# An Application of Homotopy Perturbation Method to Fractional-Order Thin Film Flow of the Johnson–Segalman Fluid Model

Authors: Mubashir Qayyum; Farnaz Ismail; Syed Inayat Ali Shah; Muhammad Sohail; Essam R. El-Zahar; K. C Gokul
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Thin film flow is an important theme in fluid mechanics and has many industrial applications. These flows can be observed in oil refinement process, laser cutting, and nuclear reactors. In this theoretical study, we explore thin film flow of non-Newtonian Johnson–Segalman fluid on a vertical belt in fractional space in lifting and drainage scenarios. Modelled fractional-order boundary value problems are solved numerically using the homotopy perturbation method along with Caputo definition of fractional derivative. In this study, instantaneous and average velocities and volumetric flux are computed in lifting and drainage cases. Validity and convergence of homotopy-based solutions are confirmed by finding residual errors in each case. Moreover, the consequences of different fractional and fluid parameters are graphically studied on the velocity profile. Analysis shows that fractional parameters have opposite effects of the fluid velocity.


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## Body

## 1. Introduction

Modelling and analysis of non-Newtonian fluids is an important and active research theme in industrial engineering. Food processing, paper production, blood flow analysis, and mud drilling are different applications areas of non-Newtonian fluids. These fluids are defined through a nonlinear relationship between rate of deformation and stress tensors, and therefore, it has several models in different scenarios. Johnson–Segalman fluid is one of the very significant fluid modals which have numerous engineering and industrial applications.Thin film flow can be detected in different natural situations, for example, movement of raindrop on window glass, tears in the eyes, and lava flow. Industrial applications of such flows include oil refining, nuclear reactors, and laser cutting[1–4]. The initial work on thin film flow was carried out in [3] for Newtonian fluids, but this study has limitations and cannot be generalized for non-Newtonian fluids such as melted plastics, gels, pastes, honey, ketchup, and blood [5]. Siddiqui et al. examined thin film of different fluids including PTT (Phan-Thien and Tanner) and third and fourth grade fluids in [6–8]. Landau [9] and Stuart [10] extended these analyses to turbulence. Ullah et al. studied the film flow under slip conditions in generalized Maxwell fluids in [11]. Ruan et al. studied thin film from a distributed source on a vertical wall in [12]. Ahmad and Xu proposed an improved nanofluid model in thin film under the action of gravity in [13].Fractional calculus has gained significant attention due to its striking application as a new modelling tool in a variety of ﬁelds, such as fluid dynamics, hydrology, system control theory, signal processing, physics, biology, and finance [14–20]. These fractional models are more appropriate for describing memory and transmissible properties of different materials than integral-order models. In the past few decades, various numerical techniques have been developed by different researchers for nonlinear BVPs. Wazwaz proposed the modified decomposition method (MDM) for BVPs in [21]. Noor and Mohyud-Din used VIM along with He’s polynomials for the solution of higher order BVPs [22]. Liu et al. used the multiscale method for nanoparticle diffusion in sheared cellular blood flow in [23]. Jafari and Gejji used adomian decomposition for the solution of system of FDEs in [24]. Hashim et al. apply OHAM to fractional-order fuzzy differential equations in [22]. Rysak and Gregorczyk proposed DTM for fractional dynamical systems in [25]. Zada et al. apply NIM to fractional PDEs in [26]. Yaghouti used radial basis functions to different families of fractional differential equations in [27]. Al-Kuze et al. used spectral quasi-linearization and irreversibility analysis of magnetized cross-fluid flow [28].In this study, we extend the theoretical study of thin film flow to fractional space in case of non-Newtonian Johnson–Segalman fluid since exact solutions of highly nonlinear fractional differential equations (FDEs) are not possible. The conventional approach is to use perturbation techniques in such scenarios, which again need small or large parameters. To avoid this, we utilize the well-known homotopy perturbation method (HPM) [29–33] along with fractional calculus for solution purpose. This method is effectively used by many scholars in [29–33]. Validation and convergence of the obtained numerical solutions are confirmed by means of finding residual errors. To the best of the authors’ knowledge, the given problem has not been attempted before in fractional space.


