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 (x,y,z). The x 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)∂Sxy∂y=∂u∂t-1+α2u,Sxy+S1∂Sxy∂t+12μ1Sxx∂u∂y=(∂u∂y+S2∂2u∂y∂t),Sxx+S1∂Sxx∂t-2S1Sxy∂u∂y=-2S2(∂u∂y)2,
where above we used the following dimensionless parameters as in [3]y=Y(gν2)1/3,x=x(gν2)1/3,h=H(x,t)y=Y(gν2)1/3,t=Ty=Y(g2ν)1/3,S1=λ1(g2ν)1/3,S2=λ2(g2ν)1/3,u=U(νg)1/3,v=V(νg)1/3,Sij=Pijμgν1/3,μ1=μ0(g2ν)1/3,α2=(νg)1/3ρg.
And here u=u(y,t),Sxy=Sxy(y,t),Sxx=Sxx(y,t) are the dimensionless velocity and the dimensionless deviatoric stress tensor; α2,S1,S2, and μ1 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,u(0,t)=0,t≥0,∂u∂y=0ony=h.
The liquid is at rest at t=0, therefore initial conditions areu(y,0)=Sxy(y,0)=Sxx(y,0)=0.
To calculate the shape of the film profile at a given time t, the thickness h is allowed to vary with x while assuming the flow is still locally parallel. Combining the material derivative at the free surfacev(x,h,t)=∂h∂t+u(x,h,t)∂h∂x,
with the equation of continuity yields a differential equation in h,-∂h∂x(∫0h∂u∂hdy+u)=∂h∂t.
Introducing the flow rate Q(h,t) per unit width across the film thickness h,Q(h,t)=∫0hu(x,y,t)dy,
differentiating Q with respect to h, and substituting the result into (2.8) and integrating givex(h,t)-x0(h)=∫0t∂Q∂hdτ,
where x0(h)=x(h,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 x as a function of h and t.

Next some mathematical notation is introduced. Denote the inner product in 𝕃2(0,h) by(f,g)=∫0hf(y)g(y)dy.
If f∈𝕃2(0,H), then Fourier sine series is defined asf(y)=∑k=1∞f̂s(k)sin(((2k-1)π2h)y),
wheref̂s(k)=2h∫0πsin((2k-1)πy2h)f(y)dyk=1,2,….
Similarly, Fourier cosine series is defined asf(y)=f̂c(0)2+∑k=1∞f̂c(k)cos(((2k-1)π2h)y),
wheref̂c(k)=2H∫0Hcos((2k-1)πy2)f(y)dyk=1,2,….
Denote by ∥∥Hm the Sobolev norm, given by‖f‖Hm2={∑k=1∞(1+|2k-12|2)m|f̂s(k)|2,|f̂c(0)|22+∑k=1∞(1+|2k-12|2)m|f̂c(k)|2.

The space of periodic Sobolev functions on the interval [0,h]is defined as the closure of the space of smooth periodic functions with respect to the Hm-norm and will be simply denoted by Hm. In particular, the space 𝕃2(0,h)with norm denoted by ∥∥𝕃2 is recovered for m=0. We now define subspaces of 𝕃2(0,h)spanned by the setDN*={2sin((2k-1)πy/2h)h,1≤k≤N},DN**={2cos((2k-1)πy/2h)h,1≤k≤N}.
The operators PN and PN* denote the orthogonal, self-adjoint projection of 𝕃2 onto DN* and DN**defined, respectively, byPNf(y)=∑k=1Nsin((2k-1)πy2h)f̂s(k),PN*f(y)=f̂c(0)2+∑k=1Ncos((2k-1)πy2h)f̂c(k).
Forf∈Hm, the estimates:‖f-PNf‖L2≤CpN-m‖∂ymf‖L2,‖f-PNf‖Hn≤CpNn-m‖∂ymf‖L2,
hold for an appropriate constant Cp 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 C([0,T],Hn). Similarly, we also consider the space C([0,T],DN*), where the topology on the finite-dimensional space DN* can be given by any norm. Finally, note the inverse inequality‖∂ymφ‖L2≤Nm‖φ‖L2,
which holds for integers m>0 and φ∈DN*. 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.

