Discrete Dynamics in Nature and Society

Volume 2011, Article ID 285809, 19 pages

http://dx.doi.org/10.1155/2011/285809

## Spectral Approximation of an Oldroyd Liquid Draining down a Porous Vertical Surface

^{1}Department of Mathematics, University of Gaziantep, 27310 Gaziantep, Turkey^{2}Department of Elementary Mathematics Education, Mevlana University, 42003 Konya, Turkey^{3}Department of Mathematics, Nigde University, 51240 Nigde, Turkey

Received 29 June 2011; Revised 5 October 2011; Accepted 10 October 2011

Academic Editor: Carlo Piccardi

Copyright © 2011 F. Talay Akyildiz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Consideration is given to the free drainage of an Oldroyd four-constant liquid from a vertical porous surface. The governing systems of quasilinear partial differential equations are solved by the Fourier-Galerkin spectral method. It is shown that Fourier-Galerkin approximations are convergent with spectral accuracy. An efficient and accurate algorithm based on the Fourier-Galerkin approximations for the governing system of quasilinear partial differential equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. The effect of the material parameters, elasticity, and porous medium constant on the centerline velocity and drainage rate is discussed.

#### 1. Introduction

Thin-film drainage down porous vertical surfaces is important in industry. Draining films occur in processes as diverse as dip coating, electroplating, enameling, emptying storage vessels, and oil recovery mechanisms [1, 2]. Spectral projection and corresponding error analysis of the system of nonlinear partial differential equations arising in the free drainage start-up flow of Oldroyd four constant liquids over a porous vertical surface is considered.

Literature review reveals that this problem is not considered. But for the case of impermeable wall, Goshawk and Waters [3] and Pennington and Waters [4] investigated the drainage of an Oldroyd four-constant liquid from a vertical surface via a finite difference method. But the problem they consider is a special case of the expended investigation in this paper, and error analysis is not explored in their work. Again, for case of steady flow (or start-up phase neglected), the literature more richer, in this case, Keeley et al. [5] investigate the drainage of thin films of non-Newtonian liquids from vertical surface, and the behavior of the Phan Thain-Tanner models are investigated in detail [6, 7]. In the present study, Galerkin’s method of the system of quasilinear partial differential equations governing the free drainage problem is investigated for a porous vertical surface. It is shown that method converges and that the convergence is not at all dependent on whether or not the physical parameters of the problem assume special values. The paper is organized as follows. The problem is defined in Section 2, and some basic results on Fourier approximations are given. A suitable Fourier-Galerkin approximation for the problem under consideration is proposed in Section 3 and error analysis given following [8–11]. Efficient and robust algorithms for the problem under consideration are constructed and numerical results presented in Section 4.

#### 2. Mathematical Formulation and Preliminary Results on Fourier Approximation

Consider a thin liquid film draining down a flat porous vertical surface defined by Cartesian coordinates . The axis points vertically downwards, the solid surface lies in the plane = 0 with the thickness of the liquid film measured in the positive direction, and the axis is positioned perpendicular to the gravitational force completing a set of right-handed axes. The nondimensionalized equations of motion and the dimensionless Oldroyd four constant constitutive model form a quasilinear system of PDEs, where (details can be found in [3, 4, 12, 13] for the interested reader) where above we used the following dimensionless parameters as in [3] And here are the dimensionless velocity and the dimensionless deviatoric stress tensor; , and represent the porous medium constant, dimensionless relaxation and retardation time constants, and a dimensionless material parameter, respectively. No slip at the wall and zero shear rate on the free surface of the liquid are assumed, The liquid is at rest at , therefore initial conditions are To calculate the shape of the film profile at a given time , the thickness is allowed to vary with while assuming the flow is still locally parallel. Combining the material derivative at the free surface with the equation of continuity yields a differential equation in , Introducing the flow rate per unit width across the film thickness , differentiating with respect to , and substituting the result into (2.8) and integrating give where is the initial profile and can be chosen to represent any suitable initial shape. Equation (2.10) effectively determines the position of the free surface as a function of and .

Next some mathematical notation is introduced. Denote the inner product in by If , then Fourier sine series is defined as where Similarly, Fourier cosine series is defined as where Denote by the Sobolev norm, given by

The space of periodic Sobolev functions on the interval is defined as the closure of the space of smooth periodic functions with respect to the -norm and will be simply denoted by . In particular, the space with norm denoted by is recovered for . We now define subspaces of spanned by the set The operators and denote the orthogonal, self-adjoint projection of onto and defined, respectively, by For, the estimates: hold for an appropriate constant and a positive integer . The reader is referred to [8] for the proof of these inequalities.

The space of continuous functions from the interval into the space is denoted by . Similarly, we also consider the space , where the topology on the finite-dimensional space can be given by any norm. Finally, note the inverse inequality which holds for integers and . A proof of this estimate can also be found in [11]. We will make use of the Sobolev lemma, which guarantees the existence of a constant such that

We now note that exactly the same estimates hold for . In the following, it will always be assumed that a solution of our problem (2.11)–(2.15) exists on some time interval with a certain amount of spatial regularity. In particular, we suppose that a solution exists in the space for some. With these preliminaries in place, we are now set to tackle the problem of defining a suitable spectral projection of (2.11)–(2.15) and proving the convergence of such a projection. First, the Fourier-Galerkin method is presented and a proof of convergence given.

