Binary chemical reaction with activation energy in rotating flow subject to nonlinear heat flux and heat source/sink
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Journal of Computational Design and Engineering, 2020, 7(3), 279–286
doi: 10.1093/jcde/qwaa023
Journal homepage: www.jcde.org
Advance Access Publication Date: 8 April 2020
RESEARCH ARTICLE
Binary chemical reaction with activation energy in
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rotating flow subject to nonlinear heat flux and heat
source/sink
M. Ijaz Khan1,4, *, Tehreem Nasir1,4 , T. Hayat1,2,4 , Niaz B. Khan3,4 and
A. Alsaedi2,4
1
Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan; 2 Nonlinear Analysis and
Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80207,
Jeddah 21589, Saudi Arabia; 3 School of Mechanical and Manufacturing Engineering, National University of
Sciences and Technology, Islamabad 44000, Pakistan and 4 Department of Mathematics, Riphah International
University, Faisalabad Campus, Satiana Road, Faisalabad 38000, Pakistan
*Corresponding author. E-mail: ijazfmg khan@yahoo.com
Abstract
Time-dependent rotating flow in presence of heat source/sink, applied magnetic field, Joule heating, thermal radiation, and
viscous dissipation is considered. Chemical reaction with Arrhenius activation energy is implemented. The governing
partial differential equations have been reduced to ordinary differential systems. Shooting scheme is implemented for the
computations of governing systems. Graphical results are arranged for velocity, temperature, and concentration, skin
friction coefficients, and heat and mass transfer rates. Main results are mentioned in conclusion portion. It is analyzed that
velocity decays in the presence of magnetic variable while temperature and concentration fields are enhanced via Eckert
number and fitted rate constant. Moreover drag force and mass and heat transfer rates decrease through higher
estimations of rotation rate variable, magnetic parameter, and Eckert number.
Keywords: rotating flow; Arrhenius activation energy; thermal radiation; heat source/sink; Joule heating; viscous dissipation
1. Introduction activation energy variable. They also concluded that heat and
mass transfer have same behavior through Weissenberg num-
Numerous chemically reacting procedures comprise the species
ber. Unsteady rotating liquid flow in the presence of binary
with Arrhenius activation energy like oil reservoir engineer-
chemical reaction is explored by Awad, Motsa, and Khumalo
ing and occurring in geothermal system. The relations between
(2014). Hayat, Ijaz Khan, Tamoor, Waqas, and Alsaedi (2018)
chemical reactions and mass transport are usually very com-
discussed Arrhenius activation energy in binary chemically re-
posite and can be examined in many reactions at different
active stagnation point flow over a stretchable surface. They
rates. Recently Khan, Hayat, Khan, and Alsaedi (2018) investi-
implemented Runge–Kutta–Fehlberg approach for the impact
gated Arrhenius activation energy impact in stagnation point
of different flow variables. The obtained computations divulge
flow with nonlinear thermal radiation and heat source/sink.
that species concentration enhances through larger estima-
They examined that concentration species increases for larger
tion of activation energy variable while it decreases for larger
Received: 28 December 2018; Revised: 14 March 2019; Accepted: 29 June 2019
C The Author(s) 2020. Published by Oxford University Press on behalf of the Society for Computational Design and Engineering. This is an Open Access
article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits
unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
279
280 Binary chemical reaction with activation energy in rotating flow
