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Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
Propulsion and Power Research 2016;5(3):223–235
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ORIGINAL ARTICLE
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt Taza Gula, Fazle Ghania, S. Islama, R.A. Shahb, I. Khanc,n, Saleem Nasira, S. Sharidand a
Department Department c Department d Department b
of of of of
Mathematics, Abdul Wali Khan University, Mardan, KPK, Pakistan Mathematics, University Engineering Technology Peshawar, KPK, Pakistan Basic Sciences, College of Engineering Majmaah University, P.O. Box 66, Majmaah 11952, Saudi Arabia Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, UTM Johor Bahru, 81310 Johor, Malaysia
Received 14 April 2015; accepted 17 July 2015 Available online 18 August 2016
KEYWORDS Unsteady flow; Thin film; Lifting and drainage; Fourth grade fluid; Adomian decomposition method (ADM); Optimal homotopy asymptotic method (OHAM)
Abstract This article studies the unsteady thin film flow of a fourth grade fluid over a moving and oscillating vertical belt. The problem is modeled in terms of non-nonlinear partial differential equations with some physical conditions. Both problems of lift and drainage are studied. Two different techniques namely the adomian decomposition method (ADM) and the optimal homotopy asymptotic method (OHAM) are used for finding the analytical solutions. These solutions are compared and found in excellent agreement. For the physical analysis of the problem, graphical results are provided and discussed for various embedded flow parameters. & 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Due to the diverse physical structures of non-Newtonian fluids, several constitutive models have been proposed in the literature to predict all of their salient features. n
Corresponding author.
E-mail address: [email protected] (I. Khan). Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.
Generally, there are three non-Newtonian fluids models. They are known as (i) the differential type, (ii) the rate type, and (iii) the integral type. However, the most famous amongst them are the first two models. In this article, we will study the first model, the differential type fluid also known as grade n fluids and consider its subclass known as the fourth grade fluid. The simplest subclass of differential type is the second grade fluid for which one can reasonably hope to obtain an analytical solution. Although a second grade fluid model also known as grade
2212-540X & 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/j.jppr.2016.07.002
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2 is able to predict the normal stress differences but it does not take into account the shear thinning and shear thickening phenomena that many fluids exhibit [1–4]. On the other hand, a third grade fluid model (grade 3 fluid) represents a more realistic description of the behavior of non-Newtonian fluids. This model is known to capture the non-Newtonian effects such as shear thinning or shear thickening as well as normal stresses [5–8]. There is another model for differential type fluids known as fourth grade fluid model (grade 4 fluid) which at one time capture most of the non-Newtonian flow properties. Means, Cauchy stress tensor appearing in this fluid model contains several parameters. Due to which the governing equation for fourth grade fluid becomes quite complicated. This model is also very capable to predict the effects of normal stresses that lead to phenomena like rod-climbing and dieswell [9]. Therefore, very few studies, which illustrates the effect of fourth grade fluids even on steady flow, have been reported in the literature. In general, it is not easy to study fourth grade fluid even for steady flow problems. However, under certain assumptions based on the flow situation, it is possible and some investigations are carried out in this direction [10–14]. On the other hand, the science of thin liquid films has developed rapidly in recent years. Thin film flow problems appear in many fields, varying from specific situations in the flow in human lunges to lubrication problems in engineering which is probably one of the largest subfield of thin film flow problems [15]. Few other applications are found in coating flows, biofluids, microfluidic engineering, and medicine [16,17]. Two well know examples from everyday life are slipping on a wet bathroom floor and aquaplaning or hydroplaning on a wet road. The study of thin film flow for practical applications is a challenging interplay between fluid mechanics, structural mechanics and theology. Based on this motivation, recently several researchers are getting interested to study the thin film flows. Amongst them, we discuss here some important contributions of the following researchers. Siddiqui et al. [18,19] discussed the analytical solution of thin film flow of third grade fluid and Oldroyed-8 constant fluid through vertical moving belt by applying adomian decomposition method (ADM) and variational iteration method (VIM). Aiyesimi et al. [20] investigated the unsteady magnetohydrodynamic (MHD) thin film flow of a third grade fluid with heat transfer and no slip boundary condition down an inclined plane. Gul et al. [21,22] studied analytically the thin film flows of second grade and third grade fluids through a vertical belt for lift and drainage problems using optimal homotopy asymptotic method (OHAM) and ADM. Few other investigations in this direction are mentioned in Refs. [23–40]. The basic theme of this article is to venture further in the regime of thin film flows and to extend this idea to the fourth grade fluid executing the unsteady motion over a
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vertical oscillating belt using two analytical techniques namely the OHAM and ADM methods. OHAM on the other hand, is a new powerful technique which is a generalized form of homotopy perturbation method (HPM) and HAM. It is a straightforward technique and does not require the existence of any small or large parameter as does the traditional perturbation method. OHAM has successfully applied to a number of nonlinear problems arising in the science and engineering by various researchers. This proves the validity and acceptability of OHAM as a useful technique. For the correctness and verification of OHAM solutions, they are compared with those obtained by ADM. The rest of the paper is arranged as follows: The constitutive equations for fourth grade fluid are given in Section 2. Using these constitutive equations for unidirectional and one dimension flow, the continuity equation is satisfied identically and the momentum equation is derived in Section 3 with some physical conditions. In Section 4, the basic concepts of ADM and OHAM are discussed. These methods are used in Section 5 where the analytical solutions for both lifting and drainage problems are obtained. The graphical results are given in Section 6. This paper ends with some conclusions in Section 7.
2. Fundamental equation The constitutive equations for fourth grade fluid model is T ¼ pI þ μA1 þ
3 X
Si ;
ð1Þ
i¼1
where T is the Cauchy stress tensor, I is identity tensor, p is fluid pressure and Si is the extra stress tensor S1 ¼ α1 A2 þ α2 A21 ; S2 ¼ β1 A3 þ β1 ðA1 A2 þ A2 A1 Þ þ β3 trA22 A1 ; S3 ¼ γ 1 A4 þ γ 2 ðA3 A1 þ A1 A3 Þ þ γ 3 A22 þγ 4 ðA2 A21 þ A21 A2 Þ þ γ 5 ðtrA2 ÞA2 þ γ 6 ðtrA2 ÞA21 þfγ 7 ðtrA3 Þ þ γ 8 ðtrA2 A1 ÞgA1 :
ð2Þ
Here μ is coefficient of viscosity, αi ði ¼ 1; 2Þ; βj ðj ¼ 1; 2; 3Þ and γ k ðk ¼ 1; 2; …; 8Þ are material constants of second, third and fourth grades respectively. Rivlin–Ericksen tensors A1 ; A2 ; A3 and A4 are defined as: A0 ¼ I; A1 ¼ ð∇V Þ þ ð∇V ÞT ; An ¼
DAn 1 þ An 1 ð∇V Þ þ ð∇V ÞT An 1 ; Dt
ð3Þ n ¼ 2; 3; 4:
ð4Þ The continuity and momentum equation for incompressible fluid are given by
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
div V ¼ 0; ρ
225
ð5Þ
DV ¼ div T þ ρg; Dt
ð6Þ
where V ¼ ðu; v; wÞ is the velocity vector, ρ is the fluid D density, Dt is the material time derivative.
3. Formulation of lift problem We consider an oscillating and moving wide flat belt place in a large bath of a fourth grade fluid. The belt is moving in vertically upward direction with velocity U and pick up a thin film of a fourth grade fluid of constant thickness δ. The x- and y-axes are chosen in such a way that the belt is parallel to y-axis and perpendicular to x-axis as shown in the physical configuration (see Figures 1 and 2). The pressure gradient is not considered here and the flow is assumed as unsteady and laminar. By taking velocity field of the form V ¼ ð0; vðx; tÞ; 0Þ;
ð7Þ
Subject to the following boundary conditions vðx; tÞ ¼ U þ U cos ωt; ∂vðx; tÞ ¼ 0; ∂x
at
at
x ¼ 0;
x ¼ δ;
Here ω is used as frequency of the oscillating belt.
ð8Þ ð9Þ
Figure 2 Geometry of drainage problem.
Using the velocity field given in Eq. (7), the continuity Eq. (5) is identically satisfied and the momentum Eq. (6) reduces to the form ρ
∂v ∂ ¼ T xy ρg: ∂t ∂x
ð10Þ
From Eq. (1) the component T xy of Cauchy stress component is ∂v ∂ ∂v ∂2 ∂v þ β1 2 T xy ¼ μ þ α1 ∂x ∂t ∂x ∂t ∂x 3 ∂v ∂3 ∂v þ2 β2 þ β3 þ γ1 3 ∂x ∂t ∂x ∂ ∂v 3 : ð11Þ þ2ð3γ 2 þ γ 3 þ γ 4 þ γ 5 þ 3γ 7 þ γ 8 Þ ∂t ∂x Inserting Eq. (11) into Eq. (10), we get ∂v 2 ∂2 v ∂v ∂2 v ∂ ∂2 v ρ ¼ μ 2 þ α1 þ 6 β2 þ β 3 ∂t ∂x ∂t ∂x2 ∂x ∂x2 ∂3 ∂2 v þγ 1 3 þ 2ð3γ 2 þ γ 3 þ γ 4 þ γ 5 ∂t ∂x2 " # ∂ ∂v 2 ∂ ∂2 v þ3γ 7 þ γ 8 Þ ρg: ð12Þ ∂x ∂x ∂t ∂x2
Figure 1
Geometry of lift problem.
Introducing the following non-dimensional variables as mentioned in Refs. [13,21,22].
