Math. Model. Nat. Phenom. 13 (2018) 2 Mathematical Modelling of Natural Phenomenahttps://doi.org/10.1051/mmnp/2018013 www.mmnp-journal.org

APPLICATIONS OF FRACTIONAL DERIVATIVES TO

NANOFLUIDS: EXACT AND NUMERICAL SOLUTIONS

Sidra Aman1, Ilyas Khan2,*, Zulkhibri Ismail1

and Mohd Zuki Salleh1

Abstract. In this article the idea of time fractional derivatives in Caputo sense is used to studymemory effects on the behavior of nanofluids because some physical processes complex visco-elasticity,behavior of mechatronic and rheology are impossible to described by classical models. In presentattempt heat and mass transfer of nanofluids (sodium alginate (SA) carrier fluid with graphene nanopar-ticles) are tackled using fractional derivative approach. Exact solutions are determined for temperature,concentration and velocity field, and Nusselt number via Laplace transform technique. The obtainedsolutions are then expressed in terms of wright function or its fractional derivatives. Numerical solutionsfor velocity, temperature, concentration and Nusselt number are obtained using finite difference scheme.It is found that these solutions are significantly controlled by the variations of parameters includingthermal Grashof number, fractional parameter and nanoparticles volume fraction. It is observed thatrate of heat transfer increases with increasing nanoparticles volume fraction and Caputo time fractionalparameters.

Mathematics Subject Classification. 35Q53, 34B20, 35G31

Received August 10, 2017. Accepted January 10, 2018.

1. Introduction

In many physical and engineering processes classical models of integer-order derivatives are not able to workrequisitely in complex dynamics. In recent years, fractional calculus is found to be convenient to describe suchcases in chemistry, biology, mechanics, signal and image processing. More clearly, in fractional calculus thederivatives of classical models are replaced by fractional order derivatives. However, we can get through a hugerange of fractional models in the literature such as used in fractional Brownian motion, fractional neutron pointkinetic model, power law, Riesz potential, computational fractional derivative equations, fractional filters, frac-tional transforms, fractional wavelets, geophysics, fluid dynamics and levy statistics. An appealing literatureof fractional ideas can be found in [4, 5, 6, 8, 18]. The most used among them are the Riemann-Liouville andCaputo fractional derivatives [11, 13, 14]. But due to some limitations in applications of Riemann-Liouvillederivatives i.e. having non-zero derivative of a constant and its Laplace transform have some terms withoutany physical implementations; Caputo fractional derivative has sort of out these limitations but still there is

Keywords and phrases: Heat and mass transfer, graphene nanoparticles, finite difference scheme, time fractional derivatives,Laplace transform.

1 Faculty of Industrial Sciences & Technology, Universiti Malaysia Pahang, 26300 Kuantan, Pahang, Malaysia.2 Basic Engineering Sciences Department, College of Engineering Majmaah University, Majmaah 11952, Saudi Arabia.

* Corresponding author: ilyaskhanqau@yahoo.com; i.said@mu.edu.sa

Article published by EDP Sciences c© EDP Sciences, 2018

2 S. AMAN ET AL.

a singular kernel in its definition. To overcome this difficulty, Caputo and Fabrizio [10] have introduced a newdefinition of fractional derivatives with no singular kernel. Atangana and Secer [7] explicated an appealingdetail on fractional order derivatives and table of fractional derivatives some functions to explicit its overallidea. Sheikh et al. [19] compared the solutions of the flow of a generalized Casson fluid via Atangana-Baleanuand Caputo-Fabrizio fractional derivative approaches. Groundwater fractal flow with fractional differentiationand Mittag-Leffler law was recently tackled by Ahokposi et al. [1]. They used three different numerical schemesincluding implicit, explicit and Crank-Nicholson method. Khan et al. [16] emphasized flow of generalized Cassonfluid with fractional derivative and found closed form solutions in terms of Wright function. In another study,Khan et al. [17] applied Caputo-Fabrizio derivatives in heat transfer analysis of a Maxwell fluid. Due to seri-ous implementations of nanofluids in engine cooling, solar water heating, electronic cooling and solar thermalengineering, this is an attractive field of research. Some notable literature regarding heat transfer and the roleof nanofluids in it can be found in [2, 3, 15, 20]. All of the above work on fractional calculus is carried outfor viscous fluids and none of them considered fractional model for the flow of nanofluids. We found very rarework for nanofluids in this area such as Cao et al. [9] and Fetecau et al. [12]. However, nanofluids are of seriousimplementations in engineering and technology in the processes of heat transfer. Thus, the need of research inthis space attracted the authors toward this problem to analyze nanofluids using Caputo-time fractional deriva-tives. Combined analysis of heat and mass transfer is considered. The problem under consideration is studiedanalytically using the Laplace transform technique and numerically using the finite difference scheme.