Schmidt number. Furthermore contrast behavior is examined Sherwood number have been examined and discussed graphi-
for heat and mass transfer rates when compared with skin fric- cally. Concluding remarks are pointed out. Ahmad, Khan, Hayat,
tion coefficient. Stagnation point flow of Carreau-nanomaterial Khan, and Alsaedi (2018), Baiguzin, Burmistrov, Kuznetsov, and
with activation energy and thermal radiation is examined by Farakhov (2019), Hayat, Khan, Alsaedi, and Khan (2017b), Hayat,
Hsiao (2017a). The problem is discussed in the presence of Khan, Farooq, Gull, and Alsaedi (2016), Hayat, Khan, Qayyum,
mixed convection, activation energy, thermal radiation, MHD, Khan, and Alsaedi (2018), Hayat, Khan, Waqas, and Alsaedi
and Ohmic dissipation. Nonlinear expressions are tackled nu- (2017b, 2017c), Hayat, Kiyani, Alsaedi, Khan, and Ahmad (2018),
merically with the help of parameters control method. The Hayat, Qayyum, Khan, and Alsaedi (2017), Hayat, Tamoor, Khan,
obtained computations investigate that temperature field en- and Alsaedi (2016), Hsiao (2016, 2017b, 2017c), Javed, Abbas,
hances for higher estimation of radiative variable. It is also ex- Sajid, and Ali (2011), Khan, Alsaedi, Qayyum, Hayat, and Khan
amined that larger efficiency system of thermal energy is ob- (2019), Khan, Hayat, Waqas, Khan, and Alsaedi (2018), Khan,
tained for higher/lower estimations of flow variables. Activa- Ibrahim, Khan, Hayat, and Javed (2018), Khan, Khan, Hayat, and
tion energy impact in nanomaterial Couette–Poiseuille flow with Alsaedi (2018), Khan, Tamoor, Hayat, and Alsaedi (2017), Nazar,
convective boundary conditions is scrutinized by Zeeshan, She- Amin, and Pop (2004), Tamoor, Waqas, Khan, Alsaedi, and Hayat
hzad, and Ellahi (2018). Here Buongiorno model is used for the (2017), Turkyilmazoglu (2016, 2017a, 2017b, 2018a,2018b), Waqas,
Downloaded from https://academic.oup.com/jcde/article/7/3/279/5818057 by guest on 11 November 2020
description of formulation. Further nonlinear expressions are Khan, Hayat, and Alsaedi (2017), Waqas, Khan, Hayat, Alsaedi,
tackled by homotopy analysis method. The obtained outcome and Khan (2017), and Yasmeen, Hayat, Khan, Imtiaz, and Alsaedi
presents that nanoparticle volume fraction is directly propor- (2016) represent the solution methodology for Newtonian and
tional to reaction and activation parameters while the impact non-Newtonian fluid models.
of Brownian variable on concentration profile gives contrast be-
havior to that of thermophoresis variable. Ahmad, Farooq, Javed,
and Anjum (2018), Boivin, Cannac, and Métayer (2019), Hayat 2. Mathematical Description
et al. (2016), Hayat, Khan, Farooq, Yasmeen, and Alsaedi (2016), Let us consider three-dimensional (3D) flow of viscous liquid
Hayat, Khan, Qayyum, Alsaedi, and Khan (2018), Hayat, Khan, by a stretchable surface. Arrhenius activation energy for chem-
Waqas, and Alsaedi (2017a), Hayat, Waqas, Khan, Alsaedi, and ical reaction process is considered. Heat transport mechanism
Shehzad (2017), Irfan, Khan, Khan, and Gulzar (2019), Khan, is examined in the presence of linear radiative flux. Further-
Khan, Waqas, Hayat, and Alsaedi (2017), Khan, Waqas, Hayat, more Joule heating and dissipation effect are considered. Fluid is
and Alsaedi (2017), Khan, Waqas, Hayat, Alsaedi, and Khan conducted electrically via applied magnetic field. Low magnetic
(2017), Khan, Yasmeen, Khan, Farooq, and Wakeel (2016), Lin Reynolds number leads to omission of induced magnetic field.
and Luo (2018), Sheikholeslami et al. (2019), and Xie, Xiao, and Let us assumed Tw and C w as surface temperature and concen-
Ren (2018) show the chemically reactive flows with different flow tration while T∞ and C ∞ being ambient temperature and con-
assumptions. centration (see Fig. 1).