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v x μt δ2 ρg α1 ; x~ ¼ ; t~ ¼ 2 ; St ¼ ; α~ ¼ 2 ; U δ k ρδ ρδ β 2 þ β3 U 2 β μ γ μ2 ; β~ 1 ¼ 12 ; γ~ 1 ¼ 13 4 ; β~ ¼ δ ρ δ μδ2 γ U2 ð13Þ γ~ ¼ 3γ 2 þ γ 3 þ γ 4 þ γ 5 þ 3γ 7 þ γ 8 1 4 : ρδ
v~ ¼
Here t is a dimensionless time parameter, α~ is dimension~ β~ 1 are less second grade non-Newtonian parameter, β; dimensionless third grade non-Newtonian parameters and γ~ ; γ~ 1 are dimensionless fourth grade non-Newtonian parameters. Inserting above non-dimensional variables into Eq. (12) and boundary conditions (8) and (9), we get: ∂v ∂2 v ∂ ∂2 v ∂2 ∂2 v ¼ St þ 2 þ α þ β1 2 ∂t ∂x ∂t ∂x2 ∂t ∂x2 2 2 3 2 ∂v ∂ v ∂ ∂ v þ6β þ γ1 3 ∂x ∂x2 ∂t ∂x2 " # ∂ ∂v 2 ∂ ∂2 v þ2γ ; ð14Þ ∂x ∂x ∂t ∂x2 vðx; tÞ ¼ 1 þ cos ωt; ∂vðx; tÞ ¼ 0; ∂x em.
at
at
x ¼ 0;
x ¼ 1:
ð15Þ ð16Þ
source term, Ruðx; tÞ is a remaining linear term and Nuðx; tÞ is non-linear analytical term expandable in the adomian polynomials An . After applying the inverse operator Lx 1 to both sides of Eq. (19), we write Lx 1 Lx uðx; tÞ ¼ Lx 1 gðx; tÞ Lx 1 Lt uðx; tÞ Lx 1 Ruðx; tÞ Lx 1 Nuðx; tÞ;
ð20Þ
uðx; tÞ ¼ f ðx; tÞ Lx 1 Lt uðx; tÞ Lx 1 Ruðx; tÞ Lx 1 Nuðx; tÞ:
ð21Þ
Here the function f ðx; tÞ represents the terms arising from Lx 1 gðx; tÞ after using the given conditions. Lx 1 ¼ ∬ ðÞdxdx is used as inverse operator for the second order partial differential equation. In this method, the series solution uðx; tÞ is defined as: X1 uðx; tÞ ¼ u ðx; tÞ; ð22Þ n¼0 n X1
X1 1 u ðx; tÞ ¼ f ðx; tÞ L L u ðx; tÞ n t x n¼0 n¼0 n X1 1 Lx R un ðx; tÞ Xn1¼ 0 1 Lx N u ðx; tÞ: n¼0 n
ð23Þ
The non-linear term is expanded in adomian polynomials as: X1 N ðvðx; tÞÞ ¼ A : ð24Þ n¼0 n where the components u0 ðx; tÞ; u1 ðx; tÞ; u2 ðx; tÞ; … are periodically derived as:
4. Solution techniques 4.1. Basic concept of adomian decomposition method (ADM) ADM is used to decompose the unknown function uðx; tÞ into a sum of an infinite number of components defined by the decomposition series. X1 uðx; tÞ ¼ u ðx; tÞ: ð17Þ n¼0 n The decomposition method is used to find the components u0 ðx; tÞ; u1 ðx; tÞ; u2 ðx; tÞ; … separately. The determination of these components can be obtained through simple integrals. To give a clear overview of ADM, we consider the partial differential equation in an operator form as Lt uðx; tÞ þ Lx uðx; tÞ þ Ruðx; tÞ þ Nuðx; tÞ ¼ gðx; tÞ;
ð18Þ
Lx uðx; tÞ ¼ gðx; tÞ Lt uðx; tÞ Ruðx; tÞ Nuðx; tÞ:
ð19Þ
where Lx ¼ ∂x∂ 2 and Lt ¼ ∂t∂ are linear operators in the partial differential equation and are easily invertible, gðx; tÞ is a 2
u0 ðx; tÞ þ u1 ðx; tÞ þ u2 ðx; tÞ þ … ¼ f ðx; tÞ Lx 1 Rðu0 ðx; tÞ þ u1 ðx; tÞ þ u2 ðx; tÞ þ…Þ Lx 1 ðA0 þ A1 þ A2 þ …Þ:
ð25Þ
To determine the components of the series u0 ðx; tÞþ u1 ðx; tÞ þ u2 ðx; tÞ þ …, it is important to note that ADM suggests that the zeroth component u0 ðx; tÞ is usually defined by the function f ðx; tÞ described above. The formal recursive relation is defined as: u0 ðx; tÞ ¼ f ðx; tÞ; u1 ðx; tÞ ¼ Lx 1 Lt ðu0 ðx; tÞÞ Lx 1 Rðu0 ðx; tÞÞ Lx 1 ðA0 Þ; u2 ðx; tÞ ¼ Lx 1 Lt ðu1 ðx; tÞÞ Lx 1 Rðu1 ðx; tÞÞ Lx 1 ðA1 Þ; u3 ðx; tÞ ¼ Lx 1 Lt ðu2 ðx; tÞÞ Lx 1 Rðu2 ðx; tÞÞ Lx 1 ðA2 Þ;
ð26Þ
and so on.
4.2. Optimal homotopy asymptotic method (OHAM) For the analysis of OHAM, we consider the boundary value problem as consider in Refs. [12,13]: ∂u~ ~ ~L ðuðxÞ ~ ~ ~ Þ þ N ðuðxÞ ~ Þ þ GðxÞ ¼ 0; B u; ~ ¼ 0; ð27Þ ∂x
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
227
Table 1 Comparison of OHAM and ADM for the lift velocity profile, by taking ω ¼ 0:02; α ¼ 0:02; β ¼ 0:2; β1 ¼ 0:1; γ ¼ 0:2; γ 1 ¼ 0:01; t ¼ 2; St ¼ 0:3; c1 ¼ 0:0517334034; c2 ¼ 0:0272742812: x
OHAM
ADM
Absolute error
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.9998000 1.7966503 1.5941974 1.3924453 1.1913952 0.9910501 0.7914146 0.5924887 0.3942759 0.1967780 1:3 10 18
1.9998000 1.7638191 1.5363173 1.3170952 1.1060142 0.9028535 0.7074496 0.5196317 0.3392280 0.1660729 1:4 10 17
0 0.0328310 0.0578801 0.0753353 0.0853805 0.0881976 0.0839647 0.0728568 0.0550479 0.0307064 1:3 10 17
where L~ a linear operator in the differential equation is, N~ is a non-linear term, x A R is an independent variable, B~ is a ~ is a source term. According to boundary operator and G Refs. [12,13] we construct a set of equation for OHAM: ~ ~ ½1 p L~ ðuðxÞ ~ Þ þ GðxÞ þ N~ ϕ~ ðx; pÞ ; ¼ Hðp; ci Þ L~ ϕ~ ðx; pÞ þ GðxÞ
∂ϕ~ B~ ϕ~ ðx; pÞ; ¼ 0: ∂x
∂u~ 2 ðxÞ ¼ 0: B~ u~ 2 ðxÞ; ∂x
ð34Þ
ð28Þ The general governing equations for u~ k ðxÞ are given by
Here pA ½0; 1 is an embedding parameter, Hðp; ci Þ is a non-zero auxiliary function for p a 0 and Hð0; ci Þ ¼ 0, ϕ~ ðx; pÞ is an unknown function. Obviously, when p ¼ 0 and p ¼ 1, it holds that: ~ ϕ~ ðx; 0Þ ¼ u~ 0 ðxÞ; ϕ~ ðx; 1Þ ¼ uðxÞ; ð29Þ Note that, when p varies from 0 to 1 then ϕ~ ðx; pÞ also ~ varies from u~ 0 ðxÞ to uðxÞ: The zeroth component solution u~ 0 ðxÞ is obtained from Eq. (28) when p ¼ 0 i.e ∂u~ 0 ðxÞ ~ L~ ðu~ 0 ðxÞÞ þ GðxÞ ¼ 0; B~ u~ 0 ðxÞ; ¼ 0; ð30Þ ∂x auxiliary function Hðp; ci Þ is chosen as: Hðp; ci Þ ¼ pc1 þ p2 c2 þ …
X
u ðx; ci Þpk ; kZ1 k
ð31Þ
i ¼ 1; 2; …:;
ð32Þ
k 1 X ci L~ ðu~ k i ðxÞÞ i¼0
þN~ k 1 ðu~ 0 ðxÞ; u~ 1 ðxÞ; …; u~ k 1 ðxÞÞ ; ∂u~ k ðxÞ ¼ 0: B~ u~ k ðxÞ; ∂x
k ¼ 2; 3 ð35Þ
Here N~ m ðu~ 0 ðxÞ; u~ 1 ðxÞ; …; u~ m 1 ðxÞ Þ is the coefficient of ~ pÞ : pm , in the expansion of N~ ϕðx; ~ p; ci Þ ¼ N~ 0 ðu~ 0 ðxÞÞ N~ ϕðx; þ
1 X
N~ k 1 ðu0 ðxÞ; u~ 1 ðxÞ; …u~ m ðxÞÞpm ;
ð36Þ
the convergence of the series in Eq. (32) depend upon the convergence control parameters c1 ; c2 ; …. If it converges at p ¼ 1 then the mth order approximation u~ is ~ c1 ; c2 ; c3 ; …; cm Þ ¼ u~ 0 ðxÞ þ uðx;
m X
u~ i ðx; c1 ; c2 ; c3 ; …; ci Þ;
ð37Þ
i¼1
Inserting Eq. (32) into Eq. (28), collecting the same powers of p, and equating each coefficient of p, the zero order problem is given in Eq. (30) and the first and second order are given in Eqs. (33) and (34). ~ L~ ðu~ 1 ðxÞÞ þ GðxÞ ¼ c1 N~ 0 ðu~ 0 ðxÞÞ; ~B u~ 1 ðxÞ; ∂u~ 1 ðxÞ ¼ 0; ∂x
L~ ðu~ k ðxÞÞ L~ ðu~ k 1 ðxÞÞ ¼ ck N~ 0 ðu~ 0 ðxÞÞ þ
m¼1
where c1 ; c2 are auxiliary constants. Marinca [12,13] uses a special procedure to expand ϕ~ ðx; pÞ with respect to p by using Taylor series. ϕ~ ðx; p; ci Þ ¼ u~ 0 ðxÞ þ
L~ ðu~ 2 ðxÞÞ L~ ðu~ 1 ðxÞÞ ¼ c2 N~ 0 ðu~ 0 ðxÞÞ þ c1 L~ ðu~ 1 ðxÞÞN~ 0 ðu~ 0 ðxÞ; u~ 1 ðxÞÞ ;
ð33Þ
inserting Eq. (36) into Eq. (27), the residual is obtained as: ~ ~ ci Þ ¼ L~ ðuðx; ~ ci ÞÞ þ N~ ðuðx; ~ ci ÞÞ þ GðxÞ; Rðx; i ¼ 1; 2; …; m;
ð38Þ
Numerous methods like Ritz method, method of least squares, Galerkin method and collocation method are used to find the optimal values of ci ; i ¼ 1; 2; 3; ….
