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 (π‘₯,𝑦,𝑧). The π‘₯ axis points vertically downwards, the solid surface lies in the plane 𝑦 = 0 with the thickness of the liquid film measured in the positive 𝑦 direction, and the 𝑧 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)πœ•π‘†π‘₯𝑦=πœ•π‘¦πœ•π‘’πœ•π‘‘βˆ’1+𝛼2𝑆𝑒,(2.1)π‘₯𝑦+𝑆1πœ•π‘†π‘₯𝑦+1πœ•π‘‘2πœ‡1𝑆π‘₯π‘₯πœ•π‘’=ξ‚΅πœ•π‘¦πœ•π‘’πœ•π‘¦+𝑆2πœ•2π‘’ξ‚Άπ‘†πœ•π‘¦πœ•π‘‘,(2.2)π‘₯π‘₯+𝑆1πœ•π‘†π‘₯π‘₯πœ•π‘‘βˆ’2𝑆1𝑆π‘₯π‘¦πœ•π‘’πœ•π‘¦=βˆ’2𝑆2ξ‚΅πœ•π‘’ξ‚Άπœ•π‘¦2,(2.3) where above we used the following dimensionless parameters as in [3]𝑔𝑦=π‘Œπœˆ21/3𝑔,π‘₯=π‘₯𝜈21/3𝑔,β„Ž=𝐻(π‘₯,𝑑)𝑦=π‘Œπœˆ21/3𝑔,𝑑=𝑇𝑦=π‘Œ2πœˆξ‚Ά1/3,𝑆1=πœ†1𝑔2πœˆξ‚Ά1/3,𝑆2=πœ†2𝑔2πœˆξ‚Ά1/3π‘ˆ,𝑒=(πœˆπ‘”)1/3𝑉,𝑣=(πœˆπ‘”)1/3,𝑆𝑖𝑗=π‘ƒπ‘–π‘—πœ‡π‘”πœˆ1/3,πœ‡1=πœ‡0𝑔2πœˆξ‚Ά1/3,𝛼2=(πœˆπ‘”)1/3.πœŒπ‘”(2.4) And here 𝑒=𝑒(𝑦,𝑑),𝑆π‘₯𝑦=𝑆π‘₯𝑦(𝑦,𝑑),𝑆π‘₯π‘₯=𝑆π‘₯π‘₯(𝑦,𝑑) are the dimensionless velocity and the dimensionless deviatoric stress tensor; 𝛼2,𝑆1,𝑆2, and πœ‡1 represent the porous medium constant, dimensionless relaxation and retardation time constants, and a dimensionless material parameter, respectively. No slip at the wall 𝑦=0 and zero shear rate on the free surface of the liquid are assumed,𝑒(0,𝑑)=0,𝑑β‰₯0,πœ•π‘’πœ•π‘¦=0on𝑦=β„Ž.(2.5) The liquid is at rest at 𝑑=0, therefore initial conditions are𝑒(𝑦,0)=𝑆π‘₯𝑦(𝑦,0)=𝑆π‘₯π‘₯(𝑦,0)=0.(2.6) To calculate the shape of the film profile at a given time 𝑑, the thickness β„Ž is allowed to vary with π‘₯ while assuming the flow is still locally parallel. Combining the material derivative at the free surface𝑣(π‘₯,β„Ž,𝑑)=πœ•β„Žπœ•π‘‘+𝑒(π‘₯,β„Ž,𝑑)πœ•β„Ž,πœ•π‘₯(2.7) with the equation of continuity yields a differential equation in β„Ž,βˆ’πœ•β„Žξ‚΅ξ€œπœ•π‘₯β„Ž0πœ•π‘’ξ‚Ά=πœ•β„Žπ‘‘π‘¦+π‘’πœ•β„Ž.πœ•π‘‘(2.8) Introducing the flow rate 𝑄(β„Ž,𝑑) per unit width across the film thickness β„Ž,π‘„ξ€œ(β„Ž,𝑑)=β„Ž0𝑒(π‘₯,𝑦,𝑑)𝑑𝑦,(2.9) differentiating 𝑄 with respect to β„Ž, and substituting the result into (2.8) and integrating giveπ‘₯(β„Ž,𝑑)βˆ’π‘₯0(ξ€œβ„Ž)=𝑑0πœ•π‘„πœ•β„Žπ‘‘πœ,(2.10) where π‘₯0(β„Ž)=π‘₯(β„Ž,0) 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 π‘₯ as a function of β„Ž and 𝑑.

Next some mathematical notation is introduced. Denote the inner product in 𝕃2(0,β„Ž) byξ€œ(𝑓,𝑔)=β„Ž0𝑓(𝑦)𝑔(𝑦)𝑑𝑦.(2.11) If π‘“βˆˆπ•ƒ2(0,𝐻), then Fourier sine series is defined as𝑓(𝑦)=βˆžξ“π‘˜=1𝑓𝑠(π‘˜)sinξ‚΅ξ‚΅(2π‘˜βˆ’1)πœ‹ξ‚Άπ‘¦ξ‚Ά,2β„Ž(2.12) where𝑓𝑠2(π‘˜)=β„Žξ€œπœ‹0ξ‚΅sin(2π‘˜βˆ’1)πœ‹π‘¦ξ‚Ά2β„Žπ‘“(𝑦)π‘‘π‘¦π‘˜=1,2,….(2.13) Similarly, Fourier cosine series is defined as𝑓𝑓(𝑦)=𝑐(0)2+βˆžξ“π‘˜=1𝑓𝑐(π‘˜)cosξ‚΅ξ‚΅(2π‘˜βˆ’1)πœ‹ξ‚Άπ‘¦ξ‚Ά,2β„Ž(2.14) where𝑓𝑐2(π‘˜)=π»ξ€œπ»0ξ‚΅cos(2π‘˜βˆ’1)πœ‹π‘¦2𝑓(𝑦)π‘‘π‘¦π‘˜=1,2,….(2.15) Denote by β€–β€–π»π‘š the Sobolev norm, given by‖𝑓‖2π»π‘š=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©βˆžξ“π‘˜=1ξ‚΅|||1+2π‘˜βˆ’12|||2ξ‚Άπ‘š||𝑓𝑠(||π‘˜)2,||𝑓𝑐||(0)22+βˆžξ“π‘˜=1ξ‚΅|||1+2π‘˜βˆ’12|||2ξ‚Άπ‘š||𝑓𝑐(||π‘˜)2.(2.16)