We now note that exactly the same estimates hold for PN*. 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 (C([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

{ek(y),kϵℕ}={2sin((2k-1)πy/2h)/h,k∈ℕ} and {fk(y),kϵℕ}={2cos((2k-1)πy/2h)/h,k∈ℕ} are chosen to be an orthonormal basis of the Hilbert space𝕃02[0,h] and 𝕃2[0,h], respectively. Then, the subspace of these Hilbert spaces spanned by the DN*={2sin((2k-1)πy/2h)/h,1≤k≤N} andDN**={2cos((2k-1)πy/2h)/h,0≤k≤N},respectively. Fourier-Galerkin approximation of (2.1)–(2.5) are find the functions uN(t)∈DN*, SxyN(t), and SxxN∈DN** for all 0≤t≤T, such that(∂tuN-1-∂YSxyN,ω1+β2u,ω1)=0,t∈[0,T],(SxyN+S1∂tSxyN+12μ1SxxN∂yuN-∂yuN-S2∂ty2uN,ω2)=0,t∈[0,T],(SxxN+S1∂tSxxN-2S1SxyN∂yuN-2S2(∂yuN)2,ω2)=0,t∈[0,T],uN(0)=0,SxyN(0)=0,SxxN(0)=0,
for all ω1∈DN* and for all ω2∈DN**. Since for each t, uN(·,t), SxyN(·,t), and SxyN(·,t) have the formuN(y,t)=∑k=1NûN(k,t)2sin((2k-1)πy/2h)h,SxyN(y,t)=∑k=1NŝxyN(k,t)2cos((2k-1)πy/2h)h,SxxN(y,t)=∑k=1NŝxxN(k,t)2cos((2k-1)πy/2h)h.

Taking ω1=2/Hsin((2k-1)πy/2h),ω2=2/Hcos((2k-1)πy/2h),1≤k≤Nin (3.1)–(3.3) yields the following system of equations for the Fourier coefficients of uN, SxyN, and SxxN:ddtûN(k,t)=-(2k-1)π2hŝxyN(k,t)+2h2hπ(2k-1),ŝxyN(k,t)+S1ddtŝxyN(k,t)=(2k-1)π2hûN(k,t)+S2(2k-1)π2hddtûN(k,t)+μ1(2k-1)π4h2h∑i,j=1NcijNŝxxN(i,t)ûN(k,t),ŝxxN(k,t)+S1ddtŝxxN(k,t)-(2k-1)π2hûN(k,t)-2S1(2k-1)π4h2h∑i,j=1NcijNŝxyN(i,t)ûN(k,t)=-2S2((2k-1)π4h)22h∑i,j=1NdijNûN(i,t)ûN(j,t),ûN(k,t)=0,ŝxyN(k,t)=0,ŝxxN(k,t)=0,cijN=∫0hcos((2i-1)πy2h)sin((2j-1)πy2h)cos((2N-1)πy2h)dy,dijN=∫0hsin((2i-1)πy2h)sin((2j-1)πy2h)cos((2N-1)πy2h)dy.

This is a nonlinear system of ordinary differential equations for the functions {ûN(k,t),ŝxyN(k,t),ŝxxN(k,t)}k=1N; 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 {uN,SxyN,SxxN} converges to the exact solution {u,Sxy,Sxx} when u is smooth enough. This is stated in the next theorem.

Theorem 3.1.

Suppose that a solution {u,Sxy,Sxx} of (2.1)–(2.5) exists in the space (C([0,T],Hm))3 for m≥1 and for some time T>0. If ∥PNu(h,t)-uN(u,t)∥≤cN1-m and ∥PN*Sxy-SxyN∥≤cN1-m, then, for large enough N, there exists a unique solution {uN,SxyN,SxxN} of the finite dimensional problem (3.1)–(3.4). Moreover, there exist constants Γ1,Γ2, and Γ3 such that
supt∈[0,T]‖u-uN‖L2≤Γ1N1-m,supt∈[0,T]‖Sxy-SxyN‖L2≤Γ2N1-m,supt∈[0,T]‖Sxx-SxxN‖L2≤Γ3N1-m.