2. Formulation of the problem

Consider Poiselliue flow of sodium alginate (SA) based-nanofluid with graphene nanoparticles in a verticalchannel. Θ0 and Θd show left and right plate temperatures while Φ0 and Φd show concentration at left and rightplate. Heat and mass transfer with mixed convection flow of nanofluid (induced by external pressure gradiantand buoyancy force) is governed by the following equations.

ρnf∂uα

∂t∗α= −∂p

∂x+ µnf

∂2u∗

∂y∗2+ (ρβΘ)nfg (Θ −Θ0) + (ρβΦ)nfg (Φ− Φ0) , (2.1)

(ρcp)nf∂αΘ

∂t∗α= knf

∂2Θ

∂y∗2+ 4α2

0 (Θ −Θ0) , (2.2)

∂αΦ

∂t∗α= Dnf

∂2Φ

∂y∗2, (2.3)

where u∗, Θ, Φ, ρnf , µnf , βΘ, βΦ, g, (ρcp)nf , knf , α0 and Dnf are respectively fluid velocity in the direction,temperature, concentration, density, the dynamic viscosity, volumetric thermal expansion coefficient, gravita-tional acceleration, heat capacitance of nanofluids, thermal conductivity of nanofluid, radiation absorption andthermal diffusion coefficient.

We consider ∂p∂x = H(t) [λ0 + λ exp(iωt)], with boundary conditions

u(0, t) = 0, u(d, t) = 0, Θ(0, t) = Θ0, Θ(d, t) = Θd, Θ(y, 0) = Θ0, Φ(0, t) = Φ0, Φ(d, t) = Cd. (2.4)

FRACTIONAL DERIVATIVES IN NANOFLUIDS 3

The density ρnf , thermal expansion coefficient (ρβ)nf , and heat capacitance (ρcp)nf are derived by using therelations:

ρnf = (1− φ)ρf + φρs, (ρβ)nf = (1− φ)(ρβ)f + φ(ρβ)s, µnf =µf

(1− φ)2.5

(ρcp)nf = (1− φ)(ρcp)f + φ(ρcp)s (2.5)

where φ is the nanoparticles volume fraction, ρf and ρs is the density of SA and nanoparticles, the volumetriccoefficient of thermal expansions of carbon nanotubes and base fluids are denoted by βs and βf respectively,(cp)s and (cp)f is the specific heat capacities of nanoparticles and base fluids at constant pressure. Expressionfor thermal conductivity:

knf = kf

(ks + 2kf − 2φ(kf − ks)ks + 2kf + φ(kf − ks)

),

where kf and ks are thermal conductivities of SA and graphene particles. Using the non-dimensional variables

u =u∗

U0, x =

x∗

d, t =

t∗U0

d, y =

y∗

d, p =

d

µU0p∗, λ0 =

λ∗0d2

µU0, λ =

λ∗d2

µU0

θ =Θ −Θ0

Θd −Θ0, C =

Φ− Φ0

Φd − Φ0, ω =

ω∗d

U0, −∂p

∂x= λ∗0 + λ∗exp (iω∗t∗) . (2.6)

The following non-dimensional differential equations are obtained:

φ1Re∂αu

∂tα= H(t) [λ0 + λ exp (iωt)] + φ2

∂2u

∂y2+ φ3Grθ + φ4GmC; y ∈ [0, 1], t ≥ 0, (2.7)

φ5Pe

λnf

∂αθ

∂tα=∂2θ

∂y2+N2

λnfθ, (2.8)

