Saqib et al. Advances in Difference Equations (2019) 2019:52 https://doi.org/10.1186/s13662-019-1988-5

R E S E A R C H Open Access

Application of fractional differentialequations to heat transfer in hybrid nanofluid:modeling and solution via integral transformsMuhammad Saqib1, Ilyas Khan2* and Sharidan Shafie1

*Correspondence:ilyaskhan@tdt.edu.vn2Faculty of Mathematics andStatistics, Ton Duc Thang University,Ho Chi Minh City, VietnamFull list of author information isavailable at the end of the article

AbstractThis article deals with the generalization of natural convection flow ofCu – Al2O3 – H2O hybrid nanofluid in two infinite vertical parallel plates. Todemonstrate the flow phenomena in two parallel plates of hybrid nanofluids, theBrinkman type fluid model together with the energy equation is considered. TheCaputo–Fabrizio fractional derivative and the Laplace transform technique are usedto developed exact analytical solutions for velocity and temperature profiles. Thegeneral solutions for velocity and temperature profiles are brought into light throughnumerical computation and graphical representation. The obtained results show thatthe velocity and temperature profiles show dual behaviors for 0 < α < 1 and 0 < β < 1where α and β are the fractional parameters. It is noticed that, for a shorter time, thevelocity and temperature distributions decrease with increasing values of thefractional parameters, whereas the trend reverses for a longer time. Moreover, it isfound that the velocity and temperature profiles oppositely behave for the volumefraction of hybrid nanofluids.

MSC: 34A08; 76R10

Keywords: Applications of fractional derivatives; Heat transfer problem; Modelingand simulation; Hybrid nanofluid

1 IntroductionIn recent decades, it was acknowledged that fractional operators are appropriate tools fordifferentiation as compared to the local differentiation particularly in physical real wordproblems. These fractional operators can be constructed by the convolutions of the localderivative as the kernel of fractional operators; various kernels for fractional operatorshave been suggested in the literature but the most common is the power law kernel (x–α),which is used in the construction of Riemann–Liouville and Caputo fractional operators(see [1], p. 65–106). However, the exponential decay law exp(–αx) was used by Caputo andFabrizio (see [2], p. 1–13). Atangana and Baleanu developed fractional operators in theCaputo and Riemann–Liouville sense using the generalized Mittag–Leffler law Eα(–φxα)as a kernel (see [3], p. 763–769). All these fractional operators have some shortcomingsand challenges but at the same time this area is growing fast, and researchers devoted theirattention to this field (see [4–10] and the references therein).

© The Author(s) 2019. 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 anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 2 of 18

It is important to mention here that fractional order calculus has many applications inalmost 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 Newtonianviscous fluid flowing over the infinite vertical plate. Markis et al. ([13], p. 1663–1679) an-alyzed the flow of a fractional Maxwell’s fluid. According to their report, the fractionalresults showed excellent agreement with experimental work by adjusting the fractionalparameter. Alkahtani and Atangana ([14], p. 106–113) used different fractional operatorsto analyze the memory effect in a potential energy field caused by a charge. They pre-sented some novel numerical approaches to the solutions of a fractional system of equa-tions. Vieru et al. ([15], p. 85–96) presented exact solutions for the time-fraction model ofviscous fluid flow near a vertical plate taking into consideration mass diffusion and New-tonian heating. Abro et al. ([16], p. 1–10) presented exact analytical solutions for the flowof an Oldroyd-B fluid in a horizontal circular pipe. Jain ([17], p. 1–11) introduced a noveland powerful numerical scheme and implemented to different fractional order differentialequations. Some other interesting and significant studies on fractional derivatives can befound in [18–26] and the references therein.

The nanofluid is an innovation of nanotechnology to overcome the problems of heattransport in many engineering and industrial sectors. A detailed discussion on nanoflu-ids 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 externalmagnetic field effect on the F3O4 – H20 nanofluid inside a porous enclosure. Hassanan etal. ([29], p. 482–488) developed exact solutions for nanofluids with different nanoparticlesfor the unsteady flow of a micropolar fluid. The literature of nanofluids has exponentiallyincreased and has reached a next level by introducing hybrid nanofluids which are the sus-pensions of two or more types of nanoparticles in the composite form with low concentra-tion. Hybrid nanofluids are introduced to overcome the drawbacks of single nanoparticlesuspensions and connect the synergetic effect of nanoparticles. The hybrid nanofluid isbranded to further improve the thermal conductivity and heat transport, which leads toindustrial and engineering applications with low cost (see [30], p. 262–273). Hussain etal. ([31], p. 1054–1066) carried out an entropy generation analysis on a hybrid nanofluidin a cavity. Farooq et al. ([32], p. 1–14) presented a numerical study on hybrid nanofluidskeeping into consideration suction/injection, entropy generation, and viscous dissipation.

In the existing literature, experimental, theoretical and numerical studies on hybridnanofluids are very limited. A study of a hybrid nanofluid fluid with exact solutions andthe Caputo fractional derivative even does not exist. So, there is an urgent need to con-tribute to the literature of hybrid nanofluids using the application of fractional differentialequations. Motivated by the above discussion, the present study focused on the heat trans-fer in hybrid nanofluid in two vertical parallel plates using fractional derivative approach.A water-based hybrid nanofluid is characterized here with composite hybrid nanoparticlesof cupper (Cu) and alumina (Al2O3). The fractional Brinkman type fluid model with phys-ical initial and boundary conditions is considered for the flow phenomena. The Laplacetransform technique is used to obtain exact analytical solutions for the velocity and tem-perature profiles. Using the properties of the Caputo–Fabrizio fractional derivative theobtained solutions are reduced to the classical form for α = 1 and β = 1. To explore the

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 3 of 18

physical aspect of the flow parameters the solutions are numerically computed and plottedin different graphs with a physical explanation.

