# On Modified Algorithm for Fourth-Grade Fluid

Authors: Farah Jabeen Awan; Asif Mehmood; Syed Tauseef Mohyud-Din; Saleh M. Hassan
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper shows the analysis of the thin film flow of fourth-grade fluid on the outer side of a vertical cylinder. Solution of the governing nonlinear equation is obtained by Rational Homotopy Perturbation Method (RHPM); comparison with exact solution reflects the reliability of the method. Analysis shows that this method is reliable for even high nonlinearity. Graphs and tables strengthen the idea.


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## Body

## 1. Introduction

Recently, a number of new and modified techniques have been introduced by various scientists which subsequently proved extremely useful to tackle various nonlinear problems of diversified physical nature. It is an established fact that most of the physical problems in nature are nonlinear and hence solutions of such problems are of utmost importance. In a similar context, Rational Homotopy Perturbation Method (RHPM) [1] is a newly developed modified form of Homotopy Perturbation Method (HPM). HPM is developed by the coupling of Homotopy and perturbation is very useful for solving nonlinear problems [2–16]. The literature reveals that Jalaal et al. applied Homotopy Perturbation Method to find the velocity profile of a spherical solid particle in plane Couette fluid flow [2]. Liao compared Homotopy Analysis and Homotopy Perturbation Method [3], and Ghorbani and Nadjafi made an elegant comparison of He’s polynomials which are obtained from Homotopy Perturbation Method (HPM) Adomian’s Polynomials [4]. They worked on new developments of the HPM [5]. Mohyud-Din applied HPM on various problems like partial differential equations, higher dimensional initial boundary value problem, and fourth-order, nonlinear higher-order, and sixth-order boundary value problems [6–10]. Xu solved boundary layer equation in unbounded domain by HPM [11]. Vázquez-Leal [1] modified the idea of Homotopy Perturbation Method. He used two power series of Homotopy parameters in a quotient, resulting in a series of linear differential equations.Siddiqui applied Homotopy Perturbation Method on thin film flow of fourth-grade fluid on the outer wall of vertical cylinder. Hayat applied Homotopy Analysis Method on the same problem and proved convergence. Afterwards, Sajid introduced the slip effect for this problem and presented solution using Homotopy Analysis Method [12–16].It is worth mentioning that Hayat et al. [17, 18] made an appropriate use of Homotopy technique and applied the same on thin film flow of an Oldroyd 6-constant fluid over a moving belt [17] and also on Couette and Poiseuille flows for fourth-grade fluid [18]. It is to be highlighted that the proposed method (RHPM) is highly suitable for a wide range of physical problems including heat transfer and thermodynamics. The detailed study of the literature reconfirms the reliability of the suggested Rational Homotopy Perturbation Method (see [19–23] and the references therein). In a similar context, Ganji et al. [20, 21] solved nonlinear equations arising in heat transfer by applying the coupling of Homotopy and perturbation. Moreover, Chowdhury and Hashim [22] and Islam et al. [23] extended modified version of Homotopy Perturbation Method for problems related to heat transfer. Inspired and motivated by the ongoing research in this area, we apply a relatively new modified version of Homotopy Perturbation Method which is called Rational Homotopy Perturbation Method (RHPM) [24] on the thin film flow of fourth-grade fluid coupled with slip effect. A convergent and reliable solution is presented with the aid of figures and tables. Analysis shows that Rational Homotopy Perturbation Method is adequate to present the flow behavior of this problem. It is observed that RHPM is very efficient for such problems. Moreover, RHPM is equally applicable on the mathematical models derived from nature like Stiff system of equations [24], transient of nonlinear circuits [19], and heat transfer problems (see [19–24] and the references therein).


## 2. Basic Idea of RHPM

Foundation of Rational Homotopy Perturbation Method and Homotopy Perturbation Method is the same. Consider the nonlinear differential equation to explain both methods:(1)Lu+Nu-gr=0,wherer∈Ω,and consider the boundary condition (2)Bu,∂u∂η,wherer∈Γ,where N and L are nonlinear and linear operators, respectively, and B is a boundary operator. g(r) is an analytic function and domain Ω has the boundary Γ. Homotopy can be written as (3)Hυ,p=1-pLυ-Lu0+pLυ+Nυ-gr=0,p∈0,1,where u0 is an initial approximation for (1) which satisfies the boundary condition and Homotopy parameter is p. For p=0 and p=1, (3) can be written as (4)Hυ,0=Lυ-Lu0=0,Hυ,1=Lυ+Nυ-gr=0.For HPM, let us take the solution for (3) as(5)υ=υ0+pυ1+p2υ2+⋯;consider p→1 to get approximate solution (6)uy=limp→1υ=υ0+υ1+υ2+⋯.Let us take the solution of (3) for RHPM as (7)υ=υ0+pυ1+p2υ2+⋯ψ0+pψ1+p2ψ2⋯,where known analytic functions ψ0,ψ1,ψ2… have independent variables and unknown functions υ0,υ1,υ2… are to be determined by the Rational Homotopy Perturbation Method. An approximate solution of (1) is obtained when limiting case of (7) is taken as p→1. Hence, (8)u=υ0+υ1+υ2+⋯ψ0+ψ1+ψ2⋯.The limiting case in the above equation is associated with existence of limits (9)limp→1⁡∑i=0∞υi,limp→1⁡∑i=0∞ψi,where∑i=0∞ψi≠0.