Sc1

(1− φ)

∂αC

∂tα=∂2C

∂y2, (2.9)

with dimensionless boundary conditions:

u(y, 0) = 0, u(0, t) = 0, u(1, t) = 0

θ(y, 0) = 0, θ(0, t) = 0, θ(1, t) = 1

C(y, 0) = 0, C(0, t) = 0, C(1, t) = 1 (2.10)

where

Re =U0d

ν, Gr =

gβT d2(Θd −Θ0)

νU0, Gm =

gβcd2 (Φd − Φ0)

νU0, N2 =

4α02d2

k, Pe = Pr Re. (2.11)

Equations (2.7)–(2.9) can be written as:

φ1ReDαt u(y, t) = H(t) [λ0 + λ exp (iωt)] + φ2

∂2u(y, t)

∂y2+ φ3Grθ(y, t) + φ4GmC(y, t), (2.12)

4 S. AMAN ET AL.

PeDαt θ(y, t) =

∂2θ(y, t)

∂y2+N2θ(y, t), (2.13)

Sc

(1− φ)Dαt C(y, t) =

∂2C(y, t)

∂y2, (2.14)

where the Caputo time-fractional derivative Dαt u(y, t) of u(y, t) is:

Dαt u(y, t) =

{1

Γ (1−α)∫ t0

1(t−τ)α

∂u(y,τ)∂τ dτ ; 0 < α < 1,

∂u(y,t)∂t ; α = 1.

(2.15)

3. Solution of the problem

3.1. Temperature field

Applying the Laplace transform to equation (2.13) and using equation (2.10), we obtain the problem intransform domain as:

(b0q

α − b21

)θ (y, q) =

∂2θ (y, q)

∂y2, (3.1)

θ (0, q) = 0, θ (1, q) =1

q, (3.2)

with solution

θ (y, q) =1

q1−αsinh

[y√b0√qα + b2

]qα sinh

[√b0√qα + b2

] , (3.3)

where b2 = − b2

1

b0. Using the convolution theorem and the formula

L−1{

1

q1−α

}=

{t−α

Γ (1−α) , 0 < α < 1,

δ(t), α = 1,(3.4)

we obtain for 0 < α < 1

θ(y, t) =

∫ t

0

(t− τ)−α

Γ (1− α)h(y√b0, τ, +b2,

√b0

)dτ

=

∫ t

0

(t− τ)−α

Γ (1− α)

∫ ∞0

τ−1f(y√b0, u, +b2,

√b0

)φ(0, −α, −uτ−α

)dudτ

=

∫ ∞0

f(y√b0, u, +b2,

√b0

)∫ t

0

(t− τ)−α

Γ (1− α)τ−1φ

(0, −α, −uτ−α

)dτdu (3.5)

The temperature equation (3.5) can be uttered in terms of Wright function.

θ(y, t) =

∫ ∞0

f(y√b0, u,+b2,

√b0

)Dαt Φ(1,−α,−ut−α)du; 0 < α < 1. (3.6)

FRACTIONAL DERIVATIVES IN NANOFLUIDS 5

For classical nanofluids α = 1,

θ(y, t) = f(y√b0, u, +b2,

√b0

). (3.7)

3.2. Concentration field

Using equations (2.10) and (2.12), we obtain the following problem in transform domain:

b3qαC(y, t) =

∂2C(y, t)

∂y2; b3 =

Sc

(1− φ)(3.8)

C (0, q) = 0, C (1, q) =1

q. (3.9)

The solution of the problem is

C (y, q) =1

q1−αsinh

[y√b3√qα]

qα sinh[√b3√qα] . (3.10)

Applying inverse Laplace transform:

C(y, t) =

∫ t

0

(t− τ)−α

Γ (1− α)

∫ ∞0

τ−1f(y√b3, τ, 0,

√b3

)φ(0, −α, −uτ−α

)dudτ

=

∫ ∞0

f(y√b3, τ, 0,

√b3

)∫ t

0

(t− τ)−α

Γ (1− α)τ−1φ

(0, −α, −uτ−α

)dτdu. (3.11)

