Sage Journals: Discover world-class research

Abstract

In this article, a free convection flow of $Cu - A l_{2} O_{3} - H_{2} O$ hybrid Maxwell nanofluids through a channel formed by two infinite vertical plates have been studied. Together with the the energy balance and heat source, a fractional model of Maxwell fluid is considered. To develop an analytical exact solution for velocity field, only the Caputo-Fabrizio definition of non-integral derivative together with application of Laplace transform method has been used. Some graphical presentation and discussion are made to see the effects of hybrid nanofluids and non-dimensional parameters on velocity boundary layer. As a result, a dual behavior of velocity was exposed due to fractional parameter for large and small times. A comparison between two kind of non-Newtonian fluids has been made and found that Brinkman fluid is more viscous than Maxwell fluid. Also, by letting Brinkman and Maxwell parameters zero, they coincides and the results obtained for Newtonian fluid showed graphically. The obtained results are realistic from the fractional model as by adjusting the values of fractional parameter can be compared with some experimental data.

Keywords

Caputo-Fabrizio hybrid nanofluid channel flow boundary layer heat source Maxwell fluid

Introduction

In the recent years, the fractional calculus has become an emerging tool in all fields of science. As compared to classical calculus, non integers derivatives play a more significant role in all scientific inquires. The functions of fractional order calculus in several fields of science and technology, like diffusion, electrochemistry, and viscoelasticity, combustion process, geophysics, fractional filters, fractional wavelets, fractional neutron point kinetic model, flow of magnetic particles in blood, formation of suspension in medicine, microelectronics are elaborated.¹ The solutions of non-integer order differential equations are found to describe the real-life situations better compared to the solutions coming from the corresponding integer order differential equations. Various fractional operators have been used efficiently in heat and mass transport phenomenons in dynamics of many fluids. Shah et al.² studied the electro-osmotic flow of Oldroyd-B fluid in micro channel and reveal the dual behavior of fluid velocity subject to Caputo-Fabrizio fractional parameter for large and small values of time. Al-Mdallal et al.³ solved model of Walter’s B fluid with the Caputo-Farizio fractional derivative without kernel. Sheikh et al.⁴ have been made a generalization of Casson fluid model and presented a compression among effects of Atangana-Baleanu and Caputo-Fabrizio. Imran et al.⁵ also gave a comparative study for fractional derivatives of singular and non singular kernal utilizing unsteady differential type fluid flow subject to Newtonian heating. Azhar et al.⁶ discussed free convection flow of fractional nanofluid over vertical flat plate with constant heat flux and heat generation. Kashif et al.^7–9 discussed the different numerical and analytical model of memristor, rotating gas flow, and RL and RC electrical circuits with different fractional approaches namely fractal fractional operators, Atangana-Baleanu, and Caputo-Fabrizio aproaches.

The intermingling of nanosized particle in base fluid like water and oil make the nanofluid. Since last two decades, the Nanofluid science has been serving and enhancing the performance of many heat transport processes in engineering and industrial sectors. Moreover, the new extension in nanotechnology is the invention of hybrid nanofluid. Basically this new product formed by mixing two or more substances of different physical and chemical properties like heat flux and thermal conductivity of substance have been exaggeratedly changed by hybrid nanofliud. Due to this high performance in thermal conductivity, nanofluid got an attention of many researchers.