Time-dependent flows in rotating frame have various ap- The governing flow expressions in the presence of Arrhenius
plications in geophysical fluid dynamics, chemical, mechani- activation energy are (Awad, Motsa, & Khumalo, 2014)
cal, and nuclear engineering systems. Viscoelastic nanomaterial
flow with convective boundary conditions due to rotating disks
is analyzed by Hayat, Javed, Imtiaz, and Alsaedi (2017). Qayyum,
Khan, Hayat, and Alsaedi (2018) examined viscous liquid flow ∂u ∂v ∂w
+ + = 0, (1)
submerged in five different nanoparticles. Kumar and Sood ∂x ∂y ∂z
(2017) explored MHD flow of copper-water nanoliquid due to ro- ∂u ∂u ∂u ∂u 1 ∂p ∂2u ∂2v ∂2w
+u +v +w + 2u = − +ν + 2 +
tating permeable medium with chemical reaction. Optimization ∂t ∂x ∂y ∂z ρ ∂x ∂ x2 ∂y ∂z2
of entropy generation with Ag-H2 O and Cu-H2 O nanoparticles
σ B02
is studied by Hayat, Khan, Qayyum, and Alsaedi (2018). Turkyil- − u, (2)
ρ
mazoglu (2012) developed an analytical technique for MHD 3D
stagnation point flow by a rotating disk. Hayat, Qayyum, Khan, ∂v ∂v ∂v ∂v 1 ∂p ∂ v
2
∂ v2
∂ v 2
+u +v +w − 2v = − +v + 2 + 2
and Alsaedi (2018) investigated MHD radiative flow by a rotating ∂t ∂x ∂y ∂z ρ ∂y ∂ x2 ∂y ∂z
disk with entropy generation and dissipative heat. Chemically σ B02
vapor deposition chamber through perforated showerhead due v, − (3)
ρ
to rotating disk is studied by Liu, Peng, Lai, Huang, and Liang 2
∂w ∂w ∂w ∂w ∂ w ∂2w ∂2w
(2017). Doh and Muthtamilselvan (2017) explored MHD flow of +u +v +w =v + + , (4)
∂t ∂x ∂y ∂z ∂x 2 ∂y2 ∂z2
non-Newtonian material with thermophoretic particle deposi- ⎫
2 2 2
tion due to rotating disk. Hayat, Khan, Alsaedi, and Khan (2017a) ∂T
∂t + u ∂∂Tx + v ∂∂Ty + w ∂∂z
T =α ∂ T + ∂ T + ∂ T − ρc1p ∂qr σ 2 2 2 ⎪
∂z + ρc p Bo u + v ⎪⎪
⎡ ∂ x2 ∂ y2 ∂z2 ⎤ ⎪
⎬
2 2 2
examined nanomaterial flow with dissipation and Joule heating. 2 ∂u + 2 ∂v + 2 ∂w +
⎢ ∂z ∂z ∂z ⎥ ⎪
Yao and Lian (2018) worked on rotationally symmetric flow by an + ρcμp ⎣ 2 2
Q
2 ⎦ + ρcop (T − T∞ ) , ⎪
⎪
⎪
∂v + ∂u ∂u + ∂w ∂v + ∂w ⎭
∂x ∂y + ∂z ∂x + ∂z ∂y
infinite rotating disk.
This article addresses impacts of binary chemical reaction (5)
with Arrhenius activation energy in time dependent flow of
rotating viscous fluid over a stretched surface. Heat trans- ∂C ∂C ∂C ∂C ∂ 2C ∂ 2C ∂ 2C
+u +v +w = D + +
port features are characterized for radiative heat flux and heat ∂t ∂x ∂y ∂z ∂ x2 ∂ y2 ∂z2
n
source/sink. Further dissipation and Joule heating are consid- T −E a
− kr2 exp (C − C ∞ ) , (6)
ered. The nonlinear coupled systems are tackled numerically T∞ κT
by built-in-shooting technique (Hayat, Khan, Imtiaz, Alseadi,
u = uw = ax, v = 0, w = 0, T = Tw , C = C w , at z = 0,
& Waqas, 2016). Rate of heat transfer, surface drag force, and (7)
u → 0, w → 0, T → T∞ , C → C ∞ as z → ∞.
Journal of Computational Design and Engineering, 2020, 7(3), 279–286 281
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Figure 1: Schematic flow analysis.
Ea
Here u, v, w indicate velocity components, p pressure, , σ, ρ source/sink variable, E (= κ T∞
) non-dimensional activation en-
density, ν(= μρ ) kinematic viscosity, α(= ρck ) thermal diffusivity, ergy, and E c(= a2 x2
) Eckert number.