The space of periodic Sobolev functions on the interval [0,β„Ž]is defined as the closure of the space of smooth periodic functions with respect to the π»π‘š-norm and will be simply denoted by π»π‘š. In particular, the space 𝕃2(0,β„Ž)with norm denoted by ‖‖𝕃2 is recovered for π‘š=0. We now define subspaces of 𝕃2(0,β„Ž)spanned by the setπ·βˆ—π‘=ξƒ―βˆš2sin((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)βˆšβ„Žξƒ°,𝐷,1β‰€π‘˜β‰€π‘π‘βˆ—βˆ—=ξƒ―βˆš2cos((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)βˆšβ„Žξƒ°.,1β‰€π‘˜β‰€π‘(2.17) The operators 𝑃𝑁 and π‘ƒβˆ—π‘ denote the orthogonal, self-adjoint projection of 𝕃2 onto π·βˆ—π‘ and π·π‘βˆ—βˆ—defined, respectively, by𝑃𝑁𝑓(𝑦)=π‘ξ“π‘˜=1ξ‚΅sin(2π‘˜βˆ’1)πœ‹π‘¦ξ‚Άξπ‘“2β„Žπ‘ π‘ƒ(π‘˜),βˆ—π‘ξπ‘“π‘“(𝑦)=𝑐(0)2+π‘ξ“π‘˜=1ξ‚΅(cos2π‘˜βˆ’1)πœ‹π‘¦ξ‚Άξπ‘“2β„Žπ‘(π‘˜).(2.18) Forπ‘“βˆˆπ»π‘š, the estimates:β€–β€–π‘“βˆ’π‘ƒπ‘π‘“β€–β€–πΏ2β‰€πΆπ‘π‘βˆ’π‘šβ€–β€–πœ•π‘šπ‘¦π‘“β€–β€–πΏ2,β€–β€–π‘“βˆ’π‘ƒπ‘π‘“β€–β€–π»π‘›β‰€πΆπ‘π‘π‘›βˆ’π‘šβ€–β€–πœ•π‘šπ‘¦π‘“β€–β€–πΏ2,(2.19) hold for an appropriate constant 𝐢𝑝 and a positive integer 𝑛. The reader is referred to [8] for the proof of these inequalities.

The space of continuous functions from the interval [0,𝑇] into the space 𝐻𝑛is denoted by 𝐢([0,𝑇],𝐻𝑛). Similarly, we also consider the space 𝐢([0,𝑇],π·βˆ—π‘), where the topology on the finite-dimensional space π·βˆ—π‘ can be given by any norm. Finally, note the inverse inequalityβ€–β€–πœ•π‘šπ‘¦πœ‘β€–β€–πΏ2β‰€π‘π‘šβ€–πœ‘β€–πΏ2,(2.20) which holds for integers π‘š>0 and πœ‘βˆˆπ·βˆ—π‘. 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 𝑐 such thatsup𝑦||||𝑓(𝑦)≀𝑐‖𝑓‖𝐻1.(2.21)

We now note that exactly the same estimates hold for π‘ƒβˆ—π‘. In the following, it will always be assumed that a solution of our problem (2.11)–(2.15) exists on some time interval[0,𝑇] with a certain amount of spatial regularity. In particular, we suppose that a solution exists in the (𝐢([0,𝑇],𝐻1))3 space for some𝑇>0. 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