3.3. Velocity field

Using equations (2.10) and (2.12), we obtain the following problem in transform domain for the u(y, q):

(φ1Reqα) u(y, q) = φ2∂2u(y, q)

∂y2+ φ3Gr

sinh[y√b0√qα + b2

]q sinh

[√b0√qα + b2

]+φ4Gm

sinh[y√b3√qα]

q sinh[√b3√qα] +

λ0q

+λ

q − iω, (3.12)

u(0, q) = 0, u(1, q) = 0. (3.13)

The particular solution of equation (3.12) is:

up(y, q) =a2

(a1 − b0) (qα + c0)

sinh[y√b0√qα + b2

]q sinh

[√b0√qα + b2

] +a3

(a1 − b3) (qα + d0)

sinh[y√b3√qα]

q sinh[√b3√qα]

+a4

a1 (qα + a0)

(λ0q

+λ

q − iω

). (3.14)

Complementary solution of the homogeneous equation associated with equation (3.12) is

uc(y, q) = C1 sinh(y√a1qα) + C2 cosh(y

√a1qα). (3.15)

6 S. AMAN ET AL.

The general solution of equation (3.12) is

u(y, q) = C1 sinh(y√a1qα) + C2 cosh(y

√a1qα) +

a4a1qα

(λ0q

+λ

q − iω

)+

a2(a1 − b0) (qα + c0)

sinh[y√b0√qα + b2

]q sinh

[√b0√qα + b2

] +a3

(a1 − b3) qαsinh

[y√b3√qα]

q sinh[√b3√qα] (3.16)

Using equation (3.13), we obtain the solution of u(y, q) as:

u(y, q) =a4a1

(λ0q

+λ

q − iω

)cosh [

√a1qα] sinh [y

√a1qα]

qα sinh [√a1qα]

− a2(a1 − b0) (qα + c0)

qα

q

sinh [y√a1qα]

qα sinh [√a1qα]

− a3(a1 − b3) q

sinh [y√a1qα]

qα sinh [√a1qα]

−(λ0q

+λ

q − iω

)a4 sinh [y

√a1qα]

a1qα sinh [√a1qα]

−a4 cosh [y√a1qα]

a1qα

(λ0q

+λ

q − iω

)+

a2q (a1 − b0)

qα

(qα + c0)

sinh[y√b0√qα + b2

]qα sinh

[√b0√qα + b2

]+

a3q (a1 − b3)

sinh[y√b3qα

]qα sinh

[√b3qα

] +a4a1qα

(λ0q

+λ

q − iω

). (3.17)

Taking inverse Laplace transform of equation (3.17):

u(y, t) = (λ0 + λeiwt)

[1

2ψ [√a1 (y + 1) , t] +

1

2ψ [√a1 (y − 1) , t]

]−a4Fα,α−1(t,−c0)

∞∑n=0

[erfc

(1−√a1y+2n

2√t

)]−erfc

(1+√a1y+2n

2√t

)− a1a2

(a1 − b0)(Fα,α−1(t,−c0))

∞∑n=0

[erfc

(1−√a1y+2n

2√t

)]−erfc

(1+√a1y+2n

2√t

) − a1a3(a1 − b3)

∞∑n=0

[erfc

(1−√a1y+2n

2√t

)]−erfc

(1+√a1y+2n

2√t

) −a4(λ0 + λeiwt)

1

2

(2− erfcy√a1

2√t

)+erfc

y√a1

2√t

+a2b0

(a1 − b0)(1 + (b2 − c0)tαEα,α+1(−c0tα))

×∫ ∞0

∞∑n=0

[erfc

(1−y+2n

2√u

)]−erfc

(1+y+2n

2√u

) e−b0b2ut−1Φ(0,−α,−b0t−α)du

+a3b3

(a1 − b3)

∞∑n=0

[erfc

(1−√b3y + 2n

2√t

)]− erfc

(1 +√b3y + 2n

2√t

)+a4a1

(λ0 + λeiwt)tα−1

Γ (α). (3.18)

3.4. Nusselt number

The entity of physical interest and applications Nusselt number is evaluated. Nusselt number the informationabout heat transfer rate of the fractional nanofluid at the walls. First, we find Nusselt number Nua(t) at wall:

Nu1(t) =d

kf

qw(Td − T0)

= − d

kf

knf(Td − T0)

∂T (y, t)

∂y

∣∣∣∣y=0

= −knfkf

∂θ(y, t)

∂y

∣∣∣∣y=0

, (3.19)

FRACTIONAL DERIVATIVES IN NANOFLUIDS 7

Table 1. Thermophysical properties of carrier fluid and graphene nanoparticles.