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Table 2 Comparison of OHAM and ADM for the lift velocity profile, by taking ω ¼ 0:02; α ¼ 0:2; β ¼ 0:2; β1 ¼ 0:3; γ ¼ 0:2; γ 1 ¼ 0:04; t ¼ 5; St ¼ 0:4; c1 ¼ 0:2661918234; c2 ¼ 0:2382524332: x
OHAM
ADM
Absolute error
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.9950041 0.9011902 0.8060385 0.7095748 0.6118278 0.5128257 0.4125973 0.3111713 0.2085768 0.1048432 1:8 10 17
0.9950041 0.9071647 0.8169358 0.7242514 0.6290402 0.5312253 0.4307243 0.3274490 0.2213048 0.1121909 6:6 10 18
0 0.00597381 0.01089723 0.01467664 0.01721239 0.01839954 0.01812704 0.01627770 0.01272798 0.00734773 1:2 10 17
We apply the method of least squares in our problem as given below: Z b 2 ð39Þ Jðc1 ; c2 ; c3 ; …; cn Þ ¼ R~ ðx; c1 ; c2 ; c3 ; …; cm Þdx: where a and b are the constant values taking from domain of the problem. Convergence control parameters ðc1 ; c2 ; c3 ; …; cn Þ can be identified from: ð40Þ
Finally, from these convergence control parameters, the approximate solution is well determined.
4.3. The ADM solution of lift problem Rewrite Eq. (14) of the form of Eq. (18), we get 2 ∂v ∂ ∂2 v ∂2 ∂2 v ∂v α β 6β 1 ∂t ∂t ∂x2 ∂x ∂t 2 ∂x2 " # 2 ∂ v ∂3 ∂2 v ∂ ∂v 2 ∂ ∂2 v γ 1 3 2γ ; 2 2 ∂x ∂x ∂x ∂t ∂x2 ∂t ∂x
n¼0
" # " # 1 1 X X x2 αLx 1 An β1 Lx 1 Bn 2 n¼0 n¼0 " # " # " # 1 1 1 X X X 1 1 1 C n γ 1 Lx Dn 2γLx En ; 6βLx
vðx; tÞ ¼ b0 þ b1 x þ St
n¼0
a
∂J ∂J ∂J ¼ ¼ ¼ … ¼ 0: ∂c1 ∂c2 ∂c3
1 X
Lx vðx; tÞ ¼ St þ
n¼0
n¼0
ð43Þ where An ; Bn ; C n ; Dn and En are adomian polynomials and define as 1 X ∂ ∂2 v An ¼ ; ∂t ∂x2 n¼0 1 X ∂2 ∂2 v Bn ¼ 2 ; ∂t ∂x2 n¼0 1 X ∂3 ∂2 v Dn ¼ 3 ; ∂t ∂x2 n¼0 2 2 1 X ∂v ∂ v Cn ¼ ; 2 ∂x ∂x n¼0 " # 1 X ∂ ∂v 2 ∂ ∂2 v ; ð44Þ En ¼ ∂x ∂x ∂t ∂x2 n¼0
ð41Þ
Using the inverse operator Lx 1 on Eq. (33), we get
2 ∂v ∂ ∂ v αLx 1 Lx 1 Lx vðx; tÞ ¼ Lx 1 St þ ∂t ∂t ∂x2 " # 2 2 ∂v ∂v 2 ∂2 v 1 ∂ 1 β1 Lx 6βL x ∂x ∂x2 ∂t 2 ∂x2 " 3 2 # ∂ v ∂v 2 ∂ ∂2 v 1 ∂ 1 ∂ ; 2γLx γ 1 Lx ∂x ∂x ∂t ∂x2 ∂t 3 ∂x2
ð42Þ
v 0 þ v 1 þ v 2 þ … ¼ b0 þ b1 x x2 þSt αLx 1 ½A0 þ A1 þ A2 þ … 2 β1 Lx 1 ½B0 þ B1 þ B2 þ … 6βLx 1 ½C0 þ C1 þ C 2 þ … γ 1 Lx 1 ½D0 þ D1 þ D2 þ … γLx 1 ½E 0 þ E1 þ E 2 þ …; ð45Þ the components of velocity profile are obtained by comparing both side of Eq. (45).
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
Zeroth component problem: v0 ðx; t Þ ¼ b0 þ b1 x þ St
x2 ; 2
p1 : ð46Þ
Solution of zeroth component problem using boundary conditions given in Eqs. (15) and (16) is: v0 ðx; t Þ ¼ 1 þ cos ½ωt St St 2 1 þ cos ½ωt þ xþ x ; 2 2
ð47Þ
229
2 ∂2 v1 ∂v0 ∂2 v0 ∂v0 þ ¼ S þ c 1 þ c þ 6βc t 1 1 1 2 2 ∂x ∂t ∂x ∂x ∂v0 ∂ ∂v0 þ4c1 γ ∂x ∂t ∂x 2 2 ! ∂ ∂ v0 ∂v0 þc1 α þ 2γ ∂t ∂x2 ∂x ∂2 ∂2 v0 ∂3 ∂2 v0 þc1 β1 2 þ γ : ð52Þ ∂t ∂x2 ∂t 3 ∂x2
First component problem: v1 ðx; t Þ ¼ St aLx 1 ½A0 β1 Lx 1 ½B0 6βLx 1 ½C0 γ 1 Lx 1 ½D0 2γLx 1 ½E 0 ;
ð48Þ
Solution of first component problem using boundary conditions given in Eqs. (15) and (16) is: hωt i3 hωt i 1 v1 ðx; t Þ ¼ St 8γ cos sin ω sin ½ωt 2 2 3St hωt i4 hωt i2 12β cos 2St cos 2 2 hωt i hωt i βS2 2γω cos þ sin t x 3 2 2 4 htωi4 1 þ ω sin ½ωt þ 12βSt cos 2 2 hωt i3 hωt i hωt i2 8γωSt cos sin þ 6βSt 2 cos 2 2 2 hωt i hωt i 3βS3 t 2γωSt 2 cos sin þ x2 2 2 4 hωt i2 1 ω sin ½ωt þ 4βSt 2 cos 6 2 hωt i hωt i 4 3 3 2 γωSt cos sin þ βSt x 3 2 2 3 βSt 4 þ ð49Þ x; 2 Second component problem: 1 ∂v1 v2 ðx; t Þ ¼ Lx aLx 1 ½A1 β1 Lx 1 ½B1 ∂t 6βLx 1 ½C 1 γ 1 Lx 1 ½D1 2γLx 1 ½E 1 ;
ð50Þ
St St 2 xþ x; v0 ðx; tÞ ¼ 1 þ cos ½ωt 1 þ cos ½ωt þ ð53Þ 2 2 hωti hωt i3 2γS hωt i 1 t 8γ cos cos v1 ðx; tÞ ¼ c1 ωSt sin þ 2 2 2 3 3 htωi4 hωt i2 3βS3 t 12βSt cos 2βSt 2 cos x 2 2 4 hωt i3 hωt i 1 þ ω sin ½ωt 1 4γSt cos 4γSt 2 cos 2 2 2
hωti4 hωti2 3βS3 2 t þ12βSt cos þ 6βSt cos þ x2 2 2 4 hωt i 1 1 þ ω sin ½ωt 4γSt 2 cos 2 2 2 3 hωt i2 βS t βS3t x3 þ 4βSt 2 cos ð54Þ x4 : 2 2
Due to massive calculation, the analytical result have been given up to first term while graphical solutions are given up to the second order term. The value of ci for the lift velocity components are c1 ¼ 0:0517334034;
c2 ¼ 0:0272742812:
Now we consider the fourth grade fluid, falling on the stationary oscillating vertical belt. The fluid flow is due to gravity in downward direction. All the remaining assumptions are similar of the lift problem but Stock number is taken positive. Using value of T xy the governing equation of drainage problem become ∂v 2 ∂2 v ∂v ∂2 v ∂ ∂2 v ¼ μ 2 þ α1 þ β þ 6 β 2 3 ∂t ∂x ∂t ∂x2 ∂x ∂x2 3 2 ∂ ∂ v þγ 1 3 þ 2ð3γ 2 þ γ 3 þ γ 4 þ γ 5 þ 3γ 7 þ γ 8 Þ ∂t ∂x2 " # ∂ ∂v 2 ∂ ∂2 v þ ρg: ð55Þ ∂x ∂x ∂t ∂x2 ρ
4.4. OHAM solution for lift problem Apply the standard form of OHAM to Eq. (14), we found zero, first and second components of velocity problem ∂2 v0 ¼ St ; ∂x2
5. Formulation of drainage problem
due to massive calculation the analytical result have been declared up to first order while graphical solutions are given up to second order.
p0 :
Solution of zero and first component problem using boundary conditions given in Eqs. (15) and (16) is:
ð51Þ
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Taza Gul et al.
Figure 3 Comparison of OHAM and ADM methods for lift velocity profile by taking ω ¼ 0:02; α ¼ 0:02; β ¼ 0:2; β1 ¼ 0:1; γ ¼ 0:2; γ 1 ¼ 0:01; t ¼ 2; St ¼ 0:3; c1 ¼ 0:0517334034; c2 ¼ 0:0272742812.