{π‘’π‘˜βˆš(𝑦),π‘˜πœ–β„•}={√2sin((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)/β„Ž,π‘˜βˆˆβ„•} and {π‘“π‘˜(𝑦),π‘˜πœ–β„•}={√√2cos((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)/β„Ž,π‘˜βˆˆβ„•} are chosen to be an orthonormal basis of the Hilbert space𝕃20[0,β„Ž] and 𝕃2[0,β„Ž], respectively. Then, the subspace of these Hilbert spaces spanned by the π·βˆ—π‘βˆš={√2sin((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)/β„Ž,1β‰€π‘˜β‰€π‘} andπ·π‘βˆ—βˆ—βˆš={√2cos((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)/β„Ž,0β‰€π‘˜β‰€π‘},respectively. Fourier-Galerkin approximation of (2.1)–(2.5) are find the functions 𝑒𝑁(𝑑)βˆˆπ·βˆ—π‘, 𝑆𝑁π‘₯𝑦(𝑑), and 𝑆𝑁π‘₯π‘₯βˆˆπ·π‘βˆ—βˆ— for all 0≀𝑑≀𝑇, such thatξ€·πœ•π‘‘π‘’π‘βˆ’1βˆ’πœ•π‘Œπ‘†π‘π‘₯𝑦,πœ”1+𝛽2𝑒,πœ”1ξ€Έ[],𝑆=0,π‘‘βˆˆ0,𝑇(3.1)𝑁π‘₯𝑦+𝑆1πœ•π‘‘π‘†π‘π‘₯𝑦+12πœ‡1𝑆𝑁π‘₯π‘₯πœ•π‘¦π‘’π‘βˆ’πœ•π‘¦π‘’π‘βˆ’π‘†2πœ•2𝑑𝑦𝑒𝑁,πœ”2[]𝑆=0,π‘‘βˆˆ0,𝑇,(3.2)𝑁π‘₯π‘₯+𝑆1πœ•π‘‘π‘†π‘π‘₯π‘₯βˆ’2𝑆1𝑆𝑁π‘₯π‘¦πœ•π‘¦π‘’π‘βˆ’2𝑆2ξ€·πœ•π‘¦π‘’π‘ξ€Έ2,πœ”2[]𝑒=0,π‘‘βˆˆ0,𝑇,(3.3)𝑁(0)=0,𝑆𝑁π‘₯𝑦(0)=0,𝑆𝑁π‘₯π‘₯(0)=0,(3.4) for all πœ”1βˆˆπ·βˆ—π‘ and for all πœ”2βˆˆπ·π‘βˆ—βˆ—. Since for each 𝑑, 𝑒𝑁(β‹…,𝑑), 𝑆𝑁π‘₯𝑦(β‹…,𝑑), and 𝑆𝑁π‘₯𝑦(β‹…,𝑑) have the form𝑒𝑁(𝑦,𝑑)=π‘ξ“π‘˜=1Μ‚π‘’π‘βˆš(π‘˜,𝑑)2sin((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)βˆšβ„Ž,𝑆𝑁π‘₯𝑦(𝑦,𝑑)=π‘ξ“π‘˜=1̂𝑠𝑁π‘₯π‘¦βˆš(π‘˜,𝑑)2cos((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)βˆšβ„Ž,𝑆𝑁π‘₯π‘₯(𝑦,𝑑)=π‘ξ“π‘˜=1̂𝑠𝑁π‘₯π‘₯√(π‘˜,𝑑)2cos((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž)βˆšβ„Ž.(3.5)

Taking πœ”1=√2/𝐻sin((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž),πœ”2=√2/𝐻cos((2π‘˜βˆ’1)πœ‹π‘¦/2β„Ž),1β‰€π‘˜β‰€π‘in (3.1)–(3.3) yields the following system of equations for the Fourier coefficients of 𝑒𝑁, 𝑆𝑁π‘₯𝑦, and 𝑆𝑁π‘₯π‘₯:𝑑𝑑𝑑̂𝑒𝑁(π‘˜,𝑑)=βˆ’(2π‘˜βˆ’1)πœ‹2β„ŽΜ‚π‘ π‘π‘₯𝑦(π‘˜,𝑑)+2β„Ž2β„Žπœ‹(2π‘˜βˆ’1),(3.6)̂𝑠𝑁π‘₯𝑦(π‘˜,𝑑)+𝑆1𝑑𝑑𝑑̂𝑠𝑁π‘₯𝑦(π‘˜,𝑑)=(2π‘˜βˆ’1)πœ‹2β„ŽΜ‚π‘’π‘(π‘˜,𝑑)+𝑆2(2π‘˜βˆ’1)πœ‹π‘‘2β„Žπ‘‘π‘‘Μ‚π‘’π‘(π‘˜,𝑑)+πœ‡1(2π‘˜βˆ’1)πœ‹ξ‚™4β„Ž2β„Žπ‘ξ“π‘–,𝑗=1𝑐𝑖𝑗𝑁̂𝑠𝑁π‘₯π‘₯(𝑖,𝑑)̂𝑒𝑁(π‘˜,𝑑),(3.7)̂𝑠𝑁π‘₯π‘₯(π‘˜,𝑑)+𝑆1𝑑𝑑𝑑̂𝑠𝑁π‘₯π‘₯(π‘˜,𝑑)βˆ’(2π‘˜βˆ’1)πœ‹2β„ŽΜ‚π‘’π‘(π‘˜,𝑑)βˆ’2𝑆1(2π‘˜βˆ’1)πœ‹ξ‚™4β„Ž2β„Žπ‘ξ“π‘–,𝑗=1𝑐𝑖𝑗𝑁̂𝑠𝑁π‘₯𝑦(𝑖,𝑑)̂𝑒𝑁(π‘˜,𝑑)=βˆ’2𝑆2ξ‚΅(2π‘˜βˆ’1)πœ‹ξ‚Ά4β„Ž2ξ‚™2β„Žπ‘ξ“π‘–,𝑗=1𝑑𝑖𝑗𝑁̂𝑒𝑁(𝑖,𝑑)̂𝑒𝑁(𝑗,𝑑),(3.8)̂𝑒𝑁(π‘˜,𝑑)=0,̂𝑠𝑁π‘₯𝑦(π‘˜,𝑑)=0,̂𝑠𝑁π‘₯π‘₯𝑐(π‘˜,𝑑)=0,(3.9)𝑖𝑗𝑁=ξ€œβ„Ž0ξ‚΅cos(2π‘–βˆ’1)πœ‹π‘¦ξ‚Άξ‚΅2β„Žsin(2π‘—βˆ’1)πœ‹π‘¦ξ‚Άξ‚΅2β„Žcos(2π‘βˆ’1)πœ‹π‘¦ξ‚Άπ‘‘2β„Žπ‘‘π‘¦,𝑖𝑗𝑁=ξ€œβ„Ž0ξ‚΅sin(2π‘–βˆ’1)πœ‹π‘¦ξ‚Άξ‚΅2β„Žsin(2π‘—βˆ’1)πœ‹π‘¦ξ‚Άξ‚΅2β„Žcos(2π‘βˆ’1)πœ‹π‘¦ξ‚Ά2β„Žπ‘‘π‘¦.(3.10)

This is a nonlinear system of ordinary differential equations for the functions {̂𝑒𝑁(π‘˜,𝑑),̂𝑠𝑁π‘₯𝑦(π‘˜,𝑑),̂𝑠𝑁π‘₯π‘₯(π‘˜,𝑑)}π‘π‘˜=1; by standard existence theory, there is a unique solution which exists on some time interval[0,𝑇𝑁), where 𝑇𝑁 possibly may be equal to𝑇. Since the argument is standard, the proof is omitted here. The main result of this paper is the fact that the Galerkin approximation {𝑒𝑁,𝑆𝑁π‘₯𝑦,𝑆𝑁π‘₯π‘₯} converges to the exact solution {𝑒,𝑆π‘₯𝑦,𝑆π‘₯π‘₯} when 𝑒 is smooth enough. This is stated in the next theorem.