p c p (Tw −T∞ )
T temperature, qr radiative heat flux, c p specific heat, Q 0 heat
source/sink coefficient, T∞ ambient temperature, C concentra-
tion, B0 strength of magnetic field, C ∞ ambient concentration, D 3. Quantities of Physical Curiosity
solutal diffusivity, kr reaction rate, E a modified Arrhenius func- Mathematically coefficient of skin friction and rates of heat (Nux )
tion, n fitted rate constant, κ = 8.61 × 10−5 eV/k Boltzmann con- and mass (Shx ) transfer are
stant, a dimensional constant, Tw surface temperature, and C w
y
surface concentration. τwx y τw xqw
C xf = , Cf = , Nux = ,
Considering ρ(ax)
2
ρ(ax)
2 k (Tw − T∞ )
√ ⎫ x jw
u = axf (ξ, η) , v = axh (ξ, η) , w = − aνξ f (ξ, η) , ⎬ Shx = , (14)
D (C w − C ∞ )
θ (ξ, η) = Tw −T∞ , φ (ξ, η) = C w −C ∞ , η = νξ z, ξ = 1 − e , τ = at.⎭
T−T∞ C −C ∞ a −τ
y
where τwx , τw , qw , and jw are defined as
(8)
∂u ∂v ∂T
τwx = μ |z=0 , τwy = μ |z=0 , qw = −k |z=0 ,
∂z ∂z ∂z
Incompressibility equation (1) trivially satisfied and remain-
ing flow expressions become ∂C
J w = −k |z=0 . (15)
∂z
η ∂f
f + (1 − ξ ) f + ξ f f − f 2 + 2hλ − M f = ξ (1 − ξ )
, (9)
2 ∂ξ Finally, one arrives
η ∂h
h + (1 − ξ ) h + ξ ( f h − f h − 2hλ − Mh) = ξ (1 − ξ ) , (10) C xf (Rex )
1/2
= ξ −1/2 f (0), (16)
2 ∂ξ
⎫
θ (1 + R) + 2η (1 − ξ ) θ + ξ f θ + ME cξ f 2 + h2 ⎬
1
= ξ −1/2 h (0),
y 1/2
Pr C f (Rex ) (17)
2 ⎭
(11)
+E c f + h2 + Q ∗ ξ θ = ξ (1 − ξ ) ∂θ ∂ξ
,
1 η Table 1: Comparative investigation of present analysis and Javed et
φ + (1 − ξ ) φ + ξ f φ
Sc 2 al. (2011) when M = 0.
−E ∂φ
− σ ξ φ (1 + δθ ) exp = ξ (1 − ξ ) , (12) λ Javed et al. (2011) Present results
1 + δθ ∂ξ
f (ξ, 0) = 0, f (ξ, 0) = 1, h (ξ, 0) = 0, θ (ξ, 0) = 1, φ (ξ, 0) = 1, ξ ≥ 0, − f (0) −h (0) − f (0) −h (0)
f (ξ, ∞) → 0, h (ξ, ∞) → 0, θ (ξ, ∞) → 0, φ (ξ, ∞) → 0, ξ ≥ 0.
0.2 1.3474169 0.37015223 1.3474202 0.3701522
(13) 0.5 1.5194131 0.76251409 1.5194194 0.76251423
σ Bo2
2.0 2.2827966 1.8485044 2.2828126 1.8485027
Here M(= ρa
) indicates magnetic variable, λ(= a
) rotation rate 5.0 3.3444338 3.0609192 3.3444607 3.0609160
kr2 16σ ∗ T∞
3
parameter, σ (= a
) chemical reaction variable, R(= 3kk∗
) radia- 10.0 4.6017220 4.3990640 4.6017645 4.3990572
(ρc ) ν
tive variable, Sc(= Dν ) Schmidt number, Pr(= kp ) Prandtl num- 50.0 10.058172 9.9668099 10.058259 9.9667986
ber, δ(= TwT−T
∞
∞
) temperature difference variable, Q ∗ (= (ρcQp0) a ) heat
282 Binary chemical reaction with activation energy in rotating flow
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Figure 4: h(η) versus λ.
Figure 2: f (η) versus λ.
Figure 5: θ (η) versus Ec.
Figure 3: f (η) versus M.
1 −1/2
(Rex ) Nux = −ξ −1/2 θ (0), (18)
1+ R
−1/2
(Rex ) Shx = −ξ −1/2 φ (0). (19)
(ax) x
where Rex (= ν
) represents the local Reynolds number.
4. Communicational Representation of
Results
In this portion the built-in-shooting technique is implemented
to solve the nonlinear coupled equations (9–12). The factors for
example fluid velocity i.e. f (η) in x-direction, h(η) in y-direction,
fluid temperature θ(η), species concentration φ(η), drag force
0.5 y 0.5
in x- and y-directions (C xf (Rex ) , C f (Rex ) ), heat transfer rate
Figure 6: θ (η) versus Pr.