Before the proof is given, note that the assumptions of the theorem encompass the existence of constants κ, κ1, and κ2 such thatsupt∈[0,T]‖u(y,t)‖Hm≤κ,supt∈[0,T]‖Sxx(y,t)‖Hm≤κ1,supt∈[0,T]‖Sxy(y,t)‖Hm≤κ2.
In particular, it follows that there are other constants ψ, ψ1, and ψ2 such thatsupt∈[0,T]‖u(y,t)‖Hm≤ψ,supt∈[0,T]‖Sxx(y,t)‖Hm≤ψ1,supt∈[0,T]‖Sxy(y,t)‖Hm≤ψ2.
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 {uN,SxyN,SxxN} of (3.1)–(3.4) exists on the time interval [0,tN*] and that supt∈[0,tN*]∥uN(y,t)∥H2≤2ψ, supt∈[0,tN*]∥SxyN(y,t)∥H2≤2ψ1, supt∈[0,tN*]∥SxxN(y,t)∥H2≤2ψ2, supt∈[0,tN*]|PNu(h,t)-uN(h,t)|≤βN1-m, and supt∈[0,tN*]|PN*Sxy(h,t)-SxyN(h,t)|≤β1N1-m, then the error estimate:
supt∈[0,tN*]‖u-uN‖L2≤Γ1N1-m,supt∈[0,tN*]‖Sxy-SxyN‖L2≤Γ2N1-m,supt∈[0,tN*]‖Sxx-SxxN‖L2≤Γ3N1-m,
holds for constant Γ1 which is the function of T,α2,Cp,κ,β, and c, constantΓ2 which is the function of T,S1,S2,κ1,ψ,ψ1,ψ2,c1,β1, andc, and constant Γ3 which is the function of T,S1,κ1,ψ,ψ1,c1,κ, and c.

Proof.