Theorem 3.1. Suppose that a solution {𝑒,𝑆π‘₯𝑦,𝑆π‘₯π‘₯} of (2.1)–(2.5) exists in the space (𝐢([0,𝑇],π»π‘š))3 for π‘šβ‰₯1 and for some time 𝑇>0. If ‖𝑃𝑁𝑒(β„Ž,𝑑)βˆ’π‘’π‘(𝑒,𝑑)‖≀𝑐𝑁1βˆ’π‘š and β€–π‘ƒβˆ—π‘π‘†π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦‖≀𝑐𝑁1βˆ’π‘š, then, for large enough 𝑁, there exists a unique solution {𝑒𝑁,𝑆𝑁π‘₯𝑦,𝑆𝑁π‘₯π‘₯} of the finite dimensional problem (3.1)–(3.4). Moreover, there exist constants Ξ“1,Ξ“2, and Ξ“3 such that sup[]π‘‘βˆˆ0,π‘‡β€–β€–π‘’βˆ’π‘’π‘β€–β€–πΏ2≀Γ1𝑁1βˆ’π‘š,sup[]π‘‘βˆˆ0,𝑇‖‖𝑆π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦‖‖𝐿2≀Γ2𝑁1βˆ’π‘š,sup[]π‘‘βˆˆ0,𝑇‖‖𝑆π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2≀Γ3𝑁1βˆ’π‘š.(3.11)

Before the proof is given, note that the assumptions of the theorem encompass the existence of constants πœ…, πœ…1, and πœ…2 such thatsup[]π‘‘βˆˆ0,𝑇(‖𝑒𝑦,𝑑)β€–π»π‘šβ‰€πœ…,sup[]π‘‘βˆˆ0,𝑇‖‖𝑆π‘₯π‘₯(‖‖𝑦,𝑑)π»π‘šβ‰€πœ…1,sup[]π‘‘βˆˆ0,𝑇‖‖𝑆π‘₯𝑦(‖‖𝑦,𝑑)π»π‘šβ‰€πœ…2.(3.12) In particular, it follows that there are other constants πœ“, πœ“1, and πœ“2 such thatsup[]π‘‘βˆˆ0,𝑇(‖𝑒𝑦,𝑑)β€–π»π‘šβ‰€πœ“,sup[]π‘‘βˆˆ0,𝑇‖‖𝑆π‘₯π‘₯(‖‖𝑦,𝑑)π»π‘šβ‰€πœ“1,sup[]π‘‘βˆˆ0,𝑇‖‖𝑆π‘₯𝑦(‖‖𝑦,𝑑)π»π‘šβ‰€πœ“2.(3.13) 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 {𝑒𝑁,𝑆𝑁π‘₯𝑦,𝑆𝑁π‘₯π‘₯} of (3.1)–(3.4) exists on the time interval [0,π‘‘βˆ—π‘] and that supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]‖𝑒𝑁(𝑦,𝑑)‖𝐻2≀2πœ“, supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]‖𝑆𝑁π‘₯𝑦(𝑦,𝑑)‖𝐻2≀2πœ“1, supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]‖𝑆𝑁π‘₯π‘₯(𝑦,𝑑)‖𝐻2≀2πœ“2, supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]|𝑃𝑁𝑒(β„Ž,𝑑)βˆ’π‘’π‘(β„Ž,𝑑)|≀𝛽𝑁1βˆ’π‘š, and supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]|π‘ƒβˆ—π‘π‘†π‘₯𝑦(β„Ž,𝑑)βˆ’π‘†π‘π‘₯𝑦(β„Ž,𝑑)|≀𝛽1𝑁1βˆ’π‘š, then the error estimate: supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘’βˆ’π‘’π‘β€–β€–πΏ2≀Γ1𝑁1βˆ’π‘š,supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘†π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦‖‖𝐿2≀Γ2𝑁1βˆ’π‘š,supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘†π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2≀Γ3𝑁1βˆ’π‘š,(3.14) holds for constant Ξ“1 which is the function of 𝑇,𝛼2,𝐢𝑝,πœ…,𝛽, and 𝑐, constantΞ“2 which is the function of 𝑇,𝑆1,𝑆2,πœ…1,πœ“,πœ“1,πœ“2,𝑐1,𝛽1, and𝑐, and constant Ξ“3 which is the function of 𝑇,𝑆1,πœ…1,πœ“,πœ“1,𝑐1,πœ…, and 𝑐.

Proof. Let𝜎1=π‘ƒπ‘π‘’βˆ’π‘’π‘,𝜎2=π‘ƒβˆ—π‘π‘†π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦,𝜎3=π‘ƒβˆ—π‘π‘†π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯. Also, from the definition of 𝑃𝑁 and π‘ƒβˆ—π‘,we have πœ•π‘¦πœŽ1=π‘ƒβˆ—π‘πœ•π‘¦π‘’βˆ’πœ•π‘¦π‘’π‘,πœ•π‘¦πœŽ2=π‘ƒπ‘πœ•π‘¦π‘†π‘₯π‘¦βˆ’πœ•π‘¦π‘†π‘π‘₯𝑦,πœ•π‘¦πœŽ3=π‘ƒπ‘πœ•π‘¦π‘†π‘₯π‘₯βˆ’πœ•π‘¦π‘†π‘π‘₯π‘₯. We apply𝑃𝑁,π‘ƒβˆ—π‘ and π‘ƒβˆ—π‘ to both sides of (2.1)–(2.3), respectively. Since 𝑃𝑁, π‘ƒβˆ—π‘ commute with derivation, we obtain πœ•π‘‘π‘ƒπ‘π‘’+𝛼2𝑒=1+πœ•π‘¦π‘ƒπ‘π‘†π‘₯𝑦,π‘ƒβˆ—π‘π‘†π‘₯𝑦+𝑆1πœ•π‘‘π‘ƒβˆ—π‘π‘†π‘₯𝑦+12πœ‡1π‘ƒβˆ—π‘ξ€·π‘†π‘₯π‘₯πœ•π‘¦π‘’ξ€Έ=π‘ƒβˆ—π‘πœ•π‘¦π‘’+𝑆2πœ•π‘‘π‘ƒβˆ—π‘πœ•π‘¦π‘ƒπ‘’,βˆ—π‘π‘†π‘₯π‘₯+𝑆1πœ•π‘‘π‘ƒβˆ—π‘π‘†π‘₯π‘₯βˆ’2𝑆1π‘ƒβˆ—π‘ξ‚€π‘†π‘₯π‘¦πœ•π‘¦π‘’ξ‚=βˆ’2𝑆2π‘ƒβˆ—π‘ξ€·πœ•π‘¦π‘’ξ€Έ2.