 Research
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Application of fractional differential equations to heat transfer in hybrid nanofluid: modeling and solution via integral transforms
 Muhammad Saqib^{1},
 Ilyas Khan^{2}Email author and
 Sharidan Shafie^{1}
https://doi.org/10.1186/s1366201919885
© The Author(s) 2019
 Received: 22 October 2018
 Accepted: 23 January 2019
 Published: 6 February 2019
Abstract
This article deals with the generalization of natural convection flow of \(Cu  Al_{2}O_{3}  H_{2}O\) hybrid nanofluid in two infinite vertical parallel plates. To demonstrate the flow phenomena in two parallel plates of hybrid nanofluids, the Brinkman type fluid model together with the energy equation is considered. The Caputo–Fabrizio fractional derivative and the Laplace transform technique are used to developed exact analytical solutions for velocity and temperature profiles. The general solutions for velocity and temperature profiles are brought into light through numerical computation and graphical representation. The obtained results show that the velocity and temperature profiles show dual behaviors for \(0 < \alpha < 1\) and \(0 < \beta < 1\) where α and β are the fractional parameters. It is noticed that, for a shorter time, the velocity and temperature distributions decrease with increasing values of the fractional parameters, whereas the trend reverses for a longer time. Moreover, it is found that the velocity and temperature profiles oppositely behave for the volume fraction of hybrid nanofluids.
Keywords
 Applications of fractional derivatives
 Heat transfer problem
 Modeling and simulation
 Hybrid nanofluid
MSC
 34A08
 76R10
1 Introduction
In recent decades, it was acknowledged that fractional operators are appropriate tools for differentiation as compared to the local differentiation particularly in physical real word problems. These fractional operators can be constructed by the convolutions of the local derivative as the kernel of fractional operators; various kernels for fractional operators have been suggested in the literature but the most common is the power law kernel (\(x^{  \alpha} \)), which is used in the construction of Riemann–Liouville and Caputo fractional operators (see [1], p. 65–106). However, the exponential decay law \(\exp (  \alpha x )\) was used by Caputo and Fabrizio (see [2], p. 1–13). Atangana and Baleanu developed fractional operators in the Caputo and Riemann–Liouville sense using the generalized Mittag–Leffler law \(E_{\alpha} (  \phi x^{\alpha} )\) as a kernel (see [3], p. 763–769). All these fractional operators have some shortcomings and challenges but at the same time this area is growing fast, and researchers devoted their attention to this field (see [4–10] and the references therein).
It is important to mention here that fractional order calculus has many applications in almost every field of science and technology which includes diffusion, relaxation process, control, electrochemistry and viscoelasticity (see [11], p. 79–85). Zafar and Fetecau ([12], p. 2789–2769) applied Caputo–Fabrizio fractional derivative to the flow of Newtonian viscous fluid flowing over the infinite vertical plate. Markis et al. ([13], p. 1663–1679) analyzed the flow of a fractional Maxwell’s fluid. According to their report, the fractional results showed excellent agreement with experimental work by adjusting the fractional parameter. Alkahtani and Atangana ([14], p. 106–113) used different fractional operators to analyze the memory effect in a potential energy field caused by a charge. They presented some novel numerical approaches to the solutions of a fractional system of equations. Vieru et al. ([15], p. 85–96) presented exact solutions for the timefraction model of viscous fluid flow near a vertical plate taking into consideration mass diffusion and Newtonian heating. Abro et al. ([16], p. 1–10) presented exact analytical solutions for the flow of an OldroydB fluid in a horizontal circular pipe. Jain ([17], p. 1–11) introduced a novel and powerful numerical scheme and implemented to different fractional order differential equations. Some other interesting and significant studies on fractional derivatives can be found in [18–26] and the references therein.
The nanofluid is an innovation of nanotechnology to overcome the problems of heat transport in many engineering and industrial sectors. A detailed discussion on nanofluids with a list of applications is reported by Wang et al. ([27], p. 1–19) in a review paper. Sheikholeslam et al. ([28], p. 71–82) numerically studied the shape effect and the external magnetic field effect on the \(F_{3}O_{4}  H_{2}0\) nanofluid inside a porous enclosure. Hassanan et al. ([29], p. 482–488) developed exact solutions for nanofluids with different nanoparticles for the unsteady flow of a micropolar fluid. The literature of nanofluids has exponentially increased and has reached a next level by introducing hybrid nanofluids which are the suspensions of two or more types of nanoparticles in the composite form with low concentration. Hybrid nanofluids are introduced to overcome the drawbacks of single nanoparticle suspensions and connect the synergetic effect of nanoparticles. The hybrid nanofluid is branded to further improve the thermal conductivity and heat transport, which leads to industrial and engineering applications with low cost (see [30], p. 262–273). Hussain et al. ([31], p. 1054–1066) carried out an entropy generation analysis on a hybrid nanofluid in a cavity. Farooq et al. ([32], p. 1–14) presented a numerical study on hybrid nanofluids keeping into consideration suction/injection, entropy generation, and viscous dissipation.
In the existing literature, experimental, theoretical and numerical studies on hybrid nanofluids are very limited. A study of a hybrid nanofluid fluid with exact solutions and the Caputo fractional derivative even does not exist. So, there is an urgent need to contribute to the literature of hybrid nanofluids using the application of fractional differential equations. Motivated by the above discussion, the present study focused on the heat transfer in hybrid nanofluid in two vertical parallel plates using fractional derivative approach. A waterbased hybrid nanofluid is characterized here with composite hybrid nanoparticles of cupper (Cu) and alumina (\(Al_{2}O_{3}\)). The fractional Brinkman type fluid model with physical initial and boundary conditions is considered for the flow phenomena. The Laplace transform technique is used to obtain exact analytical solutions for the velocity and temperature profiles. Using the properties of the Caputo–Fabrizio fractional derivative the obtained solutions are reduced to the classical form for \(\alpha = 1\) and \(\beta = 1\). To explore the physical aspect of the flow parameters the solutions are numerically computed and plotted in different graphs with a physical explanation.
2 Problem’s description
3 Thermophysical properties of hybrid nanofluid
This section demonstrates the modification of thermophysical properties of a conventional nanofluid and a hybrid nanofluid \(Cu  Al_{2}O_{3}  H_{2}O\) in a spherical shape.
3.1 The effective density
3.2 The effective dynamic viscosity
3.3 The effective volumetric thermal expansion and heat capacitance
3.4 The effective thermal conductivity
Numerical values of thermophysical properties of base fluid and nanoparticles
Material  Base fluid  Nanoparticles  