Letσ1=PNu-uN,σ2=PN*Sxy-SxyN,σ3=PN*Sxx-SxxN. Also, from the definition of PN and PN*,we have ∂yσ1=PN*∂yu-∂yuN,∂yσ2=PN∂ySxy-∂ySxyN,∂yσ3=PN∂ySxx-∂ySxxN. We applyPN,PN* and PN* to both sides of (2.1)–(2.3), respectively. Since PN, PN* commute with derivation, we obtain
∂tPNu+α2u=1+∂yPNSxy,PN*Sxy+S1∂tPN*Sxy+12μ1PN*(Sxx∂yu)=PN*∂yu+S2∂tPN*∂yu,PN*Sxx+S1∂tPN*Sxx-2S1PN*(Sxy∂yu)=-2S2PN*(∂yu)2.
We multiply these equations with test functions σ1∈DN*,σ2∈DN**, and σ3∈DN**, respectively, integrate over [0,h], and subtract the resulting expressions from (3.1), (3.2), and (3.3) to get
12ddt‖σ1‖L22+α2‖σ1‖L22=(PN1-1N,σ1)+(PN∂y(Sxy-SxyN),σ1),‖σ2‖L22+S112ddt‖σ2‖L22+12μ1(PN*(Sxx∂yu)-SxxN∂yuN,σ2)=(PN*∂y(u-uN),σ2)+S2(PN*∂ty2(u-uN),σ2),‖σ2‖L22+S112ddt‖σ2‖L22+12μ1(PN*(Sxx∂yu)-SxxN∂yuN,σ2)=(PN*∂y(u-uN),σ2)+S2(∂tPN*∂y(u-uN),σ2),‖σ3‖L22+S112ddt‖σ3‖L22-2S1(PN*(Sxy∂yu)-SxyN∂yuN,σ3)=-2S2(PN*(∂yu)2-(∂yuN)2,σ3).
Since σ1∈DN*,(σ2,σ3)∈DN**,
12ddt‖σ1‖L22+α2‖σ1‖L22=(∂yσ2,σ1)≤‖∂yσ2‖L2‖σ1‖L2,
then we have
12ddt‖σ1‖L22+α2‖σ1‖L22≤‖∂yσ2‖L2≤|σ2(H,t)|.
Consequently, from hypothesis and Gronwall’s inequality, we obtain
supt∈[0,tN*]‖σ1‖L2≤Γ1N1-m,
where Γ1 is a function of T,β,α2, andc:
‖σ2‖L22+S112ddt‖σ2‖L22+12μ1(Sxx∂yu-SxxN∂yuN,σ2)=(∂yσ1,σ2)+S2(∂ty2σ1,σ2).
Hence, we get
‖σ2‖L22+S112ddt‖σ2‖L22+12μ1(Sxx∂yu-SxxN∂yuN,σ2)≤‖∂yσ1‖L2‖σ2‖L2+S2‖∂ty2σ1‖L2‖σ2‖L2,‖σ3‖L22+S112ddt‖σ3‖L22-2S1(Sxy∂yu-SxyN∂yuN,σ3)=-2S2((∂yu)2-(∂yuN)2,σ3).
Let us estimate third term on the left-hand side of (3.20) in the time interval [0,TN):
(Sxx∂yu-SxxN∂yuN,σ2)=((Sxx-SxxN)(∂y(u+uN))+SxxN(∂y(u-uN))+∂yuN((Sxx-SxxN),σ2)).
Thus,
|(Sxx∂yu-SxxN∂yuN,σ2)|≤supy|∂y(u+uN)|‖Sxx-SxxN‖L2‖σ2‖L2+supy|SxxN|‖∂y(u-uN)‖L2‖σ2‖L2+supy|∂yuN|‖Sxx-SxxN‖L2‖σ2‖L2≤(c‖u+uN‖H2+c‖uN‖H2)(‖Sxx-PNSxx‖L2+‖PNSxx-SxxN‖L2)‖σ2‖L2+c‖SxxN‖H1(‖u-PNu‖H1+‖PNu-uN‖H1)‖σ2‖L2≤5cψ×(CpN-m‖Sxx‖Hm+‖σ2‖L2)‖σ2‖L2+2c⋀1(CpN1-m‖u‖Hm+‖σ2‖L2)‖σ2‖L2.