First component problem:
Along with drainage boundary conditions vðx; tÞ ¼ U cos ωt;
x ¼ 0;
at
∂vðx; tÞ ¼ 0; at x ¼ δ: ∂x The dimensionless form of Eq. (55) is given as: ∂v ∂2 v ∂ ∂2 v ∂2 ∂2 v ¼ St þ 2 þ α þ β1 2 ∂t ∂x ∂t ∂x2 ∂t ∂x2 2 2 3 2 ∂v ∂ v ∂ ∂ v þ6β þ γ1 3 2 ∂x ∂x ∂t ∂x2 " 2 # 2 ∂ ∂v ∂ ∂ v þ2γ ; ∂x ∂x ∂t ∂x2
ð56Þ
∂vðx; tÞ ¼ 0; ∂x
at
at
x ¼ 0;
x ¼ 1:
v1 ðx; t Þ ¼ St aLx 1 ½A0 β1 Lx 1 ½B0 6βLx 1 ½C0 γ 1 Lx 1 ½D0 2γLx 1 ½E 0 :
ð63Þ
ð57Þ
ð58Þ
with boundary conditions regarding in Eqs. (56) and (57) vðx; tÞ ¼ cos ωt;
Figure 4 Comparison of OHAM and ADM methods for drainage velocity profile by taking ω ¼ 0:02; α ¼ 0:2; β ¼ 0:2; β1 ¼ 0:3; γ ¼ 0:2; γ 1 ¼ 0:04; t ¼ 5; St ¼ 0:4; c1 ¼ 0:2661918234; c2 ¼ 0:2382524332.
ð59Þ ð60Þ
5.1. The ADM solution of drainage problem From Eq. (58), we are applying the modified adomian decomposition method (MADM). For both lifting and drainage problems the modified adomian polynomial are same. After simplification zeroth, first and second velocity components problem are.
Solution of first component drainage problem using boundary condition in Eqs. (52) and (53) is: 1 v1 ðx; t Þ ¼ 3βSt cos ½ωt 2 þ ω sin ½ωt γSt 2 1 3 βS2 γωSt sin ½2ωt βSt 2 cos ½ωt þ t x 4 1 þ ω sin ½ωt 1 γSt 2 3βSt cos ½ωt 2 2 3βS3t 2 þγωSt sin ½2ωt þ 3βSt 2 cos ½ωt x 4 1 1 þ ω sin ½ωt 2γSt 2 3 2 3 βSt 4 3 3 2 2βSt cos ½ωt þ βSt x ð64Þ x : 2 Due to the lengthy calculation the solution of second component of velocity distribution using boundary conditions in Eqs. (59) and (60) is too large. Therefore, derivation is given up to the first order term while graphical solutions are given up to second order term.
5.2. OHAM solution of drainage problem
Zeroth component problem: x2 : ð61Þ 2 Solution of zeroth component problem using boundary conditions in Eqs. (52) and (53) is: St St 2 v0 ðx; t Þ ¼ cos ½ωt 1 þ cos ½ωt þ x x : ð62Þ 2 2 v0 ðx; t Þ ¼ b0 þ b1 x St
Apply the standard form of OHAM on Eq. (58), we found zero, first and second components of velocity problem. Zero component problem: p0 :
∂ 2 v0 ¼ St ; ∂x2
ð65Þ
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
Figure 5 In 3D graphs, effect of time level on lift velocity profile.
Figure 6 In 3D graphs, effect of time level on drainage velocity profile.
Figure 7 Lift velocity distribution of fluid when ω ¼ 0:02; α ¼ 0:2; β ¼ 0:4; β1 ¼ 0:5; γ ¼ 0:2; γ 1 ¼ 0:2; St ¼ 0:1:
Figure 8 Drainage velocity distribution of fluid when ω ¼ 0:02; α ¼ 0:2; β ¼ 0:4; β1 ¼ 0:5; γ ¼ 0:2; γ 1 ¼ 0:2; St ¼ 0:1:
v1 ðx; t Þ ¼ c1
First component problem: p1 :
∂2 v1 ∂v0 ¼ St þ c 1 St c 1 ∂x2 ∂t
2 ∂2 v0 ∂v0 þ 2 1 þ c1 þ 6βc1 ∂x ∂x ∂v0 ∂ ∂v0 þ4c1 γ ∂x ∂t ∂x 2 2 ! ∂ ∂ v0 ∂v0 α þ 2γ þc1 ∂t ∂x2 ∂x 2 2 3 2 ∂ ∂ v0 ∂ ∂ v0 þc1 β1 2 þγ 3 : 2 ∂t ∂x ∂t ∂x2
231
1 ω sin ½ωt γS2t 1 6γSt cos ½ωt 3
3βS3t þ3βSt cos ½ωt cos ½ωt þ x 4 1 þ ω sin ½ωt γS2t þ 2γSt cos ½ωt 2 3βS3t 2 3βSt cos ½ωt 2 þ 3βS2t cos ½ωt x 4 1 1 þ ω sin ½ωt 2γS2t 2βS2t cos ½ωt 3 2 3 βSt 4 3 3 þβSt x þ ð68Þ x : 2 2
ð66Þ
Solution of zeroth and first component problem using boundary conditions given in Eqs. (59) and (60): St St 2 v0 ðx; t Þ ¼ cos ½ωt 1 þ cos ½ωt x x : ð67Þ 2 2
βS2t
Due to massive calculation the analytical result have been given up to first term while graphical solutions are given up to second order. The value of ci for the drainage velocity components are: c1 ¼ 0:2661918234;
c2 ¼ 0:2382524332:
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Figure 9 Effect of non-Newtonian parameter β1 on lift velocity profile when ω ¼ 5; α ¼ 0:2; β ¼ 0:4; γ ¼ 0:2; γ 1 ¼ 0:2; St ¼ 0:1.
Figure 10 Effect of β1 on drainage velocity when ω ¼ 5; α ¼ 0:2; β ¼ 0:4; γ ¼ 0:2; γ 1 ¼ 0:2; St ¼ 0:1.
Figure 12 Effect of β on drainage velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:7; γ ¼ 2; γ 1 ¼ 0:2; St ¼ 0:2.
Figure 11 Effect of third grade parameter β on lift velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:7; γ ¼ 2; γ 1 ¼ 0:2; St ¼ 0:2.
Figure 13 Effect of fourth grade parameter γ on lift velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:7; t ¼ 3; γ 1 ¼ 0:2; St ¼ 0:2.
6. Results and discussion
Tables 1 and 2 and graphically in Figures 3 and 4 for different values of embedded parameters. Whereas, the effects of other physical parameters t; St ; β; γ; β1 and γ 1 on velocity profile for both lift and drainage problems are studied in Figures 5–18. All results for the thin film of fourth grade fluid near the belt are illustrated in the x-coordinate only for a selected domain x A ½0; 1. In Figures 5–8, the velocity profiles are studied versus time for different values of independent variable x: Due to the no-slip condition, the fluid near the belt oscillates
Analytical solutions for unsteady thin film flow of a fourth grade fluid through a moving and oscillating vertical belt are obtained. The physical configuration of the problem is shown in Figures 1 and 2. Two different techniques namely ADM and OHAM are used for the solutions of lift and drainage problems. The results of velocity obtained from both methods for lift and drainage problems are compared numerically in
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
Figure 14 Effect of fourth grade γ on drainage velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:7; t ¼ 3; γ 1 ¼ 0:2; St ¼ 0:2.
Figure 15 Effect of the parameter γ 1 on lift velocity when ω ¼ 3; α ¼ 0:2; β1 ¼ 0:7; γ ¼ 0:8; t ¼ 2; St ¼ 0:2.
Figure 16 Effect of the parameter γ 1 on drainage velocity when ω ¼ 3; α ¼ 0:2; β1 ¼ 0:7; γ ¼ 0:8; t ¼ 2; St ¼ 0:2.
jointly with the belt in the same period and oscillation reduces gradually towards the free surface shown in Figures 5 and 6. Increasing domain from (0,1) velocity amplitude raises gradually towards the surface of the fluid layer shown in Figures 7 and 8. It is found that velocity accepts and oscillating behavior. Generally, due to larger apparent viscosity, non-Newtonian fluid exhibits much thicker boundary layer than that of Newtonian fluid. With the increase in fourth
233
Figure 17 Effect of the stock number St on lift velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:3; γ ¼ 0:5; γ 1 ¼ 0:3; β ¼ 0:4; t ¼ 5.
Figure 18 Effect of the St on drainage velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:3; γ ¼ 0:5; γ 1 ¼ 0:3; β ¼ 0:4; t ¼ 5.
grade parameters the apparent viscosity of the fluid become more bulky (viscous forces dominate), then the flow simultaneously adjust to the present driving force and oscillate closely with the similar phase in the whole flow domain. Therefore, increasing non-Newtonian parameters α; β; γ; β1 and γ 1 of second, third and fourth grade fluids causes further thickening of the boundary layer. So, increase in these higher order parameter increases velocity profiles of both lift and drainage problems shown in Figures 9–16. Figures 17 and 18 display the effects of St on lift and drainage velocities. From Figure 17, we observed that by increasing the value of St the lift velocity decreases whereas Figure 18 shows that drainage velocity increases by increasing the St : The reason is that the friction force cause the effect of gravity and it seems to be smaller near the belt. While it increasing gradually towards the free surface.
7. Conclusion In this article, we consider the unsteady flow of fourth grade fluid on moving and oscillating vertical belt. The analytical solution of lift and drainage velocity field achieved by applying two techniques namely ADM and OHAM. The result obtained from both techniques compared numerically
234
and graphically. The result of ADM and OHAM close associated to each other. The effects of model parameters are presented graphically on lift and drainage velocity field. 1. If the apparent viscosity of the fluid is more bulky (viscous forces dominate), then the flow simultaneously adjust to the present driving force and oscillate closely with the similar phase in the whole flow domain. 2. Due to oscillation of the belt, we observed that the fluid motion is maximum at the surface of the belt and minimum at the surface of the fluid. 3. The higher order parameters are more efficient in the increase of fluid motion as one go from belt towards free surface.
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Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt [32] F. Mabood, W.A. Khan, A.I.M. Ismail, Optimal homotopy asymptotic method for flow and heat transfer of a viscoelastic fluid in an axisymmetric channel with a porous wall, PLoS One 8 (12) (2013) e83581, http://dx.doi.org/10.1371/journal. pone.0083581. [33] J.A. Khan, M. Mustafa, T. Hayat, A. Alsaedi, On threedimensional flow and heat transfer over a non-linearly stretching sheet analytical and numerical solutions, PLoS One 9 (9) (2014) e107287, http://dx.doi.org/10.1371/journal. pone.0107287. [34] T. Aziz, R.J. Moitsheki, A. Fatima, F.M. Mahomed, Shock wave solutions for some nonlinear flow models arising in the study of a non-Newtonian third grade fluid Article ID 602902, Math. Probl. Eng. (2013) 1–6, http://dx.doi.org/ 10.1155/2013/602902. [35] M. Ahmad, H. Zaman, N. Rehman, Effects of hall current on unsteady MHD flows of a second grade fluid, Cent. Eur. J. Phys. 8 (2010) 422–431.