Physical properties Sodium alginate Graphene nanoparticles

ρ (kg/m3) 989 2250cp (J/kg K) 4175 2100k (W/m K) 0.6376 2500

Table 2. Nusselt number Nu1 variation with different values of fractional parameter andvolume fraction.

Fractional parameter, α φ = 0 φ = 0.01 φ = 0.02 φ = 0.04

0.01 1.128 1.162 1.196 1.2650.2 1.154 1.188 1.222 1.2910.4 1.184 1.217 1.251 1.3220.6 1.211 1.245 1.279 1.351 1.217 1.249 1.283 1.351

Table 3. Nusselt number Nu2 variation with different values of fractional parameter andvolume fraction.

Fractional parameter, α φ = 0 φ = 0.01 φ = 0.02 φ = 0.04

0.01 0.753 0.777 0.802 0.8530.2 0.704 0.728 0.753 0.8050.4 0.645 0.669 0.694 0.7450.6 0.573 0.579 0.621 0.6711 0.573 0.579 0.621 0.671

Nu1(t) = −knfkf

L−1

{√b0√qα + b2q

1

sinh(√b0√qα + b2

)}

= −knfkf

L−1

{√b0√qα + b2q

∞∑n=0

exp(−√b0 (2n+ 1)

√qα + b2

)√qα + b2

}. (3.20)

Let

Cn(q) =

∞∑n=0

exp(−√b0 (2n+ 1)

√qα + b2

)√qα + b2

,

Dn(q) =√b0qα + b2

q.

Their inverse Laplace transform are:

Cn(t) =

∫ ∞0

1√πz

exp

(−b0(2n+ 1)

2

4z− b0z

)t−1φ

(0,−α;−zt−α

)dz,

Dn(t) =√b0

[t−α

Γ (1− α)+ b2

]0 < α < 1,

8 S. AMAN ET AL.

Figure 1. Temperature profile for different values of volume fraction φ for fractional nanofluid.

Figure 2. Temperature profile for different values of volume fraction φ for ordinary nanofluid.

Thus the Nusselt number is:

Nu1(t) = −knfkf

(Dn(t) ∗

∞∑n=0

Cn(t)

)= −knf

kf

∞∑n=0

∫ ∞0

Dn(t− τ)Cn(τ)dτ. (3.21)

Now we find Nusselt number Nu2(t) at wall y = 0, which is given by

Nu2(t) = −knfkf

L−1

{√b0qα + b2

q

cosh(√b0√qα + b2

)sinh

(√b0√qα + b2

)}

= −knfkf

L−1

{√b0qα + b2

q

∞∑n=0

exp(−2n√b0√qα + b2

)+ exp

(−2(n+ 1)

√b0√qα + b2

)√qα + b2

}. (3.22)

FRACTIONAL DERIVATIVES IN NANOFLUIDS 9

Figure 3. Concentration profile for different values of fractional parameter α.

Figure 4. Velocity profile for different values of fractional parameter α.

The inverse Laplace transform of

Gn(q) =

∞∑n=0

exp(−2n√b0√qα + b2

)+ exp

(−2(n+ 1)

√b0√qα + b2

)√qα + b2

,

is

Gn(t) =

∫ ∞0

1√πz

[exp

(−n2b0z− b0z

)+ exp

(−(n+ 1)

2b0

z− b0z

)]t−1φ

(0,−α;−zt−α

)dz.

The Nusselt number is:

Nu2(t) = −knfkf

(Dn(t) ∗

∞∑n=0

Gn(t)

)= −knf

kf

∞∑n=0

∫ ∞0

Dn(t− τ)Gn(τ)dτ. (3.23)

10 S. AMAN ET AL.

Figure 5. Velocity profile for different values of Grashoff number Gr.