## 2. Preliminaries

### 2.1. Fractional Calculus

Few basic definitions of fractional calculus are given below.Definition 1.
Letft,t>0, be a real function in the space Cµ,µ∈R, if ∃ a real number p>µ∍ft=tpf1t,wheref1t∈C0,∞, and it is in the space(1)Cµm⇔fm∈Cµ,m∈N.Definition 2.
The Caputo fractional derivativeDα is defined as(2)Dαft=1Γm−α∫0tt−τm−α−1fmdτ,form−1<α<m,m∈N,t>0.Definition 3.
α≥0 order Riemann–Liouville fractional integration is defined as(3)Jαft=1Γα∫0tt−τα−1fτdτ,k−1<α<k,andk∈N.


### 2.2. Basic Equations

The basic equations of incompressible fluid of Johnson–Segalman fluid are(4)divV=0,(5)ρDVDt=divσ+ρf,where V is the velocity vector, ρ is the constant density, f is the body force per unit mass, and σ is the Cauchy stress tensor, where(6)σ=−pI+T,(7)T=S+2μD,(8)S+mW−aDTS+SW−aD+DSDt=2Dη.


## 2.1. Fractional Calculus

Few basic definitions of fractional calculus are given below.Definition 1.
Letft,t>0, be a real function in the space Cµ,µ∈R, if ∃ a real number p>µ∍ft=tpf1t,wheref1t∈C0,∞, and it is in the space(1)Cµm⇔fm∈Cµ,m∈N.Definition 2.
The Caputo fractional derivativeDα is defined as(2)Dαft=1Γm−α∫0tt−τm−α−1fmdτ,form−1<α<m,m∈N,t>0.Definition 3.
α≥0 order Riemann–Liouville fractional integration is defined as(3)Jαft=1Γα∫0tt−τα−1fτdτ,k−1<α<k,andk∈N.


## 2.2. Basic Equations

The basic equations of incompressible fluid of Johnson–Segalman fluid are(4)divV=0,(5)ρDVDt=divσ+ρf,where V is the velocity vector, ρ is the constant density, f is the body force per unit mass, and σ is the Cauchy stress tensor, where(6)σ=−pI+T,(7)T=S+2μD,(8)S+mW−aDTS+SW−aD+DSDt=2Dη.


## 3. Mathematical Formulation in Lifting Case

The belt is passing through the container filled with Johnson–Segalman fluid, and it is moving vertically with constant speedU0. A uniform thickness δ of the thin film is taken up by the belt, but due to gravity fluid, it is draining down. Let pressure be assumed to be atmospheric, and the flow is uniform, steady, and laminar. Also, the x-axis is considered to be normal, while the y-axis is along the belt.The boundary conditions are(9)x=0,v=U0,x=δ,Txy=0.Velocity is of the form(10)V=0,vx,0,(11)T=TxandTxz,where the extra and shear stress tensor is T=Tx and Txz, respectively.Substitution of (8) in (3) and (4) shows that (3) is satisfied, while (4) has the form(12)0=dTxxdx+ρf1,(13)0=dTxydx+ρf2,where f1 and f2 are the components of body force.Since gravitational force is in downward whiley-axis is in upward directions, hence, (12) and (13) become(14)0=dTxxdx,0=dTxydx−ρg.By using equations (8), (10), and (11) in (7), components of S are(15)Sxx=−ηm1−adv/dx21+m21−a2dv/dx2,(16)Sxy=ηdv/dx1+m21−a2dv/dx2=Syx,(17)Syy=ηm1+adv/dx21+m21−a2dv/dx2.Using (15)–(17), stress tensor T in (6) is(18)Txx=−ηm1−adv/dx21+m21−a2dv/dx2,(19)Txy=μdvdx+ηdv/dx1+m21−a2dv/dx2=Tyx,(20)Tyy=ηm1+adv/dx21+m21−a2dv/dx2.SubstitutingTxy in (14) gives(21)ddxμdvdx+ηdv/dx1+m21−a2dv/dx2=ρg.Boundary conditions become(22)dvdx=0atx=δ,(23)v=U0atx=0.Introduce nondimensional parameters as(24)x∗=xδ,v∗=vU0andφ=μμ+η,where φ is the ratios of viscosities. Equations (21)–(23) become(25)ddxφdvdx+1−φdv/dx1+We21−a2dv/dx2=St.(26)Atx=1,dvdx=0.(27)Atx=0,v=1,where St=ρgδ2/μeffU0 and We=mU0/δ are Stokes and Weissenberg numbers, respectively, and μeff=μ+η.Simplification of equation (25) gives(28)d2vdx2+We41−a22φd2vdx2−Stdvdx4−We21−a2d2vdx2+φd2vdx2+2Stdvdx2−St=0,with(29)dvdx=0atx=1,v=1atx=0.Now, using definition of fractional calculus given in Section2.1, the fractional form of equation (28) is(30)d2vxdx2+We41−a22φd2vxdx2−StDαvx4−We21−a2d2vxdx2+φd2vxdx2+2StDαvx2−St=0,with(31)v0=1,v′1=0,0<α<1.