## 3. Solution by RHPM

Let us consider the dimensionless equation representing thin film flow of fourth-grade fluid which is falling on the outer side of a vertical cylinder having an infinite length of radiusR [13]:(10)xd2udx2+dudx+Kx+2βdudx3+3dudx2d2udx2=0,where β is nondimensional fluid parameter, K is dimensionless constant which corresponds to gravity, and M corresponds to partial slip effect. Boundary condition is (11)ux-MK2l2-1=0atx=1,duxdx=0atx=l.Exact solution is (12)ux=∫1xgx^dx^+MK2l2-1,1≤x≤l,where (13)gx=25/331/3βx2+-18β2Kx2x2-l2+332β3x6+108β4K2x4x2-l222/324/332/3βx-18β2Kx2x2-l2+332β3x6+108β4K2x4x2-l221/3.Integral in (12) can be calculated numerically.Consider the following Homotopy equation:(14)1-pxυ′′+υ′-u0′+xu0′′+pxυ′′+υ′+kx+2βυ′3+3υ′2υ′′,where u0=(MK/2)l2-1+(K/4)1-x2+2l2Log[x] is initial approximation. Let us consider the solution of (14) with [2,2] order of approximation as (15)υ=υ0x+pυ1x+p2υ2x1+apx2+bp2x4,where a and b are the adjustment parameters. By considering (14) and (15) and regrouping according to the p-powers, we get the following.Zeroth-Order Problem.The zeroth-order problem is as follows:(16)-kl2x+kx+υ0′+xυ0′′=0,subject to the boundary conditions (17)υ0x=1atx=0,dυ0xdx=0atx=1.First-Order Problem.The first-order problem is as follows:(18)-kl2x-4axυ0-4ax2υ0′+2βυ0′3+υ1′+6xβυ0′2υ0′′+xυ1′′=0,and relevant boundary conditions are (19)υ1x=0atx=0,dυ1xdx=0atx=1.Second-Order Problem.The second-order problem is as follows:(20)8a2x3υ0-16bx3υ0-4axυ1-8bx4υ0′-24axβυ0υ0′2-24ax2βυ0′3-4ax2υ1′+6βυ0′2υ1′+υ2′-24ax2βυ0υ0′υ0′′+12xβυ0′υ1′υ0′′+6xβυ0′2υ1′′+xυ2′′=0,and relevant boundary conditions are (21)υ2x=0atx=0,dυ2xdx=0atx=1.Solving these problems with the corresponding boundary conditions, we have (22)υ0=14K-2KM+2Kl2M-Kx2+2Kl2Logx-4Kl2LoglLogx+2Kl2Logx2,υ1=116x2-aKx2+4aKl2x2+8aKMx2-8aKl2Mx2+4aKx2-4aKl2x4-8aKMx4+8aKl2Mx4-3aKx6+32K3l6β-K3x2β-32K3l6x2β+K3x6β-60K3l6βLogl-12K3l2x2βLogl+60K3l6x2βLogl+⋯,υ2=15760x4280a2Kx4-160β1Kx4-1440al2x4-270a2K0l2x4+360βKl2x4-2160a2KMx4+2880βKMx4+2160a2Kl2Mx4-2880βKl2Mx4-360a2Kx6⋯.Substituting (22) into (15) and assuming p→1 result in (23)ux=14K-2KM+2Kl2M-Kx2+2Kl2Logx-4Kl2LoglLogx+2Kl2Logx2+116x2-aKx2+4aKl2x2+8aKMx2-8aKl2Mx2+4aKx2-4aKl2x4-8aKMx4+8aKl2Mx4-3aKx6+32K3l6β-K3x2β-32K3l6x2β+K3x6β-60K3l6βLogl-12K3l2x2βLogl+60K3l6x2βLogl+⋯+15760x4280a2Kx4-160β1Kx4-1440al2x4-270a2K0l2x4+360βKl2x4-2160a2KMx4+2880βKMx4+2160a2Kl2Mx4-2880βKl2Mx4-360a2Kx6⋯+⋯1+ax2+bx4-1.Now, let us discuss the solution for different parameters.Case 1 (k=1, M=0.1, β=0.1, and l=1.5).
Choosing the adjustment parameters for (23) as a=0.00487593 and b=-0.0013628, we have (24)ux=14K-2KM+2Kl2M-Kx2+2Kl2Logx-4Kl2LoglLogx+2Kl2Logx2+116x2-aKx2+4aKl2x2+8aKMx2-8aKl2Mx2+4aKx2-4aKl2x4-8aKMx4+8aKl2Mx4-3aKx6+32K3l6β-K3x2β-32K3l6x2β+K3x6β-60K3l6βLogl-12K3l2x2βLogl+60K3l6x2βLogl+⋯+15760x4280a2Kx4-160β1Kx4-1440al2x4-270a2K0l2x4+360βKl2x4-2160a2KMx4+2880βKMx4+2160a2Kl2Mx4-2880βKl2Mx4-360a2Kx6⋯+⋯1+0.00487593x2-0.0013628x4-1.Case 2 (k=1, M=0.3, β=0.5, and l=1.5).
Choosing the adjustment parameters for (23) as a=-0.0011041205 and b=0.001978206018, we have (25)ux=14K-2KM+2Kl2M-Kx2+2Kl2Logx-4Kl2LoglLogx+2Kl2Logx2+116x2-aKx2+4aKl2x2+8aKMx2-8aKl2Mx2+4aKx2-4aKl2x4-8aKMx4+8aKl2Mx4-3aKx6+32K3l6β-K3x2β-32K3l6x2β+K3x6β-60K3l6βLogl-12K3l2x2βLogl+60K3l6x2βLogl+⋯+15760x4280a2Kx4-160β1Kx4-1440al2x4-270a2K0l2x4+360βKl2x4-2160a2KMx4+2880βKMx4+2160a2Kl2Mx4-2880βKl2Mx4-360a2Kx6⋯+⋯1-0.0011041205x2+0.0019782060x4-1.