Figure 6. Velocity profile for different values of radiation parameter N .

4. Numerical simulations, graphs and discussion

In present study Sodium Alginate is taken as non-Newtonian base fluid with graphene nanoparticles. Thermo-physical properties of base fluid and nanoparticles are given in Table 1. The problem is now solved numericallyusing the finite difference scheme. Numerical results for Nusselt number are computed in Tables 2 and 3 andin Figure 8, whereas numerical results for velocity, temperature, concentration to study the effect of differentparameters on the considered flow are depicted in Figures 1–7. Figure 1 shows that temperature is an increasingfunction of volume fraction φ for fractional nanofluids. Physically, by increasing the amount of nanoparticles tothe base fluid enhance its thermal conductivity which increases fluids temperature. Figure 2 shows that tempera-ture of the nanofluid increases with the radiation parameter N . It is due to an increase in heat energy transfer tothe fluid. Figure 3 depicted that concentration of the nanofluid is decreasing with increasing values of fractionalparameter α. Concentration decreases for both ordinary and fractional nanofluid. Figure 4 elaborates that fluidflow minimizes with increasing fractional parameter α. It is clear from the graph that the fractional nanofluidshave greater flow than the ordinary nanofluid (α = 1). We can see a unique behavior of ordinary nanofluidnear the boundary (y = 1). Figure 5, illustrates that Gr contributes an increase on the flow of nanofluid as anincrease in value of Gr favors enhancement in temperature gradient and buoyancy force. Figure 6 illustrates

FRACTIONAL DERIVATIVES IN NANOFLUIDS 11

Figure 7. Velocity profile for different values of volume fraction φ.

Figure 8. Nusselt number variation for different values of volume fraction α.

that increasing radiation parameter N turns the flow of fractional nanofluid faster. Figure 7 shows the influenceof amount of graphene nanoparticles φ on the velocity. It is clear that the velocity of nanofluid slow downwith increase in φ as the fluid becomes denser. Figures 8a and 8b show the effect of fractional parameter αon Nusselt number with time t. Nusselt number is increasing at start and then acquires a constant position.Smaller the value of greater is the heat transfer rate, also showing fractional nanofluids exhibit higher Nusseltnumber compared to classical ordinary nanofluid α = 1. Finally, a numerical evaluation is carried out for theimpact of φ and α in Table 2 for Nusselt numbers Nu1 and Nu2 of fractional nanofluid α ∈ (0, 1) and ordinarynanofluid (α = 1). It is spotted that the rate of heat transfer Nu1 and Nu2 get maximize with increasing φwhile decreases with increasing values of α. Numerical values spotted that fractional nanofluids have elevatedrate of heat transfer than that of Classical nanofluids. Thus, fractional nanofluids can be considered the bestfor the purpose of heat transfer rate enhancement.

12 S. AMAN ET AL.

5. Conclusion

In this attempt, the exact solutions for heat and mass transfer of fractional Sodium Alginate based nanofluidare obtained using Caputo-time fractional derivatives. Laplace transform method is used to figure out theexact expressions of velocity, concentration and temperature and then revealed graphically in detail. Further-more, numerical solutions are obtained using the finite difference scheme and the results are shown in tableand graphs. Graphene nanoparticles were chosen as nanoparticles inside Sodium Alginate as base fluids. Someimportant outcomes are:

– Velocity increases with increase in Gr and N while decreases with increasing values of φ and α.– Fractional nanofluids have higher velocity and rate of heat transfer than that of ordinary nanofluid.

Increase in radiation parameter, increases the heat transfer rate.– The present work has useful applications in engineering, physics and many other fields. However, addition

of nanoparticles to the base fluid of fractional problem bestows attraction to this research.

Acknowledgements. The authors gratefully acknowledge the financial support received from Universiti Malaysia Pahang(UMP) under RDU170354 and RDU170358. The first author acknowledge with thanks the Deanship of Scientific Research(DSR) at Majmaah University, Majmaah Saudi Arabia for technical and financial support through vote number 37/97for this research project.

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