(3.15) We multiply these equations with test functions 𝜎1βˆˆπ·βˆ—π‘,𝜎2βˆˆπ·π‘βˆ—βˆ—, and 𝜎3βˆˆπ·π‘βˆ—βˆ—, respectively, integrate over [0,β„Ž], and subtract the resulting expressions from (3.1), (3.2), and (3.3) to get 12π‘‘β€–β€–πœŽπ‘‘π‘‘1β€–β€–2𝐿2+𝛼2β€–β€–πœŽ1β€–β€–2𝐿2=𝑃𝑁1βˆ’1𝑁,𝜎1ξ€Έ+ξ€·π‘ƒπ‘πœ•π‘¦ξ€·π‘†π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦,𝜎1ξ€Έ,β€–β€–πœŽ2β€–β€–2𝐿2+𝑆112π‘‘β€–β€–πœŽπ‘‘π‘‘2β€–β€–2𝐿2+12πœ‡1ξ€·π‘ƒβˆ—π‘ξ€·π‘†π‘₯π‘₯πœ•π‘¦π‘’ξ€Έβˆ’π‘†π‘π‘₯π‘₯πœ•π‘¦π‘’π‘,𝜎2ξ€Έ=ξ‚€π‘ƒβˆ—π‘πœ•π‘¦ξ€·π‘’βˆ’π‘’π‘ξ€Έ,𝜎2+𝑆2ξ€·π‘ƒβˆ—π‘πœ•2π‘‘π‘¦ξ€·π‘’βˆ’π‘’π‘ξ€Έ,𝜎2ξ€Έ,β€–β€–πœŽ2β€–β€–2𝐿2+𝑆112π‘‘β€–β€–πœŽπ‘‘π‘‘2β€–β€–2𝐿2+12πœ‡1ξ€·π‘ƒβˆ—π‘ξ€·π‘†π‘₯π‘₯πœ•π‘¦π‘’ξ€Έβˆ’π‘†π‘π‘₯π‘₯πœ•π‘¦π‘’π‘,𝜎2ξ€Έ=ξ‚€π‘ƒβˆ—π‘πœ•π‘¦ξ€·π‘’βˆ’π‘’π‘ξ€Έ,𝜎2+𝑆2ξ€·πœ•π‘‘π‘ƒβˆ—π‘πœ•π‘¦ξ€·π‘’βˆ’π‘’π‘ξ€Έ,𝜎2ξ€Έ,β€–β€–πœŽ3β€–β€–2𝐿2+𝑆112π‘‘β€–β€–πœŽπ‘‘π‘‘3β€–β€–2𝐿2βˆ’2𝑆1ξ€·π‘ƒβˆ—π‘ξ€·π‘†π‘₯π‘¦πœ•π‘¦π‘’ξ€Έβˆ’π‘†π‘π‘₯π‘¦πœ•π‘¦π‘’π‘,𝜎3ξ€Έ=βˆ’2𝑆2ξ‚€π‘ƒβˆ—π‘ξ€·πœ•π‘¦π‘’ξ€Έ2βˆ’ξ€·πœ•π‘¦π‘’π‘ξ€Έ2,𝜎3.(3.16) Since 𝜎1βˆˆπ·βˆ—π‘,(𝜎2,𝜎3)βˆˆπ·π‘βˆ—βˆ—, 12π‘‘β€–β€–πœŽπ‘‘π‘‘1β€–β€–2𝐿2+𝛼2β€–β€–πœŽ1β€–β€–2𝐿2=ξ€·πœ•π‘¦πœŽ2,𝜎1ξ€Έβ‰€β€–β€–πœ•π‘¦πœŽ2‖‖𝐿2β€–β€–πœŽ1‖‖𝐿2,(3.17) then we have 12π‘‘β€–β€–πœŽπ‘‘π‘‘1β€–β€–2𝐿2+𝛼2β€–β€–πœŽ1β€–β€–2𝐿2β‰€β€–β€–πœ•π‘¦πœŽ2‖‖𝐿2≀||𝜎2||.(𝐻,𝑑)(3.18) Consequently, from hypothesis and Gronwall’s inequality, we obtain supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–πœŽ1‖‖𝐿2≀Γ1𝑁1βˆ’π‘š,(3.19) where Ξ“1 is a function of 𝑇,𝛽,𝛼2, and𝑐: β€–β€–πœŽ2β€–β€–2𝐿2+𝑆112π‘‘β€–β€–πœŽπ‘‘π‘‘2β€–β€–2𝐿2+12πœ‡1𝑆π‘₯π‘₯πœ•π‘¦π‘’βˆ’π‘†π‘π‘₯π‘₯πœ•π‘¦π‘’π‘,𝜎2ξ€Έ=ξ€·πœ•π‘¦πœŽ1,𝜎2ξ€Έ+𝑆2ξ€·πœ•2π‘‘π‘¦πœŽ1,𝜎2ξ€Έ.(3.20) Hence, we get β€–β€–πœŽ2β€–β€–2𝐿2+𝑆112π‘‘β€–β€–πœŽπ‘‘π‘‘2β€–β€–2𝐿2+12πœ‡1𝑆π‘₯π‘₯πœ•π‘¦π‘’βˆ’π‘†π‘π‘₯π‘₯πœ•π‘¦π‘’π‘,𝜎2ξ€Έβ‰€β€–β€–πœ•π‘¦πœŽ1‖‖𝐿2β€–β€–πœŽ2‖‖𝐿2+𝑆2β€–β€–πœ•2π‘‘π‘¦πœŽ1‖‖𝐿2β€–β€–πœŽ2‖‖𝐿2,β€–β€–πœŽ(3.21)3β€–β€–2𝐿2+𝑆112π‘‘β€–β€–πœŽπ‘‘π‘‘3β€–β€–2𝐿2βˆ’2𝑆1𝑆π‘₯π‘¦πœ•π‘¦π‘’βˆ’π‘†π‘π‘₯π‘¦πœ•π‘¦π‘’π‘,𝜎3ξ€Έ=βˆ’2𝑆2ξ‚€ξ€·πœ•π‘¦π‘’ξ€Έ2βˆ’ξ€·πœ•π‘¦π‘’π‘ξ€Έ2,𝜎3.(3.22) Let us estimate third term on the left-hand side of (3.20) in the time interval [0,𝑇𝑁): 𝑆π‘₯π‘₯πœ•π‘¦π‘’βˆ’π‘†π‘π‘₯π‘₯πœ•π‘¦π‘’π‘,𝜎2ξ€Έ=𝑆π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯πœ•ξ€Έξ€·π‘¦ξ€·π‘’+𝑒𝑁+𝑆𝑁π‘₯π‘₯ξ€·πœ•π‘¦ξ€·π‘’βˆ’π‘’π‘ξ€Έξ€Έ+πœ•π‘¦π‘’π‘π‘†ξ€·ξ€·π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯ξ€Έ,𝜎2.