Homotopy analysis method with application to thin-film flow of couple stress fluid through a vertical cylinder

Abstract

This work solves the problem of thin-film withdrawal and drainage of a steady incompressible couple stress fluid on the outer surface of a vertical cylinder. The governing equations for velocity and temperature distributions are subjected to the boundary conditions and solved with the help of homotopy analysis method. The obtained expressions for flow profile, temperature profile, average velocity, volume flow rate, and shear stress confirmed that the thin-film flow of couple stress fluid highly depends on involved parameters say Stokes number St
, vorticity parameter λ, couple stress parameter η, and Brinkman number Br presented in the graphical description as well.

Full text

Research Article
Muhammad Farooq, Zia Ullah, Muhammad Zeb, Hijaz Ahmad, Muhammad Ayaz,
Muhammad Sulaiman, Chutarat Tearnbucha*, and Weerawat Sudsutad
Homotopy analysis method with application to
thin-film flow of couple stress fluid through a
vertical cylinder
https://doi.org/10.1515/phys-2022-0056
received November 07, 2021; accepted March 20, 2022
Abstract: This work solves the problem of thin-film with-
drawal and drainage of a steady incompressible couple
stress fluid on the outer surface of a vertical cylinder. The
governing equations for velocity and temperature distri-
butions are subjected to the boundary conditions and
solved with the help of homotopy analysis method. The
obtained expressions for flow profile, temperature profile,
average velocity, volume flow rate, and shear stress con-
firmed that the thin-film flow of couple stress fluid highly
depends on involved parameters say Stokes number S
t
,
vorticity parameter λ, couple stress parameter η, and
Brinkman number Br presented in the graphical descrip-
tion as well.
Keywords: thin film flow, withdrawal, drainage, couple
stress fluid, vertical cylinder, differential equations, homo-
topy analysis method
1 Introduction
Non-Newtonian fluids have a vigorous place in the pre-
sent research due their prevalent importance in the
food, chemical, construction, pharmaceutical industries
etc.Non-Newtonian fluids cover production of synthetic
items, all types of automobiles, food products, biotic
fluids, cable and filament layer, sheets manufacturing,
denitrification freezing, gassy flow, penetrating sludge,
heat up tubes, etc.
All non-Newtonian fluids are not of the same kind, so
for better understanding different types of non-Newtonian
fluids, these fluids are categories in different fluid models
such as differential type and rate type fluids models.
Couple stress fluid is one among these fluids and its equa-
tion is centered on solid theoretic grounds, whenever
relationship among stress and strain are not linear. The
behavior of couple stress fluid has been studied in many
research works which have come up with fruitful results.
The couple stress fluids model introduced by Stokes [1],
has distinct characteristics, such as the presence of couple
stresses, non-symmetric stress tensor and body couples.
Many scholars have scrutinized the flow behavior of
couple stress fluid, like Devakar et al. [2]studied the
couple stress fluid with slip boundary conditions in par-
allel plates. Jangili et al. [3]have demonstrated the
irreversibility rate for the couple stress fluid under the
effect of variable viscosity and thermal conductivity.
Farooq et al. [4]have investigated the behavior of
couple stress fluid with temperature dependent variable
viscosity on inclined plate. The importance of this theory
is consideration of the rotational effects, which is not
considered in Navier–Stokes equations. Applications of
couple stress fluids are in industries such as extrusion for
polymer fluids, colloidal solution, and the lubrication of
engine and bearings [5]. Its applications in biomechanics
are discussed in refs [6–8].
Muhammad Farooq, Zia Ullah: Department of Mathematics, Abdul
Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan
Muhammad Zeb: Department of Mathematics, COMSATS University
Islamabad, Attock Campus, Punjab, Pakistan
Hijaz Ahmad: Section of Mathematics, International Telematic
University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma,
Italy, e-mail: hijaz555@gmail.com
Muhammad Ayaz, Muhammad Sulaiman: School of Science, Xi’an
University of Architecture and Technology, Xi’an 710055, China

* Corresponding author: Chutarat Tearnbucha, Department of
General Education, Faculty of Science and Health Technology,
Navamindradhiraj University, Bangkok, 10300 Thailand,
e-mail: chutaratt@nmu.ac.th
Weerawat Sudsutad: Department of Statistics, Faculty of Science,
Ramkhamhaeng University, Bangkok, 10240 Thailand
Open Physics 2022; 20: 705–714
Open Access. © 2022 Muhammad Farooq et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0
International License.