−0.5 −0.5
(Nux (Rex ) ), and mass transfer rate (Shx (Rex ) ) have been
plotted by taking estimation of different flow variables as mag-
netic variable M = 0.5, rotation rate parameter λ = 0.2, chemi-
leads to monotonic exponential decrease in f (η) and h(η). Phys-
cal reaction variable σ = 0.3, radiative variable R = 0.5, Schmidt
ically magnetic field creates resistive force, called Lorentz force
number Sc = 0.9, Prandtl number Pr = 1.5, temperature differ-
to the liquid flow. That is why liquid velocity decays. The same
ence variable δ = 1.0, heat source/sink variable Q ∗ = 0.5, non-
observation has been presented by Nazar et al. (2004). Impact
dimensional activation energy E = 1.5, fitted rate constant n =
of Eckert number on θ(η) is plotted in Fig. 5. Here thermal field
0.5, and Eckert number E c = 1.0. The present results compared
is an increasing function of Eckert number. It is due to the fact
with Javed et al. (2011) and found good agreement (See Table 1).
that internal energy of liquid particles increases. Therefore tem-
Figures 2–4 elucidate the significance of rotation parameter
perature field enhances. Figure 6 highlights the impact of Prandtl
(λ) and magnetic parameter (M) on velocity fields f (η) and h(η).
number on θ(η). Larger estimation of Prandtl yield less diffusivity
From Figs 2–4, it is examined that an enhancement in λ and M
Journal of Computational Design and Engineering, 2020, 7(3), 279–286 283
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Figure 7: θ(η) versus R. Figure 10: φ(η) versus Sc.
Figure 8: θ(η) versus δ.
Figure 11: φ(η) versus σ .
tion. Figure 10 provides influences for Schmidt number on con-
centration field. Here φ(η) shows decreasing impact for larger
Schmidt number. Physically Schmidt number contains Brown-
ian diffusivity and enhancement in Schmidt number provides
lower Brownian diffusivity. That is why concentration field de-
cays. Figure 11 illustrates the impact of chemical reaction vari-
able (σ ) on concentration. It is examined that (σ ) decays both
concentration field and solutal layer. Physically higher estima-
tion of chemical reaction variation causes a thickening of solutal
layer. Therefore concentration profile decays.
Figures 12–15 elaborate analysis of drag forces
0.5 y 0.5
(C xf (Rex ) , C f (Rex ) ) and heat and mass transfer rates
−0.5 −0.5
(Nux (Rex ) , Shx (Rex ) ). From these figures, it is examined
that magnitude of surface drag forces in x- and y-directions
decrease versus rotation rate parameter (see Figs 12 and 13).
Figure 9: φ(η) versus n.
From Figs 14 and 15, it is analyzed that both heat and mass
transfer rates diminish for higher estimations of rotation rate
which decays the temperature and associated layer thickness. parameter and Eckert number.
Figure 7 highlights R characteristics versus temperature. It is ex-
amined that both θ(η) and thermal layer have been increased via
higher estimation of radiation variable. Physically due to a rise 5. Concluding Remarks
in radiative variable the kinetic energy of inside fluid particles Unsteady 3D flow of viscous rotating liquid by a sheet is consid-
enhances and as a result θ(η) increases. Figure 8 addresses be- ered. Main results are as follows:
havior of temperature difference variable on concentration field.
From Fig. 8 it is observed that temperature is gradually enhanced r f (ζ ) decays when magnetic variable is increased.
for higher estimation of temperature ratio variable (δ). Figure 9 r θ(η) is an increasing function of radiative variable.
is plotted to investigate the fitted rate constant variable (n). Here r Higher estimation of Eckert number enhance the tempera-
larger estimations of fitted rate constant give rise to concentra- ture field.
284 Binary chemical reaction with activation energy in rotating flow
r Schmidt number and chemical reaction parameter corre-
spond to a decay in concentration field.
r Skin friction decays in both x- and y-directions for higher es-
timation of rotation variable.
r Temperature and concentration gradients decrease via mag-
netic parameter, rotation parameter, and Eckert number.
Conflict of interest statement
The authors declared that they have no conflict of interest and
the paper presents their own work that does not been infringe
any third-party rights, especially authorship of any part of the
article is an original contribution, not published before and not
being under consideration for publication elsewhere.
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Figure 12: Combines effects of M and λ on C xf Re0.5
x .
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