## 4. Homotopy Solution of Johnson–Segalman Fluid in the Lifting Case

HomotopyΩ×0,1⟶ℜ for (30) is(32)1−pd2vxdx2+pd2vxdx2+We41−a22φd2vxdx2−StDαvx4−We21−a2d2vxdx2+φd2vxdx2+2StDαvx2−St=0.Using (30) and (31), we obtain the following.Zeroth-order problem:(33)v0″x=0,v00=1,v0′1=0.First-order problem:(34)−St−2StWe2Dαv0x2+2a2StWe2Dαv0x2−StWe4Dαv0x4+2a2StWe4Dαv0x4−a4StWe4Dαv0x4−We2Dαv0x2v0″x+a2WeDαv0x2v0″x−We2ϕDαv0x2v0″x+a2We2ϕDαv0x2v0″x+We4ϕDαv0x4v0″x−2a2We4ϕDαv0x4v0″x+a4We4ϕDαv0x4v0″x+v1″x=0,v10=0,v1′1=0.Second-order problem:(35)We2Dαv0xDαv1x+4a2StWe2Dαv0x−4StDαv1x−4StWe4Dαv0x3Dαv1x8a2StWe4Dαv0x3Dαv1x−4a4StWe4Dαv0x3Dαv1x−2We2Dαv0xDαv1xv0″x+2a2We2Dαv0xDαv1xv0″x−2We2ϕDαv0xDαv1xv0″x+2a2We2ϕDαv0xDαv1xv0″x+4We4ϕDαv0x3Dαv1xv0″x−8a2We4ϕDαv0x3Dαv1xv0″x+4a4We4ϕDαv0x3Dαv1xv0″x−We2Dαv0x2v1″x+a2We2Dαv0x2v1″x−We2ϕDαv0x2v1″x+a2We2ϕDαv0x2v1″x+We4ϕDαv0x4v1″x−2a2We4ϕDαv0x4v1″x+a4We4ϕDαv0x4v1″x+v2″x=0,v20=0,v2′1=0.Similarly, we can find higher order problems and their solutions.Third-order approximate solution, after applying Caputo definition and keepingα=0.99,ϕ=0.1,St=0.01, a=0.01,andWe=0.1 fixed, is(36)Vx=1+12−0.02t+0.01x2+0.0796711−8.24342×10−8x2.98+1.23219×10−7x4−8.24098×10−8x5+2.04999×10−8x6x1.98+0.0398355−9.06776×10−8x2.98+1.35541×10−7x4−9.06508×10−8x5+2.25499×10−8x6x1.98.The residual is denoted byR and is defined as(37)R=d2Vxdx2+We41−a22φd2Vxdx2−StDαVx4−We21−a2d2Vxdx2+φd2Vxdx2+2StDαVx2−St.


## 5. Flow Rate and Average Velocity in the Lifting Case

Flow rate in this case is(38)Q=1615−16α+4α2Γ4−α2−3+2α−1−7+2α3−1+a2St3We2−3+α2−16+α25+α−13+2α3+ϕ−2−3+St−5+2αΓ4−α2,and average velocity is(39)V¯=1615−16α+4α2Γ4−α2−3+2α−1−7+2α3−1+a2St3We2−3+α2−16+α25+α−13+2α3+ϕ−2−3+St−5+2αΓ4−α2.


