Cattaneo–Christov double-diffusion model for viscoelastic nanofluid with activation energy and nonlinear thermal radiation
Authors: Ayub Muhammad , Malik Muhammad Yousaf , Ijaz Misbah , Alqarni Marei Saeed , Alqahtani Ali Saeed 
Published: January 2020

Purpose
The purpose of this paper is to explore the novel aspects of activation energy in the nonlinearly convective flow of Walter-B nanofluid in view of Cattaneo–Christov double-diffusion model over a permeable stretched sheet. Features of nonlinear thermal radiation, dual stratification, non-uniform heat generation/absorption, MHD and binary chemical reaction are also evaluated for present flow problem. Walter-B nanomaterial model is employed to describe the significant slip mechanism of Brownian and thermophoresis diffusions. Generalized Fourier’s and Fick’s laws are examined through Cattaneo–Christov double-diffusion model. Modified Arrhenius formula for activation energy is also implemented.

Design/methodology/approach
Several techniques are employed for solving nonlinear differential equations. The authors have used a homotopy technique (HAM) for our nonlinear problem to get convergent solutions. The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear coupled ordinary/partial differential equations. The capability of the HAM to naturally display convergence of the series solution is unusual in analytical and semi-analytic approaches to nonlinear partial differential equations. This analytical method has the following great advantages over other techniques:

It provides a series solution without depending upon small/large physical parameters and applicable for not only weakly but also strongly nonlinear problems.
It guarantees the convergence of series solutions for nonlinear problems.
It provides us a great choice to select the base function of the required solution and the corresponding auxiliary linear operator of the homotopy.
Brief mathematical description of HAM technique (Liao, 2012; Mabood et al., 2016) is as follows. For a general nonlinear equation:(1)

N

[

u

(
x
)

]

=
0
,

where N denotes a nonlinear operator, x the independent variables and u(x) is an unknown function, respectively. By means of generalizing the traditional homotopy method, Liao (1992) creates the so-called zero-order deformation equation:(2)

(

1
−
q

)

L

[

u
ˆ

(

x
;
q

)

−

u

o

(
x
)

]

=
q
h
H

(
x
)

N

[

u
ˆ

(

x
;
q

)

]

,

here q∈[0, 1] is the embedding parameter, H(x) ≠ 0 is an auxiliary function, h(≠ 0) is a nonzero parameter, L is an auxiliary linear operator, uo(x) is an initial guess of u(x) and

u
ˆ

(

x
;
q

)

is an unknown function, respectively. It is significant that one has great freedom to choose auxiliary things in HAM. Noticeably, when q=0 and q=1, following holds:(3)

u
ˆ

(

x
;
0

)

=

u

o

(
x
)

and
u
ˆ

(

x
;
1

)

=
u

(
x
)

,

Expanding

u
ˆ

(

x
;
q

)

in Taylor series with respect to (q), we have:(4)

u
ˆ

(

x
;
q

)

=

u

o

(
x
)

+

∑

m
=
1

∞

u

m

(
x
)

q

m

,

where

u

m

(
x
)

=

1

m
!

∂

m

u
ˆ

(

x
;
q

)

∂

q

m

|

q
=
0

.

If the initial guess, the auxiliary linear operator, the auxiliary h and the auxiliary function are selected properly, then the series (4) converges at q=1, then we have:(5)

u

(
x
)

=

u

o

(
x
)

+

∑

m
=
1

+
∞

u

m

(
x
)

.

By defining a vector

u
→

=

(

u

o

(
x
)

,

u

1

(
x
)

,

u

2

(
x
)

,

…
,

u

n

(
x
)

)

,

and differentiating Equation (2) m-times with respect to (q) and then setting q=0, we obtain the mth-order deformation equation:(6)

L

[

u
ˆ

m

(
x
)

−

χ

m

u

m
−
1

(
x
)

]

=
h
H

(
x
)

R

m

[

u
→

m
−
1

]

,

where:(7)

R

m

[

u
→

m
−
1

]

=

1

(

m
−
1

)

!

∂

m
−
1

N

[

u

(

x
;
q

)

]

∂

q

m
−
1

|

q
=
0

and

χ

m

=

|

0

m
⩽
1
1

m
>
1

.

Applying L−1 on both sides of Equation (6), we get:(8)

u

m

(
x
)

=

χ

m

u

m
−
1

(
x
)

+
h

L

−
1

[

H

(
x
)

R

m

[

u
→

m
−
1

]

]

.

In this way, we obtain um for m ⩾ 1, at mth-order, we have:(9)

u

(
x
)

=

∑

m
=
1

M

u

m

(
x
)

.

Findings
It is evident from obtained results that the nanoparticle concentration field is directly proportional to the chemical reaction with activation energy. Additionally, both temperature and concentration distributions are declining functions of thermal and solutal stratification parameters (P1) and (P2), respectively. Moreover, temperature Θ(Ω1) enhances for greater values of Brownian motion parameter (Nb), non-uniform heat source/sink parameter (B1) and thermophoresis factor (Nt). Reverse behavior of concentration ϒ(Ω1) field is remarked in view of (Nb) and (Nt). Graphs and tables are also constructed to analyze the effect of different flow parameters on skin friction coefficient, local Nusselt number, Sherwood numbers, velocity, temperature and concentration fields.

Originality/value
The novelty of the present problem is to inspect the Arrhenius activation energy phenomena for viscoelastic Walter-B nanofluid model with additional features of nonlinear thermal radiation, non-uniform heat generation/absorption, nonlinear mixed convection, thermal and solutal stratification. The novel aspect of binary chemical reaction is analyzed to characterize the impact of activation energy in the presence of Cattaneo–Christov double-diffusion model. The mathematical model of Buongiorno is employed to incorporate Brownian motion and thermophoresis effects due to nanoparticles.