## 4. Conclusion

In this section, we will give analysis of the present attempt. Figure1 shows that solution found by Rational Homotopy Perturbation Method overlaps with exact solution. Figure 2 represents the notion that there exists a gradual shift of velocity with varying slip effect. By increasing the slip effect, velocity also increases. Figure 3 shows that, by increasing the nonlinearity and slip effect, Rational Homotopy Perturbation Method and exact solutions are in good agreement. Figure 4 represents the effect of β on the velocity. Tables 1-2 show that error is negligible. Table 3 shows that, for different values of slip effects, Rational Homotopy Perturbation Method efficiently represents the flow behavior. Table 4 reflects the reliability of the method for different values of β. Tables 3-4 demand that a and b be determined individually by the method described in Cases 1 and 2. Mathematica 5.2 is used in all calculations.Table 1
Flow behavior and error estimation of fluid forK=1, M=0.1, β=0.1, and l=1.5.
x
RHPM
Numerical RHPM
1.0
0.0622812
2.18802
E
-
4
1.1
0.114139
2.46544
E
-
4
1.2
0.153569
-
2.6404
E
-
5
1.3
0.180729
-
5.56143
E
-
5
1.4
0.196448
8.225052
E
-
6
1.5
0.201573
-
4.88900
E
-
6Table 2
Flow behavior and error estimation forK=1, M=0.3, β=0.5, and l=1.5.
x
RHPM
Numerical RHPM
1.0
0.187336
1.63748
E
-
4
1.1
0.230755
2.06236
E
-
3
1.2
0.268424
1.31266
E
-
4
1.3
0.295047
-
7.2133
E
-
4
1.0
0.310211
-
4.2611
E
-
4
1.5
0.314603
2.72582
E
-
4Table 3
Error table fork=1, β=0.1, and l=1.5
x
M
=
0.4
M
=
0.8
M
=
1.0
a
=
0.0016161
a
=
0.000470222
a
=
0.0002958
b
=
0.0000202
b
=
0.000244044
b
=
0.0002376
1.0
4.08412
E
-
4
3.56878
E
-
4
3.33297
E
-
4
1.1
1.35926
E
-
4
6.35029
E
-
5
4.41049
E
-
5
1.2
-
2.85247
E
-
4
-
3.51118
E
-
4
-
3.65604
E
-
4
1.3
-
3.1902
E
-
4
-
3.49445
E
-
4
-
3.57517
E
-
4
1.4
-
1.30714
E
-
4
-
9.74395
E
-
5
-
9.70654
E
-
5
1.5
8.70222
E
-
5
2.09943
E
-
4
2.21282
E
-
4Table 4
Error table fork=1, M=0.1, and l=1.5.
x
β
=
0.2
β
=
0.6
β
=
0.8
a
=
0.006301675
a
=
-
0.002555156
a
=
-
0.01257134
b
=
-
0.00121253
b
=
0.0042851561
b
=
0.009340074
1.0
3.1646
E
-
4
1.07938
E
-
4
2.02609
E
-
4
1.1
6.06636
E
-
4
2.68093
E
-
3
3.84367
E
-
3
1.2
-
3.04017
E
-
5
3.02243
E
-
4
5.24388
E
-
4
1.3
-
1.72439
E
-
4
-
8.45452
E
-
4
-
1.24596
E
-
3
1.4
-
1.32375
E
-
5
-
5.49597
E
-
4
-
8.79013
E
-
4
1.5
1.11499
E
-
4
3.11275
E
-
4
5.13033
E
-
4Figure 1
Comparison of RHPM with exact solution.Figure 2
Effect ofM on velocity profile.Figure 3
Comparison of RHPM with exact solution.Figure 4
Effect ofβ on velocity profile.



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