Das et al.¹⁰ gave a numerical study that how nanofluids are efficient in thermal conduction and illustrated the affects of several factors like particle size, different base fluids, types of nanoparticles, solid volume fraction, temperature, pH etc. Xian et al.¹¹ presented an experimental study for performance of hybrid nanofluid in heat transfer in automobile radiator system. The hybrid nanofluids an amazing outcome of current advancement in heat transfer capabilities of conventional fluids. Babar et al.¹² gave a brief comparison between the nano and hybrid nanofluids and concluded that water-based nanofluids show the better performance for ethylene glycol-based nanofluids. Farhana et al.¹³ analyzed the flow of nano and hybrid nanofluid in tube of solar collector for three different designed models. Uysal et al.¹⁴ discussed the heat transfer due to conduction with entropy generation for a hybrid nanofluid flow through a minichannel and have been concluded that for a small addition of hybrid nanoparticle to pure water there is considerable increase in heat transfer coefficient and flux at boundary. Hayat et al.¹⁵ presented a comparison among the simple and hybrid nanofluids and found that even in the existence of heat source, chemical reaction, and radiations, the heat flux of Hybrid nanofluid is higher than the conventional nanofluid. Many other researchers validated the better performance of hybrid nanofluid by theoretical and experimental studies in heat transport problems for different situations and geometries using fractional calculus.^16–24 Ahokposi et al.²⁵ presented fractional flow with Mittag-Leffler law for ground water. Farooq et al.²⁶ have discussed the motion of hybrid nanofluid flow in a circular stretched disc under the influence of radiation, entropy generation and viscous dissipation. Hussanan et al.²⁷ investigated a convectional heat transfer for nanofluid flow of micropolar fluid along with the mixture of water, kerosene, and engine oil. Aminossadati et al.²⁸ have discussed a free conventional cooling process of localized heat source at the bottom of an enclosure filled by a nanofluid. Bhattad et al.²⁹ worked on model and presented an experimental study of hybrid nanofluid flow with heat transfer and pressure drop over a flate plate heat exchanger. Gul et al.³⁰ discussed Koo and Kleinstreuer model and elaborated the effects of heat generation and radiation upon an magnetohydrodynamic (MHD) flow of nanofluids induced by mix convection through a vertical channel. Ahmed et al.³¹ depicted the effects of radiation and chemical reaction on nanofluid flow through couple of Riga plates. Khan et al.³² considered a Casson nanofluid and discussed the unsteady flow of Casson nanofluid over a vertical plate with isothermal conditions for Newtonian heating and have been directed the effects of heat generation. Asif et al.³³ examined the motion of Brinkman type fluid in an open rectangular channel. In³⁴ Fetecau studied unsteady flow of an Oldroyd-B fluid generating by a constantly accelerating plate between two side walls. Sajad et al.³⁵ discussed the applied thermodynamics concepts for magnetohydrodynamic Brinkman fluid flow in porous medium.

Brinkman type fluid model utilized by Saqib et al.³⁶ presents a generalization of free convection flow of $Cu - A l_{2} O_{3} H_{2} O$ hybrid nanofluid through two infinite vertical parallel plates. In this piece of research, Saqib’s work has been extended to bigger class of fluid and a fractional Maxwell type fluid model of water based hybrid nano fluid subject to appropriates thermal and geometric conditions. Under consideration, nanofluid is the composition of copper (Cu) and alumina $(A l_{2} O_{3})$ hybrid nanoparticles. Energy momentum balance has been solved analytically and obtained the exact solutions for velocity and temperature fields. For $α = 1, and β = 1$ the obtained solutions are coincided to the solutions performed by classical derivatives of integer order. To discover the physical aspects of the flow and material parameters on the obtained results, some graphs for velocity profiles are plotted.

Mathematical statement of physical model

Consider a fractional Maxwell hybrid nanofluid presented in two infinite vertical parallel plates separated by a distance $d$ . The plates are situated in $xz -$ plane and the $y -$ axis is orthogonal to the plane of plates as shown the Figure 1. At first sight, the plates and fluid both are in equilibrium at atmosphere temperature $T_{0}$ . For the time $t = 0^{+},$ the temperature of one plate situated at $y = d$ rises or lowers from $T_{0}$ to $T_{W}$ as a result the natural convection of fluid takes place, consequently, the fluid begins to move along the $x -$ direction. In this situation, the body forces develop in the form of buoyancy forces due to the temperature gradient. Under these assumptions, the governing equations of the Maxwell $Cu - A l_{2} O_{3} - H_{2} O$ hybrid nanofluid are given by³⁶

\begin{matrix} ρ_{hnf} (1 + λ_{1} \frac{\partial}{\partial t}) \frac{\partial u (y, t)}{\partial t} = μ_{hnf} \frac{\partial^{2} u (y, t)}{\partial y^{2}} \\ + g (ρ β_{T})_{hnf} (1 + λ_{1} \frac{\partial}{\partial t}) (T (y, t) - T_{0}), \end{matrix}

(1)

(ρ Cp)_{hnf} \frac{\partial T (y, t)}{\partial t} = k_{hnf} \frac{\partial^{2} T (y, t)}{\partial y^{2}} + Q_{0} (T (y, t) - T_{0}),

(2)

with associated initial and boundary conditions

u (y, 0) = 0, T (y, 0) = T_{0} \forall y \geq 0,

(3)

u (0, t) = 0, T (0, t) = T_{0} for t > 0,

(4)

u (d, t) = 0, T (d, t) = T_{W} for t > 0,

(5)

where $u (y, t)$ is fluid velocity field function, $λ$ is relaxation time, $T (y, t)$ is the temperature, $Q_{0}$ is the heat source. and $ρ_{hnf}$ , $μ_{hnf}$ , $β_{hnf}$ , $(Cp)_{hnf}$ , $k_{hnf}$ are density, dynamic viscosity, volumetric thermal expansion, specific heat, and thermal conduction parameter for the hybrid nanofluid respectively as defined in³⁶ as:

\begin{matrix} ρ_{hnf} = (1 - ϕ_{hnf}) ρ_{f} + ϕ_{A l_{2} O_{3}} ρ_{A l_{2} O_{3}} + ϕ_{Cu} ρ_{Cu}, \\ μ_{hnf} = \frac{μ_{f}}{{1 - (ϕ_{A l_{2} O_{3}} + ϕ_{Cu})}^{2.5}}, \\ (ρ β_{T})_{hnf} = (1 - ϕ_{hnf}) (ρ β_{T})_{f} + ϕ_{A l_{2} O_{3}} (ρ β_{T})_{A l_{2} O_{3}} + ϕ_{Cu} (ρ β_{T})_{Cu}, \\ (ρ Cp)_{hnf} = (1 - ϕ_{hnf}) (ρ Cp)_{f} + ϕ_{A l_{2} O_{3}} (ρ Cp)_{A l_{2} O_{3}} + ϕ_{Cu} (ρ Cp)_{Cu}, \end{matrix}

(6)

where $ρ_{hnf}$ is the density of the hybrid nanofluid, $ρ_{f}$ is the density of the base fluid, $ϕ_{hnf}$ is the volume concentration of solid particles such that $ϕ_{hnf} = ϕ_{A l_{2} O_{3}} + ϕ_{Cu}$ , $ϕ_{A l_{2} O_{3}}$ is the volume concentration of alumina, $ρ_{A l_{2} O_{2}}$ is the density of alumina, $ϕ_{Cu}$ is the volume concentration of copper and $ρ_{Cu}$ is the density of copper. The thermophysical properties for base fluid and respective nanoparticles are define in Table 1.³⁶

Figure 1.

Physical model.

Table 1.

Numerical values of thermophysical properties of base fluid and nanoparticles.

Material	Base fluid	Nanoparticles	Cu
	$H_{2} O$	$A l_{2} O_{3}$
$ρ (kg m^{- 3})$	997.1	3970	8933
$C_{p} (Jk g^{- 1} K^{- 1})$	4179	765	385
$k (W m^{- 1} K^{- 1})$	0.613	40	400
$β_{T} \times 10^{- 5} (K^{- 1})$	21	0.85	1.87
Pr	6.2	-	-

The Caputo-Fabrizio fractional derivative and Laplace transform

The Caputo-Fabrizio fractional operator is defined by⁵

{{}^{CF}D}_{t}^{δ} f (t) = \frac{N (δ)}{1 - δ} \int_{0}^{t} \exp (- \frac{δ (t - τ)}{1 - δ}) \frac{\partial f (τ)}{\partial τ} d τ, 0 < δ < 1,

(7)

which is the convolution product of the functions $\frac{N (δ)}{1 - δ} \exp (\frac{- δ (t)}{1 - δ})$ and $f (t)$ of fractional order $δ$ , where $N (δ)$ is normalization function and define as $N (δ) = \frac{2}{2 - δ}$ .

In this paper, the following Laplace transform of Caputo-Fabrizio fractional operator will be utilized

£ {^{CF} D_{t}^{δ} f (t)} (q) = \frac{q \bar{f} (q) - f (0)}{(1 - δ) q + δ}, 0 < δ < 1,

(8)

such that

lim_{δ \to 1} [£ {^{CF} D_{t}^{δ} f (t)} (q)] = q \bar{f} (q) - f (0) = £ {\frac{\partial f (t)}{\partial t}} .

(9)

Modeling with Caputo-Fabrizio time fractional derivative

The dimensional model is renovated to a dimensionless model to lessen the involved parameters and unit free system by inserting the following non-dimensional variables

u = \frac{d}{ν_{f}} u, ξ = \frac{y}{d}, τ = \frac{ν_{f}}{d^{2}} t, θ = \frac{T - T_{0}}{T_{W} - T_{0}} .

(10)

Equation (1) leads to the following dimensionless form, after utilizing the dimensionless relations given in equation (10):

r_{0} (1 + λ \frac{\partial}{\partial t}) u (ξ, t) = r_{1} \frac{\partial^{2} u (ξ, t)}{\partial ξ^{2}} + r_{2} (1 + λ \frac{\partial}{\partial t}) Gr θ (ξ, t),

(11)

where

λ = \frac{λ_{1} ν_{f}}{d^{2}}, Gr = \frac{d^{3} g (β_{T}) f}{ν_{f}^{2}} (T_{W} - T_{0}),

are the Grashof number, and the relaxation parameter respectively, while

\begin{matrix} r_{0} = (1 - ϕ) + \frac{ϕ_{A l_{2} O_{3}} ρ_{A l_{2} O_{3}} + ϕ_{Cu} ρ_{Cu}}{ρ_{f}}, \\ r_{1} = \frac{1}{{1 - (ϕ_{A l_{2} O_{3}} + ϕ_{Cu})}^{2.5}}, \\ and r_{2} = (1 - ϕ) + \frac{ϕ_{A l_{2} O_{3}} {(ρ β_{T})}_{A l_{2} O_{3}} + ϕ_{Cu} {(ρ β_{T})}_{Cu}}{{(ρ β_{T})}_{f}}, \end{matrix}

are the constant parameters appearing in mathematical calculations.