## 6. Mathematical Formulation in the Drainage Case

Let fluid be draining down due to gravity on the infinite stationary belt. Equations (4)–(8) become(40)0=dTxydx+ρg.Using equation (19) in (14) gives the form(41)ddxμdvdx+ηdv/dx1+m21−a2dv/dx2=−ρg,with(42)dvdx=0atx=δ,v=0atx=0.Equation (40) in the dimensionless form is(43)ddx1−φdv/dx1+We21−a2dv/dx2+φdvdx=−St.(44)Atx=1,dvdx=0,Atx=0,v=0.Simplification of (43) gives(45)d2vdx2+We41−a22φd2vdx2+Stdvdx4−We21−a2d2vdx2+φd2vdx2−2Stdvdx2+St=0.Three cases as fractional boundary value problems are obtained.Case 1.
(46)d2vxdx2+We41−a22φd2vxdx2+StDαvx4−We21−a2d2vxdx2+φd2vxdx2−2StDαvx2+St=0,with(47)v0=0,v′1=0,0<α<1.Case 2.
(48)Dγvx+We41−a22φDγvx+Stdvxdx4−We21−a2Dγvx+Dγvx−2Stdvxdx2+St=0.(49)v0=0,v′1=0,1<γ<2.Case 3.
Here both first- and second-order derivatives are replaced by the noninteger order derivative as follows:(50)Dγvx+We41−a22φDγvx+StDαvx4−We21−a2Dγvx+Dγvx−2StDαvx2+St=0,with(51)v0=0,v1=0,1<γ<2,0<α<1.