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ORIGINAL ARTICLE
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt Taza Gula, Fazle Ghania, S. Islama, R.A. Shahb, I. Khanc,n, Saleem Nasira, S. Sharidand a
Department Department c Department d Department b
of of of of
Mathematics, Abdul Wali Khan University, Mardan, KPK, Pakistan Mathematics, University Engineering Technology Peshawar, KPK, Pakistan Basic Sciences, College of Engineering Majmaah University, P.O. Box 66, Majmaah 11952, Saudi Arabia Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, UTM Johor Bahru, 81310 Johor, Malaysia
Received 14 April 2015; accepted 17 July 2015 Available online 18 August 2016
KEYWORDS Unsteady flow; Thin film; Lifting and drainage; Fourth grade fluid; Adomian decomposition method (ADM); Optimal homotopy asymptotic method (OHAM)
Abstract This article studies the unsteady thin film flow of a fourth grade fluid over a moving and oscillating vertical belt. The problem is modeled in terms of non-nonlinear partial differential equations with some physical conditions. Both problems of lift and drainage are studied. Two different techniques namely the adomian decomposition method (ADM) and the optimal homotopy asymptotic method (OHAM) are used for finding the analytical solutions. These solutions are compared and found in excellent agreement. For the physical analysis of the problem, graphical results are provided and discussed for various embedded flow parameters. & 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Due to the diverse physical structures of non-Newtonian fluids, several constitutive models have been proposed in the literature to predict all of their salient features. n
Corresponding author.
E-mail address: [email protected] (I. Khan). Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.
Generally, there are three non-Newtonian fluids models. They are known as (i) the differential type, (ii) the rate type, and (iii) the integral type. However, the most famous amongst them are the first two models. In this article, we will study the first model, the differential type fluid also known as grade n fluids and consider its subclass known as the fourth grade fluid. The simplest subclass of differential type is the second grade fluid for which one can reasonably hope to obtain an analytical solution. Although a second grade fluid model also known as grade
2212-540X & 2016 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/j.jppr.2016.07.002
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2 is able to predict the normal stress differences but it does not take into account the shear thinning and shear thickening phenomena that many fluids exhibit [1–4]. On the other hand, a third grade fluid model (grade 3 fluid) represents a more realistic description of the behavior of non-Newtonian fluids. This model is known to capture the non-Newtonian effects such as shear thinning or shear thickening as well as normal stresses [5–8]. There is another model for differential type fluids known as fourth grade fluid model (grade 4 fluid) which at one time capture most of the non-Newtonian flow properties. Means, Cauchy stress tensor appearing in this fluid model contains several parameters. Due to which the governing equation for fourth grade fluid becomes quite complicated. This model is also very capable to predict the effects of normal stresses that lead to phenomena like rod-climbing and dieswell [9]. Therefore, very few studies, which illustrates the effect of fourth grade fluids even on steady flow, have been reported in the literature. In general, it is not easy to study fourth grade fluid even for steady flow problems. However, under certain assumptions based on the flow situation, it is possible and some investigations are carried out in this direction [10–14]. On the other hand, the science of thin liquid films has developed rapidly in recent years. Thin film flow problems appear in many fields, varying from specific situations in the flow in human lunges to lubrication problems in engineering which is probably one of the largest subfield of thin film flow problems [15]. Few other applications are found in coating flows, biofluids, microfluidic engineering, and medicine [16,17]. Two well know examples from everyday life are slipping on a wet bathroom floor and aquaplaning or hydroplaning on a wet road. The study of thin film flow for practical applications is a challenging interplay between fluid mechanics, structural mechanics and theology. Based on this motivation, recently several researchers are getting interested to study the thin film flows. Amongst them, we discuss here some important contributions of the following researchers. Siddiqui et al. [18,19] discussed the analytical solution of thin film flow of third grade fluid and Oldroyed-8 constant fluid through vertical moving belt by applying adomian decomposition method (ADM) and variational iteration method (VIM). Aiyesimi et al. [20] investigated the unsteady magnetohydrodynamic (MHD) thin film flow of a third grade fluid with heat transfer and no slip boundary condition down an inclined plane. Gul et al. [21,22] studied analytically the thin film flows of second grade and third grade fluids through a vertical belt for lift and drainage problems using optimal homotopy asymptotic method (OHAM) and ADM. Few other investigations in this direction are mentioned in Refs. [23–40]. The basic theme of this article is to venture further in the regime of thin film flows and to extend this idea to the fourth grade fluid executing the unsteady motion over a
Taza Gul et al.
vertical oscillating belt using two analytical techniques namely the OHAM and ADM methods. OHAM on the other hand, is a new powerful technique which is a generalized form of homotopy perturbation method (HPM) and HAM. It is a straightforward technique and does not require the existence of any small or large parameter as does the traditional perturbation method. OHAM has successfully applied to a number of nonlinear problems arising in the science and engineering by various researchers. This proves the validity and acceptability of OHAM as a useful technique. For the correctness and verification of OHAM solutions, they are compared with those obtained by ADM. The rest of the paper is arranged as follows: The constitutive equations for fourth grade fluid are given in Section 2. Using these constitutive equations for unidirectional and one dimension flow, the continuity equation is satisfied identically and the momentum equation is derived in Section 3 with some physical conditions. In Section 4, the basic concepts of ADM and OHAM are discussed. These methods are used in Section 5 where the analytical solutions for both lifting and drainage problems are obtained. The graphical results are given in Section 6. This paper ends with some conclusions in Section 7.
2. Fundamental equation The constitutive equations for fourth grade fluid model is T ¼ pI þ μA1 þ
3 X
Si ;
ð1Þ
i¼1
where T is the Cauchy stress tensor, I is identity tensor, p is fluid pressure and Si is the extra stress tensor S1 ¼ α1 A2 þ α2 A21 ; S2 ¼ β1 A3 þ β1 ðA1 A2 þ A2 A1 Þ þ β3 trA22 A1 ; S3 ¼ γ 1 A4 þ γ 2 ðA3 A1 þ A1 A3 Þ þ γ 3 A22 þγ 4 ðA2 A21 þ A21 A2 Þ þ γ 5 ðtrA2 ÞA2 þ γ 6 ðtrA2 ÞA21 þfγ 7 ðtrA3 Þ þ γ 8 ðtrA2 A1 ÞgA1 :
ð2Þ
Here μ is coefficient of viscosity, αi ði ¼ 1; 2Þ; βj ðj ¼ 1; 2; 3Þ and γ k ðk ¼ 1; 2; …; 8Þ are material constants of second, third and fourth grades respectively. Rivlin–Ericksen tensors A1 ; A2 ; A3 and A4 are defined as: A0 ¼ I; A1 ¼ ð∇V Þ þ ð∇V ÞT ; An ¼
DAn 1 þ An 1 ð∇V Þ þ ð∇V ÞT An 1 ; Dt
ð3Þ n ¼ 2; 3; 4:
ð4Þ The continuity and momentum equation for incompressible fluid are given by
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
div V ¼ 0; ρ
225
ð5Þ
DV ¼ div T þ ρg; Dt
ð6Þ
where V ¼ ðu; v; wÞ is the velocity vector, ρ is the fluid D density, Dt is the material time derivative.
3. Formulation of lift problem We consider an oscillating and moving wide flat belt place in a large bath of a fourth grade fluid. The belt is moving in vertically upward direction with velocity U and pick up a thin film of a fourth grade fluid of constant thickness δ. The x- and y-axes are chosen in such a way that the belt is parallel to y-axis and perpendicular to x-axis as shown in the physical configuration (see Figures 1 and 2). The pressure gradient is not considered here and the flow is assumed as unsteady and laminar. By taking velocity field of the form V ¼ ð0; vðx; tÞ; 0Þ;
ð7Þ
Subject to the following boundary conditions vðx; tÞ ¼ U þ U cos ωt; ∂vðx; tÞ ¼ 0; ∂x
at
at
x ¼ 0;
x ¼ δ;
Here ω is used as frequency of the oscillating belt.
ð8Þ ð9Þ
Figure 2 Geometry of drainage problem.
Using the velocity field given in Eq. (7), the continuity Eq. (5) is identically satisfied and the momentum Eq. (6) reduces to the form ρ
∂v ∂ ¼ T xy ρg: ∂t ∂x
ð10Þ
From Eq. (1) the component T xy of Cauchy stress component is ∂v ∂ ∂v ∂2 ∂v þ β1 2 T xy ¼ μ þ α1 ∂x ∂t ∂x ∂t ∂x 3 ∂v ∂3 ∂v þ2 β2 þ β3 þ γ1 3 ∂x ∂t ∂x ∂ ∂v 3 : ð11Þ þ2ð3γ 2 þ γ 3 þ γ 4 þ γ 5 þ 3γ 7 þ γ 8 Þ ∂t ∂x Inserting Eq. (11) into Eq. (10), we get ∂v 2 ∂2 v ∂v ∂2 v ∂ ∂2 v ρ ¼ μ 2 þ α1 þ 6 β2 þ β 3 ∂t ∂x ∂t ∂x2 ∂x ∂x2 ∂3 ∂2 v þγ 1 3 þ 2ð3γ 2 þ γ 3 þ γ 4 þ γ 5 ∂t ∂x2 " # ∂ ∂v 2 ∂ ∂2 v þ3γ 7 þ γ 8 Þ ρg: ð12Þ ∂x ∂x ∂t ∂x2
Figure 1
Geometry of lift problem.
Introducing the following non-dimensional variables as mentioned in Refs. [13,21,22].