Thermal balance equation is:

(ρ Cp)_{hnf} \frac{\partial T (y, t)}{\partial t} = - \frac{\partial q_{1} (y, t)}{\partial y} + Q_{0} (T (y, t) - T_{0}),

(12)

where $q_{1}$ is the heat flux.

The generalized fractional Fouriers law suggested by Hristov³⁷ and Henry et al.³⁸ through Caputo-Fabrizio fractional derivative:

q_{1} (y, t) = - k_{1 (hnf)} \frac{\partial T (y, t)}{\partial y} - k_{2 (hnf)} {{}^{CF}D}_{t}^{α} (1 - α) \frac{\partial T (y, t)}{\partial y},

(13)

where $α$ is the fractional parameter, $k_{1 (hnf)}$ and $k_{2 (hnf)}$ are the effective thermal conductivity and the elastic conductivity of hybrid nanofluid.

Using equation (13) in equation (12) and making the result the respective result dimensionless by using the relations from equation (10):

\begin{matrix} r_{3} \frac{\partial θ (ξ, τ)}{\partial t} = \frac{λ_{1 (hnf)}}{\Pr_{1}} \frac{\partial^{2} θ (ξ, τ)}{\partial ξ^{2}} + \frac{λ_{2 (hnf)}}{\Pr_{2}} {{}^{CF}D}_{t}^{α} (1 - α) \\ \frac{\partial^{2} θ (ξ, τ)}{\partial ξ^{2}} + Q θ (ξ, τ), \end{matrix}

(14)

where

\begin{matrix} Q = \frac{d^{2} Q_{0}}{k_{f}}, \Pr_{1} = \frac{(μ Cp) f}{k_{1 (f)}}, \Pr_{2} = \frac{(μ Cp) f}{k_{2 (f)}}, \\ λ_{1 (hnf)} = \frac{k_{1 (hnf)}}{k_{1 (f)}}, λ_{2 (hnf)} = \frac{k_{2 (hnf)}}{k_{2 (f)}}, \end{matrix}

r_{3} = (1 - ϕ) + \frac{ϕ_{A l_{2} O_{3}} {(ρ Cp)}_{A l_{2} O_{3}} + ϕ_{Cu} {(ρ Cp)}_{Cu}}{{(ρ Cp)}_{f}},

with non-dimensional initial and boundary conditions:

u (ξ, 0) = 0, θ (ξ, 0) = 0, \forall ξ \geq 0,

(15)

u (0, t) = 0, θ (0, t) = 0, for t > 0,

(16)

u (1, t) = 0, θ (1, t) = 1, for t > 0 .

(17)

Solution of the problem

To solve equations (11) and (14), the Laplace transform method will be used and closed form solutions will be developed for the velocity and temperature respectively subject to the corresponding initial and boundary conditions from equations (15) to (17).

Solutions of the temperature field

Applying the Laplace transform to equation (14), keeping in mind the Laplace transform of Caputo-Fabrizio fractional operator defined in equation (8), and using the corresponding initial condition from equation (15) yield the following transformed energy balance:

\begin{matrix} r_{3} q \bar{θ} (ξ, q) = \frac{λ_{1 (hnf)}}{\Pr_{1}} \frac{\partial^{2} \bar{θ} (ξ, q)}{\partial ξ^{2}} + \frac{λ_{2 (hnf)}}{\Pr_{2}} (1 - α) \\ [\frac{q}{(1 - α) q + α}] \frac{\partial^{2} \bar{θ} (ξ, q)}{\partial ξ^{2}} + Q \bar{θ} (ξ, q), 0 < α < 1, \end{matrix}

(18)

and after further simplification of equation (18), we have:

\frac{\partial^{2} \bar{θ} (ξ, q)}{\partial ξ^{2}} - \frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})} \bar{θ} (ξ, q) = 0, 0 < α < 1 .