ξ€Έξ€Έ(3.23) Thus, ||𝑆π‘₯π‘₯πœ•π‘¦π‘’βˆ’π‘†π‘π‘₯π‘₯πœ•π‘¦π‘’π‘,𝜎2ξ€Έ||≀sup𝑦||πœ•π‘¦ξ€·π‘’+𝑒𝑁||‖‖𝑆π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2β€–β€–πœŽ2‖‖𝐿2+sup𝑦||𝑆𝑁π‘₯π‘₯||β€–β€–πœ•π‘¦ξ€·π‘’βˆ’π‘’π‘ξ€Έβ€–β€–πΏ2β€–β€–πœŽ2‖‖𝐿2+sup𝑦||πœ•π‘¦π‘’π‘||‖‖𝑆π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2β€–β€–πœŽ2‖‖𝐿2≀𝑐‖‖𝑒+𝑒𝑁‖‖𝐻2‖‖𝑒+𝑐𝑁‖‖𝐻2‖‖𝑆π‘₯π‘₯βˆ’π‘ƒπ‘π‘†π‘₯π‘₯‖‖𝐿2+‖‖𝑃𝑁𝑆π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2ξ€Έβ€–β€–πœŽ2‖‖𝐿2‖‖𝑆+𝑐𝑁π‘₯π‘₯‖‖𝐻1ξ€·β€–β€–π‘’βˆ’π‘ƒπ‘π‘’β€–β€–π»1+β€–β€–π‘ƒπ‘π‘’βˆ’π‘’π‘β€–β€–π»1ξ€Έβ€–β€–πœŽ2‖‖𝐿2×𝐢≀5π‘πœ“π‘π‘βˆ’π‘šβ€–β€–π‘†π‘₯π‘₯β€–β€–π»π‘š+β€–β€–πœŽ2‖‖𝐿2ξ€Έβ€–β€–πœŽ2‖‖𝐿2+2𝑐1𝐢𝑝𝑁1βˆ’π‘šβ€–π‘’β€–π»π‘š+β€–β€–πœŽ2‖‖𝐿2ξ€Έβ€–β€–πœŽ2‖‖𝐿2.(3.24) Hence, β€–β€–πœŽ2β€–β€–2𝐿2+𝑆1π‘‘β€–β€–πœŽπ‘‘π‘‘2‖‖𝐿2β‰€πœ“2β€–β€–πœŽ2‖‖𝐿2+5π‘πœ“πœ…π‘βˆ’π‘š+2π‘πœ…1πœ“1𝐢𝑝𝑁1βˆ’π‘š+β€–β€–πœ•π‘¦πœŽ1‖‖𝐿2β‰€ξ€·πœ“2ξ€Έβ€–β€–πœŽβˆ’12‖‖𝐿2+5π‘πœ“πΆπ‘πœ…π‘βˆ’π‘š+2π‘πœ…1πœ“1𝐢𝑝𝑁1βˆ’π‘š+β€–β€–πœ•π‘¦πœŽ1‖‖𝐿2≀𝑐1β€–β€–πœŽ2‖‖𝐿2+𝑐𝑁1βˆ’π‘š+β€–β€–πœ•π‘¦πœŽ1‖‖𝐿2+𝑆2β€–β€–πœ•2π‘‘π‘¦πœŽ1‖‖𝐿2.(3.25) Using the hypothesis and Gronwall’s inequality, supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–πœŽ2‖‖𝐿2≀Γ2𝑁1βˆ’π‘š,(3.26)Ξ“1 is a function of 𝑇,𝑆1,𝑆2,πœ…1,πœ“,πœ“1,πœ“2,𝑐1,𝛽1, and𝑐. Estimating the RHS of (3.22) in the time interval [0,𝑇𝑁), ξ‚€ξ€·πœ•π‘¦π‘’ξ€Έ2βˆ’ξ€·πœ•π‘¦π‘’π‘ξ€Έ2,𝜎3=ξ€·πœ•π‘¦ξ€½ξ€·π‘’+π‘’π‘ξ€Έξ€·π‘’βˆ’π‘’π‘ξ€Έξ€Ύ,𝜎3ξ€Έ=ξ€·πœ•π‘¦ξ€·π‘’+π‘’π‘ξ€Έξ€·π‘’βˆ’π‘’π‘ξ€Έ,𝜎3ξ€Έ+𝑒+π‘’π‘ξ€Έπœ•π‘¦ξ€·π‘’βˆ’π‘’π‘ξ€Έ,𝜎3ξ€Έ=ξ€·πœ•π‘¦ξ€·π‘’+π‘’π‘ξ€Έξ€·π‘’βˆ’π‘’π‘ξ€Έ,𝜎3ξ€Έ+𝑒+π‘’π‘ξ€Έπœ•π‘¦ξ€·π‘’βˆ’π‘ƒπ‘π‘’ξ€Έ,𝜎3ξ€Έ+𝑒+π‘’π‘ξ€Έπœ•π‘¦ξ€·π‘ƒπ‘π‘’βˆ’π‘’π‘ξ€Έ,𝜎3ξ€Έ,|||ξ€·πœ•(3.27)𝑦𝑒2βˆ’ξ€·πœ•π‘¦π‘’π‘ξ€Έ2,𝜎3|||≀sup𝑦||πœ•π‘¦ξ€·π‘’+𝑒𝑁||β€–β€–π‘’βˆ’π‘’π‘β€–β€–πΏ2β€–β€–πœŽ3‖‖𝐿2+sup𝑦||πœ•π‘¦ξ€·π‘’+𝑒𝑁||β€–β€–πœ•π‘¦ξ€·π‘’βˆ’π‘ƒπ‘π‘’ξ€Έβ€–β€–πΏ2β€–β€–πœŽ3‖‖𝐿2+||||ξ€œπ»0𝑒+π‘’π‘ξ€Έπœ•π‘¦ξ€·πœŽ3ξ€ΈπœŽ3||||‖‖𝑑𝑦≀𝑐𝑒+𝑒𝑁‖‖𝐻2ξ€·β€–β€–π‘’βˆ’π‘ƒπ‘π‘’β€–β€–πΏ2+β€–β€–π‘ƒπ‘π‘’βˆ’π‘’π‘β€–β€–πΏ2ξ€Έβ€–β€–πœŽ3‖‖𝐿2β€–β€–+𝑐𝑒+𝑒𝑁‖‖𝐻1β€–β€–π‘’βˆ’π‘ƒπ‘π‘’β€–β€–π»1β€–β€–πœŽ3‖‖𝐿2+12ξ€œπ»0𝜎23||πœ•π‘¦ξ€·π‘’+𝑒𝑁||𝐢𝑑𝑦≀3π‘πœ“π‘π‘βˆ’π‘šβ€–π‘’β€–π»π‘š+β€–β€–πœŽ3‖‖𝐿2ξ€Έβ€–β€–πœŽ3‖‖𝐿2+3π‘πœ“πΆπ‘π‘1βˆ’π‘šβ€–π‘’β€–π»π‘šβ€–β€–πœŽ3‖‖𝐿2+12sup𝑦||πœ•π‘¦ξ€·π‘’+𝑒𝑁||ξ€œπ»0𝜎23𝑑𝑦.(3.28) Noting that the last integral is bounded by(1/2)3π‘πœ“, the estimate is |||ξ€·πœ•π‘¦π‘’ξ€Έ2βˆ’ξ€·πœ•π‘¦π‘’π‘ξ€Έ2,𝜎3|||β€–β€–πœŽβ‰€3π‘πœ“3‖‖𝐿2ξ‚€32β€–β€–πœŽ3‖‖𝐿2+πΆπ‘β€–π‘’β€–π»π‘šξ€·π‘βˆ’π‘š+𝑁1βˆ’π‘šξ€Έξ‚.(3.29)(𝑆π‘₯π‘¦πœ•π‘¦π‘’βˆ’π‘†π‘π‘₯π‘¦πœ•π‘¦π‘’π‘,𝜎3)can be estimated in exactly the same way as (𝑆π‘₯π‘₯πœ•π‘¦π‘’βˆ’π‘†π‘π‘₯π‘₯πœ•π‘¦π‘’π‘,𝜎2). Then, (3.22) as a whole is estimated as β€–β€–πœŽ3‖‖𝐿2+𝑆1π‘‘β€–β€–πœŽπ‘‘π‘‘3‖‖𝐿2β‰€ξ€·πœ“2β€–β€–πœŽ3‖‖𝐿2+5π‘πœ“πΆπ‘πœ…π‘βˆ’π‘š+2π‘πœ…1πœ“1𝐢𝑝𝑁1βˆ’π‘šξ€Έξ‚€3+3π‘πœ“2β€–β€–πœŽ3‖‖𝐿2+πΆπ‘πœ…ξ€·π‘βˆ’π‘š+𝑁1βˆ’π‘šξ€Έξ‚.(3.30) Therefore, we get 𝑐1β€–β€–πœŽ3‖‖𝐿2+𝑆1π‘‘β€–β€–πœŽπ‘‘π‘‘3‖‖𝐿2≀𝑐𝑁1βˆ’π‘š.