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

Muhammad Farooq; Zia Ullah; Muhammad Zeb; Hijaz Ahmad; Muhammad Ayaz; Muhammad Sulaiman; Chutarat Tearnbucha; Weerawat Sudsutad

doi:10.1515/phys-2022-0056

Open Access Published by De Gruyter Open Access July 21, 2022

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

Muhammad Farooq , Zia Ullah , Muhammad Zeb , Hijaz Ahmad , Muhammad Ayaz , Muhammad Sulaiman , Chutarat Tearnbucha and Weerawat Sudsutad

From the journal Open Physics

https://doi.org/10.1515/phys-2022-0056

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 S_t , vorticity parameter λ, couple stress parameter η, and Brinkman number Br presented in the graphical description as well.

Keywords: thin film flow; withdrawal; drainage; couple stress fluid; vertical cylinder; differential equations; homotopy analysis method

1 Introduction

Non-Newtonian fluids have a vigorous place in the present 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 equation 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 parallel 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 [6,7,8].

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 process 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]. Moreover, Nayak et al. [13] have presented heat and mass transfer phenomenon in the boundary layer flow of electrically 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 permeable boundary upgrade the stream nearby the plate. In ref. [14], Gowda et al. have presented heat and mass transfer briefly in ferromagnetic fluid on the surface of a stretchable 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 convection 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 literature review shows that approximate analytical techniques [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40] 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 analyzed 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 thickness δ of 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 R of the cylinder as shown in Figure 1(a). Assuming that the flow is steady and has no change with respect to θ the velocity and temperature fields are:

(1) V = [ 0 , 0 , w ( r ) ] , θ = θ ( r ) .

Figure 1

(a) Geometry of the lifting problem. (b) Geometry of the drainage.

The following are the basic equations that regulate the flow of an incompressible non-isothermal couple stress fluid:

(2) ∇ ⋅ V = 0 ,

(3) ρ d V d t = ρ f − ∇ p + ∇ ⋅ T − η ∇ 4 V ,

(4) ρ C p d θ d t = k ∇ 2 θ + trc ( T ⋅ L ) ,

where V represents the velocity vector and d d t is material time derivative defined as

(5) d d t ( ⋅ ) = ∂ ∂ t + V ⋅ ∇ ( ⋅ ) ,

k is the thermal conductivity, ρ is the constant density, C _p is the specific heat constant, f is the body force, θ is the temperature, p is the dynamic pressure, and T is the stress tensor taken as:

(6) T = μ A 1 ,

where μ is the viscosity of the fluid, and A ₁ is the first Rivlin–Erickson tensor.

(7) A 1 = L + L T , and L = ∇ V .

The velocity field (1), balances Eq. (2), and reduces Eq. (3), in the following components:

r-component:

(8) ∂ p ∂ r = 0 .

θ-component:

(9) 1 r ∂ p ∂ θ = 0 ,

z-component:

(10) ∂ p ∂ z = − ρ g + μ 1 r d d r r d w d r − η 1 r d d r r d d r 1 r d d r r d w d r .

Using Eqs. (8) and (9), Eq. (10) becomes

(11) d p d z = − ρ g + μ 1 r d d r r d w d r − η 1 r d d r r d d r 1 r d d r r d w d r .

Consider that the pressure is atmospheric, then d p d z = 0 , and so Eq. (11) gets the form:

(12) d 4 w d r 4 + 2 r d 3 w d r 3 + 1 r 2 − μ η d 2 w d r 2 + 1 r 3 − μ η r d w d r = − ρ g η .

The velocity and temperature fields simplify Eq. (4) as:

(13) κ d 2 θ d r 2 + 1 r d θ d r + μ d w d r 2 = 0 .

For the fourth order (12) and second order (13) nonlinear differential equations, the following boundary conditions can be set from the geometry as given:

(14) w = U , d 2 w d r 2 = 0 , θ = θ 0 at r = R , d w d r = 0 , d 3 w d r 3 = 0 , d θ d r = 0 at r = R + δ .

Using the non-dimensional parameters

(15) w ∗ = w U , r ∗ = r δ , λ = μ δ 2 η , S t = ρ g δ 2 μ U , Br = μ eff U 2 ( θ 1 − θ 0 ) k , R ∗ = R δ .

After dropping asterisks Eqs. (12–14) become

(16) d 4 w d r 4 + 2 r d 3 w d r 3 + 1 r 2 − λ d 2 w d r 2 + 1 r 3 − λ r d w d r = − λ S t ,

(17) d 2 θ d r 2 + 1 r d θ d r + μ U 2 κ ( θ 1 − θ 0 ) d w d r 2 = 0 .