(19)

Equation (19) satisfies the following transformed boundary conditions:

\begin{matrix} \bar{u} (0, q) = 0, \bar{θ} (0, q) = 0, \\ \bar{u} (1, q) = 0, \bar{θ} (1, q) = \frac{1}{q}, \end{matrix}

(20)

where

\begin{matrix} a_{1} = \frac{1}{1 - α}, m_{1} = \frac{λ_{1 (hnf)}}{r_{3} \Pr_{1}}, m_{2} = \frac{λ_{2 (hnf)}}{r_{3} \Pr_{2}}, \\ m_{3} = \frac{Q}{r_{3}}, m_{4} = \frac{a_{1} m_{1}}{m_{1} + m_{2}} \end{matrix}

The transformed solution of equation (19), using the corresponding boundary conditions from equation (20) is given by:

\bar{θ} (ξ, q) = \frac{1}{q} \frac{\sinh ξ \sqrt{\frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})}}}{\sinh \sqrt{\frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})}}}, 0 < α < 1,

(21)

In summation notation

\begin{matrix} \sum_{n = 0}^{\infty} [\frac{1}{q} e^{- (2 n + 1 - ξ) \sqrt{\frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})}}} \\ - \frac{1}{q} e^{- (2 n + 1 + ξ) \sqrt{\frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})}}}] . \end{matrix}

(22)

The inversion of Laplace transform is made numerically with the help of Stehfest’s and Tzou’s algorithms.^39,40

Solution of momentum equation

Taking the Laplace transform on equation (11), we obtain:

r_{0} (1 + λ q) q \bar{u} (ξ, q) = r_{1} \frac{\partial^{2} \bar{u} (ξ, q)}{\partial ξ^{2}} + r_{2} (1 + λ q) Gr \bar{θ} (ξ, q),

(23)

which can simplify as

\frac{\partial^{2} \bar{u} (ξ, q)}{\partial ξ^{2}} - \frac{r_{0} q (1 + λ q)}{r_{1}} \bar{u} (ξ, q) = - \frac{r_{2} Gr (1 + λ q)}{r_{1}} \bar{θ} (ξ, q),

(24)

after further simplification

\frac{\partial^{2} \bar{u} (ξ, q)}{\partial ξ^{2}} - m_{5} q (q + m_{7}) \bar{u} (ξ, q) = - m_{6} Gr (q + m_{7}) \bar{θ} (ξ, q),

(25)

where

m_{5} = \frac{r_{0} λ}{r_{1}}, m_{6} = \frac{r_{0} λ}{r_{1}}, m_{7} = \frac{1}{λ},

Equation (25) satisfies the following transformed conditions

\bar{u} (0, q) = 0, \bar{u} (1, q) = 0 .

(26)

The solution of equation (25) subject to condition (26) is:

\begin{matrix} \bar{u} (ξ, q) = G (q) [\frac{1}{q} \frac{\sinh ξ \sqrt{m_{5} q (q + m_{7})}}{\sinh \sqrt{m_{5} q (q + m_{7})}} \\ - \frac{1}{q} \frac{\sinh ξ \sqrt{\frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})}}}{\sinh \sqrt{\frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})}}}], \end{matrix}

(27)

where

\begin{matrix} G (q) = \\ \frac{m_{6} Gr (m_{1} + m_{2}) (q + m_{4}) (q + m_{7})}{(q + a_{1}) (q - m_{3}) - m_{5} (m_{1} + m_{2}) q (q + m_{7}) (q + m_{4})}, \end{matrix}

(28)

the summation form of above equation (27) is

\begin{matrix} \bar{F} (ξ, q) = G (q) \sum_{n = 0}^{\infty} [\frac{1}{q} e^{- (2 n + 1 - ξ) \sqrt{m_{5} q (q + m_{7})}} \\ - \frac{1}{q} e^{- (2 n + 1 + ξ) \sqrt{m_{5} q (q + m_{7})}}] \\ - G (q) \sum_{n = 0}^{\infty} [[\frac{1}{q} e^{- (2 n + 1 - ξ) \sqrt{\frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})}}} \\ - \frac{1}{q} e^{- (2 n + 1 + ξ) \sqrt{\frac{(q + a_{1}) (q - m_{3})}{(m_{1} + m_{2}) (q + m_{4})}}}] . \end{matrix}

(29)

Since equation (29) is very complex and not possible to obtain the inverse Laplace transform. Therefore, inverse Laplace transform is obtained numerically.

Results and discussion

In this paper, the fractional model of Maxwell hyrid nanofluid is considered and a generalized free convection flow between parallel plates is presented as shown in Figure 1. Momentum balance along with energy balance are fractionalized by introducing the Caputo-Fabrizio definition of time fractional operator. Generalized fractional partial differential equation for momentum and energy balance are solved for velocity and temperature by Laplace transform method. Results for velocity and temperature are presented in graphs to depict the influence of flow and material parameters like Prendlt Pr and Grashof Gr numbers, fractional parameter $α$ , heat generation parameter Q, volume fraction of hybrid nanofluid $ϕ_{hnf}$ , and relaxation time $λ$ on fluid flow. The inverse Laplace transform is obtained by using Stehfest’s algorithm as well as Tzou’s algorithm to check the validity of our obtained results.