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v x μt δ2 ρg α1 ; x~ ¼ ; t~ ¼ 2 ; St ¼ ; α~ ¼ 2 ; U δ k ρδ ρδ β 2 þ β3 U 2 β μ γ μ2 ; β~ 1 ¼ 12 ; γ~ 1 ¼ 13 4 ; β~ ¼ δ ρ δ μδ2 γ U2 ð13Þ γ~ ¼ 3γ 2 þ γ 3 þ γ 4 þ γ 5 þ 3γ 7 þ γ 8 1 4 : ρδ
v~ ¼
Here t is a dimensionless time parameter, α~ is dimension~ β~ 1 are less second grade non-Newtonian parameter, β; dimensionless third grade non-Newtonian parameters and γ~ ; γ~ 1 are dimensionless fourth grade non-Newtonian parameters. Inserting above non-dimensional variables into Eq. (12) and boundary conditions (8) and (9), we get: ∂v ∂2 v ∂ ∂2 v ∂2 ∂2 v ¼ St þ 2 þ α þ β1 2 ∂t ∂x ∂t ∂x2 ∂t ∂x2 2 2 3 2 ∂v ∂ v ∂ ∂ v þ6β þ γ1 3 ∂x ∂x2 ∂t ∂x2 " # ∂ ∂v 2 ∂ ∂2 v þ2γ ; ð14Þ ∂x ∂x ∂t ∂x2 vðx; tÞ ¼ 1 þ cos ωt; ∂vðx; tÞ ¼ 0; ∂x em.
at
at
x ¼ 0;
x ¼ 1:
ð15Þ ð16Þ
source term, Ruðx; tÞ is a remaining linear term and Nuðx; tÞ is non-linear analytical term expandable in the adomian polynomials An . After applying the inverse operator Lx 1 to both sides of Eq. (19), we write Lx 1 Lx uðx; tÞ ¼ Lx 1 gðx; tÞ Lx 1 Lt uðx; tÞ Lx 1 Ruðx; tÞ Lx 1 Nuðx; tÞ;
ð20Þ
uðx; tÞ ¼ f ðx; tÞ Lx 1 Lt uðx; tÞ Lx 1 Ruðx; tÞ Lx 1 Nuðx; tÞ:
ð21Þ
Here the function f ðx; tÞ represents the terms arising from Lx 1 gðx; tÞ after using the given conditions. Lx 1 ¼ ∬ ðÞdxdx is used as inverse operator for the second order partial differential equation. In this method, the series solution uðx; tÞ is defined as: X1 uðx; tÞ ¼ u ðx; tÞ; ð22Þ n¼0 n X1
X1 1 u ðx; tÞ ¼ f ðx; tÞ L L u ðx; tÞ n t x n¼0 n¼0 n X1 1 Lx R un ðx; tÞ Xn1¼ 0 1 Lx N u ðx; tÞ: n¼0 n
ð23Þ
The non-linear term is expanded in adomian polynomials as: X1 N ðvðx; tÞÞ ¼ A : ð24Þ n¼0 n where the components u0 ðx; tÞ; u1 ðx; tÞ; u2 ðx; tÞ; … are periodically derived as:
4. Solution techniques 4.1. Basic concept of adomian decomposition method (ADM) ADM is used to decompose the unknown function uðx; tÞ into a sum of an infinite number of components defined by the decomposition series. X1 uðx; tÞ ¼ u ðx; tÞ: ð17Þ n¼0 n The decomposition method is used to find the components u0 ðx; tÞ; u1 ðx; tÞ; u2 ðx; tÞ; … separately. The determination of these components can be obtained through simple integrals. To give a clear overview of ADM, we consider the partial differential equation in an operator form as Lt uðx; tÞ þ Lx uðx; tÞ þ Ruðx; tÞ þ Nuðx; tÞ ¼ gðx; tÞ;
ð18Þ
Lx uðx; tÞ ¼ gðx; tÞ Lt uðx; tÞ Ruðx; tÞ Nuðx; tÞ:
ð19Þ
where Lx ¼ ∂x∂ 2 and Lt ¼ ∂t∂ are linear operators in the partial differential equation and are easily invertible, gðx; tÞ is a 2
u0 ðx; tÞ þ u1 ðx; tÞ þ u2 ðx; tÞ þ … ¼ f ðx; tÞ Lx 1 Rðu0 ðx; tÞ þ u1 ðx; tÞ þ u2 ðx; tÞ þ…Þ Lx 1 ðA0 þ A1 þ A2 þ …Þ:
ð25Þ
To determine the components of the series u0 ðx; tÞþ u1 ðx; tÞ þ u2 ðx; tÞ þ …, it is important to note that ADM suggests that the zeroth component u0 ðx; tÞ is usually defined by the function f ðx; tÞ described above. The formal recursive relation is defined as: u0 ðx; tÞ ¼ f ðx; tÞ; u1 ðx; tÞ ¼ Lx 1 Lt ðu0 ðx; tÞÞ Lx 1 Rðu0 ðx; tÞÞ Lx 1 ðA0 Þ; u2 ðx; tÞ ¼ Lx 1 Lt ðu1 ðx; tÞÞ Lx 1 Rðu1 ðx; tÞÞ Lx 1 ðA1 Þ; u3 ðx; tÞ ¼ Lx 1 Lt ðu2 ðx; tÞÞ Lx 1 Rðu2 ðx; tÞÞ Lx 1 ðA2 Þ;
ð26Þ
and so on.
4.2. Optimal homotopy asymptotic method (OHAM) For the analysis of OHAM, we consider the boundary value problem as consider in Refs. [12,13]: ∂u~ ~ ~L ðuðxÞ ~ ~ ~ Þ þ N ðuðxÞ ~ Þ þ GðxÞ ¼ 0; B u; ~ ¼ 0; ð27Þ ∂x
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
227
Table 1 Comparison of OHAM and ADM for the lift velocity profile, by taking ω ¼ 0:02; α ¼ 0:02; β ¼ 0:2; β1 ¼ 0:1; γ ¼ 0:2; γ 1 ¼ 0:01; t ¼ 2; St ¼ 0:3; c1 ¼ 0:0517334034; c2 ¼ 0:0272742812: x
OHAM
ADM
Absolute error
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.9998000 1.7966503 1.5941974 1.3924453 1.1913952 0.9910501 0.7914146 0.5924887 0.3942759 0.1967780 1:3 10 18
1.9998000 1.7638191 1.5363173 1.3170952 1.1060142 0.9028535 0.7074496 0.5196317 0.3392280 0.1660729 1:4 10 17
0 0.0328310 0.0578801 0.0753353 0.0853805 0.0881976 0.0839647 0.0728568 0.0550479 0.0307064 1:3 10 17
where L~ a linear operator in the differential equation is, N~ is a non-linear term, x A R is an independent variable, B~ is a ~ is a source term. According to boundary operator and G Refs. [12,13] we construct a set of equation for OHAM: ~ ~ ½1 p L~ ðuðxÞ ~ Þ þ GðxÞ þ N~ ϕ~ ðx; pÞ ; ¼ Hðp; ci Þ L~ ϕ~ ðx; pÞ þ GðxÞ
∂ϕ~ B~ ϕ~ ðx; pÞ; ¼ 0: ∂x
∂u~ 2 ðxÞ ¼ 0: B~ u~ 2 ðxÞ; ∂x
ð34Þ
ð28Þ The general governing equations for u~ k ðxÞ are given by
Here pA ½0; 1 is an embedding parameter, Hðp; ci Þ is a non-zero auxiliary function for p a 0 and Hð0; ci Þ ¼ 0, ϕ~ ðx; pÞ is an unknown function. Obviously, when p ¼ 0 and p ¼ 1, it holds that: ~ ϕ~ ðx; 0Þ ¼ u~ 0 ðxÞ; ϕ~ ðx; 1Þ ¼ uðxÞ; ð29Þ Note that, when p varies from 0 to 1 then ϕ~ ðx; pÞ also ~ varies from u~ 0 ðxÞ to uðxÞ: The zeroth component solution u~ 0 ðxÞ is obtained from Eq. (28) when p ¼ 0 i.e ∂u~ 0 ðxÞ ~ L~ ðu~ 0 ðxÞÞ þ GðxÞ ¼ 0; B~ u~ 0 ðxÞ; ¼ 0; ð30Þ ∂x auxiliary function Hðp; ci Þ is chosen as: Hðp; ci Þ ¼ pc1 þ p2 c2 þ …
X
u ðx; ci Þpk ; kZ1 k
ð31Þ
i ¼ 1; 2; …:;
ð32Þ
k 1 X ci L~ ðu~ k i ðxÞÞ i¼0
þN~ k 1 ðu~ 0 ðxÞ; u~ 1 ðxÞ; …; u~ k 1 ðxÞÞ ; ∂u~ k ðxÞ ¼ 0: B~ u~ k ðxÞ; ∂x
k ¼ 2; 3 ð35Þ
Here N~ m ðu~ 0 ðxÞ; u~ 1 ðxÞ; …; u~ m 1 ðxÞ Þ is the coefficient of ~ pÞ : pm , in the expansion of N~ ϕðx; ~ p; ci Þ ¼ N~ 0 ðu~ 0 ðxÞÞ N~ ϕðx; þ
1 X
N~ k 1 ðu0 ðxÞ; u~ 1 ðxÞ; …u~ m ðxÞÞpm ;
ð36Þ
the convergence of the series in Eq. (32) depend upon the convergence control parameters c1 ; c2 ; …. If it converges at p ¼ 1 then the mth order approximation u~ is ~ c1 ; c2 ; c3 ; …; cm Þ ¼ u~ 0 ðxÞ þ uðx;
m X
u~ i ðx; c1 ; c2 ; c3 ; …; ci Þ;
ð37Þ
i¼1
Inserting Eq. (32) into Eq. (28), collecting the same powers of p, and equating each coefficient of p, the zero order problem is given in Eq. (30) and the first and second order are given in Eqs. (33) and (34). ~ L~ ðu~ 1 ðxÞÞ þ GðxÞ ¼ c1 N~ 0 ðu~ 0 ðxÞÞ; ~B u~ 1 ðxÞ; ∂u~ 1 ðxÞ ¼ 0; ∂x
L~ ðu~ k ðxÞÞ L~ ðu~ k 1 ðxÞÞ ¼ ck N~ 0 ðu~ 0 ðxÞÞ þ
m¼1
where c1 ; c2 are auxiliary constants. Marinca [12,13] uses a special procedure to expand ϕ~ ðx; pÞ with respect to p by using Taylor series. ϕ~ ðx; p; ci Þ ¼ u~ 0 ðxÞ þ
L~ ðu~ 2 ðxÞÞ L~ ðu~ 1 ðxÞÞ ¼ c2 N~ 0 ðu~ 0 ðxÞÞ þ c1 L~ ðu~ 1 ðxÞÞN~ 0 ðu~ 0 ðxÞ; u~ 1 ðxÞÞ ;
ð33Þ
inserting Eq. (36) into Eq. (27), the residual is obtained as: ~ ~ ci Þ ¼ L~ ðuðx; ~ ci ÞÞ þ N~ ðuðx; ~ ci ÞÞ þ GðxÞ; Rðx; i ¼ 1; 2; …; m;
ð38Þ
Numerous methods like Ritz method, method of least squares, Galerkin method and collocation method are used to find the optimal values of ci ; i ¼ 1; 2; 3; ….