2 Problem’s descriptionLet us consider the unsteady free convection flow of a generalized incompressible hybridnanofluid in two infinite vertical parallel plates at a distance d. The plates are taken alongthe x-axis and the y-axis is chosen normal to it. At t ≤ 0, the plates and fluid are at restwith ambient temperature T0. After t = 0+, the temperature of the plate at y = d rises orlowers from T0 to TW due to which the free convection takes place. At this moment, thefluid starts motion in the x– direction due to the temperature gradient which gives rise tothe buoyancy forces. The Brinkman type fluid model is utilized to describe flow phenom-ena of the hybrid nanofluid. Under the assumptions of ([33], p. 1472–1488) the governingequations of the Cu – Al2O3 – H2O hybrid nanofluid are given by

ρhnf

(∂u(y, t)

∂t+ β∗

b u(y, t))

= μhnf∂2u(y, t)

∂y2 + g(ρβT )hnf(T(y, t) – T0

), (1)

(ρCp)hnf∂T(y, t)

∂t= khnf

∂2T(y, t)∂y2 + Q0(T – T0), (2)

together with the following appropriate initial and boundary conditions:

u(y, 0) = 0, T(y, 0) = T0, ∀y ≥ 0, (3)

u(0, t) = 0, T(0, t) = T0 for t > 0u(d, t) = 0, T(d, t) = TW for t > 0

}, (4)

where ρhnf is the density, u(y, t) is the velocity, β∗b is the Brinkmann parameter, μhnf is the

dynamic viscosity, βhnf is the volumetric thermal expansion, (Cp)hnf is the specific heat,T(y, t) is the temperature, khnf is the thermal conductivity and Q0 is the heat generationof the hybrid nanofluid.

3 Thermophysical properties of hybrid nanofluidThis section demonstrates the modification of thermophysical properties of a conven-tional nanofluid and a hybrid nanofluid Cu – Al2O3 – H2O in a spherical shape.

3.1 The effective densityThe effective density ρnf of conventional nanofluid is defined by Aminossadati andGhasemi (see [34], p. 630–640) and can be expressed as

ρnf = (1 – φ)ρf + φρs, (5)

where φ is the volume concentration of the nanoparticles, ρf and ρs are the densities of thebase fluid and solid particles respectively. The mathematical expression for the effectivedensity of the hybrid nanofluid can be obtained by modifying Eq. (5) (see [31], p. 1054–1066):

ρhnf = (1 – φhnf )ρf + φAl2O3ρAl2O3 + φCuρCu, (6)

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 4 of 18

where ρhnf is the density of the hybrid nanofluid, φhnf is the volume concentration of solidparticles such that φhnf = φAl2O3 + φCu, ρf is the density of the base fluid, φAl2O3 is thevolume concentration of alumina, ρAl2O2 is the density of alumina, φCu is the volume con-centration of cupper and ρCu is the density of cupper.

3.2 The effective dynamic viscosityThe dynamics viscosity μnf of an ordinary nanofluid is expressed by Brinkman (see [35],p. 571) by

μnf =μf

(1 – φ)2.5 , (7)

which leads to the following modified form for a hybrid nanofluid:

μhnf =μf

{1 – (φAl2O3 + φCu)}2.5 . (8)

3.3 The effective volumetric thermal expansion and heat capacitanceThe thermal expansion and heat capacitance are, respectively, defined by Bourantas andLoukopoulos (see [36], p. 35–41) in the form

(βTρ)nf = (1 – φ)(βTρ)f + φ(βTρ)s, (9)

(ρCp)nf = (1 – φ)(ρCp)f + φ(ρCp)s, (10)

with the following altered form for a hybrid nanofluid:

(ρβT )hnf = (1 – φhnf )(ρβT )f + φAl2O3 (ρβt)Al2O3 + φCu(ρβT )Cu, (11)

(ρCp)hnf = (1 – φhnf )(ρCp)f + φAl2O3 (ρCp)Al2O3 + φCu(ρCp)Cu. (12)

3.4 The effective thermal conductivityThe effective thermal conductivity for a conventional nanofluid is based on Maxwell’smodel (see [37], p. 87–92), which is defined by

Knf

Kf=

ks + 2kf – 2φ(ks – kf )ks + 2kf + 2φ(ks – kf )

, (13)

where Knf is the thermal conductivity of the nanofluid, Ks is the thermal conductivity ofsolid nanometer-sized particles and Kf is the thermal conductivity of the base fluid. Forthe hybrid nanofluid, Maxwell’s model can be modified:

khnf

kf=

φAl2O3 kAl2O3 +φCukCuφhnf

+ 2kf + 2(φAl2O3 kAl2O3 + φCukCu) – 2kf φhnf

φAl2O3 kAl2O3 +φCukCuφhnf

+ 2kf – (φAl2O3 kAl2O3 + φCukCu) – 2kf φhnf

. (14)

It is important to highlight here that by making φAl2O3 = 0 or φCu = 0 the effective ther-mophysical properties of the hybrid nanofluid presented in Eqs. (8), (11), (12) and (14)can be reduced to the effective thermophysical properties of a conventional nanofluid pre-sented in Eqs. (7), (9), (10) and (13), respectively. Furthermore, the typo mistake made in

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 5 of 18

Table 1 Numerical values of thermophysical properties of base fluid and nanoparticles

Material Base fluid Nanoparticles

H2O Al2O3 Cu

ρ (kg/m3) 997.1 3970 8933Cp (J/kgK) 4179 765 385K (W/mK) 0.613 40 400βT × 10–5 (K–1) 21 0.85 1.67Pr 6.2 – –

[31], p. 1054–1066) and [32], p. 1–14 has been corrected here in the expression of thethermal conductivity for the hybrid nanofluid. The numerical values of the base fluid andnanoparticles are given in Table 1.