## 7. Results and Discussion

In this paper, the fractional study of thin film flow of Johnson–Segalman fluid is carried out in lifting and drainage cases. The obtained boundary value problems are solved for different values of involved parameters and results mentioned in Tables1–4 for lifting while Tables 5–10 in drainage situation. Tables 1 and 5 present solutions and corresponding errors for different α. Tables 2 and 6 show solutions along with errors for ϕ. Tables 3 and 7 show case solutions and errors for various St. Table 4 demonstrates solutions and errors for various We in the lifting case.Table 1
Solution and residual error forα keeping a=0.1,We=0.01,St=0.001,andϕ=0.1 fixed in lifting.
α=0.2α=0.6α=0.99xVxRes errorVxRes errorxRes error0.10.905−8.76128 × 10−250.905−9.08606 × 10−240.905−6.59422 × 10−230.20.82−2.5981 × 10−240.82−1.45747 × 10−230.82−4.249 × 10−230.30.745−4.78151 × 10−240.745−1.74003 × 10−230.745−2.54931 × 10−230.40.68−7.11425 × 10−240.68−1.79924 × 10−230.68−1.41813 × 10−240.50.625−9.29068 × 10−240.625−1.69071 × 10−230.625−6.9968 × 10−240.60.58−1.10441 × 10−230.58−1.4743 × 10−230.58−3.0318 × 10−240.70.545−1.2179 × 10−230.545−1.20434 × 10−230.545−1.0542 × 10−240.80.52−1.25923 × 10−230.52−9.26204 × 10−240.52−2.52839 × 10−250.90.505−1.22773 × 10−230.505−6.71129 × 10−240.505−2.91563 × 10−271.0.5−1.13159 × 10−230.5−4.57323 × 10−240.51.0211 × 10−26Table 2
Solution and residual error forϕ keeping α=0.99,We=0.01,St=0.001,anda=0.1 fixed in lifting.
ϕ=0.1ϕ=0.6ϕ=0.9xVxRes errorVxRes errorVxRes error0.10.905−6.59422 × 10−230.905−9.09923 × 10−230.905−1.08443 × 10–220.20.82−4.249 × 10−230.82−5.8628 × 10−230.82−6.98245 × 10−230.30.745−2.54931 × 10−230.745−3.51797 × 10−230.745−4.19438 × 10−230.40.68−1.41813 × 10−230.68−1.95548 × 10−230.68−2.32819 × 10−230.50.625−6.9968 × 10−240.625−9.6702 × 10−240.625−1.15297 × 10−230.60.58−3.0318 × 10−240.58−4.18483 × 10−240.58−4.96745 × 10−240.70.545−1.0542 × 10−240.545−1.45024 × 10−240.545−1.7253 × 10−240.80.52−2.52839 × 10−250.52−3.53713 × 10−250.52−4.20658 × 10−250.90.505−2.91563 × 10−270.505−1.76326 × 10−260.505−2.24163 × 10−261.0.51.0211 × 10−260.55.3895 × 10−270.59.2139 × 10−27Table 3
Solution and residual error forSt keeping α=0.98,We=0.001,ϕ=0.1,anda=0.1 fixed in lifting.
St=0.001St=0.01St=0.1xVxRes errorVxRes errorVxRes error0.10.999905−7.23543 × 10−270.99905−6.29219 × 10−220.9905−6.30818 × 10−170.20.99982−3.63823 × 10−270.9982−4.18657 × 10−220.982−4.18849 × 10−170.30.999745−2.86435 × 10−270.99745−2.5807 × 10−220.9745−2.57976 × 10−170.40.99968−1.51698∗ × 10−270.9968−1.46438 × 10−220.968−1.46541 × 10−180.50.999625−8.70705 × 10−280.99625−7.53253 × 10−230.9625−7.52476 × 10−180.60.99958−4.61286 × 10−280.9958−3.40645 × 10−230.958−3.37798 × 10−180.70.999545−4.34662 × 10−280.99545−1.28478 × 10−230.9545−1.25942 × 10−180.80.99952−4.00347 × 10−290.9952−3.70343 × 10−240.952−3.64878 × 10−190.90.999505−1.66647 × 10−290.99505−1.07941 × 10−240.9505−8.14999 × 10−201.0.99959.95937 × 10−300.995−8.78476 × 10−260.95−8.71776 × 10−21Table 4
Solution and residual error forWe keeping α=0.95,St=0.001,ϕ=0.1,anda=0.1 fixed in lifting.
We=0.001We=0.01We=0.1xVxRes errorVxRes errorVxRes error0.10.905−5.48732 × 10−270.905−5.51525 × 10−230.905−5.4982 × 10−190.20.82−3.75182 × 10−270.82−3.98956 × 10−230.82−3.98946 × 10−190.30.745−2.5638 × 10−270.745−2.63606 × 10−230.745−2.63525 × 10−190.40.68−1.17028 × 10−270.68−1.60262 × 10−230.68−1.6049 × 10−190.50.625−7.01495 × 10−280.625−8.9237 × 10−240.625−8.93613 × 10−200.60.58−4.80416 × 10−280.58−4.47664 × 10−240.58−4.46647 × 10−200.70.545−3.2225 × 10−280.545−1.95569 × 10−240.545−1.9563 × 10−200.80.527.79 × 10−300.52−7.49229 × 10−250.52−7.37223 × 10−210.90.5051.3884 × 10−280.505−2.49356 × 10−250.505−2.42499 × 10−211.0.59.40717 × 10−290.5−6.13229 × 10−260.5−5.63892 × 10−22Table 5