Using the previously defined dimensionless quantities, Eq. (17) becomes

(18) d 2 θ d r 2 + 1 r d θ d r + Br d w d r 2 = 0 .

The dimensionless form of the boundary conditions given in (14) take the form:

(19) w = 1 , d 2 w d r 2 = 0 , θ = θ 0 at r = R , d w d r = 0 , d 3 w d r 3 = 0 , d θ d r = 0 at r = R + 1 .

3 Solution of lifting problem using HAM

The approximate analytical technique, HAM, is used to solve Eqs. (16) and (18), together with Eq. (19). Fundamental roots of the model equations via HAM are given below in detail.

(20) L w ⌢ ( w ⌢ ) = w i v , L θ ⌢ ( θ ) = θ ″ ,

linear operators L w ⌢ are signified as

(21) L w ⌢ ( ( e 2 R − e 2 δ ) ( e 2 δ e − r + e r ) ) = 0 , L θ ⌢ ( e 2 R + e 2 δ e r ) = 0 .

The consistent non-linear operators are reasonably selected as N w ⌢ and N θ ⌢ , and recognized in system as:

(22) N w ⌢ [ w ⌢ ( r ; ζ ) ] = [ w ⌢ ] r r r r + 2 r [ w ⌢ ] r r r + 1 r 2 − λ [ w ⌢ ] r r + 1 r 3 − λ r [ w ⌢ ] r + λ S t ,

(23) N θ ⌢ [ w ⌢ ( r ; ζ ) , θ ⌢ ( r ; ζ ) ] = [ θ ⌢ ] r r + 1 r [ θ ⌢ ] r + Br [ w ⌢ r ] 2 .

For Eqs. (20) and (21), the 0th-order system can be written as:

(24) ( 1 − ζ ) L w ⌢ [ w ⌢ ( r ; ζ ) − w ⌢ 0 ( r ) ] = p ℏ w ⌢ N w ⌢ [ w ⌢ ( r ; ζ ) ] ,

(25) ( 1 − ζ ) L θ ⌢ [ θ ⌢ ( r ; ζ ) − θ ⌢ 0 ( r ) ] = p ℏ θ ⌢ N θ ⌢ [ w ⌢ ( r ; ζ ) , θ ⌢ ( r ; ζ ) ] .

While BCs are:

(26) w ⌢ ( r ; ζ ) ∣ r = 0 = 1 , d 2 w ⌢ ( r ; ζ ) d r 2 r = 0 = 0 , θ ⌢ ( r ; ζ ) ∣ r = 0 = θ ⌢ 0 ( r ; ζ ) ∣ r = 0 d w ⌢ ( r ; ζ ) d r r = 1 = 0 , d 3 w ⌢ ( r ; ζ ) d r 3 r = 1 = 0 , d θ ⌢ ( r ; ζ ) d r r = 1 = 0 .

Here ζ is the embedding parameter ζ ∈ [0, 1], to regulate for the solution convergence of ℏ w ⌢ , ℏ θ ⌢ is used. When ζ = 0 and ζ = 1 we have:

(27) w ⌢ ( r ; 1 ) = w ⌢ ( r ) ,

(28) θ ⌢ ( r ; 1 ) = θ ⌢ ( r ) ,

Expand the w ⌢ ( r ; ζ ) , θ ⌢ ( r ; ζ ) through Taylor’s series for ζ = 0

(29) w ⌢ ( r ; ζ ) = w ⌢ 0 ( r ) + ∑ n = 1 ∞ w ⌢ n ( r ) ζ n , θ ⌢ ( r ; ζ ) = θ ⌢ 0 ( r ) + ∑ n = 1 ∞ θ ⌢ n ( r ) ζ n .

(30) w ⌢ n ( r ) = 1 n ! ∂ w ⌢ ( r ; ζ ) ∂ r p = 0 , θ ⌢ n ( r ) = 1 n ! ∂ θ ⌢ ( r ; ζ ) ∂ r p = 0 .

While BCs are:

(31) w ⌢ ( 0 ) = 1 , w ⌢ ″ ( 0 ) = 0 , θ ⌢ ( 0 ) = θ ⌢ 0 ( 0 ) w ⌢ ′ ( δ ) = 0 , w ⌢ ″ ′ ( δ ) = 0 , θ ⌢ ′ ( δ ) = 0 .