Several researchers have studied and addressed the
physical relevance of thin film. Miladinova et al. [9]
described, for example, the thin film power law model
fluid flowing on a tilted plate and found a nonlinear
interaction saturation observed with permanent wave
of finite amplitude. Alam et al. [10]studied Johnson–
Segalman thin film flow for lifting and drainage on a
vertical surface and have discussed the effects of involved
parameters on film flow. Shah et al. [11]presented the
film flow of second grade in a straight annular die and
developed the governing equations to analyze the pro-
cess of wire coating. The magnetohydrodynamics (MHD)
boundary layer flow of mass and heat transfer marvels
with radiation and dissipative impacts past a permeable
stretchable surface was explored by Kar et al. [12].More-
over, Nayak et al. [13]have presented heat and mass
transfer phenomenon in the boundary layer flow of elec-
trically conducting viscoelastic fluid in the presence of
source/sink, chemical reaction, and normal magnetic
field, and found that magnetic field is very useful in
promoting the velocity and concentration distribution.
Also, the investigation shows that the expanding upsides of
the heat source boundary and the shortfall of the perme-
able boundary upgrade the stream nearby the plate. In ref.
[14],Gowdaet al. have presented heat and mass transfer
briefly in ferromagnetic fluid on the surface of a stretch-
able sheet. Gul et al. [15]have studied third grade fluid
thin film flow for lifting and drainage with the impact of
MHD and heat subordinate thickness. On the other hand,
Ahmad et al. [16]investigated the unsteady free convec-
tion flow of a second grade fluid.
Siddiqui et al. [17]have studied the flow of thin films
of third-and fourth-grade fluids falling on an inclined plane
and vertical cylinder via homotopy analysis method (HAM).
Farooq et al. [18,19]investigated the withdrawal and
drainage of generalized second-grade fluid with and
without slip conditions on a vertical cylinder. The litera-
ture review shows that approximate analytical techniques
[20–40]are very strong tools to solve highly nonlinear
differential equations, and one among these is HAM [41].
In this work we have solved the problem of thin film flow
of couple stress fluid on the outer surface of vertical
cylinder, and the developed mathematical model is ana-
lyzed with the help of HAM.
2 Lifting problem
Consider incompressible, non-isothermal couple stress
fluid in a container, and a vertical cylinder passing
through the container, moving upward contacting fluid
from the container, develops a thin film of constant thick-
ness δof the fluid on the outer surface symmetrically. The
cylindrical coordinates are fixed as the axial axis is
located at the center of the cylinder and the radial axis
is kept along the radius Rof the cylinder as shown in
Figure 1(a). Assuming that the flow is steady and has
no change with respect to θthe velocity and temperature
fields are: [()]()==Vwrθθr0, 0, , . (1)
The following are the basic equations that regulate
the flow of an incompressible non-isothermal couple
stress fluid:
∇
⋅=V0,
(2)
Figure 1: (a)Geometry of the lifting problem. (b)Geometry of the drainage.
706 Muhammad Farooq et al.

∇∇=−+⋅−∇
VfTV
ρ
tρp η
d
d,
4
(3)
()=∇+ ⋅TL
ρ
Cθ
tkθ
d
dtrc
,
p2
(4)
where Vrepresents the velocity vector and
t
d
d
is material
time derivative defined as
() ⎛⎝⎞⎠()⋅=∂∂+⋅∇⋅
tt
V
d
d,(5)
kis the thermal conductivity, ρis the constant density, C
p
is the specific heat constant, fis the body force, θis the
temperature, pis the dynamic pressure, and Tis the
stress tensor taken as: =TμA
,
1
(6)
where μis the viscosity of the fluid, and A
1
is the first
Rivlin–Erickson tensor.
