results. Figure 1 shows the behavior of radiation parameter on temperature. The Caputo fractional model has a smaller temperature as compared to Caputo–Fabrizio. The enhancement of radiation parameter enhances the fluid temperature. The variation of time on temperature is shown in Figure 2. Figure 2 presents the same behavior of fractional models like Figure 1. The fluid temperature increases with increasing time. Due to this, the boundary layer increases with increasing time. The impact of is indicated in Figure 3. It is stated that incrementing the Prandtl number decrements the temperature. Physically, the higher the value of the higher the fluid viscosity and the lower the thermal conductivity. Because of this, the thickness of the boundary layer falls. Figure 4 shows the comparison between the fractional model and ordinary model, i.e., . The temperature of the ordinary model is higher than that of the fractional model.
Tables 1–3 show some basic findings of the given work. Table 4 depicts a numerical solution for temperature and velocity profiles calculated using the CF and Caputo time derivatives for different values of fractional factor . It indicates that increasing the fractional factor’s value enhances the fluid’s temperature and velocity. It suggests that CF has a greater velocity and temperature than Caputo. Table 2 shows the comparisons between various numerical inverse LT algorithms like Stehfest’s algorithm and Tzou’s algorithm with the exact solution. It verifies the validity and correctness of solutions up to the desired level of precision. Table 3 presents the relationships between the solutions obtained for temperature, velocity, and shear stress using various embedded factors. From this, we can see that the increase of various embedded parameters increases the fluid temperature, velocity, and shear stress except for the Prandtl number which shows the inverse effects from other parameters.
The effect of the velocity curve for the Grashof number is presented in Figure 5. It shows that the enhancement of the value of enhances the velocity of both the models. This behavior is due to the rise in buoyancy force because of temperature gradient. Figure 6 shows the influence of radiation factor on velocity. Clearly, Figure 6 shows that the velocity of fluid with C and CF variants of fractional derivatives increases due to increasing value of the radiation factor. Figure 7 portrays the behavior of Maxwell fluid factor on velocity field. It shows the same effect as Figure 6. Incrementing the value of the Maxwell fluid factor increments the velocity of the Caputo and Caputo–Fabrizio models. In Figure 8, the impact of time is shown for velocity. It is noted that the fluid velocity increases with the increase in time. The temperature is greater near the plate and decreases as we go away from the plate and finally becomes zero in the free stream region. From Figure 9, we can see that the fluid velocity decreases when the value of Prandtl number increases. Moreover, the enhancement of decreases the thickness of the boundary layer. Figure 10 shows the comparison between the fractional model and ordinary fluid model. It shows that the velocity of the ordinary fluid model, i.e., , is greater than the velocity of the fractional model.
15. Conclusion
The objective of the present work is to conduct a comparative study of the natural convection flow of fractional Maxwell fluid in the presence of radiation and uniform heat flux. The two fractional derivative definitions are used (C and CF) in the formulation of the problem. The solutions for heat and velocity are obtained through the Laplace transform method. The following are the study’s key findings:(i)The temperature of the fluid increases with increase in the embedded factors like (ii)Increasing the Prandtl number reduces the fluid’s temperature.(iii)The enhancement of the fluid parameters like enhances the fluid’s velocity for both models, while the Prandtl number shows the adverse effects from other factors.(iv)Increasing the value of increases the shear stress of the fluid, while the Prandtl number has the opposite effect.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This study was supported by the Key Scientific Research Projects of Chaohu University (subject nos. XLZ-201804 and XLZ-201808) and the Industry University Research Project of Chaohu University (subject nos. hxkt20200013 and hxkt20200059).