Hence,
‖σ2‖L22+S1ddt‖σ2‖L2≤ψ2‖σ2‖L2+5cψκN-m+2cκ1ψ1CpN1-m+‖∂yσ1‖L2≤(ψ2-1)‖σ2‖L2+5cψCpκN-m+2cκ1ψ1CpN1-m+‖∂yσ1‖L2≤c1‖σ2‖L2+cN1-m+‖∂yσ1‖L2+S2‖∂ty2σ1‖L2.
Using the hypothesis and Gronwall’s inequality,
supt∈[0,tN*]‖σ2‖L2≤Γ2N1-m,Γ1 is a function of T,S1,S2,κ1,ψ,ψ1,ψ2,c1,β1, andc. Estimating the RHS of (3.22) in the time interval [0,TN),
((∂yu)2-(∂yuN)2,σ3)=(∂y{(u+uN)(u-uN)},σ3)=(∂y(u+uN)(u-uN),σ3)+((u+uN)∂y(u-uN),σ3)=(∂y(u+uN)(u-uN),σ3)+((u+uN)∂y(u-PNu),σ3)+((u+uN)∂y(PNu-uN),σ3),|(∂yu)2-(∂yuN)2,σ3|≤supy|∂y(u+uN)|‖u-uN‖L2‖σ3‖L2+supy|∂y(u+uN)|‖∂y(u-PNu)‖L2‖σ3‖L2+|∫0H(u+uN)∂y(σ3)σ3dy|≤c‖u+uN‖H2(‖u-PNu‖L2+‖PNu-uN‖L2)‖σ3‖L2+c‖u+uN‖H1‖u-PNu‖H1‖σ3‖L2+12∫0Hσ32|∂y(u+uN)dy|≤3cψ(CpN-m‖u‖Hm+‖σ3‖L2)‖σ3‖L2+3cψCpN1-m‖u‖Hm‖σ3‖L2+12supy|∂y(u+uN)|∫0Hσ32dy.
Noting that the last integral is bounded by(1/2)3cψ, the estimate is
|(∂yu)2-(∂yuN)2,σ3|≤3cψ‖σ3‖L2(32‖σ3‖L2+Cp‖u‖Hm(N-m+N1-m)).(Sxy∂yu-SxyN∂yuN,σ3)can be estimated in exactly the same way as (Sxx∂yu-SxxN∂yuN,σ2). Then, (3.22) as a whole is estimated as
‖σ3‖L2+S1ddt‖σ3‖L2≤(ψ2‖σ3‖L2+5cψCpκN-m+2cκ1ψ1CpN1-m)+3cψ(32‖σ3‖L2+Cpκ(N-m+N1-m)).
Therefore, we get
c1‖σ3‖L2+S1ddt‖σ3‖L2≤cN1-m.
Then, using the Gronwall’s inequality,
supt∈[0,tN*]‖σ3‖L2≤Γ3N1-m,
where Γ3 is a function of T,S1,κ1,ψ,ψ1,c1, andc. Since
‖u-uN‖L2=‖u+PNu-PNu-uN‖L2≤‖u-PNu‖L2+‖PNu-uN‖L2,‖Sxy-SxyN‖L2=‖Sxy+PN*Sxy-PN*Sxy-SxyN‖L2≤‖Sxy-PN*Sxy‖L2+‖PN*Sxy-SxyN‖L2,‖Sxx-SxxN‖L2=‖Sxx+PN*Sxx-PN*Sxx-SxxN‖L2≤‖Sxx-PN*Sxx‖L2+‖PN*Sxx-SxxN‖L2.
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
supt∈[0,tN*]‖u-uN‖L2≤Γ1N1-m,supt∈[0,tN*]‖Sxy-SxyN‖L2≤Γ2N1-m,supt∈[0,tN*]‖Sxx-SxxN‖L2≤Γ3N1-m,
where Γ1,Γ2, and Γ3 are constants, functions of (T,α2,Cp,κ,β, and c), (T,S1,S2,κ1,ψ,ψ1,ψ2,c1,β1, and c), and of (T,S1,κ1,ψ,ψ1,c1,κ, andc), respectively.