=+ =∇ALL L V,and
.
T
1(7)
The velocity field (1), balances Eq. (2), and reduces
Eq. (3), in the following components:
r-component: ∂∂=
p
r0
.
(8)
θ-component: ∂∂=
rp
θ
10, (9)
z-component: ⎡
⎣⎤
⎦
⎡
⎣
⎢⎛⎝⎞⎠⎤
⎦
⎥⎡
⎣
⎢⎛⎝⎞⎠⎤
⎦
⎥
∂∂=− +
−
p
zρg μrrrw
r
ηrr
rrrr
rw
r
1d
dd
d
1d
dd
d1d
dd
d.
(10)
Using Eqs. (8)and (9), Eq. (10)becomes
⎡
⎣⎤
⎦
⎡
⎣
⎢⎛⎝⎞⎠⎤
⎦
⎥⎡
⎣
⎢⎛⎝⎞⎠⎤
⎦
⎥
=− +
−
p
zρg μrrrw
r
ηrr
rrrr
rw
r
d
d1d
dd
d
1d
dd
d1d
dd
d.
(11)
Consider that the pressure is atmospheric, then =
0
p
z
d
d,
and so Eq. (11)gets the form:
⎜⎟
⎜⎟
⎛⎝⎞⎠
⎛⎝⎞⎠
++−
+− =−
w
rr
w
rr
μ
ηw
r
rμ
ηr w
rρg
η
d
d2d
d1d
d
1d
d.
4
4
3
32
2
2
3
(12)
The velocity and temperature fields simplify Eq. (4)as:
⎜⎟
⎛⎝⎞⎠⎛⎝⎞⎠
++ =
κ
θ
rr
θ
rμw
r
d
d1d
dd
d0
.
2
2
2
(13)
For the fourth order (12)and second order (13)non-
linear differential equations, the following boundary con-
ditions can be set from the geometry as given:
====
====+
wU w
rθθ rR
w
rw
rθ
rrRδ
,d
d0, at ,
d
d0, d
d0, d
d0at .
2
20
3
3
(14)
Using the non-dimensional parameters
()
====
=−=
∗∗∗
ww
Urr
δλμδ
ηSρgδ
μU
μU
θθk
RR
δ
,, , ,
Br , .
t
22
eff 2
10
(15)
After dropping asterisks Eqs. (12–14)become
⎛⎝⎞⎠⎛⎝⎞⎠
++−+−
=−
w
rr
w
rr
λw
rr
λ
rw
r
λS
d
d2d
d1d
d1d
d
,
t
4
4
3
32
2
23 (16)
⎜
⎟
⎛⎝
⎞⎠()
⎛⎝⎞⎠
++−=
θ
rr
θ
rμU
κθ θ w
r
d
d1d
dd
d0
.
2
2
2
10
2
(17)
Using the previously defined dimensionless quanti-
ties, Eq. (17)becomes
⎜
⎟
⎛⎝
⎞⎠⎛⎝⎞⎠
++ =
θ
rr
θ
rw
r
d
d1d
dBr d
d0
.
2
2
2
(18)
The dimensionless form of the boundary conditions
given in (14)take the form:
====
====+
ww
rθθ rR
w
rw
rθ
rrR
1, d
d0, at ,
d
d0, d
d0, d
d0at 1.
2
20
3
3
(19)
3 Solution of lifting problem
using HAM
The approximate analytical technique, HAM, is used to
solve Eqs. (16)and (18), together with Eq. (19). Funda-
mental roots of the model equations via HAM are given
below in detail.