4 Generalization of local modelIn this section, the dimensional system is first transformed to dimensionless form usingnon-similarity variables to reduce the number of variables and get rid of units. The dimen-sionless system is then artificially converted to time-fractional form or generalized formusing the Caputo–Fabrizio fractional operator (see [2], p. 1–13). It is worth to mentionhere that the fractional models are more general and convenient in the description of flowbehavior and memory effect. Moreover, the results obtained from the fractional modelare additionally realistic because by adjusting the fractional parameter the obtained re-sults can be compared with experimental data to reach excellent agreement as obtainedby Markis et al. (see [13], p. 1663–1679). Now introducing the following non-similaritydimensionless variables:

v =dνf

u, ξ =yd

, τ =νf

d2 t, θ =T – T0

TW – T0,

into Eqs. (1)–(4) yields the following:

a0

(∂v(ξ , τ )

∂τ+ βbv(ξ , τ )

)= a1

∂2v(ξ , τ )∂ξ 2 + a2 Gr θ (ξ , τ ), (15)

a3 Pr∂θ (ξ , τ )

∂τ= λhnf

∂2θ (ξ , τ )∂ξ 2 + Qθ (ξ , τ ), (16)

v(ξ , 0) = 0, θ (ξ , 0) = 0, ∀ξ ≥ 0, (17)

v(0, τ ) = 0, θ (0, τ ) = 0 for τ > 0v(1, τ ) = 0, θ (1, τ ) = 1 for t > 0

}, (18)

where

βb =d2β∗

bν2

f, Gr =

d3g(βT )f

ν2f

(TW – T0), Pr =(μCp)f

kf,

Q =d2Q0

kf, λhnf =

khnf

kf, a0 = (1 – φ) +

φAl2O3ρAl2O3 + φCuρCu

ρf,

a1 =1

{1 – (φAl2O3 + φCu)}2.5 a2 = (1 – φ) +φAl2O3 (ρβt)Al2O3 + φCu(ρβT )Cu

(ρβT )f,

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 6 of 18

a3 = (1 – φ) +φAl2O3 (ρCp)Al2O3 + φCu(ρCp)Cu

(ρCp)f,

is the dimensionless Brinkman type fluid parameter, the thermal Grashof number, thePrandtl number and heat generation parameter, respectively. Here λhnf , a0, a1, a2 and a3

are the constant terms produced during the calculation. The time-fractional form ofEqs. (15) and (16) in terms of Caputo–Fabrizio fractional operator is given by

a0

a1

CF Dατ v(ξ , τ ) +

a0

a1βbv(ξ , τ ) =

∂2v(ξ , τ )∂ξ 2 +

a2

a1Gr θ (ξ , τ ), (19)

a3 Pr CF Dβτ θ (ξ , τ ) = λhnf

∂2θ (ξ , τ )∂ξ 2 + Qθ (ξ , τ ), (20)

where CF Dατ v(ξ , τ ), and CF Dβ

τ θ (ξ , τ ) is for the Caputo–Fabrizio fractional operators of frac-tional order α and β . Equations (19) and (20) are the Caputo–Fabrizio generalized formof Eqs. (15) and (16), while the initial and boundary conditions will remain the same as inEqs. (17) and (18). The Caputo–Fabrizio fractional operator is defined by (see [2], p. 1–13)

CF Dδt f (t) =

N(δ)1 – δ

∫ t

0exp

(–

δ(t – τ )1 – δ

)∂f (τ )∂τ

dτ , 0 < δ < 1, (21)

which is the convolution product of the function N(δ)1–δ

exp(– δt1–δ

) and f (t) of fractional orderδ. In this study the following two properties of Caputo–Fabrizio fractional operator willbe utilized.

1. Property 1: According to Losanda and Nieto (see [38], p. 87–92) N(δ) is thenormalization function such that

N(1) = N(0) = 1. (22)

2. Property 2: taking into consideration Eq. (22), the Laplace transform of Eq. (21) yields

L{CF Dδ

t f (t)}

(q) =qf̄ (q) – f (0)(1 – δ)q + δ

, 0 < δ < 1, (23)

such that

limδ→1

[L{CF Dδ

t f (t)}

(q)]

= limδ→1

{qf̄ (q) – f (0)(1 – δ)q + δ

}= qf̄ (q) – f (0) = L

{∂f (t)∂t

}, (24)

where f̄ (q) is the Laplace transform of f (t) and f (0) is the initial value of the function.

5 Solution of the problemTo solve Eqs. (19) and (20) the Laplace transform method L{f (t)}(q) = f̄ (q) =

∫ ∞0 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.