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Table 2 Comparison of OHAM and ADM for the lift velocity profile, by taking ω ¼ 0:02; α ¼ 0:2; β ¼ 0:2; β1 ¼ 0:3; γ ¼ 0:2; γ 1 ¼ 0:04; t ¼ 5; St ¼ 0:4; c1 ¼ 0:2661918234; c2 ¼ 0:2382524332: x
OHAM
ADM
Absolute error
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.9950041 0.9011902 0.8060385 0.7095748 0.6118278 0.5128257 0.4125973 0.3111713 0.2085768 0.1048432 1:8 10 17
0.9950041 0.9071647 0.8169358 0.7242514 0.6290402 0.5312253 0.4307243 0.3274490 0.2213048 0.1121909 6:6 10 18
0 0.00597381 0.01089723 0.01467664 0.01721239 0.01839954 0.01812704 0.01627770 0.01272798 0.00734773 1:2 10 17
We apply the method of least squares in our problem as given below: Z b 2 ð39Þ Jðc1 ; c2 ; c3 ; …; cn Þ ¼ R~ ðx; c1 ; c2 ; c3 ; …; cm Þdx: where a and b are the constant values taking from domain of the problem. Convergence control parameters ðc1 ; c2 ; c3 ; …; cn Þ can be identified from: ð40Þ
Finally, from these convergence control parameters, the approximate solution is well determined.
4.3. The ADM solution of lift problem Rewrite Eq. (14) of the form of Eq. (18), we get 2 ∂v ∂ ∂2 v ∂2 ∂2 v ∂v α β 6β 1 ∂t ∂t ∂x2 ∂x ∂t 2 ∂x2 " # 2 ∂ v ∂3 ∂2 v ∂ ∂v 2 ∂ ∂2 v γ 1 3 2γ ; 2 2 ∂x ∂x ∂x ∂t ∂x2 ∂t ∂x
n¼0
" # " # 1 1 X X x2 αLx 1 An β1 Lx 1 Bn 2 n¼0 n¼0 " # " # " # 1 1 1 X X X 1 1 1 C n γ 1 Lx Dn 2γLx En ; 6βLx
vðx; tÞ ¼ b0 þ b1 x þ St
n¼0
a
∂J ∂J ∂J ¼ ¼ ¼ … ¼ 0: ∂c1 ∂c2 ∂c3
1 X
Lx vðx; tÞ ¼ St þ
n¼0
n¼0
ð43Þ where An ; Bn ; C n ; Dn and En are adomian polynomials and define as 1 X ∂ ∂2 v An ¼ ; ∂t ∂x2 n¼0 1 X ∂2 ∂2 v Bn ¼ 2 ; ∂t ∂x2 n¼0 1 X ∂3 ∂2 v Dn ¼ 3 ; ∂t ∂x2 n¼0 2 2 1 X ∂v ∂ v Cn ¼ ; 2 ∂x ∂x n¼0 " # 1 X ∂ ∂v 2 ∂ ∂2 v ; ð44Þ En ¼ ∂x ∂x ∂t ∂x2 n¼0
ð41Þ
Using the inverse operator Lx 1 on Eq. (33), we get
2 ∂v ∂ ∂ v αLx 1 Lx 1 Lx vðx; tÞ ¼ Lx 1 St þ ∂t ∂t ∂x2 " # 2 2 ∂v ∂v 2 ∂2 v 1 ∂ 1 β1 Lx 6βL x ∂x ∂x2 ∂t 2 ∂x2 " 3 2 # ∂ v ∂v 2 ∂ ∂2 v 1 ∂ 1 ∂ ; 2γLx γ 1 Lx ∂x ∂x ∂t ∂x2 ∂t 3 ∂x2
ð42Þ
v 0 þ v 1 þ v 2 þ … ¼ b0 þ b1 x x2 þSt αLx 1 ½A0 þ A1 þ A2 þ … 2 β1 Lx 1 ½B0 þ B1 þ B2 þ … 6βLx 1 ½C0 þ C1 þ C 2 þ … γ 1 Lx 1 ½D0 þ D1 þ D2 þ … γLx 1 ½E 0 þ E1 þ E 2 þ …; ð45Þ the components of velocity profile are obtained by comparing both side of Eq. (45).
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
Zeroth component problem: v0 ðx; t Þ ¼ b0 þ b1 x þ St
x2 ; 2
p1 : ð46Þ
Solution of zeroth component problem using boundary conditions given in Eqs. (15) and (16) is: v0 ðx; t Þ ¼ 1 þ cos ½ωt St St 2 1 þ cos ½ωt þ xþ x ; 2 2
ð47Þ
229
2 ∂2 v1 ∂v0 ∂2 v0 ∂v0 þ ¼ S þ c 1 þ c þ 6βc t 1 1 1 2 2 ∂x ∂t ∂x ∂x ∂v0 ∂ ∂v0 þ4c1 γ ∂x ∂t ∂x 2 2 ! ∂ ∂ v0 ∂v0 þc1 α þ 2γ ∂t ∂x2 ∂x ∂2 ∂2 v0 ∂3 ∂2 v0 þc1 β1 2 þ γ : ð52Þ ∂t ∂x2 ∂t 3 ∂x2
First component problem: v1 ðx; t Þ ¼ St aLx 1 ½A0 β1 Lx 1 ½B0 6βLx 1 ½C0 γ 1 Lx 1 ½D0 2γLx 1 ½E 0 ;
ð48Þ
Solution of first component problem using boundary conditions given in Eqs. (15) and (16) is: hωt i3 hωt i 1 v1 ðx; t Þ ¼ St 8γ cos sin ω sin ½ωt 2 2 3St hωt i4 hωt i2 12β cos 2St cos 2 2 hωt i hωt i βS2 2γω cos þ sin t x 3 2 2 4 htωi4 1 þ ω sin ½ωt þ 12βSt cos 2 2 hωt i3 hωt i hωt i2 8γωSt cos sin þ 6βSt 2 cos 2 2 2 hωt i hωt i 3βS3 t 2γωSt 2 cos sin þ x2 2 2 4 hωt i2 1 ω sin ½ωt þ 4βSt 2 cos 6 2 hωt i hωt i 4 3 3 2 γωSt cos sin þ βSt x 3 2 2 3 βSt 4 þ ð49Þ x; 2 Second component problem: 1 ∂v1 v2 ðx; t Þ ¼ Lx aLx 1 ½A1 β1 Lx 1 ½B1 ∂t 6βLx 1 ½C 1 γ 1 Lx 1 ½D1 2γLx 1 ½E 1 ;
ð50Þ
St St 2 xþ x; v0 ðx; tÞ ¼ 1 þ cos ½ωt 1 þ cos ½ωt þ ð53Þ 2 2 hωti hωt i3 2γS hωt i 1 t 8γ cos cos v1 ðx; tÞ ¼ c1 ωSt sin þ 2 2 2 3 3 htωi4 hωt i2 3βS3 t 12βSt cos 2βSt 2 cos x 2 2 4 hωt i3 hωt i 1 þ ω sin ½ωt 1 4γSt cos 4γSt 2 cos 2 2 2
hωti4 hωti2 3βS3 2 t þ12βSt cos þ 6βSt cos þ x2 2 2 4 hωt i 1 1 þ ω sin ½ωt 4γSt 2 cos 2 2 2 3 hωt i2 βS t βS3t x3 þ 4βSt 2 cos ð54Þ x4 : 2 2
Due to massive calculation, the analytical result have been given up to first term while graphical solutions are given up to the second order term. The value of ci for the lift velocity components are c1 ¼ 0:0517334034;
c2 ¼ 0:0272742812:
Now we consider the fourth grade fluid, falling on the stationary oscillating vertical belt. The fluid flow is due to gravity in downward direction. All the remaining assumptions are similar of the lift problem but Stock number is taken positive. Using value of T xy the governing equation of drainage problem become ∂v 2 ∂2 v ∂v ∂2 v ∂ ∂2 v ¼ μ 2 þ α1 þ β þ 6 β 2 3 ∂t ∂x ∂t ∂x2 ∂x ∂x2 3 2 ∂ ∂ v þγ 1 3 þ 2ð3γ 2 þ γ 3 þ γ 4 þ γ 5 þ 3γ 7 þ γ 8 Þ ∂t ∂x2 " # ∂ ∂v 2 ∂ ∂2 v þ ρg: ð55Þ ∂x ∂x ∂t ∂x2 ρ
4.4. OHAM solution for lift problem Apply the standard form of OHAM to Eq. (14), we found zero, first and second components of velocity problem ∂2 v0 ¼ St ; ∂x2
5. Formulation of drainage problem
due to massive calculation the analytical result have been declared up to first order while graphical solutions are given up to second order.
p0 :
Solution of zero and first component problem using boundary conditions given in Eqs. (15) and (16) is:
ð51Þ
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Figure 3 Comparison of OHAM and ADM methods for lift velocity profile by taking ω ¼ 0:02; α ¼ 0:02; β ¼ 0:2; β1 ¼ 0:1; γ ¼ 0:2; γ 1 ¼ 0:01; t ¼ 2; St ¼ 0:3; c1 ¼ 0:0517334034; c2 ¼ 0:0272742812.