() ()==″
⌢
⌢
⌢
L
wwLθθ,,
wiv θ
(20)
linear operators
⌢
L
ware signified as
HAM with application to thin-film flow of couple stress fluid 707

(( )( ))
()
−+=
+=
−
⌢
⌢
Le eee e
Le ee
0,
0.
wRδδrr
θRδr
222
22
(21)
The consistent non-linear operators are reasonably
selected as
⌢
Ν
w
and
⌢
Ν
θ, and recognized in system as:
[()][] [] ⎛⎝⎞⎠[]
⎛⎝⎞⎠[]
=++−
+− +
⌢⌢⌢ ⌢
⌢
⌢
Νwrζ w rwrλw
rλ
rwλS
;21
1,
w rrrr rrr rr
rt
2
3
(22)
[( ) ( )][] [] []
⌢=⌢+⌢+
⌢⌢
⌢
N
wrζ θrζ θ rθw;, ; 1Br
.
θrr r r 2(23)
For Eqs. (20)and (21), the 0th-order system can be
written as:
( )[() ()] [()]−−=ℏ
⌢⌢ ⌢
⌢⌢⌢
ζL wrζ wr p N wrζ1; ;
,
www0(24)
( ) [ ( ) ()] [ ( ) ( )]−⌢−⌢=ℏ ⌢
⌢
⌢⌢⌢
ζL θrζ θr p N wrζ θrζ1; ;,;
.
θθθ
0(25)
While BCs are:
()∣ ()
()∣ ()∣
() () ()
==
⌢=⌢==
⌢=
⌢⌢
⌢⌢
==
==
===
wrζ wrζ
r
θrζ θ rζ
wrζ
rwrζ
rθrζ
r
;1,
d;
d0,
;;
d;
d0, d;
d0, d;
d0.
rr
rr
rrr
0
2
20
00 0
1
3
311
(26)
Here ζis the embedding parameter ζ∈[0, 1], to reg-
ulate for the solution convergence of
ℏ
ℏ
⌢
⌢
,
wθis used.
When ζ=0 and ζ=1 we have:
() ()=
⌢⌢
w
rwr;1
,
(27)
() ()
⌢
=⌢
θr θr;1
,
(28)
Expand the
()(
)
⌢
⌢
w
rζ θ rζ;, ;
through Taylor’s series for
ζ=0
( ) () ()
( ) () ()
∑
∑
=+
⌢=⌢+⌢
⌢⌢⌢
=
∞
=
∞
wrζ w r w rζ
θrζ θ r θ rζ
;,
;.
nnn
nnn
01
01
(29)
() () () ()
=!∂∂⌢=!∂⌢∂
⌢⌢
==
w
rnwrζ
rθr nθrζ
r
1;,1;.
npn
p
00
(30)
While BCs are:
() () () ()
() () ()
=″=⌢=⌢
′= ″′= ⌢′=
⌢⌢
⌢⌢
ww θθ
wδ w δ θδ
01, 00, 0 0
0, 0, 0.
0(31)
Now
() [] [ ] ⎛⎝⎞⎠[]
⎛⎝⎞⎠[]
R=+′″+−″
+−′+
⌢⌢ ⌢
⌢
−−−
−
⌢
rw rwrλw
rλ
rwλS
21
1,
n
wnnn
nt
1
iv 121
31
(32)
() [ ] [ ] [ ]R=⌢″+⌢′+″
⌢
⌢−−−
rθ rθw
1Br
,
n
θnn n
11 1
2(33)
and
⎧
⎨
⎩
=≤>
χn
n
0, if 1,
1, if 1.
n
4 Drainage problem
Consider incompressible, non-isothermal couple stress fluid
in a container, and a vertical cylinder, moving downward
contacting the fluid from the container, develops a thin film
of constant thickness δof the fluid on the outer surface
symmetrically. The cylindrical coordinates are fixed as the
axial axis is located at the center of the cylinder and the
radial axis is kept along the radius Rof the cylinder as
showninFigure1(b).Assumingthattheflow is steady
and has no change with respect to θ,thevelocityfield
and temperature distribution are:
[()]()==Vwrθθr0, 0, ,
.
(34)
For the drainage problem, we take the dimensionless
form of the governing equations after using dimension-
less parameters given in Eq. (15)as under:
⎛⎝⎞⎠⎛⎝⎞⎠
++−+−=
w
rr
w
rr
λw
rr
λ
rw
rλS
d
d2d
d1d
d1d
d
,
t
4
4
3
32
2
23 (35)
⎜
⎟
⎛⎝
⎞⎠⎛⎝⎞⎠
++ =
θ
rr
θ
rw
r
d
d1d
dBr d
d0,
2
2
2
(36)
the boundary conditions are:
====
====+
ww
rθθ rR
dw
dr w
rθ
rrR
0, d
d0, at ,
0, d
d0, d
d0at 1.