Solution and residual error forα keeping a=0.1,We=0.01,St=0.001,andϕ=0.1 fixed in Case 1 of drainage.
α=0.3α=0.6α=0.9xVxRes errorVxRes errorVxRes error0.10.0000951.61655 × 10−240.0000959.08344 × 10−240.0000956.30751 × 10−230.20.000184.16166 × 10−240.000181.45689 × 10−230.000184.18362 × 10−230.30.0002556.96312 × 10−240.0002551.73923 × 10−230.0002552.57623 × 10−230.40.000329.57269 × 10−240.000321.79837 × 10−230.000321.46516 × 10−230.50.0003751.16444 × 10−230.0003751.6899 × 10−230.0003757.51521 × 10−240.60.000421.29537 × 10−230.000421.47362 × 10−230.000423.40636 × 10−240.70.0004551.34057 × 10−230.0004551.20384 × 10−230.0004551.25929 × 10−240.80.000481.30281 × 10−230.000489.25865 × 10−240.000483.70727 × 10−250.90.0004951.19467 × 10−230.0004956.70931 × 10−240.0004958.09182 × 10−261.0.00051.0354 × 10−230.00054.57227 × 10−240.00051.89895 × 10−26Table 6
Solution and residual error forϕ keeping α=0.98,We=0.01,St=0.001,anda=0.1 fixed for Case 1 of drainage.
ϕ=0.01ϕ=0.1ϕ=0.3xVxRes errorVxRes errorVxRes error0.10.0000955.92844 × 10−230.0000956.30751 × 10−230.0000958.18541 × 10−230.20.000183.93158 × 10−230.000184.18362 × 10−230.000185.42952 × 10−230.30.0002552.42117 × 10−230.0002552.57623 × 10−230.0002553.34316 × 10−230.40.000321.37711 × 10−230.000321.46516 × 10−230.000321.90143 × 10−230.50.0003757.0636 × 10−240.0003757.51521 × 10−240.0003759.74926 × 10−240.60.000423.20361 × 10−240.000423.40636 × 10−240.000424.41229 × 10−240.70.0004551.18469 × 10−240.0004551.25929 × 10−240.0004551.62912 × 10−240.80.000483.49409 × 10−250.000483.70727 × 10−250.000484.82038 × 10−250.90.0004957.66394 × 10−260.0004958.09182 × 10−260.0004951.05556 × 10−251.0.00051.87118 × 10−260.00051.89895 × 10−260.00052.15706 × 10−24Table 7
Solution and residual error forSt keeping α=0.99,We=0.01,ϕ=0.01,anda=0.1 fixed in Case 1 of drainage.
St=0.001St=0.01St=0.1xVxRes errorVxRes errorVxRes error0.10.0000955.82872 × 10−230.0000955.82511 × 10−180.0000955.8252 × 10–−130.20.000183.7525 × 10−230.000183.75136 × 10−180.000183.75135 × 10–−130.30.0002552.24807 × 10−230.0002552.25476 × 10−180.0002552.25477 × 10–−130.40.000321.25172 × 10−230.000321.24919 × 10−180.000321.2492 × 10–−130.50.0003756.177 × 10−240.0003756.22301 × 10−190.0003756.22317 × 10–−140.60.000422.69667 × 10−240.000422.67581 × 10−190.000422.67572 × 10–−140.70.0004559.14662 × 10−250.0004559.26852 × 10−200.0004559.2681 × 10–−150.80.000482.38577 × 10−250.000482.30937 × 10−200.000482.3095 × 10–−150.90.0004952.25046 × 10−260.0004953.77223 × 10−210.0004953.76414 × 10–−161.0.00053.98217 × 10−270.00051.84694 × 10−220.00051.88184 × 10−17Table 8
Solution and residual error forγ keeping St=0.01,ϕ=0.1,We=0.001,anda=0.1 fixed in Case 2 of drainage.
γ=1.2γ=1.6γ=1.99xVxRes errorVxRes errorVxRes error0.15.72659 × 10–−4−1.32727 × 10−181.75703 × 10–−43.197 × 10−185.1638 × 10–−51.29417 × 10−170.21.31563 × 10–−3−2.02422 × 10−185.3263 × 10–−33.56035 × 10−182.05125 × 10–−41.22623 × 10−170.32.14014 × 10–−3−1.98946 × 10−181.01899 × 10–−33.88346 × 10−184.59664 × 10–−41.24389 × 10−170.43.02251 × 10–−3−2.9885 × 10−191.61463 × 10–−33.46852 × 10−188.14833 × 10–−41.35961 × 10−170.53.95057 × 10–−3−1.84225 × 10−182.30744 × 10–−33.68767 × 10−181.27034 × 10–−31.31157 × 10−170.64.91675 × 10–−3−2.46976 × 10−183.08901 × 10–−33.57668 × 10−181.82595 × 10–−31.37891 × 10−170.75.91581 × 10–−3−2.09831 × 10−183.95307 × 10–−33.36789 × 10−182.4815 × 10–−31.30025 × 10−170.86.94391 × 10–−3−1.35899 × 10−184.89465 × 10–−33.06616 × 10−183.23682 × 10–−31.40449 × 10−170.97.99811 × 10–−3−5.26745 × 10−195.90971 × 10–−34.003 × 10−184.09177 × 10–−31.29653 × 10−171.9.07604 × 10–−3−1.50223 × 10−186.99484 × 10–−32.56045 × 10−185.04625 × 10–−31.32103 × 10−17Table 9
Solution and residual error forα keeping γ=1.95,St=0.001,ϕ=0.1,We=0.01,anda=0.1 fixed in Case 3 of drainage.