Now

(32) ℜ n w ⌢ ( r ) = [ w ⌢ ] n − 1 iv + 2 r [ w ⌢ ′ ″ ] n − 1 + 1 r 2 − λ [ w ⌢ ″ ] n − 1 + 1 r 3 − λ r [ w ⌢ ′ ] n − 1 + λ S t ,

(33) ℜ n θ ⌢ ( r ) = [ θ ⌢ ″ ] n − 1 + 1 r [ θ ⌢ ′ ] n − 1 + Br [ w ⌢ ″ ] n − 1 2 ,

and χ n = 0 , if n ≤ 1, 1 , if n > 1 .

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 δ of 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 R of the cylinder as shown in Figure 1(b). Assuming that the flow is steady and has no change with respect to θ, the velocity field and temperature distribution are:

(34) V = [ 0 , 0 , w ( r ) ] , θ = θ ( r ) .

For the drainage problem, we take the dimensionless form of the governing equations after using dimensionless parameters given in Eq. (15) as under:

(35) d 4 w d r 4 + 2 r d 3 w d r 3 + 1 r 2 − λ d 2 w d r 2 + 1 r 3 − λ r d w d r = λ S t ,

(36) d 2 θ d r 2 + 1 r d θ d r + Br d w d r 2 = 0 ,

the boundary conditions are:

(37) w = 0 , d 2 w d r 2 = 0 , θ = θ 0 at r = R , d w d r = 0 , d 3 w d r 3 = 0 , d θ d r = 0 at r = R + 1 .

5 Solution of drainage problem using HAM

The approximate analytical technique, HAM, is used to solve Eqs. (35) and (36), together with Eq. (37). Fundamental roots of the model equations via HAM are given below in detail:

(38) L w ⌢ ( w ⌢ ) = w iv , L θ ⌢ ( θ ) = θ ″ .

Linear operators L w ⌢ are signified as

(39) L w ⌢ ( ( e 2 R − e 2 δ ) ( e 2 δ e − r + e r ) ) = 0 , L θ ⌢ ( e 2 R + e 2 δ e r ) = 0 .

The consistent nonlinear operators are reasonably selected as N w ⌢ and N θ ⌢ , and recognize in system as:

(40) N w ⌢ [ w ⌢ ( r ; ζ ) ] = [ w ⌢ ] r r r r + 2 r [ w ⌢ ] r r r + 1 r 2 − λ [ w ⌢ ] r r + 1 r 3 − λ r [ w ⌢ ] r − λ S t ,

N θ ⌢ [ w ⌢ ( r ; ζ ) , θ ⌢ ( r ; ζ ) ] = [ θ ⌢ ] r r + 1 r [ θ ⌢ ] r + Br [ w ⌢ r ] 2 .

For Eq. (39), the 0th-order system is written as

(41) ( 1 − ζ ) L w ⌢ [ w ⌢ ( r ; ζ ) − w ⌢ 0 ( r ) ] = p ℏ w ⌢ N w ⌢ [ w ⌢ ( r ; ζ ) ] , ( 1 − ζ ) L θ ⌢ [ θ ⌢ ( r ; ζ ) − θ ⌢ 0 ( r ) ] = p ℏ θ ⌢ N θ ⌢ [ w ⌢ ( r ; ζ ) , θ ⌢ ( r ; ζ ) ] .

While BCs are:

(42) w ⌢ ( r ; ζ ) ∣ r = 0 = 1 , d 2 w ⌢ ( r ; ζ ) d r 2 r = 0 = 0 , θ ⌢ ( r ; ζ ) ∣ r = 0 = θ ⌢ 0 ( r ; ζ ) ∣ r = 0 d w ⌢ ( r ; ζ ) d r r = 1 = 0 , d 3 w ⌢ ( r ; ζ ) d r 3 r = 1 = 0 , d θ ⌢ ( r ; ζ ) d r r = 1 = 0 .

Here ζ is the embedding parameter ζ ∈ [0, 1], to regulate for the solution convergence of ℏ w ⌢ , ℏ θ ⌢ is used. When ζ = 0 and ζ = 1 we have:

(43) w ⌢ ( r ; 1 ) = w ⌢ ( r ) ,

(44) θ ⌢ ( r ; 1 ) = θ ⌢ ( r ) .