\(H_{2}O\)  \(Al_{2}O_{3}\)  Cu  
\(\rho\ ( \mathrm{kg} / \mathrm{m}^{3} )\)  997.1  3970  8933 
\(C_{p}\ ( \mathrm{J} / \mathrm{kg} \mathrm{K} )\)  4179  765  385 
K (W/mK)  0.613  40  400 
\(\beta_{T} \times 10^{  5}\ (K^{  1})\)  21  0.85  1.67 
Pr  6.2  –  – 
4 Generalization of local model
 1.Property 1: According to Losanda and Nieto (see [38], p. 87–92) \(N ( \delta )\) is the normalization function such that$$ N ( 1 ) = N ( 0 ) = 1. $$(22)
 2.Property 2: taking into consideration Eq. (22), the Laplace transform of Eq. (21) yieldssuch that$$ L \bigl\{ {}^{CF}D_{t}^{\delta} f ( t ) \bigr\} ( q ) = \frac{q\bar{f} ( q )  f ( 0 )}{ ( 1  \delta )q + \delta},\quad 0 < \delta < 1, $$(23)where \(\bar{f} ( q )\) is the Laplace transform of \(f ( t )\) and \(f ( 0 )\) is the initial value of the function.$$ \lim_{\delta \to 1} \bigl[ L \bigl\{ {}^{CF}D_{t}^{\delta} f ( t ) \bigr\} ( q ) \bigr] = \lim_{\delta \to 1} \biggl\{ \frac{q\bar{f} ( q )  f ( 0 )}{ ( 1  \delta )q + \delta} \biggr\} = q\bar{f} ( q )  f ( 0 ) = L \biggl\{ \frac{\partial f ( t )}{\partial t} \biggr\} , $$(24)
5 Solution of the problem
To solve Eqs. (19) and (20) the Laplace transform method \(L \{ f ( t ) \} ( q ) = \bar{f} ( q ) = \int_{0}^{\infty} f ( t )e^{  qt}\,dt\), will be applied by using the corresponding initial and boundary conditions from Eqs. (17) and (18) to develop exact analytical solutions for the velocity and temperature profiles.
5.1 Solutions of the energy equation
To reduce the solutions obtained in Eq. (33) to classical or local form, Eq. (24) is used for the following.
5.2 Solution of momentum equation
The term \(v_{1} ( \xi,\tau )\) is numerically obtained using Zakian’s algorithm. In the literature, it is proven that the Zakian algorithm is a stable way for the inverse Laplace transform because the truncated error for five multiple terms is negligible ([40], p. 83).
6 Results and discussion
In this article, the idea of the fractional derivative is used for the generalization of the free convection flow of the hybrid nanofluid. The governing equations of the Brinkman type fluid along with the energy equation is fractionalized using the Caputo–Fabrizio fractional derivative. The fractional PDEs are more general and are known as master PDEs. The momentum and energy equations are solved analytically using the Laplace transform technique. The obtained results are displayed in various graphs to study the influence of the pertinent corresponding parameters, such as the fractional parameters \(\alpha \mbox{ and } \beta\), the volume fraction of hybrid nanofluid \(\phi_{hnf}\), the heat generation parameter Q, the Brinkman parameter \(\beta_{b}\) and the thermal Grashof number Gr on velocity and temperature profiles.
7 Concluding remarks

The velocity and temperature profiles show an increasing behavior for increasing values \(\alpha \mbox{ and } \beta\) being most dominant for \(\alpha,\beta = 1\) for a larger time. But this effect reverses for a shorter time.

The fractional velocity and temperature are more general. Hence, the numerical values for \(v ( \xi,\tau )\) and \(\theta ( \xi,\tau )\) can be calculated for any value of \(\alpha \mbox{ and } \beta\) such that \(0 < \alpha,\beta < 1\).

The temperature distribution shows a very similar variation for different shapes of the hybrid nanoparticles, so the density of the nanoparticles is a significant factor as compared to thermal conductivity.

The velocity profile decreases with increasing values of \(\phi_{hnf}\) but this effect is opposite in the case of the temperature profile.

With increasing values of Gr, the free convection became dominant, increasing the nanofluid velocity for positive values but this trend reverses for negative values of Gr.

The velocity retards for larger values of \(\beta_{b}\) due the enhancement in the drag forces.
Declarations
Acknowledgements
The authors would like to acknowledge Ministry of Higher Education (MOHE) and Research Management CentreUTM, Universiti Teknologi Malaysia UTM, for the financial support through grant numbers 15H80 and 13H74 for this research.
Availability of data and materials
Not applicable.
Funding
No specific funding was received for this project.
Authors’ contributions
IK formulated the problem and drafted the manuscript. MS solved the problem and drafted the manuscript. SS performed the numerical simulation and plotted the results. All authors read and approved the final manuscript.
Competing interests
The authors do not have any competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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