Lemma 3.3.

Suppose that the solution {uN,SxyN,SxxN} of (3.1)–(3.3) exists on the time interval [0,tN*] and that supt∈[0,tN*]∥uN(y,t)∥H2≤2ψ, supt∈[0,tN*]∥SxyN(y,t)∥H2≤2ψ1, supt∈[0,tN*]∥SxxN(y,t)∥H2≤2ψ2, supt∈[0,tN*]|PNu(h,t)-uN(h,t)|≤βN1-m, and supt∈[0,tN*]|PN*Sxy(h,t)-SxyN(h,t)|≤β1N1-m, then the error estimate:
supt∈[0,tN*]‖u-uN‖H2≤Γ1N3-m,supt∈[0,tN*]‖Sxy-SxyN‖H2≤Γ2N3-m,supt∈[0,tN*]‖Sxx-SxxN‖H2≤Γ3N3-m,
holds for the constants Γ1,Γ2, and Γ3. 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 <xref ref-type="statement" rid="thm3.1">3.1</xref>.

To extend the estimate of the first inequality in (3.14) to the time interval[0,T]tN* unspecified in Lemma 3.2 is defined as
tN*=sup{t∈[0,T]∣forallt′≤t,‖uN(y,t′)‖H2≤2ψ}.
Thus, the time tN* corresponds to the largest time in [0,T] for which the H2-norm of uN is uniformly bounded by 2ψ. Since ∥uN(y,0)∥H2=∥PN(y,0)∥H2,
‖uN(y,0)‖H2≤‖u(y,0)‖H2≤ψ,
therefore tN*>0 for all N. Note that tN* is smaller than the maximum time of existence of the solution TN. Now, we need to show that there exists NL such that
tN*=T∀N≥NL,
and therefore the supremum in (3.14) holds on [0,T]. From the definition (3.38), we either have tN*=T or tN*<T in which case∥uN(y,t′)∥H2=2ψ. Now assume that tN*<T, then
2ψ=‖uN(y,tN*)‖H2≤‖uN(y,tN*)-u(y,t)‖H2+‖u(y,t)‖H2=‖uN(y,tN*)-u(y,t)‖H2+ψ.
Hence, we obtain
ψ≤‖uN(y,tN*)-u(y,t)‖H2.
On the other hand, Lemma 3.3 implies
ψ≤Γ1N3-m
or
N≤(Γ1ψ)1/m-3.
In conclusion, forNL>(Γ1/ψ)1/m-3, we cannot have tN*<T and claim (3.40) holds. It follows that N≥NL the solution uN of (3.1) is defined on[0,T], since as noted before tN*<TN, and, from (3.14),
supt∈[0,T]‖u(y,t)-uN(y,t)‖L2≤Γ1N1-m.

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
supt∈[0,T]‖Sxy(y,t)-SxyN(y,t)‖L2≤Γ2N1-m,supt∈[0,T]‖Sxx(y,t)-SxxN(y,t)‖L2≤Γ3N1-m.

4. Numerical Results and Discussion

The system of differential equations (3.6)–(3.9) is of the following formddtûN(k,t)=G1(ŝxyN(k,t)),ddtŝxyN(k,t)=G2(ŝxxN(i,t),ŝxyN(k,t),ûN(k,t),S1,S2),ddtŝxxN(k,t)=G3(ŝxxN(i,t),ŝxyN(k,t),ûN(k,t),S1,S2),ûN(k,t)=0,ŝxyN(k,t)=0,ŝxxN(k,t)=0.
Runge-Kutta method is applied to this system. The integrals in equations (2.8) and (2.9) are calculated analytically and numerically, respectively, with ∂Q/∂happroximated 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 (μ1=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(∥uN-u∥L∞[0,h])at t=1 of the Fourier-Galerkin approximations with increasing number of nodes for the drainage of Oldroyd-B liquid is listed in Table 1.

N (number of modes)

log10(error) impermeable wall

log10(error) permeable wall (α=1)

10

−5.1

−5.3

15

−6.45

−6.55

20

−7.12

−7.14

Comparison of exact analytical and numerical solutions of centerline velocity field for μ1=0, S1=2, S2=1, and h=1 for both permeable wall (α=1) and impermeable wall.

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 μ1 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 μ1 on the centerline velocity when the wall is impermeable α=0, and S1=2, S2=1, and h=1. Clearly, the overshoot gradually disappears as numerical values of the nonlinear parameter μ1 increase. In addition, the steady centerline velocity increases with increasing values of μ1. Figure 3 explores the effect of porosity on the centerline velocity for μ1=10. Increasing the porosity parameter triggers a decrease in the value of the centerline velocity. The difference between the relaxation and retardation times,S1-S2, 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 μ1 and porous medium parameter α2 on the drainage rate is examined in Figures 6 and 7, respectively. Since in all cases the nonlinear parameter μ1 has an increasing effect on the velocity, we expect increasing μ1 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 asμ1 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.

The effect of the nonlinear parameter μ1 on the centerline velocity profile for S1=2, S2=1, α=0, and h=1.

The effect of the porous medium parameter α2 on the centerline velocity profile for S1=2, S2=1, μ1=10, and h=1.

The effect of the elasticityon the centerline velocity profile for μ1=10 and h=1 for permeable wall (α=1).

The effect of the elasticityon the centerline velocity profile for μ1=10 and h=1 for impermeable wall.

The effect of the material parameter μ1 on the drainage rate for S1=2, S2=1 for impermeable wall.

The effect of porous medium parameter α2on the drainage rate for S1=2, S2=1, and μ1=10.

The effect of elasticity on the drainage rate for α=1 and μ1=1.

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