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 7 of 18

5.1 Solutions of the energy equationApplying the Laplace transform to Eq. (20) keeping in mind the definition and propertiesof the Caputo–Fabrizio fractional operator defined in Eq. (21)–(24) and using the corre-sponding initial condition from Eq. (17) yield

a3 Prqθ (ξ , τ ) – θ (ξ , 0)

(1 – β)q + β= λhnf

d2θ̄ (ξ , q)∂ξ 2 + Qθ̄ (ξ , q), 0 < β < 1, (25)

and after further simplification of Eq. (25)

d2θ̄ (ξ , q)∂ξ 2 –

b4q – b1b3

q + b1= 0, 0 < β < 1, (26)

with transformed boundary conditions

v̄(0, q) = 0, θ̄ (0, q) = 0 for q > 0v̄(1, q) = 0, θ̄ (1, q) = 1

q for q > 0

}, (27)

where

b0 =1

1 – β, b1 = b0β , b2 =

a3 Pr

λhnf, b3 =

Qλhnf

, b4 = b0b2 – b3.

The exact solution of Eq. (26) using the corresponding boundary conditions fromEq. (27) is given by

θ̄ (ξ , q) =1q

(sinh ξ

√b4q – b1b3

q + b1

)(sinh

√b4q – b1b3

q + b1

)–1

, 0 < β < 1. (28)

Equation (28) represents the solutions of the energy equation in the Laplace transformeddomain. In order to invert the Laplace transform, this equation can be written in a moresuitable and simplified form:

θ̄ (ξ , q) =∞∑

n=0

(1q

e–(1+2n–ξ )

√b4q–b1b3

q+b1 –1q

e–(1+2n+ξ )

√b4q–b1b3

q+b1

), 0 < β < 1. (29)

Let us consider

θ̄ (ξ , q) = θ̄1(ξ , q) – θ̄2(ξ , q), 0 < β < 1, (30)

where

θ̄1(ξ , q) =∞∑

n=0

1q

e–(1+2n–ξ )

√b4q–b1b3

q+b1 , 0 < β < 1, (31)

θ̄2(ξ , q) =∞∑

n=0

1q

e–(1+2n+ξ )

√b4q–b1b3

q+b1 , 0 < β < 1. (32)

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 8 of 18

Upon taking the inverse Laplace transform, Eq. (30) yields

θ (ξ , τ ) = θ1(ξ , τ ) – θ2(ξ , τ ), 0 < β < 1, (33)

and to derive the functions θ1(ξ , τ ) and θ2(ξ , τ ), the compound formula for the Laplaceinversion is used. The function Φ̄(ξ , q) is chosen as

Φ̄(ξ , q) = e–(1+2n–ξ )

√b4q–b1b3

q+b1 = e(1+2n–ξ )√

W1(q). (34)

According to Khan (see [39], p. 397–401), the inverse Laplace transform of the functionsΦ̄(ξ , q) can be obtained as

Φ(ξ , τ ) = L–1{Φ̄(ξ , q)}

=∫ ∞

0f((1 + 2n – ξ ), u

)g(u, τ ) dτ , (35)

where

f((1 + 2n – ξ ), u

)=

(1 + 2n – ξ )2u

√πu

e– (1+2n–ξ )4u , (36)

g(u, τ ) = e–b4uδ(τ ) – e–b4u√

puτ

I1√

puτe–b1τ . (37)

and

p = –b1(b3 + b4).

The values of functions f ((1 + 2n – ξ ), u) and g(u, τ ), defined in Eqs. (36) and (37), areused in Eq. (35) yielding the following simplified form:

Φ(ξ , τ ) = e–(1+2n–ξ )√

b4δ(τ ) –(1 + 2n – ξ )√p

2√

πτe–b1τ

∫ ∞

0

1u

e– (1+2n–ξ )24u –b4uI1

√puτ du. (38)

To evaluate the function θ1(ξ , τ ), we need to find the convolution product of L–1{ 1q } = 1

and the function Φ̄(ξ , q) which yields

θ1(ξ , τ ) =∞∑

n=0

(e–(1+2n–ξ )

√b4 –

∫ ∞

0

∫ τ

0

(1 + 2n – ξ )√p2√

πse–b1s

× 1u

e– (1+2n–ξ )24u –b4uI1

√pus du ds

). (39)

Similarly, the function θ2(ξ , τ ) is given by

θ1(ξ , τ ) =∞∑

n=0

(e–(1+2n+ξ )

√b4 –

∫ ∞

0

∫ τ

0

(1 + 2n + ξ )√p2√

πse–b1s

× 1u

e– (1+2n+ξ )24u –b4uI1

√pus du ds

). (40)

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 9 of 18

To reduce the solutions obtained in Eq. (33) to classical or local form, Eq. (24) is usedfor the following.

For β → 1, Eq. (24) is used which reduced Eq. (25) to the following form:

d2θ̄ (ξ , q)∂ξ 2 – (b2q + b2)θ̄ (ξ , q) = 0. (41)

With the solutions in the Laplace transform domain one finds

θ̄ (ξ , q) =1q

sinh ξ√

b2q – b3

sinh√

b2q – b3. (42)

After further simplification, Eq. (42) takes the following form:

θ̄ (ξ , q) =∞∑

n=0

(1q

e–(1+2n–ξ )√

b2q–b3 –1q

e–(1+2n+ξ )√

b2q–b3

), β = 1. (43)

Taking the inverse Laplace transform, Eq. (43) gives the following local solutions for thetemperature profile:

θ (ξ , τ ) = A1(ξ , τ ) – A2(ξ , τ ), β = 1, (44)

where

A1(ξ , τ ) =12

∞∑n=0

⎛⎜⎜⎝

e– (1+2n–ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

τ–

√– b3

b2τ )

+ e(1+2n–ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

τ+

√– b3

b2τ )