2
20
3
3
(37)
5 Solution of drainage problem
using HAM
The approximate analytical technique, HAM, is used to
solve Eqs. (35)and (36), together with Eq. (37). Funda-
mental roots of the model equations via HAM are given
below in detail: () ()==″
⌢
⌢
⌢
L
wwLθθ,
.
wθ
iv (38)
708 Muhammad Farooq et al.

Linear operators
⌢
L
ware signified as
(( )( ))
()
−+=
+=
−
⌢
⌢
Le eee e
Le ee
0,
0.
wRδδrr
θRδr
222
22
(39)
The consistent nonlinear operators are reasonably
selected as
⌢
N
w
and
⌢
N
θ, and recognize in system as:
[()][] [] ⎛⎝⎞⎠[]
⎛⎝⎞⎠[]
=++−
+− −
⌢⌢⌢ ⌢
⌢
⌢
Nwrζ w rwrλw
rλ
rwλ
;21
1S,
w rrrr rrr rr
r
2
3t
(40)
[( ) ( )][] [] []
⌢=⌢+⌢+
⌢⌢
⌢
N
wrζ θrζ θ rθw;, ; 1Br
.
θrr r r 2
For Eq. (39), the 0th-order system is written as
( )[() ()] [()]
( ) [ ( ) ()] [ ( ) ( )]
−−=ℏ
−⌢−⌢=ℏ ⌢
⌢⌢ ⌢
⌢
⌢⌢⌢
⌢⌢⌢
ζL wrζ w r p Ν wrζ
ζL θrζ θr p Ν wrζ θrζ
1; ;,
1; ;,;.
www
θθθ
0
0
(41)
While BCs are:
()∣ ()
()∣ ()∣
() ()
()
==
⌢=⌢
==
⌢=
⌢⌢
⌢⌢
==
==
==
=
wrζ wrζ
dr
θrζ θ rζ
wrζ
rwrζ
r
θrζ
r
;1,
d; 0,
;;
d;
d0, d;
d0,
d;
d0.
rr
rr
rr
r
0
2
20
00 0
1
3
31
1
(42)
Here ζis the embedding parameter ζ∈[0, 1], to reg-
ulate for the solution convergence of
ℏ
ℏ
⌢
⌢
,
wθ
is used.
When ζ=0 and ζ=1 we have:
() ()=
⌢⌢
w
rwr;1
,
(43)
() ()
⌢
=⌢
θr θr;1
.
(44)
Expand the
()(
)
⌢
⌢
w
rζ θ rζ;, ;
through Taylor’s series for
ζ=0( ) () ()
( ) () ()
∑
∑
=+
⌢=⌢+⌢
⌢⌢⌢
=
∞
=
∞
wrζ w r w rζ
θrζ θ r θ rζ
;
;,
nnn
nnn
01
01
(45)
() () () ()
=!∂∂⌢=!∂⌢∂
⌢⌢
==
w
rnwrζ
rθr nθrζ
r
1;,1;.
npn
p
00
(46)
While BCs are:
() () () ()
() () ()
=″=⌢=⌢
′= ″′=⌢′=
⌢⌢
⌢⌢
ww θθ
wδ w δ θδ
01, 00, 0 0
0, 0, 0.