(3.31) Then, using the Gronwall’s inequality, supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–πœŽ3‖‖𝐿2≀Γ3𝑁1βˆ’π‘š,(3.32) where Ξ“3 is a function of 𝑇,𝑆1,πœ…1,πœ“,πœ“1,𝑐1, and𝑐. Since β€–β€–π‘’βˆ’π‘’π‘β€–β€–πΏ2=‖‖𝑒+π‘ƒπ‘π‘’βˆ’π‘ƒπ‘π‘’βˆ’π‘’π‘β€–β€–πΏ2β‰€β€–β€–π‘’βˆ’π‘ƒπ‘π‘’β€–β€–πΏ2+β€–β€–π‘ƒπ‘π‘’βˆ’π‘’π‘β€–β€–πΏ2,‖‖𝑆(3.33)π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦‖‖𝐿2=‖‖𝑆π‘₯𝑦+π‘ƒβˆ—π‘π‘†π‘₯π‘¦βˆ’π‘ƒβˆ—π‘π‘†π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦‖‖𝐿2≀‖‖𝑆π‘₯π‘¦βˆ’π‘ƒβˆ—π‘π‘†π‘₯𝑦‖‖𝐿2+β€–β€–π‘ƒβˆ—π‘π‘†π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦‖‖𝐿2,‖‖𝑆(3.34)π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2=‖‖𝑆π‘₯π‘₯+π‘ƒβˆ—π‘π‘†π‘₯π‘₯βˆ’π‘ƒβˆ—π‘π‘†π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2≀‖‖𝑆π‘₯π‘₯βˆ’π‘ƒβˆ—π‘π‘†π‘₯π‘₯‖‖𝐿2+β€–β€–π‘ƒβˆ—π‘π‘†π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2.(3.35) 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 supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘’βˆ’π‘’π‘β€–β€–πΏ2≀Γ1𝑁1βˆ’π‘š,supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘†π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦‖‖𝐿2≀Γ2𝑁1βˆ’π‘š,supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘†π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐿2≀Γ3𝑁1βˆ’π‘š,(3.36) where Ξ“1,Ξ“2, and Ξ“3 are constants, functions of (𝑇,𝛼2,𝐢𝑝,πœ…,𝛽, and 𝑐), (𝑇,𝑆1,𝑆2,πœ…1,πœ“,πœ“1,πœ“2,𝑐1,𝛽1, and 𝑐), and of (𝑇,𝑆1,πœ…1,πœ“,πœ“1,𝑐1,πœ…, and𝑐), respectively.

Lemma 3.3. Suppose that the solution {𝑒𝑁,𝑆𝑁π‘₯𝑦,𝑆𝑁π‘₯π‘₯} of (3.1)–(3.3) exists on the time interval [0,π‘‘βˆ—π‘] and that supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]‖𝑒𝑁(𝑦,𝑑)‖𝐻2≀2πœ“, supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]‖𝑆𝑁π‘₯𝑦(𝑦,𝑑)‖𝐻2≀2πœ“1, supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]‖𝑆𝑁π‘₯π‘₯(𝑦,𝑑)‖𝐻2≀2πœ“2, supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]|𝑃𝑁𝑒(β„Ž,𝑑)βˆ’π‘’π‘(β„Ž,𝑑)|≀𝛽𝑁1βˆ’π‘š, and supπ‘‘βˆˆ[0,π‘‘βˆ—π‘]|π‘ƒβˆ—π‘π‘†π‘₯𝑦(β„Ž,𝑑)βˆ’π‘†π‘π‘₯𝑦(β„Ž,𝑑)|≀𝛽1𝑁1βˆ’π‘š, then the error estimate: supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘’βˆ’π‘’π‘β€–β€–π»2≀Γ1𝑁3βˆ’π‘š,supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘†π‘₯π‘¦βˆ’π‘†π‘π‘₯𝑦‖‖𝐻2≀Γ2𝑁3βˆ’π‘š,supξ€Ίπ‘‘βˆˆ0,π‘‘βˆ—π‘ξ€»β€–β€–π‘†π‘₯π‘₯βˆ’π‘†π‘π‘₯π‘₯‖‖𝐻2≀Γ3𝑁3βˆ’π‘š,(3.37) holds for the constants Ξ“1,Ξ“2, and Ξ“3. 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[0,𝑇]π‘‘βˆ—π‘ unspecified in Lemma 3.2 is defined as π‘‘βˆ—π‘ξ€½[]=supπ‘‘βˆˆ0,π‘‡βˆ£forallπ‘‘ξ…žβ€–β€–π‘’β‰€π‘‘,𝑁𝑦,π‘‘ξ…žξ€Έβ€–β€–π»2≀2πœ“.(3.38) Thus, the time π‘‘βˆ—π‘ corresponds to the largest time in [0,𝑇] for which the 𝐻2-norm of 𝑒𝑁 is uniformly bounded by 2πœ“. Since ‖𝑒𝑁(𝑦,0)‖𝐻2=‖𝑃𝑁(𝑦,0)‖𝐻2, ‖‖𝑒𝑁‖‖(𝑦,0)𝐻2≀‖𝑒(𝑦,0)‖𝐻2β‰€πœ“,(3.39) therefore π‘‘βˆ—π‘>0 for all 𝑁. Note that π‘‘βˆ—π‘ is smaller than the maximum time of existence of the solution 𝑇𝑁. Now, we need to show that there exists 𝑁𝐿 such that π‘‘βˆ—π‘=π‘‡βˆ€π‘β‰₯𝑁𝐿,(3.40) and therefore the supremum in (3.14) holds on [0,𝑇]. From the definition (3.38), we either have π‘‘βˆ—π‘=𝑇 or π‘‘βˆ—π‘<𝑇 in which case‖𝑒𝑁(𝑦,π‘‘ξ…ž)‖𝐻2=2πœ“. Now assume that π‘‘βˆ—π‘<𝑇, then ‖‖𝑒2πœ“=𝑁𝑦,π‘‘βˆ—π‘ξ€Έβ€–β€–π»2≀‖‖𝑒𝑁𝑦,π‘‘βˆ—π‘ξ€Έβ€–β€–βˆ’π‘’(𝑦,𝑑)𝐻2β€–+‖𝑒(𝑦,𝑑)𝐻2=‖‖𝑒𝑁𝑦,π‘‘βˆ—π‘ξ€Έβ€–β€–βˆ’π‘’(𝑦,𝑑)𝐻2+πœ“.