In Figure 2(a) and (b), the subjectivity of fractional parameter $α$ on temperature profile is highlighted for large and small times. Figure 2(a) indicates that temperature field is an increasing function of fractional parameter $α$ while from Figure 2(b) that temperature field is decreasing function of $α$ .

Figure 2.

Temperature Profile due $α$ variation for large and small times.

Figure 3(a) and (b) are plotted to see the influence of $\underset{1}{\Pr}$ , $\underset{2}{\Pr}$ and Q respectively. For enhancing value of $\Pr_{1}$ and $\Pr_{2}$ , temperature profile lowers down as clear from the Figure 3(a). The outlines of Figure 3(b) indicates that temperature raises with the increasing values of Q.

Figure 3.

Temperature profile due $\Pr_{1}, \Pr_{2}$ and Q variation.

Figure 4(a) and (b) are drawn to see the impact of fractional parameter $α$ and are exposing the dual behavior of velocity subject to the fractional parameter $α$ for larger and smaller times respectively. For the longer time, velocity profile shows an increasing trend and hence the boundary layers are increased with the increasing values of $α$ . For the short time, behavior of the velocity due to $α$ variation is totally opposite to that for large time.

Figure 4.

Velocity profile due $α$ variation for large and small times.

Figure 5(a) shows the influence of Gr on velocity fields. As the thermal Grashof number is responsible for heat transfer in convection problems. Physically, in heat transfer problems heating corresponds to the positive values of Gr, while negative values are for cooling. It is clear from Figure 2 that trend in velocity profiles is increasing for positive values of Gr. It is due to the fact that larger values of Gr means greater buoyancy forces hence ultimately convection increases the velocity profiles. But for negative values of Gr, velocity behaves in opposite manner due to cooling effect of plate. Figure 5(b) shows the influence of heat absorption parameter Q. In heat transfer problem, convection can be enhanced by adding heat source in energy equation. By assigning bigger values to Q means some of the heat observed from the system which effects the intermolecular forces and make them weak, as a consequence velocity increases.

Figure 5.

Velocity profile due Gr and Q variation.

The effect of $\Pr_{1}$ and $\Pr_{2}$ is presented in Figure 6(a) on velocity field. For larger values of $\Pr_{1}$ and $\Pr_{2}$ , the fluid velocity has shown decreasing trend. Since $\Pr_{1}$ and $\Pr_{2}$ are the ratio of momentum diffusivity to thermal conductivity. Due to increase in $\underset{1}{\Pr}$ and $\underset{2}{\Pr}$ , the momentum diffusivity increases making the fluid more thick and consequently decreases its velocity. Figure 6(b) shows the velocity profiles due to Maxwell parameter $λ$ , it is clear that nanofluids flow speedily by increasing values of $λ$ .

Figure 6.

Velocity profile due Pr and $λ$ variation.

In Figure 7(a), effect of volume fraction of nanofluid is shown. The velocity for the hybrid nanofluid decreases with an increasing values of $ϕ_{hnf}$ , the physical justification of such behavior of velocity is that by increasing volume fraction $ϕ_{hnf}$ of hybrid nanoparticle the fluid becomes more viscous which results decrease in the nano fluid velocity. Figure 7(b) shows a comparison between the velocities of different nanofluids and pure water. Graphic trend of velocity profile shows that the velocity of $A l_{2} O_{3} - H_{2} O$ has greater values followed by $Cu - A l_{2} O_{3} - H_{2} O$ , $Cu - H_{2} O$ , nanofluids and pure water. Physically, it is due to the reason that $A l_{2} O_{3}$ conducting heavy heat because of effective thermal conductivity but less dense. Therefore, $A l_{2} O_{3} - H_{2} O$ has the greater velocity amongst all the fluids which are considered here. $H_{2} O$ are more stable than the nanofluid this is because $A l_{2} O_{3} - H_{2} O$ is less dense due to higher thermal conductivity.

Figure 7.

Velocity profile due $ϕ_{hnf}$ variation and differnt nanofluids.

In Figure 8(a), a comparison between the profiles of the Brinkman and Maxwell fluids is made. By assuming the fixed values to the Brinkman and Maxwell parameters respectively, we found that Maxwell heat transfer model has high values of velocity in comparison with Brinkman model. Further, to check the validation of the present results by letting $β_{b}$ Brinkman parameter and $λ$ Maxwell parameter approaching to zero, both heat transfer flow model reduce to viscous fluid and it has been shown graphically that they are in good agreement and presented in Figure 8(b).