⎞⎟⎟⎠ , (45)

A2(ξ , τ ) =12

∞∑n=0

⎛⎜⎜⎝

e– (1+2n+ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

τ–

√– b3

b2τ )

+ e(1+2n+ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

τ+

√– b3

b2τ )

⎞⎟⎟⎠ . (46)

5.2 Solution of momentum equationApplying the Laplace transform on Eq. (19) using the corresponding initial condition fromEq. (15) yields

a0

a1

qv̄(ξ , q) – v(ξ , 0)(1 – α)q + α

+a0

a1βbv̄(ξ , q) =

d2v̄(ξ , q)dξ 2 +

a2

a1Gr θ̄ (ξ , τ ). (47)

After further simplification of Eq. (47)

d2v̄(ξ , q)dξ 2 –

d5q + d1d3

q + d1v̄(ξ , q) = –d4θ̄ (ξ , q), (48)

where

d0 =1

1 – α, d1 = αd0, d2 =

a0

a1, d3 = d2βb,

d4 =a2

a1Gr, d5 = d0d2 + d3.

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 10 of 18

In the Laplace transform domain, the exact solution of Eq. (48) is given by

v̄(ξ , q) =(q + b1)(q + b1)

(d6q2 + d7q + d8)q

sinh ξ

√d5q+d1d3

q+d1

sinh√

d5q+d1d3q+d1

–(q + b1)(q + b1)

(d6q2 + d7q + d8)q

sinh ξ

√b4q–b1b3

q+b1

sinh√

b4q–b1b3q+b1

, (49)

where

W1(q) =d5q + d1d3

q + d1, d6 = b4 – d5,

d7 = b4d1 – b1(b3 + d5), d8 = –b1d1(b3 + d3).

In order to find the inverse Laplace transform, Eq. (49) can be written in a more conve-nient and simplified form as

v̄(ξ , q) =(q + b1)(q + b1)

(d6q2 + d7q + d8)q

∞∑n=0

(e

–(1+2n–ξ )√

d5q+d1d3q+d1 – e

–(1+2n+ξ )√

d5q+d1d3q+d1

)

–(q + b1)(q + b1)

(d6q2 + d7q + d8)q

∞∑n=0

(e

–(1+2n–ξ )√

b4q–b1b3q+b1 – e

–(1+2n+ξ )√

b4q–b1b3q+b1

),

0 < α,β < 1. (50)

Let us consider

v̄(ξ , q) = v̄1(ξ , q) × {v̄2(ξ , q) – v̄3(ξ , q)

}– v̄1(ξ , q)

× {v̄4(ξ , q) – v̄5(ξ , q)

}, 0 < α,β < 1. (51)

Here

v̄1(ξ , q) =(q + b1)(q + b1)

(d6q2 + d7q + d8), (52)

v̄2(ξ , q) =1q

∞∑n=0

e–(1+2n–ξ )

√d5q+d1d3

q+d1 , (53)

v̄3(ξ , q) =1q

∞∑n=0

e–(1+2n+ξ )

√d5q+d1d3

q+d1 , (54)

v̄4(ξ , q) =1q

∞∑n=0

e–(1+2n–ξ )

√b4q–b1b3

q+b1 , (55)

v̄5(ξ , q) =1q

∞∑n=0

e–(1+2n+ξ )

√b4q–b1b3

q+b1 , (56)

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 11 of 18

and taking the inverse Laplace transform yields

v(ξ , τ ) = v1(ξ , τ ) ∗ {v2(ξ , τ ) – v3(ξ , τ )

}– v1(ξ , τ ) ∗ {

v4(ξ , τ ) – v5(ξ , τ )}

, 0 < α,β < 1, (57)

where ∗ represents a convolution product and the terms v2(ξ , τ ), v3(ξ , τ )v4(ξ , τ ) andv5(ξ , τ ) are given by

v2(ξ , τ ) =∞∑

n=0

{e–(1+2n–ξ )

√d5δ(τ )

–(1 + 2n – ξ )√p2

2√

πτe–d1τ

∫ ∞

0

1u

e– (1+2n–ξ )24u –b4uI1

√p2uτ du

}, (58)

v3(ξ , τ ) =∞∑

n=0

{e–(1+2n+ξ )

√d5δ(τ )

–(1 + 2n + ξ )√p2

2√

πτe–d1τ

∫ ∞

0

1u

e– (1+2n+ξ )24u –b4uI1

√p2uτ du

}, (59)

v4(ξ , τ ) =∞∑

n=0

{e–(1+2n–ξ )

√b4δ(τ )

–(1 + 2n – ξ )√p2

2√

πτe–b1τ

∫ ∞

0

1u

e– (1+2n–ξ )24u –b4uI1

√p2uτ du

}, (60)

v(ξ , τ ) =∞∑

n=0

{e–(1+2n+ξ )

√b4δ(τ )

–(1 + 2n + ξ )√p2

2√

πτe–b1τ

∫ ∞

0

1u

e– (1+2n+ξ )24u –b4uI1

√p2uτ du

}, (61)

where

p2 =d1d3 – d5d1

d21

.

The term v1(ξ , τ ) is numerically obtained using Zakian’s algorithm. In the literature, it isproven that the Zakian algorithm is a stable way for the inverse Laplace transform becausethe truncated error for five multiple terms is negligible ([40], p. 83).