0(47)
Now
() [] [ ] ⎛⎝⎞⎠[]
⎛⎝⎞⎠[]
R=+″′+−″
+−′−
⌢⌢ ⌢
⌢
−−−
−
⌢
rw rwrλw
rλ
rwλS
21
1,
n
wnnn
nt
1
iv 121
31
(48)
() [ ] [ ] [ ]R=⌢″+⌢′+″
⌢
⌢−−−
rθ rθBw
1,
n
θnnr
n
11 1
2(49)
here
⎧
⎨
⎩
=≤>
χn
n
0, if 1
1, if 1.
n(50)
6 Results and discussion
In this work, we have analyzed the thin film flow cases of
lifting (Figure 1a)and drainage (Figure 1b)of a steady,
incompressible, non-isothermal couple stress fluid flow
on the outer surface of a vertical cylinder. The problem
formulation and modeling of phenomena gave nonlinear
ordinary differential equations. Due to nonlinearity, exact
solutions of the problems seem to be difficult so an ana-
lytical technique, HAM, is used to obtain the required
solutions. The behavior of the fluid to the involved para-
meter is studied with the help of tables and graphical
representations.
6.1 Tabular description
Tables 1–4 are produced for different values of Stokes
number, λ, Br, and ηfor the case of lifting. Tables 5–8
are produced for different values of Stokes number, λ, Br,
Table 1: Effect of S
t
number on velocity profile w(r), keeping λ=0.5,
Br =0.7, η=0.8
rS
t
=0.4 S
t
=0.7 S
t
=0.9 S
t
=1.1
1 2.5359104 2.3195123 1.9303426 1.2283493
1.1 2.4875231 2.2174542 1.9158603 1.2093829
1.2 2.3919041 2.2046721 1.8660642 1.1836035
1.3 2.2570631 2.1031704 1.7318253 1.108333 6
1.4 1.8045716 2.0518034 1.6125352 1.0730631
1.5 1.9709216 1.7143519 1.5071462 0.9830613
1.6 1.7036087 1.5341518 1.4582441 0.9423721
1.7 1.5405933 1.5039821 1.2309302 0.9108622
1.8 1.4381075 1.3793046 1.1560128 0.8062925
1.9 1.3381075 1.2793046 1.0560128 0.7062925
HAM with application to thin-film flow of couple stress fluid 709

and ηfor the case of drainage. The behavior of involved
parameters on velocity profile and temperature distribu-
tion depicted in the tables is discussed thoroughly.
Table 1 shows the effects of Stokes number on the
velocity profile during lifting, it is observed that increase
in the value of Stokes number slows down the velocity
profile.
Table 2, gives the effects of ηon the velocity profile
during lifting, it is observed that increase in the values of
ηslows down the velocity profile. The values of velocity
profiles are taken in the interval 1.6 ≤w(r)≤0.
Table 3 describes the effects of λon the velocity pro-
file during lifting, it is observed that increase in the
values of ηslows down the velocity profile.
Table 4: Effect of Br on temperature distribution θ(r), where
S
t
=0.6, η=0.4, λ=0.7
rBr =0.2 Br =0.4 Br =0.6 Br =0.8
1 1.6504601 1.6435363 1.0635783 0.7835623
1.1 1.8261462 1.7257324 1.1285423 0.8238604
1.2 1.8363803 1.7381013 1.2071319 0.9021039
1.3 1.8445933 1.7636814 1.3536304 1.2839102
1.4 1.8937181 1.8734032 1.4157031 1.4040637
1.5 1.9908316 1.8917317 1.5317317 1.5209833
1.6 2.1785033 2.0156204 1.6156234 1.6241804
1.7 2.2390337 2.1039136 1.6534138 1.7053714
1.8 2.3797361 2.2685769 1.7673263 1.7235239
1.9 2.4903695 2.3790603 1.7732613 1.7713026
Table 5: Effect of S
t
on velocity profile w(r), keeping λ=0.5, Br =0.7,