α=0.3α=0.7α=0.99xVxRes errorVxRes errorVxRes error0.15.87208 × 10–−6−9.076 × 10−205.87208 × 10–−61.16701 × 10−195.87208 × 10–−66.48938 × 10−210.22.26882 × 10–−56.62623 × 10−202.26882 × 10–−5−3.50514 × 10−202.26882 × 10–−5−1.64872 × 10−190.35.0024 × 10–−57.16925 × 10−205.0024 × 10–−5−5.38787 × 10−205.0024 × 10–−5−5.48153 × 10−200.48.76616 × 10–−56.17061 × 10−208.76616 × 10–−54.81294 × 10−208.76616 × 10–−5−2.07791 × 10−190.51.35451 × 10–−4−7.89358 × 10−201.35451 × 10–−48.88535 × 10−201.35451 × 10–−46.49314 × 10−200.61.9328 × 10–−45.62103 × 10−201.9328 × 10–−4−8.1769 × 10−201.9328 × 10–−4−1.40755 × 10−200.72.61056 × 10–−42.13652 × 10−202.61056 × 10–−42.46311 × 10−192.61056 × 10–−48.01669 × 10−210.83.38702 × 10–−44.3926 × 10−203.38702 × 10–−4−3.74249 × 10−203.38702 × 10–−4−1.55483 × 10−190.94.26153 × 10–−4−1.25871 × 10−194.26153 × 10–−4−4.78429 × 10−204.26153 × 10–−4−1.57049 × 10−191.5.2335 × 10–−47.27423 × 10−205.2335 × 10–−43.94566 × 10−205.2335 × 10–−4−5.88772 × 10−20Table 10
Solution and residual error forγ keeping α=0.95,St=0.001,ϕ=0.1,We=0.01,anda=0.1 fixed in Case 3 of drainage.
γ=1.2γ=1.6γ=1.99xVxRes errorVxRes errorVxRes error0.15.72659 × 10–−4−8.85316 × 10−201.75703 × 10–−5−9.8013 × 10−205.1638 × 10–−64.95955 × 10−210.21.31563 × 10–−4−1.35853 × 10−195.3263 × 10–−5−5.72227 × 10−202.0512 × 10–−51.06505 × 10−200.32.14014 × 10–−41.63841 × 10−201.01899 × 10–−43.02498 × 10−224.5966 × 10–−52.98796 × 10−200.43.02251 × 10–−4−1.77002 × 10−191.61463 × 10–−42.74106 × 10−198.1483 × 10–−58.85672 × 10−200.53.95057 × 10–−46.75864 × 10−202.30744 × 10–−4−2.9077 × 10−201.2703 × 10–−4−1.25093 × 10−190.64.91675 × 10–−4−1.24341 × 10−193.08901 × 10–−42.8962 × 10−191.8259 × 10–−47.08777 × 10−210.75.91581 × 10–−43.33152 × 10−203.95307 × 10–−44.4664 × 10−202.4815 × 10–−43.73471 × 10−200.86.94391 × 10–−4−5.4743 × 10−204.89465 × 10–−4−4.42716 × 10−203.2368 × 10–−4−2.48835 × 10−200.97.99811 × 10–−4−4.85706 × 10−205.90971 × 10–−4−1.45847 × 10−194.0917 × 10–−41.10885 × 10−191.9.07604 × 10–−4−2.09174 × 10−196.99484 × 10–−41.98754 × 10−205.0462 × 10–−42.57162 × 10−19Tables8–10 exhibit solutions and residual errors for different values of fractional parameters in Cases 2 and 3, respectively. Exploration of these tables confirms that solutions are consistent. Influence of different parameters on Vx in lifting and drainage cases are seen graphically. Figures 1–8show the effect of different parameters on Vx in the lifting case. Figure 1 shows the effect of α on Vx. It is seen that increasing α reduces Vx. Figure 2 shows effect of a on Vx, showing that a has direct relationship with Vx. Figures 3–5 depict the consequence of ϕ, St, and We on Vx, respectively. It is observed that Weissenberg number is the dimensionless number while this parameter is used to measure the flow behavior into motion regarding particles. It is physically defined as ratio among viscous force and elastic force. Viscous force is enhanced against higher values of Weissenberg number, whereas higher viscous force brings a declination into motion of fluid particles. Momentum layers are also the reducing function versus implanted higher values of the Weissenberg number. It is seen that Vx has inverse relationship with mentioned parameters in all cases. Figure 6 indicates the impact of increasing a and St simultaneously on Vx. It is observed that Vx decreases with an increase in a and St, and hence, St is a more influential parameter then a. Figure 7 shows the combine effect of a and ϕ. Motion into fluid particles is reduced versus higher impact of ratio of viscosity number. For higher values of ϕ make a declination into motion of fluid. So, fluid becomes more viscous and thick. It is seen that Vx increases with increase in a and ϕ, and hence, a is more influential than ϕ. Figures 8–14 show the effect of different parameters on Vx in Case 1 of drainage. Figure 8 depicts the effect of α on Vx. It is seen that