First component problem:
Along with drainage boundary conditions vðx; tÞ ¼ U cos ωt;
x ¼ 0;
at
∂vðx; tÞ ¼ 0; at x ¼ δ: ∂x The dimensionless form of Eq. (55) is given as: ∂v ∂2 v ∂ ∂2 v ∂2 ∂2 v ¼ St þ 2 þ α þ β1 2 ∂t ∂x ∂t ∂x2 ∂t ∂x2 2 2 3 2 ∂v ∂ v ∂ ∂ v þ6β þ γ1 3 2 ∂x ∂x ∂t ∂x2 " 2 # 2 ∂ ∂v ∂ ∂ v þ2γ ; ∂x ∂x ∂t ∂x2
ð56Þ
∂vðx; tÞ ¼ 0; ∂x
at
at
x ¼ 0;
x ¼ 1:
v1 ðx; t Þ ¼ St aLx 1 ½A0 β1 Lx 1 ½B0 6βLx 1 ½C0 γ 1 Lx 1 ½D0 2γLx 1 ½E 0 :
ð63Þ
ð57Þ
ð58Þ
with boundary conditions regarding in Eqs. (56) and (57) vðx; tÞ ¼ cos ωt;
Figure 4 Comparison of OHAM and ADM methods for drainage velocity profile by taking ω ¼ 0:02; α ¼ 0:2; β ¼ 0:2; β1 ¼ 0:3; γ ¼ 0:2; γ 1 ¼ 0:04; t ¼ 5; St ¼ 0:4; c1 ¼ 0:2661918234; c2 ¼ 0:2382524332.
ð59Þ ð60Þ
5.1. The ADM solution of drainage problem From Eq. (58), we are applying the modified adomian decomposition method (MADM). For both lifting and drainage problems the modified adomian polynomial are same. After simplification zeroth, first and second velocity components problem are.
Solution of first component drainage problem using boundary condition in Eqs. (52) and (53) is: 1 v1 ðx; t Þ ¼ 3βSt cos ½ωt 2 þ ω sin ½ωt γSt 2 1 3 βS2 γωSt sin ½2ωt βSt 2 cos ½ωt þ t x 4 1 þ ω sin ½ωt 1 γSt 2 3βSt cos ½ωt 2 2 3βS3t 2 þγωSt sin ½2ωt þ 3βSt 2 cos ½ωt x 4 1 1 þ ω sin ½ωt 2γSt 2 3 2 3 βSt 4 3 3 2 2βSt cos ½ωt þ βSt x ð64Þ x : 2 Due to the lengthy calculation the solution of second component of velocity distribution using boundary conditions in Eqs. (59) and (60) is too large. Therefore, derivation is given up to the first order term while graphical solutions are given up to second order term.
5.2. OHAM solution of drainage problem
Zeroth component problem: x2 : ð61Þ 2 Solution of zeroth component problem using boundary conditions in Eqs. (52) and (53) is: St St 2 v0 ðx; t Þ ¼ cos ½ωt 1 þ cos ½ωt þ x x : ð62Þ 2 2 v0 ðx; t Þ ¼ b0 þ b1 x St
Apply the standard form of OHAM on Eq. (58), we found zero, first and second components of velocity problem. Zero component problem: p0 :
∂ 2 v0 ¼ St ; ∂x2
ð65Þ
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
Figure 5 In 3D graphs, effect of time level on lift velocity profile.
Figure 6 In 3D graphs, effect of time level on drainage velocity profile.
Figure 7 Lift velocity distribution of fluid when ω ¼ 0:02; α ¼ 0:2; β ¼ 0:4; β1 ¼ 0:5; γ ¼ 0:2; γ 1 ¼ 0:2; St ¼ 0:1:
Figure 8 Drainage velocity distribution of fluid when ω ¼ 0:02; α ¼ 0:2; β ¼ 0:4; β1 ¼ 0:5; γ ¼ 0:2; γ 1 ¼ 0:2; St ¼ 0:1:
v1 ðx; t Þ ¼ c1
First component problem: p1 :
∂2 v1 ∂v0 ¼ St þ c 1 St c 1 ∂x2 ∂t
2 ∂2 v0 ∂v0 þ 2 1 þ c1 þ 6βc1 ∂x ∂x ∂v0 ∂ ∂v0 þ4c1 γ ∂x ∂t ∂x 2 2 ! ∂ ∂ v0 ∂v0 α þ 2γ þc1 ∂t ∂x2 ∂x 2 2 3 2 ∂ ∂ v0 ∂ ∂ v0 þc1 β1 2 þγ 3 : 2 ∂t ∂x ∂t ∂x2
231
1 ω sin ½ωt γS2t 1 6γSt cos ½ωt 3
3βS3t þ3βSt cos ½ωt cos ½ωt þ x 4 1 þ ω sin ½ωt γS2t þ 2γSt cos ½ωt 2 3βS3t 2 3βSt cos ½ωt 2 þ 3βS2t cos ½ωt x 4 1 1 þ ω sin ½ωt 2γS2t 2βS2t cos ½ωt 3 2 3 βSt 4 3 3 þβSt x þ ð68Þ x : 2 2
ð66Þ
Solution of zeroth and first component problem using boundary conditions given in Eqs. (59) and (60): St St 2 v0 ðx; t Þ ¼ cos ½ωt 1 þ cos ½ωt x x : ð67Þ 2 2
βS2t
Due to massive calculation the analytical result have been given up to first term while graphical solutions are given up to second order. The value of ci for the drainage velocity components are: c1 ¼ 0:2661918234;
c2 ¼ 0:2382524332:
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Figure 9 Effect of non-Newtonian parameter β1 on lift velocity profile when ω ¼ 5; α ¼ 0:2; β ¼ 0:4; γ ¼ 0:2; γ 1 ¼ 0:2; St ¼ 0:1.
Figure 10 Effect of β1 on drainage velocity when ω ¼ 5; α ¼ 0:2; β ¼ 0:4; γ ¼ 0:2; γ 1 ¼ 0:2; St ¼ 0:1.
Figure 12 Effect of β on drainage velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:7; γ ¼ 2; γ 1 ¼ 0:2; St ¼ 0:2.
Figure 11 Effect of third grade parameter β on lift velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:7; γ ¼ 2; γ 1 ¼ 0:2; St ¼ 0:2.
Figure 13 Effect of fourth grade parameter γ on lift velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:7; t ¼ 3; γ 1 ¼ 0:2; St ¼ 0:2.
6. Results and discussion
Tables 1 and 2 and graphically in Figures 3 and 4 for different values of embedded parameters. Whereas, the effects of other physical parameters t; St ; β; γ; β1 and γ 1 on velocity profile for both lift and drainage problems are studied in Figures 5–18. All results for the thin film of fourth grade fluid near the belt are illustrated in the x-coordinate only for a selected domain x A ½0; 1. In Figures 5–8, the velocity profiles are studied versus time for different values of independent variable x: Due to the no-slip condition, the fluid near the belt oscillates
Analytical solutions for unsteady thin film flow of a fourth grade fluid through a moving and oscillating vertical belt are obtained. The physical configuration of the problem is shown in Figures 1 and 2. Two different techniques namely ADM and OHAM are used for the solutions of lift and drainage problems. The results of velocity obtained from both methods for lift and drainage problems are compared numerically in
Unsteady thin film flow of a fourth grade fluid over a vertical moving and oscillating belt
Figure 14 Effect of fourth grade γ on drainage velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:7; t ¼ 3; γ 1 ¼ 0:2; St ¼ 0:2.
Figure 15 Effect of the parameter γ 1 on lift velocity when ω ¼ 3; α ¼ 0:2; β1 ¼ 0:7; γ ¼ 0:8; t ¼ 2; St ¼ 0:2.
Figure 16 Effect of the parameter γ 1 on drainage velocity when ω ¼ 3; α ¼ 0:2; β1 ¼ 0:7; γ ¼ 0:8; t ¼ 2; St ¼ 0:2.
jointly with the belt in the same period and oscillation reduces gradually towards the free surface shown in Figures 5 and 6. Increasing domain from (0,1) velocity amplitude raises gradually towards the surface of the fluid layer shown in Figures 7 and 8. It is found that velocity accepts and oscillating behavior. Generally, due to larger apparent viscosity, non-Newtonian fluid exhibits much thicker boundary layer than that of Newtonian fluid. With the increase in fourth
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Figure 17 Effect of the stock number St on lift velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:3; γ ¼ 0:5; γ 1 ¼ 0:3; β ¼ 0:4; t ¼ 5.
Figure 18 Effect of the St on drainage velocity when ω ¼ 5; α ¼ 0:2; β1 ¼ 0:3; γ ¼ 0:5; γ 1 ¼ 0:3; β ¼ 0:4; t ¼ 5.
grade parameters the apparent viscosity of the fluid become more bulky (viscous forces dominate), then the flow simultaneously adjust to the present driving force and oscillate closely with the similar phase in the whole flow domain. Therefore, increasing non-Newtonian parameters α; β; γ; β1 and γ 1 of second, third and fourth grade fluids causes further thickening of the boundary layer. So, increase in these higher order parameter increases velocity profiles of both lift and drainage problems shown in Figures 9–16. Figures 17 and 18 display the effects of St on lift and drainage velocities. From Figure 17, we observed that by increasing the value of St the lift velocity decreases whereas Figure 18 shows that drainage velocity increases by increasing the St : The reason is that the friction force cause the effect of gravity and it seems to be smaller near the belt. While it increasing gradually towards the free surface.
7. Conclusion In this article, we consider the unsteady flow of fourth grade fluid on moving and oscillating vertical belt. The analytical solution of lift and drainage velocity field achieved by applying two techniques namely ADM and OHAM. The result obtained from both techniques compared numerically
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and graphically. The result of ADM and OHAM close associated to each other. The effects of model parameters are presented graphically on lift and drainage velocity field. 1. If the apparent viscosity of the fluid is more bulky (viscous forces dominate), then the flow simultaneously adjust to the present driving force and oscillate closely with the similar phase in the whole flow domain. 2. Due to oscillation of the belt, we observed that the fluid motion is maximum at the surface of the belt and minimum at the surface of the fluid. 3. The higher order parameters are more efficient in the increase of fluid motion as one go from belt towards free surface.
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