(3.41) Hence, we obtain β€–β€–π‘’πœ“β‰€π‘ξ€·π‘¦,π‘‘βˆ—π‘ξ€Έβ€–β€–βˆ’π‘’(𝑦,𝑑)𝐻2.(3.42) On the other hand, Lemma 3.3 implies πœ“β‰€Ξ“1𝑁3βˆ’π‘š(3.43) or Γ𝑁≀1πœ“ξ‚Ά1/π‘šβˆ’3.(3.44) In conclusion, for𝑁𝐿>(Ξ“1/πœ“)1/π‘šβˆ’3, we cannot have π‘‘βˆ—π‘<𝑇 and claim (3.40) holds. It follows that 𝑁β‰₯𝑁𝐿 the solution 𝑒𝑁 of (3.1) is defined on[0,𝑇], since as noted before π‘‘βˆ—π‘<𝑇𝑁, and, from (3.14), sup[]π‘‘βˆˆ0,𝑇‖‖𝑒(𝑦,𝑑)βˆ’π‘’π‘β€–β€–(𝑦,𝑑)𝐿2≀Γ1𝑁1βˆ’π‘š.(3.45)
In exactly the same way, we can extend the estimate of the second and third inequalities in (3.14) to the time interval[0,𝑇] and show that sup[]π‘‘βˆˆ0,𝑇‖‖𝑆π‘₯𝑦(𝑦,𝑑)βˆ’π‘†π‘π‘₯𝑦‖‖(𝑦,𝑑)𝐿2≀Γ2𝑁1βˆ’π‘š,sup[]π‘‘βˆˆ0,𝑇‖‖𝑆π‘₯π‘₯(𝑦,𝑑)βˆ’π‘†π‘π‘₯π‘₯β€–β€–(𝑦,𝑑)𝐿2≀Γ3𝑁1βˆ’π‘š.(3.46)

4. Numerical Results and Discussion

The system of differential equations (3.6)–(3.9) is of the following form𝑑𝑑𝑑̂𝑒𝑁(π‘˜,𝑑)=𝐺1̂𝑠𝑁π‘₯𝑦,𝑑(π‘˜,𝑑)𝑑𝑑̂𝑠𝑁π‘₯𝑦(π‘˜,𝑑)=𝐺2̂𝑠𝑁π‘₯π‘₯(𝑖,𝑑),̂𝑠𝑁π‘₯𝑦(π‘˜,𝑑),̂𝑒𝑁(π‘˜,𝑑),𝑆1,𝑆2ξ€Έ,𝑑𝑑𝑑̂𝑠𝑁π‘₯π‘₯(π‘˜,𝑑)=𝐺3̂𝑠𝑁π‘₯π‘₯(𝑖,𝑑),̂𝑠𝑁π‘₯𝑦(π‘˜,𝑑),̂𝑒𝑁(π‘˜,𝑑),𝑆1,𝑆2ξ€Έ,̂𝑒𝑁(π‘˜,𝑑)=0,̂𝑠𝑁π‘₯𝑦(π‘˜,𝑑)=0,̂𝑠𝑁π‘₯π‘₯(π‘˜,𝑑)=0.(4.1) Runge-Kutta method is applied to this system. The integrals in equations (2.8) and (2.9) are calculated analytically and numerically, respectively, with πœ•π‘„/πœ•β„Žapproximated 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 (πœ‡1=0) 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 𝑁=5 nodes only for both permeable and impermeable wall. The exact and approximate solutions are indistinguishable in the figure. The error log10(β€–π‘’π‘βˆ’π‘’β€–πΏβˆž[0,β„Ž])at 𝑑=1 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 𝑁=20. The aim of this paper is to elaborate the effects of the nonlinear parameter πœ‡1 and porous medium parameter 𝛼2 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 πœ‡1 on the centerline velocity when the wall is impermeable 𝛼=0, and 𝑆1=2, 𝑆2=1, and β„Ž=1. Clearly, the overshoot gradually disappears as numerical values of the nonlinear parameter πœ‡1 increase. In addition, the steady centerline velocity increases with increasing values of πœ‡1. Figure 3 explores the effect of porosity on the centerline velocity for πœ‡1=10. Increasing the porosity parameter triggers a decrease in the value of the centerline velocity. The difference between the relaxation and retardation times,𝑆1βˆ’π‘†2, 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 πœ‡1 and porous medium parameter 𝛼2 on the drainage rate is examined in Figures 6 and 7, respectively. Since in all cases the nonlinear parameter πœ‡1 has an increasing effect on the velocity, we expect increasing πœ‡1 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 asπœ‡1 is increased from zero. Since the porous medium parameter 𝛼2 has a decreasing effect on the velocity in all cases (both permeable and impermeable wall), we expect increasing 𝛼2 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.