η=0.8
rS
t
=0.4 S
t
=0.7 S
t
=0.9 S
t
=1.1
1 0.4145093 1.6390924 2.4263639 2.6213414
1.1 0.5172042 1.7193835 2.5184721 2.721573 6
1.2 0.6523218 1.7475802 2.6069014 2.8145302
1.3
0
.7113425
1.7670335 2.919463 7 3.0043533
1.4 0.8073451 1.7783413 2.9531423 3.1413732
1.5 0.8561582 1.8211802 2.9671814 3.3726331
1.6 0.9053348 1.8454321 2.9743103 3.4690103
1.7 1.9113507 1.9127422 2.9804937 3.5328372
1.8 1.9305286 2.1433426 2.9910495 3.5683403
1.9 1.97254703 2.2124862 2.9984973 3.573 8937
Table 2: Effect of ηon velocity profile w(r), keeping λ=0.5, Br =0.7,
S
t
=0.4
rη=0.4 η=0.5 η=0.8 η=0.9
1 2.8548671 1.9494123 1.4383301 0.9572148
1.1 2.7187092 1.9272341 1.3365036 0.9237491
1.2 2.6710937 1.8191301 1.3030933 0.8335634
1.3 2.6010789 1.7519503 1.1103403 0.8112309
1.4 2.4510816 1.7312308 1.0387041 0.7038902
1.5 2.4170813 1.6403341 1.1933503 0.6201835
1.6 2.3497514 1.5135418 1.0380607 0.5325718
1.7 2.2083877 1.4186503 1.0097323 0.4578215
1.8 2.1541146 1.2049317 1.6730236 0.3618403
1.9 2.0686101 1.3857039 1.1545227 0.2179192
Table 3: Effect of λon velocity profile w(r), keeping S
t
=0.6, η=0.4,
Br =0.7
rλ=0.3 λ=0.5 λ=0.7 λ=0.9
1 1.974703 7 1.6864217 0.9674132 0.4516702
1.1 1.9473901 1.6253188 0.8524841 0.4134976
1.2 1.9230121 1.6048171 0.7345154 0.4081541
1.3 1.8834926 1.4756839 0.7044018 0.3810132
1.4 1.8716702 1.4537014 0.6547839 0.3711843
1.5 1.7520807 1.3924819 0.6291736 0.2781001
1.6 1.7201745 1.3453906 0.6171425 0.2501423
1.7 1.6536785 1.2601767 0.4186143 0.2012461
1.8 1.6453206 1.1843036 0.4451758 0.1965806
1.9 1.5274021 1.0158531 0.1646151 0.0410333
Table 6: Effect of ηon velocity profile w(r), keeping λ=0.5, Br =0.7,
S
t
=0.4
rη=0.4 η=0.5 η=0.8 η=0.9
1 0.8753037 0.5383921 0.0893624 0.0875105
1.1 0.8524853 0.5172315 0.0864305 0.0854136
1.2 0.8433541 0.5063602 0.0852642 0.0820248
1.3 0.8125602 0.4146318 0.0713532 0.0704109
1.4 0.8023367 0.4053431 0.0702823 0.0602573
1.5 0.6940716 0.5751134 0.3912141 0.0560248
1.6 0.5300337 0.3123405 0.2235173 0.0425613
1.7 0.5186526 0.2175421 0.1135137 0.0396518
1.8 0.4780402 0.3726403 0.1160152 0.0230457
1.9 0.4313723 0.2004273 0.1113046 0.0186237
Table 7: Effect of λon velocity profile w(r), where S
t
=0.6, η=0.4,
Br =0.7
rλ=0.3 λ=0.5 λ=0.7 λ=0.9
1 3.423840 5 2.3973714
1
.8323324 0.9365204
1.1 3.3287324 2.1369601 1.7364732 0.8165125
1.2 3.1130282 2.0101581 1.7101322 0.7194276
1.3 3.0561301 1.5893402 1.6350164 0.6541441
1.4 2.6743152 1.5936538 1.6014932 0.4672452
1.5 2.6424101 1.6450571 1.4908379 0.4210752
1.6 2.5697234 1.6673909 1.4023246 0.3617614
1.7 2.5316782 1.6803621 1.3136239 0.3401589
1.8 2.4743341 1.7680219 1.1163218 0.3203567
1.9 2.3563132 1.8197368 1.0430135 0.2103675
710 Muhammad Farooq et al.

Table 4 indicates the heat transfer during lifting of
the fluid, it is noted that heat transfer is higher for high
values of Brinkman number Br.
Table 5 is carried out for various values of S
t
number
and it is found that drainage of the fluid can be increased
by increaing the Stokes number.