Expand the w ⌢ ( r ; ζ ) , θ ⌢ ( r ; ζ ) through Taylor’s series for ζ = 0

(45) w ⌢ ( r ; ζ ) = w ⌢ 0 ( r ) + ∑ n = 1 ∞ w ⌢ n ( r ) ζ n θ ⌢ ( r ; ζ ) = θ ⌢ 0 ( r ) + ∑ n = 1 ∞ θ ⌢ n ( r ) ζ n ,

(46) w ⌢ n ( r ) = 1 n ! ∂ w ⌢ ( r ; ζ ) ∂ r p = 0 , θ ⌢ n ( r ) = 1 n ! ∂ θ ⌢ ( r ; ζ ) ∂ r p = 0 .

While BCs are:

(47) w ⌢ ( 0 ) = 1 , w ⌢ ″ ( 0 ) = 0 , θ ⌢ ( 0 ) = θ ⌢ 0 ( 0 ) w ⌢ ′ ( δ ) = 0 , w ⌢ ″ ′ ( δ ) = 0 , θ ⌢ ′ ( δ ) = 0 .

Now

(48) ℜ n w ⌢ ( r ) = [ w ⌢ ] n − 1 iv + 2 r [ w ⌢ ″ ′ ] n − 1 + 1 r 2 − λ [ w ⌢ ″ ] n − 1 + 1 r 3 − λ r [ w ⌢ ′ ] n − 1 − λ S t ,

(49) ℜ n θ ⌢ ( r ) = [ θ ⌢ ″ ] n − 1 + 1 r [ θ ⌢ ′ ] n − 1 + B r [ w ⌢ ″ ] n − 1 2 ,

here

(50) χ n = 0 , if n ≤ 1 1 , if n > 1 .

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 analytical technique, HAM, is used to obtain the required solutions. The behavior of the fluid to the involved parameter 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 η for the case of lifting. Tables 5–8 are produced for different values of Stokes number, λ, Br, and η for the case of drainage. The behavior of involved parameters on velocity profile and temperature distribution depicted in the tables is discussed thoroughly.

Table 1

Effect of S_t number on velocity profile w(r), keeping λ = 0.5, Br = 0.7, η = 0.8

r	S_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

Table 2

Effect of η on velocity profile w(r), keeping λ = 0.5, Br = 0.7, S_t = 0.4

r	η = 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 λ on velocity profile w(r), keeping S_t = 0.6, η = 0.4, Br = 0.7

r	λ = 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 4

Effect of Br on temperature distribution θ(r), where S_t = 0.6, η = 0.4, λ = 0.7

r	Br = 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

r	S_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 6

Effect of η on velocity profile w(r), keeping λ = 0.5, Br = 0.7, S_t = 0.4

r	η = 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 λ on velocity profile w(r), where S_t = 0.6, η = 0.4, Br = 0.7

r	λ = 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

Table 8

Effect of Br on temperature distribution θ(r), where S_t = 0.6, η = 0.4, λ = 0.7

r	Br = 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

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 η on the velocity profile during lifting, it is observed that increase in the values of η slows 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 λ on the velocity profile during lifting, it is observed that increase in the values of η slows down the velocity profile.

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.

Table 7 shows that drainage of the fluid slows down for higher values of parameter λ.

Table 8 describes that heat transfer procedure increases 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 distribution. 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 λ for both lifting and drainage cases. Figures 6 and 7 are sketched for both lifting and drainage cases for different 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 drainage cases. It is observed that temperature distribution increases for both lifting and drainage cases, for higher values of Brinkman number.

Figure 2

Influence of S_t on velocity, for lifting problem.

Figure 3

Influence of S_t on velocity, for drainage problem.

Figure 4

Impact of λ on velocity, for lifting problem.

Figure 5

Impact of λ on velocity, for drainage problem.

Figure 6

Impact of η on velocity, for lifting problem.

Figure 7

Influence of η on velocity, for drainage problem.

Figure 8

Effect of Br on velocity, for lifting problem.

Figure 9

Impact of Br on velocity, for drainage problem.

7 Conclusion

The current study discusses the problem of thin film withdrawal 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 nonlinear 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 λ slows 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 drainage of the fluid.

It is observed that the involved parameters have a vital role in the flow and heat transfer of the fluid.

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 responsibility 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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Received: 2021-11-07

Revised: 2022-01-29

Accepted: 2022-03-20

Published Online: 2022-07-21

This work is licensed under the Creative Commons Attribution 4.0 International License.

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

Abstract

1 Introduction

2 Lifting problem

3 Solution of lifting problem using HAM

4 Drainage problem

5 Solution of drainage problem using HAM

6 Results and discussion

6.1 Tabular description

6.2 Graphical description

7 Conclusion

Acknowledgment

References

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