#### 3. The Fourier-Galerkin Method

and are chosen to be an orthonormal basis of the Hilbert space and , respectively. Then, the subspace of these Hilbert spaces spanned by the andrespectively. Fourier-Galerkin approximation of (2.1)–(2.5) are find the functions , , and for all , such that for all and for all . Since for each , , , and have the form

Taking in (3.1)–(3.3) yields the following system of equations for the Fourier coefficients of , , and :

This is a nonlinear system of ordinary differential equations for the functions ; by standard existence theory, there is a unique solution which exists on some time interval, where possibly may be equal to. Since the argument is standard, the proof is omitted here. The main result of this paper is the fact that the Galerkin approximation converges to the exact solution when is smooth enough. This is stated in the next theorem.

Theorem 3.1. *Suppose that a solution of (2.1)–(2.5) exists in the space for and for some time . If and , then, for large enough , there exists a unique solution of the finite dimensional problem (3.1)–(3.4). Moreover, there exist constants , and such that
*

Before the proof is given, note that the assumptions of the theorem encompass the existence of constants , , and such that In particular, it follows that there are other constants , , and such that The main ingredient in the proof of the theorem is a local error estimate which will be established by the following lemma.

Lemma 3.2. *Suppose that the solution of (3.1)–(3.4) exists on the time interval and that , , , , and , then the error estimate:
**
holds for constant which is the function of , and , constant which is the function of , and, and constant which is the function of , and . *

*Proof. *Let. Also, from the definition of and we have . We apply and to both sides of (2.1)–(2.3), respectively. Since , commute with derivation, we obtain
We multiply these equations with test functions , and , respectively, integrate over , and subtract the resulting expressions from (3.1), (3.2), and (3.3) to get
Since ,
then we have
Consequently, from hypothesis and Gronwall’s inequality, we obtain
where is a function of , and:
Hence, we get
Let us estimate third term on the left-hand side of (3.20) in the time interval :
Thus,
Hence,
Using the hypothesis and Gronwall’s inequality,
is a function of , and. Estimating the RHS of (3.22) in the time interval ,
Noting that the last integral is bounded by, the estimate is
can be estimated in exactly the same way as . Then, (3.22) as a whole is estimated as
Therefore, we get
Then, using the Gronwall’s inequality,
where is a function of , and. Since
Using (2.18) and (3.19) in (3.33), (2.18) and (3.26) in (3.34), (2.18) and (3.32) in (3.35), respectively, we obtain
where , and are constants, functions of (, and ), (, and ), and of (, and), respectively.

Lemma 3.3. *Suppose that the solution of (3.1)–(3.3) exists on the time interval and that , , , , and , then the error estimate:
**
holds for the constants , and . The proof of the Lemma follows from (3.19), (3.26), and (3.32) after application of the triangle inequality and the inverse inequality (2.19).*

*Proof of Theorem 3.1. *To extend the estimate of the first inequality in (3.14) to the time interval unspecified in Lemma 3.2 is defined as
Thus, the time corresponds to the largest time in for which the -norm of is uniformly bounded by . Since ,
therefore for all . Note that is smaller than the maximum time of existence of the solution . Now, we need to show that there exists such that
and therefore the supremum in (3.14) holds on . From the definition (3.38), we either have or in which case. Now assume that , then
Hence, we obtain
On the other hand, Lemma 3.3 implies
or
In conclusion, for, we cannot have and claim (3.40) holds. It follows that the solution of (3.1) is defined on, since as noted before , and, from (3.14),

In exactly the same way, we can extend the estimate of the second and third inequalities in (3.14) to the time interval and show that

#### 4. Numerical Results and Discussion

The system of differential equations (3.6)–(3.9) is of the following form Runge-Kutta method is applied to this system. The integrals in equations (2.8) and (2.9) are calculated analytically and numerically, respectively, with approximated by a central difference formula. To illustrate the spectral accuracy, the time step is chosen to be sufficiently small so that the error is dominated by the spatial discretization. The free drainage of the Oldroyd-B liquid () for which an exact analytical solution is possible is considered first [14]. Figure 1 compares the exact analytical solution in [14] with the approximate solution with nodes only for both permeable and impermeable wall. The exact and approximate solutions are indistinguishable in the figure. The error at of the Fourier-Galerkin approximations with increasing number of nodes for the drainage of Oldroyd-B liquid is listed in Table 1.