Table 6 depicts that drainage of the fluid slows down
for higher values of η.
Figure 2: Influence of S
t
on velocity, for lifting problem.
Table 8: Effect of Br on temperature distribution θ(r), where
S
t
=0.6, η=0.4, λ=0.7
rBr=0.2 Br =0.4 Br =0.6 Br =0.8
1 3.7534307 2.6813513 1.8124572 1.7635781
1.1 3.7733108 2.6921257 1.8306481 1.7608344
1.2 3.8303437 2.7831408 1.8592126 1.7832408
1.3 3.8624387 2.7851633 1.9175208 1.8574312
1.4 3.9521792 2.7884974 1.9745186 1.8739174
1.5 3.9721452 2.8361307 2.6317534 1.9616437
1.6 3.9853713 2.8518008 2.6780063 1.9713636
1.7 3.9873402 2.8823537 2.7041501 1.9815156
1.8 3.9976241 2.9512359 2.8007186 1.9916342
1.9 3.9986452 2.9967435 2.9014631 1.9981624
Figure 3: Influence of S
t
on velocity, for drainage problem.
Figure 4: Impact of λon velocity, for lifting problem.
Figure 5: Impact of λon velocity, for drainage problem.
Figure 6: Impact of ηon velocity, for lifting problem.
HAM with application to thin-film flow of couple stress fluid 711

Table 7 shows that drainage of the fluid slows down
for higher values of parameter λ.
Table 8 describesthatheattransferprocedureincreases
during drainage of the fluid with the increase in the values
of Brinkman number.
6.2 Graphical description
Figures 2–9 are sketched for different values of Stokes
number S
t
, vorticity parameter λ,couple stress parameter
η, and Brinkman number Br, considering both the cases
of lifting and drainage of the fluid to note the effects of
these parameters on velocity profile and temperature dis-
tribution. Figures 2 and 3 show that for increase in the
values of Stokes number S
t
, velocity profile slows down
for lifting case and enhances for drainage case. Figures 4
and 5 are plotted for various values of vorticity parameter
λ. Both the figures depict fluid share thickening behavior
as velocity profile decreases for the increase in the values
of λfor both lifting and drainage cases. Figures 6 and 7
are sketched for both lifting and drainage cases for dif-
ferent values of couple stress parameter. It is observed
that like stokes number and vorticity parameter, couple
stress parameter also slows down the fluid flow. Figures 8
and 9 are drawn to check the effect of Brinkman number
on the temperature distribution in both lifting and drai-
nage cases. It is observed that temperature distribution
increases for both lifting and drainage cases, for higher
values of Brinkman number.
7 Conclusion
The current study discusses the problem of thin film with-
drawal and drainage flow of a steady incompressible,
non-isothermal couple stress fluid on the outer surface
of a vertical cylinder, which is modeled using the non-
linear ordinary differential equations and solved with the
help of HAM, to calculate the expressions for velocity
profile and temperature distribution.
The findings are as below:
•The increase in the value of vorticity parameter λslows
down the velocity field for both lifting and drainage
cases.
•The increase in the values of Stokes number S
t
,decreases
the velocity profiles in case of lifting, while increases in
case of drainage.
•The increasing values of couple stress parameter η,
decrease velocity profiles for both lifting and drainage
cases.
•The increasing values of Brinkman number Br increases
the temperature distribution for both lifting and drai-
nage of the fluid.
It is observed that the involved parameters have a
vital role in the flow and heat transfer of the fluid.
Figure 7: Influence of ηon velocity, for drainage problem.
Figure 8: Effect of Br on velocity, for lifting problem.
Figure 9: Impact of Br on velocity, for drainage problem.
712 Muhammad Farooq et al.

Acknowledgment: Chutarat Tearnbucha would like to
acknowledge financial support by Navamindradhiraj
University through the Navamindradhiraj University
Research Fund (NURF).
Funding information: The authors state no funding
involved.
Author contributions: All authors have accepted respon-
sibility for the entire content of this manuscript and
approved its submission.
Conflict of interest: The authors state no conflict of
interest.
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