Vx has direct association with α. Figure 9 shows the impact of a on Vx. It is seen that Vx has inverse association with a. Figures 10–12 show the effect of ϕ,St,andWe on Vx, respectively. It is realized that Vx has direct connection with these parameters. Figure 13 shows the combined effect of increasing St and a. It is observed that Vx increases with increase in St and a, and hence, St is more influential than a. Stoke number St is a dimensionless number. Physically, it is used to know characterization of fluid particles during the flow of fluid. Basically, it is defined as the ratio among characteristics time and characteristics time during the flow regarding particle. So, it is noticed that flow of particles is directly proportional against velocity of fluid. Furthermore, Figure 14 shows the effect of increasing ϕ and a on Vx. Investigation shows that a is a more influential than ϕ. Figure 15 shows the effect of γ on Vx in Case 2 of drainage. It is seen that Vx has inverse association with γ. Figures 16 and 17 indicate the effect of α and γ on Vx in Case 3 of drainage. It is observed that α and γ exhibit opposite behaviors on Vx in this case. In addition to the above results, volumetric flow and average velocities are also computed in lifting and drainage cases.Figure 1
Effect ofα on Vx for a=0.1,We=1,St=1,andϕ=0.1 fixed in the lifting case.Figure 2
Effect ofa on Vx for α=0.95,St=1,ϕ=0.01,andWe=0.1 fixed in the lifting case.Figure 3
Effect ofϕ on Vx for α=0.98,a=0.1,We=1,andSt=1 fixed in the lifting case.Figure 4
The effect ofSt on Vx for α=0.95,a=0.1,We=1,andϕ=0.1 fixed in the lifting case.Figure 5
Effect ofWe on Vx for α=0.98,a=0.1,St=1,andϕ=0.1 fixed in the lifting case.Figure 6
Effect of increasingStanda simultaneously on Vx for α=0.99,We=0.1,andϕ=0.1 fixed in the lifting case.Figure 7
Effect of increasingϕanda simultaneously on Vx for α=0.95,We=1,St=0.1 fixed in the lifting case.Figure 8
Effect ofα on Vx for a=0.1,We=1,St=1,andϕ=0.1 fixed for Case 1 in the drainage case.Figure 9
Effect ofa on Vx for α=0.95,We=0.1,St=1,andϕ=0.01 fixed for Case 1 in the drainage case.Figure 10
The effect ofϕ on Vx for α=0.97,a=0.1,We=1,andSt=1 fixed for Case 1 in the drainage case.Figure 11
The effect ofSt on Vx for α=0.95,a=0.2,We=1,andϕ=0.1 fixed for Case 1 in the drainage case.Figure 12
The effect ofWe on Vx for α=0.95,a=0.1,St=1,andϕ=0.1 fixed for Case 1 in the drainage case.Figure 13
Effect of increasingStanda simultaneously on Vx for α=0.95,We=1,andϕ=0.01 fixed for Case 1 in the drainage case.Figure 14
Effect of increasingϕanda simultaneously on Vx for α=0.95,We=1,andSt=1 fixed for Case 1 in the drainage case.Figure 15
The effect ofγ on Vx for a=0.1,We=1,St=1,andϕ=0.1 fixed for Case 2 in the drainage case.Figure 16
The effect ofα on Vx for γ=1.95,a=0.1,We=1,St=1,andϕ=0.1 fixed for Case 3 in the drainage case.Figure 17
The effect ofγ on Vx for α=99,a=0.1,We=1,St=1,andϕ=0.1 fixed for Case 3 in the drainage case.


## 8. Conclusions

In this study, thin film of Johnson–Segalman fluid flow is investigated in fractional space for the case of lifting and drainage. Solutions of boundary value problems are obtained using fractional calculus and HPM. Validity of obtained solutions is established through residual errors’ computation. Few important results related to different parameters are established in the fractional space. It is comprehended that fractional parameterα has shown reverse effect in drainage and lifting. The study also discloses that fractional state Stokes and Weissenberg numbers showed similar effects on the fluid velocity in drainage and lifting cases. It is further seen that the Stokes number is the most influential parameter in both lifting and drainage situations. [34–39].



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*Source: 1019810-2022-02-23.xml*