This shows that numerical results are at least accurate up to the seventh decimal for . The aim of this paper is to elaborate the effects of the nonlinear parameter and porous medium parameter on the centerline velocity and drainage rate. The effect of these parameters on the velocity field is shown in Figures 2 and 3. Figure 2 displays the effect of the nonlinear parameter on the centerline velocity when the wall is impermeable , and , , and . Clearly, the overshoot gradually disappears as numerical values of the nonlinear parameter increase. In addition, the steady centerline velocity increases with increasing values of . Figure 3 explores the effect of porosity on the centerline velocity for . Increasing the porosity parameter triggers a decrease in the value of the centerline velocity. The difference between the relaxation and retardation times,, is a measure of the elasticity of the Oldroyd four-constant liquid, the greater the difference is the more elastic the liquid is. The effect of elasticity on the centerline velocity of constant viscosity Oldroyd four-constant liquids is shown in Figures 4 and 5 for permeable and impermeable walls, respectively. In either case, the centerline velocity increases with an increase in elasticity. The effect of the nonlinear parameter and porous medium parameter on the drainage rate is examined in Figures 6 and 7, respectively. Since in all cases the nonlinear parameter has an increasing effect on the velocity, we expect increasing will lead to a thinner film over either type of wall, impermeable or permeable. That is evident in Figure 6, which shows that liquid drain more rapidly as is increased from zero. Since the porous medium parameter has a decreasing effect on the velocity in all cases (both permeable and impermeable wall), we expect increasing will lead to a thicker film over either impermeable or permeable wall, Figure 7. The effect of the elasticity on the drainage rate is shown in Figure 8 for three liquids of which liquid 1 is the most elastic and liquid 3 is the least elastic. Liquid 1 drains more rapidly than liquid 2, which in turn drains more rapidly than liquid 3.

#### References

- J. Q. Xu, “Modeling unsteady-state gravity-driven flow in porous media,”
*Journal of Petroleum Science and Engineering*, vol. 62, no. 3-4, pp. 80–86, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. A. Goshawk, N. D. Waters, G. K. Rennie, and E. J. Staples, “Enhancement of the drainage of non-Newtonian liquid films by oscillation,”
*Journal of Non-Newtonian Fluid Mechanics*, vol. 51, no. 1, pp. 21–60, 1994. View at Publisher · View at Google Scholar · View at Scopus - J. A. Goshawk and N. D. Waters, “The effect of oscillation on the drainage of an elastico-viscous liquid,”
*Journal of Non-Newtonian Fluid Mechanics*, vol. 54, pp. 449–464, 1994. View at Publisher · View at Google Scholar · View at Scopus - S. V. Pennington, N. D. Waters, and E. W. Williams, “The numerical simulation of an oldroyd liquid draining down a vertical surface,”
*Journal of Non-Newtonian Fluid Mechanics*, vol. 34, no. 2, pp. 221–239, 1990. View at Publisher · View at Google Scholar · View at Scopus - A. M. Keeley, G. K. Rennie, and N. D. Waters, “Draining thin films-part 1,”
*Journal of Non-Newtonian Fluid Mechanics*, vol. 28, no. 2, pp. 213–226, 1988. View at Publisher · View at Google Scholar · View at Scopus - Y. Dimakopoulos and J. Tsamopoulos, “On the gas-penetration in straight tubes completely filled with a viscoelastic fluid,”
*Journal of Non-Newtonian Fluid Mechanics*, vol. 117, no. 2-3, pp. 117–139, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Pavlidis, Y. Dimakopoulos, and J. Tsamopoulos, “Fully developed flow of a viscoelastic film down a vertical cylindrical or planar wall,”
*Rheologica Acta*, vol. 48, no. 9, pp. 1031–1048, 2009. View at Publisher · View at Google Scholar · View at Scopus - Y. Maday and A. Quarteroni, “Error analysis for spectral approximation of the Korteweg-de Vries equation,”
*RAIRO Modélisation Mathématique et Analyse Numérique*, vol. 22, no. 3, pp. 499–529, 1988. View at Google Scholar · View at Zentralblatt MATH - H. Kalisch and X. Raynaud, “Convergence of a spectral projection of the Camassa-Holm equation,”
*Numerical Methods for Partial Differential Equations*, vol. 22, no. 5, pp. 1197–1215, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. Quarteroni and A. Valli,
*Numerical Approximation of Partial Differential Equations*, Springer, Berlin, Germany, 1997. - C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang,
*Spectral Methods in Fluid Dynamics*, Springer, New York, NY, USA, 1988. - F. T. Akyildiz, K. Vajravelu, and H. Ozekes, “Fourier-Galerkin domain truncation method for Stokes' first problem with Oldroyd four-constant liquid,”
*Computers and Mathematics with Applications*, vol. 55, no. 11, pp. 2452–2457, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - N. D. Waters and A. M. Keeley, “Start-up of an elastico-viscous liquid draining from a vertical surface,”
*Journal of Non-Newtonian Fluid Mechanics*, vol. 22, no. 3, pp. 325–334, 1987. View at Publisher · View at Google Scholar · View at Scopus - F. T. Akyildiz and D. A. Siginer, “Start-up of an elastico-viscous liquid draining from a porous vertical surface,”
*Journal of Non-Newtonian Fluid Mechanics*. In press. View at Publisher · View at Google Scholar