Figure 8.

Comperisions amung the Maxwell, Brinkman, and ordinary fluids.

Figure 9(a) and (b) are sketched to see the validity of inversion algorithms namely Stehfest and Tzou’s algorithms, that are applied to obtain the numerical solutions of velocity and temperature fields. The coinciding curves of velocity profiles in Figure 9(a) and conceding curves of temperature profile in Figure 9(b) indicate the validation of inversion algorithms.

Figure 9.

Validity of the inversion algorithms for velocity and temperature profiles.

Conclusion

A generalization is made for free convection flow of Maxwell nanofluid by introducing the definition of Caputo-Fabrizio of fractional order derivatives. Fluid motion in two vertical parallel plates is discussed. The explicit expressions for velocity and temperature are derived by applying Laplace transform. The effects of involved parameter on velocity profile are studied by plotting various graphs. Followings are the remarkable extractions of this research study:

Nanofluid velocity is an increasing function of Gr, Q, and $λ$ .

For increasing values of Pr and $ϕ_{hnf}$ , fluid motion slows down.

Fluid velocity exhibits the dual behavior due to $α$ for longer and shorter times. For longer time fluid velocity increases by increasing value of $α$ but behaves in opposite manner for shorter time.

Footnotes

Appendix

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

M Ahmad

References

Milici

Drgnescu

Machado

JT.

Introduction to fractional differential equations. 1st ed. Heidelberg, Germany: Springer Verlag, 2019.

Shah

Wang

, et al. Transient electro-osmatic slip flow of an Oldroyd-B fluid with time fractional Caputo-Fabrizio derivative. J Appl Comput Mech 2019; 5: 779–790.

Al-Mdallal

Abro

Khan

Analytical solutions of fractional Walters-B fluid with applications. Complexity 2018; 3: 1–10.

Sheikh

Ali

Saqib

, et al. A comparative study of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid. Phys Fluids 2017; 132: 54–62.

Imran

Shah

Aleem

, et al. Heat transfer analysis of fractional second-grade fluid subject to Newtonian heating with Caputo and Caputo-Fabrizio fractional derivatives: a comparison. Phys Fluids 2017; 132: 340–358.

Azhar

Vieru

Fetecau

Free convection flow of some fractional nanofluids over a moving vertical plate with uniform heat flux and heat source. Phys Fluids 2017; 29: 8–20.

Abro

Atangana

Mathematical analysis of memristor through fractal-fractional differential operators: a numerical study. Math Methods Appl Sci 2020; 43: 6378–6395.

Abro

Mirbhar

Gomez-Aguilar

JF.

Functional application of Fourier sine transform in radiating gas flow with non singular and non local kernel

Journal of the Brazilian Society of Mechanical Sciences and Engineering 2019; 41: 400–407.

Abro

Memon

Uqaili

MA.

A comparative mathematical analysis of RL and RCelectrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. Eur Phys J Plus 2018; 133: 113–122.

10.

Das

PK.

A review based on the effect and mechanism of thermal conductivity of normal nanofluids and hybrid nanofluids. J Mol Liq 2017; 240: 420–446.

11.

Xian

Azwadi

Sidik

, et al. Review on preparation techniques, properties and performance of hybrid hanofluid in recent engineering applications. J Adv Res Fluid Mech Therm Sci 2018; 45: 1–13.

12.

Babar

Sajid

Ali

HM.

Viscosity of hybrid nanofluids: a critical review. Therm Sci 2019; 23: 1713–1754.

13.

Farhana

CFD modelling of different properties of nanofluids in header and riser tube of flat plate solar collector. Mater Sci Eng 2019; 46: 12–41.

14.

Uysal

Gedik

Chamkha

AJ.

A numerical analysis of laminar forced convection and entropy generation of a diamond-Fe3O4/water hybrid nanofluid in a rectangular minichannel. J Appl Fluid Mech 2019; 12: 391–402.

15.

Hayat

Nadeem

Heat transfer enhancement with Ag-CuO/water hybrid nanofluid. Results in Physics 2017; 7: 2317–2324.

16.

Manikandan

Rajan

KS.

New hybrid nanofluid containing encapsulated paraffin wax and sand nanoparticles in propylene glycol-water mixture: potential heat transfer fluid for energy management. Energy Convers Manag 2017; 13: 74–85.

17.

Azwadi

Sidik

Jamilb

, et al. A review on preparation methods, stability and applications of hybrid nanofluids. Renew Sust Energ Rev 2017; 80: 1112–1122.

18.