For the velocity profile, the local solutions can be recovered by making α,β → 1 inEq. (47) which leads to the following solutions in the Laplace transform domain:

v̄(ξ , q) =d4

(b2 – d3)q – (b3 + d3)1q

sinh ξ√

d2q + d3

sinh√

d2q + d3

–d4

(b2 – d3)q – (b3 + d3)1q

sinh ξ√

b2q – b3

sinh√

b2q – b33, (62)

with the following simplified form:

v̄(ξ , q) =d9

q2 + qd10

∞∑n=0

(e–(1+2n–ξ )

√d2q+d3 – e–(1+2n+ξ )

√d2q+d3

)

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 12 of 18

–d9

q2 + qd10

∞∑n=0

(e–(1+2n–ξ )

√b2q–b3 – e–(1+2n+ξ )

√b2q–b3

), α,β = 1, (63)

where

d9 =d4

b2 – d2, d10 =

b3 + d3

b2 – d2.

The inverse Laplace transform of Eq. (63) yields

v(ξ , τ ) = B1(ξ , τ ) – B2(ξ , τ ) – B3(ξ , τ ) + B4(ξ , τ ), α,β = 1, (64)

where

B1(ξ , τ ) =d10

2

∞∑n=0

∫ τ

0e–d7(τ–s)

⎛⎜⎜⎝

e– (1+2n–ξ )√

b2

√d3d2 erfc( (1+2n–ξ )

2√

s –√

d3d2

s)

+ e(1+2n–ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

s +√

d3d2

s)

⎞⎟⎟⎠ ds, (65)

B2(ξ , τ ) =d10

2

∞∑n=0

∫ τ

0e–d7(τ–s)

⎛⎜⎜⎝

e– (1+2n+ξ )√

b2

√d3d2 erfc( (1+2n–ξ )

2√

s –√

d3d2

s)

+ e(1+2n+ξ )√

b2

√– d3

d2 erfc( (1+2n–ξ )2√

s +√

d3d2

s)

⎞⎟⎟⎠ ds, (66)

B3(ξ , τ ) =d10

2

∞∑n=0

∫ τ

0e–d7(τ–s)

⎛⎜⎜⎝

e– (1+2n–ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

s –√

– b3b2

s)

+ e(1+2n–ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

s +√

– b3b2

s)

⎞⎟⎟⎠ ds, (67)

B4(ξ , τ ) =d10

2

∞∑n=0

∫ τ

0e–d7(τ–s)

⎛⎜⎜⎝

e– (1+2n+ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

s –√

– b3b2

s)

+ e(1+2n+ξ )√

b2

√– b3

b2 erfc( (1+2n–ξ )2√

s +√

– b3b2

s)

⎞⎟⎟⎠ ds. (68)

6 Results and discussionIn this article, the idea of the fractional derivative is used for the generalization of thefree convection flow of the hybrid nanofluid. The governing equations of the Brinkmantype fluid along with the energy equation is fractionalized using the Caputo–Fabrizio frac-tional derivative. The fractional PDEs are more general and are known as master PDEs.The momentum and energy equations are solved analytically using the Laplace transformtechnique. The obtained results are displayed in various graphs to study the influence ofthe pertinent corresponding parameters, such as the fractional parameters α and β , thevolume fraction of hybrid nanofluid φhnf , the heat generation parameter Q, the Brinkmanparameter βb and the thermal Grashof number Gr on velocity and temperature profiles.

Figures 1(a) and (b) and 2(a) and (b) depict the impact of the fractional parameter α

and β on the velocity and temperature profiles. From these figures, it is noticed that thevelocity and the temperature profiles show the same trend for variations in the fractionalparameters. The velocity and temperature profiles exhibited increasing behavior for in-creasing values of α, β for a longer time. When α,β are increased, the thickness of ther-mal and momentum boundary layers are increased and become thickest at α,β = 1, whichcorresponds to the increasing performance of the velocity and temperature profiles. Thistrend of the fractional parameter is the same here for velocity and temperature profiles

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 13 of 18

Figure 1 Variation in temperature profile against ξdue to β

Figure 2 Variation in velocity profile against ξ dueto α and β

as reported by ([25], p. 7) for fractional nanofluids using the Caputo–Fabrizio fractionalderivatives. But this effect reverses for a shorter time in the case of fractional velocity andtemperature distributions.

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 14 of 18

Figure 3 Variation in temperature profile against ξdue to φhnf

Figure 4 Variation in velocity profile against ξ dueto φhnf

Figure 5 Compression of the temperature profileagainst ξ for different nanofluids

The influence of φhnf on the velocity and temperature profiles is studied in Figs. 3 and4. The trends of velocity and temperature profiles are opposite to each other. The hybridnanofluid velocity decreases with increasing φhnf . This can be physically justified as thehybrid nanofluid became more viscous with increasing φhnf , which leads to a decreasein the nanofluid velocity. Nevertheless, the temperature profile increases with increasein φhnf when the temperature is less than 180°C. The is due to the thermal conductivityenhancing with the enhancement of φhnf and the hybrid nanofluid conducting more heatas a result of heat transfer increases, which leads to an increase in the temperature profile.

In Figs. 5 and 6 the temperature and velocity profiles are compared for Cu–Al2O3 –H2O,Cu – H2O, Al2O3 – H2O and pure water. It is noticed that the temperature profile is higherfor Cu – H2O followed by Cu – Al2O3 – H2O, Al2O3 – H2O and pure water. This due to thefact that the thermal conductivity of Cu is higher than hybrid nanoparticles, alumina, andpure water. But the hybrid nanofluid is more stable. In the case of the velocity profile, the

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 15 of 18

Figure 6 Compression of the temperature profileagainst ξ for different nanofluids

Figure 7 Variation in temperature profile against ξdue to Q

Figure 8 Variation in velocity profile against ξ dueto Q

velocity of Al2O3 – H2O is higher, followed by the velocity of Cu – Al2O3 – H2O, Cu – H2Oand pure water. It can be physically justified as the Al2O3 conducting a high quantity ofheat due to the effective thermal conductivity but being less dense. Therefore, the velocityof Al2O3 – H2O is higher among all the fluids under consideration.

Figures 7 and 8 depict the influence of Q on velocity and temperature profiles. It is foundthat the velocity and temperature distributions increase with increasing values of Q. Whena larger value is assigned to Q this means that the system absorbed more heat due to whichthe intermolecular attractive force became weaker; as a result, the temperature and ve-locity profiles increase. The effect of the Brinkman parameter is presented in Fig. 9. It isnoticed that the velocity distribution decreases with increasing values of βb. The highervalues βb correspond to stronger drag forces, which lead to the retardation of the velocityprofile. The same effect of βb is reported by [33].

The effect of Gr is studied in Fig. 10 for negative and positive values. Positive values of Gr

correspond to heating of the plate, while negative values correspond to cooling of the plate.

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 16 of 18

Figure 9 Variation in velocity profile against ξ dueto βb

Figure 10 Variation in velocity profile against ξ dueto Gr

In this figure, it is noticed that for greater values of Gr the velocity profile shows an increas-ing trend. This is because when Gr is increased the buoyancy forces become stronger dueto which more convection takes place; as a result, the velocity profile increases. But thiseffect reverses for negative values of Gr due to cooling of the plate.

7 Concluding remarksIn this article, the idea of free convection is generalized using the Caputo–Fabrizio frac-tional derivative. The natural convection flow of a hybrid nanofluid in two vertical infiniteparallel plates is studied. Exact analytical solutions are developed for temperature andvelocity profiles via the Laplace transform technique. The effects of various pertinent pa-rameters are numerically studied through graphs and discuss physically. The major pointsextracted from this study are as follows:

• The velocity and temperature profiles show an increasing behavior for increasingvalues α and β being most dominant for α,β = 1 for a larger time. But this effectreverses for a shorter time.

• The fractional velocity and temperature are more general. Hence, the numericalvalues for v(ξ , τ ) and θ (ξ , τ ) can be calculated for any value of α and β such that0 < α,β < 1.

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

• The velocity profile decreases with increasing values of φhnf but this effect is oppositein the case of the temperature profile.

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 17 of 18

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

• The velocity retards for larger values of βb due the enhancement in the drag forces.

AcknowledgementsThe authors would like to acknowledge Ministry of Higher Education (MOHE) and Research Management Centre-UTM,Universiti Teknologi Malaysia UTM, for the financial support through grant numbers 15H80 and 13H74 for this research.

FundingNo specific funding was received for this project.

Availability of data and materialsNot applicable.

Competing interestsThe authors do not have any competing interests.

Authors’ contributionsIK formulated the problem and drafted the manuscript. MS solved the problem and drafted the manuscript. SSperformed the numerical simulation and plotted the results. All authors read and approved the final manuscript.

Author details1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Skudai, Malaysia. 2Faculty ofMathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 22 October 2018 Accepted: 23 January 2019

References1. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential

Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam (1998)2. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2),

1–13 (2015)3. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to

heat transfer model. Therm. Sci. 4(2), 763–769 (2016)4. Baleanu, D., Agheli, B., Darzi, R.: An optimal method for approximating the delay differential equations of noninteger

order. Adv. Differ. Equ. 2018(1), 284 (2018)5. Saqib, M., Khan, I., Shafie, S.: Natural convection channel flow of cmc-based cnts nanofluid. Eur. Phys. J. Plus 133(12),

549 (2018)6. Khalil, R., Al Horani, M., Anderson, D.: Undetermined coefficients for local fractional differential equations. J. Math.

Comput. Sci. 16, 140–146 (2016)7. Sayevand, K.: A fresh view on numerical correction and optimization of Monte Carlo algorithm and its application for

fractional differential equation. J. Math. Comput. Sci. 15(3), 209–215 (2015)8. Albadarneha, R.B., Batihab, I.M., Zurigatb, M.: Numerical solutions for linear fractional differential equations of order

1 < α < 2 using finite difference method (ffdm). Int. J. Math. Comput. Sci. 16(1), 103–111 (2016)9. Baleanu, D., Jajarmi, A., Bonyah, E., Hajipour, M.: New aspects of poor nutrition in the life cycle within the fractional

calculus. Adv. Differ. Equ. 2018(1), 230 (2018)10. Baleanu, D., Mousalou, A., Rezapour, S.: The extended fractional Caputo–Fabrizio derivative of order 0 ≤ σ < 1 on

CR[0, 1] and the existence of solutions for two higher-order series-type differential equations. Adv. Differ. Equ. 2018(1),255 (2018)

11. Saqib, M., Khan, I., Shafie, S.: Application of Atangana–Baleanu fractional derivative to mhd channel flow ofcmc-based-cnt’s nanofluid through a porous medium. Chaos Solitons Fractals 116, 79–85 (2018)

12. Zafar, A.A., Fetecau, C.: Flow over an infinite plate of a viscous fluid with non-integer order derivative without singularkernel. Alex. Eng. J. 55(3), 2789–2796 (2016)

13. Makris, N., Dargush, G.F., Constantinou, M.C.: Dynamic analysis of generalized viscoelastic fluids. J. Eng. Mech. 119(8),1663–1679 (1993)