Aman

Khan

Ismail

, et al. Applications of fractional derivatives to nanofluids: exact and numerical solutions. Math Model Nat Phenom 2018; 13: 1–12.

19.

Babu

JAR

Kumar

Rao

SS.

State-of-art review on hybrid nanofluids. Renew Sust Energ Rev 2017; 77: 551–565.

20.

Esfe

Emami

MRS

Amiri

MK.

Experimental investigation of effective parameters on MWCNTTiO2/ SAE50 hybrid nanofluid viscosity. J Therm Anal Calorim 2019; 137: 743–757.

21.

Takabi

Salehi

Augmentation of the heat transfer performance of a sinusoidal corrugated enclosure by employing hybrid nanofluid. Adv Mech Eng 2014; 6: 147059.

22.

Xian

Azwadi

Sidik

, et al. Heat transfer performance of tybrid nanofluid as nanocoolant in automobile radiator system beriache. J Adv Res 2018; 51: 14–25.

23.

Esfe

Amiri

Alirezaie

Thermal conductivity of a hybrid nanofluid. J Therm Anal Calorim 2018; 134: 1113–1122.

24.

Sharma

Tiwari

, et al. Viscosity of hybrid nanofluids: measurement and comparison. J Mech Eng Sci 2018; 12: 3614–3623.

25.

Ahokposi

Atangana

Vermeulen

DP.

Modelling groundwater fractal flow with fractional differentiation via Mittag-Leffler law. Eur Phys J Plus 2017; 132: 165–177.

26.

Farooq

Afridi

Qasim

, et al. Transpiration and viscous dissipation effects on entropy generation in hybrid nanofluid flow over a nonlinear radially stretching disk. Entropy 2018; 20: 668–682.

27.

Hussanan

Salleh

Khan

, et al. Convection heat transfer in micropolar nanofluids with oxide nanoparticles in water, kerosene and engine oil. J Mol Liq 2017; 229: 482–488.

28.

Aminossadati

Ghasemi

Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure. Eur J Mech B Fluids 2009; 28: 630–640.

29.

Bhattad

Sarkar

Ghosh

Discrete phase numerical model and experimental study of hybrid nanofluid heat transfer and pressure drop in plate heat exchanger. Int J Heat Mass Transf 2018; 91: 262–273.

30.

Gul

Khan

Shafie

Radiation and heat generation effects in mhd mixed convection flow of nanofluids. Therm Sci 2018; 22: 51–62.

31.

Ahmed

Saba

Khan

, et al. Nanofluid flow squeezed between two riga plates with nonlinear thermal radiation and chemical reaction effects. Energies 2019; 12: 76.

32.

Khan

, et al. MHD flow of sodium alginate based Casson type nanofluid passing through a porous medium with Newtonian heating. Sci Rep 2018; 8:8645.

33.

Asif

Haq

Islam

, et al. Exact solution of non-Newtonian fluid motion between side walls. Results Phys 2018; 11: 534–539.

34.

Fetecau

Nazar

Unsteady flow of an Oldroyd-B fluid generating by a constantily accelerating plate between two side wall perpendicular to the plates. Int J Non Linear Mech 2009; 44: 1039–1047.

35.

Siyal

Abro

Solangi

MA.

Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium: applications to thermal science. J Therm Anal Calorim 2019; 136: 2295–2304.

36.

Saqib

Khan

Shafie

Application of fractional differential equations to heat transfer in hybrid nanofluid: modeling and solution via integral transforms. Adv Differ Equ 2019; 2019: 52.

37.

Hristov

Transient heat diffusion with a non-singular fading memory from the Cattaneo constitutive equation with Jeffreyis kernel to the Caputo-Fabrizio time fractional derivative. Therm Sci 2016; 20: 557–562.

38.

Henry

Langlands

Straka

An introduction to fractional diffusion. In: Dewar

Detering

(eds) Complex physical, biophysical and econophysical systems. Canberra: The Australian National University, 2002, pp.8–19.

39.

Stehfest

Algorithm 368: numerical inversion of Laplace transforms. Commun ACM 1970; 13: 47–54.

40.

Tzou

DY.

Macro-to microscale heat transfer: the lagging behavior. Washington, DC: Taylor and Francis, 1997.

Mathematical modeling of ( Cu − A l 2 O 3 ) water based Maxwell hybrid nanofluids with Caputo-Fabrizio fractional derivative

Abstract

Keywords

Introduction

Mathematical statement of physical model

The Caputo-Fabrizio fractional derivative and Laplace transform

Modeling with Caputo-Fabrizio time fractional derivative

Solution of the problem

Solutions of the temperature field

Solution of momentum equation

Results and discussion

Conclusion

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iD

References