14. Alkahtani, B.S.T., Atangana, A.: Modeling the potential energy field caused by mass density distribution with etonapproach. Open Phys. 14(1), 106–113 (2016)

15. Vieru, D., Fetecau, C., Fetecau, C.: Time-fractional free convection flow near a vertical plate with Newtonian heatingand mass diffusion. Therm. Sci. 19(suppl. 1), 85–98 (2015)

16. Abro, K.A., Khan, I., Gómez-Aguilar, J.F.: A mathematical analysis of a circular pipe in rate type fluid via Hankeltransform. Eur. Phys. J. Plus 133(10), 397 (2018)

17. Jain, S.: Numerical analysis for the fractional diffusion and fractional buckmaster equation by the two-step LaplaceAdam–Bashforth method. Eur. Phys. J. Plus 133(1), 19 (2018)

18. Agarwal, P., Dragomir, S.S., Jleli, M., Samet, B.: Advances in Mathematical Inequalities and Applications. Springer, Berlin(2018)

Saqib et al. Advances in Difference Equations (2019) 2019:52 Page 18 of 18

19. Ruzhansky, M., Cho, Y.J., Agarwal, P., Area, I.: Advances in Real and Complex Analysis with Applications. Springer, Berlin(2017)

20. Saad, K.M., Iyiola, O.S., Agarwal, P.: An effective homotopy analysis method to solve the cubic isothermalauto-catalytic chemical system. AIMS Math. 3(1), 183–194 (2018)

21. Baltaeva, U., Agarwal, P.: Boundary-value problems for the third-order loaded equation with noncharacteristictype-change boundaries. Math. Methods Appl. Sci. 41(9), 3307–3315 (2018)

22. Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation methods for solving fractionaldiffusion equation. Phys. A, Stat. Mech. Appl. 500, 40–49 (2018)

23. Salahshour, S., Ahmadian, A., Senu, N., Baleanu, D., Agarwal, P.: On analytical solutions of the fractional differentialequation with uncertainty: application to the Basset problem. Entropy 17(2), 885–902 (2015)

24. Agarwal, R.A., Jain, S., Agarwal, R.P., Baleanu, D.: A remark on the fractional integral operators and the image formulasof generalized Lommel–Wright function. Front. Phys. 6, 79 (2018)

25. Azhar, W.A., Vieru, D., Fetecau, C.: Free convection flow of some fractional nanofluids over a moving vertical plate withuniform heat flux and heat source. Phys. Fluids 29(8), 082001 (2017)

26. Jain, S., Atangana, A.: Analysis of lassa hemorrhagic fever model with non-local and non-singular fractionalderivatives. Int. J. Biomath. 11(08), 0850100 (2018)

27. Wang, X.-Q., Mujumdar, A.S.: Heat transfer characteristics of nanofluids: a review. Int. J. Therm. Sci. 46(1), 1–19 (2007)28. Sheikholeslami, M., Shamlooei, M., Moradi, R.: Numerical simulation for heat transfer intensification of nanofluid in a

porous curved enclosure considering shape effect of fe3o4 nanoparticles. Chem. Eng. Process. 124, 71–82 (2018)29. Hussanan, A., Salleh, M.Z., Khan, I., Shafie, S.: Convection heat transfer in micropolar nanofluids with oxide

nanoparticles in water, kerosene and engine oil. J. Mol. Liq. 229, 482–488 (2017)30. Bhattad, A., Sarkar, J., Ghosh, P.: Discrete phase numerical model and experimental study of hybrid nanofluid heat

transfer and pressure drop in plate heat exchanger. Int. Commun. Heat Mass Transf. 91, 262–273 (2018)31. Hussain, S., Ahmed, S.E., Akbar, T.: Entropy generation analysis in mhd mixed convection of hybrid nanofluid in an

open cavity with a horizontal channel containing an adiabatic obstacle. Int. J. Heat Mass Transf. 114, 1054–1066(2017)

32. Farooq, U., Afridi, M., Qasim, M., Lu, D.: Transpiration and viscous dissipation effects on entropy generation in hybridnanofluid flow over a nonlinear radially stretching disk. Entropy 20(9), 668 (2018)

33. Jan, S.A.A., Ali, F., Sheikh, N.A., Khan, I., Saqib, M., Gohar, M.: Engine oil based generalized Brinkman-type nano-liquidwith molybdenum disulphide nanoparticles of spherical shape: Atangana–Baleanu fractional model. Numer.Methods Partial Differ. Equ. 34(5), 1472–1488 (2018)

34. Aminossadati, S.M., Ghasemi, B.: Natural convection cooling of a localised heat source at the bottom of ananofluid-filled enclosure. Eur. J. Mech. B, Fluids 28(5), 630–640 (2009)

35. Brinkman, H.C.: The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 20(4), 571 (1952)36. Bourantas, G.C., Loukopoulos, V.C.: Modeling the natural convective flow of micropolar nanofluids. Int. J. Heat Mass

Transf. 68, 35–41 (2014)37. Maxwell, J.C.: Electricity and Magnetism, vol. 2. Dover, New York (1954)38. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2),

87–92 (2015)39. Khan, I.: A note on exact solutions for the unsteady free convection flow of a Jeffrey fluid. Z. Naturforsch. A 70(6),

397–401 (2015)40. Zakian, V., Littlewood, R.K.: Numerical inversion of Laplace transforms by weighted least-squares approximation.

Comput. J. 16(1), 66–68 (1973)