Conclusiones

In this subsection, we present our numerical results for Kalliadasis’s model solved using MAT-LAB’s fsolve function. We show nine test situations with trenches of widths 1, 5 and 10 and depth D=1 in Figures 5.1, 5.2 and 5.3, respectively. Different numbers of grid points were used as well. The topography s is defined by Kalliadasis et al. in [76] and afterwards by Gaskell et al. in a similar way in [53] and presented in Equation (5.1).

Figures 5.2b and 5.3b display non-physical solutions of the nonlinear system Equations (5.3) and (5.5) which implies that these systems may have multiple solutions. The solutions shown in these figures are solutions of this system but are not the ones of physical interest. Figures 5.1c, 5.2c and 5.3c depict the results of Kalliadasis’s model with large values of δ (i.e. a shallow sloping wall). It is clear from these figures that this model can work well with a fine grid for

67 5.2. Steady-State Thin Film Flow Solving in 1D

large values of δ. However, according to the results in Figures 5.1, 5.2 and 5.3 we have also found that Kalliadasis’s model does not work well in every situation, which means that Kalli-adasiss model does not give satisfactory accuracy in every situation. This may be due to the presence of the third derivative sxxx term in Equation (2.26). This term is not approximated smoothly, and it requires more grid points in order to be accurate. It is worth to note that this model can be accurate but needs increasing numbers of grid points for a small value of δ.

In order to provide further details of the continuation strategy employed to obtain the results in Figures 5.1 to 5.3, we show in Table 5.1 the intermediate cases for solving Kalliadasis’s model with 401 mesh points, D = 1 and W = 1. We solve this model with the starting value of λ = 1 and δ = 0.5 and then we reduce both. We introduce a better initial guess at each stage through continuation and then solve the discrete system for this model.

Table 5.2 shows the numerical calculations based on Kalliadasis’ model with 401 mesh points and different values of λ, δ and W = 1, 5 and 10. Each row corresponds to the equivalent row in Table 5.1 and the ”iterations” column gives the number of iterations taken by fsolve.

Table 5.3 shows the equivalent results of numerical calculations for Kalliadasis’s model with 801 mesh points. It is apparent that it is very difficult to get convergence as λ→ 0 for small δ.

Consequently, it can be seen from the data displayed in Table 5.4 that the model gives much better results if δ = 0.5 (and so the bed topography is artificially smooth).

To summarise, the scheme introduced in [76] proved to be expensive, unreliable and also can converge to non-physical solutions as well. Therefore, we now consider an alternative model to solve the same problem more efficiently, without the problematic third derivative terms.

Chapter 5. Thin Film Flow System 68

(a)N = 400 and δ = 0.025.

(b)N = 800 and δ = 0.025.

(c)N = 800 and δ = 0.5.

Figure 5.1: The bed shape s is shown in blue (bottom curve), and the numerical solution in green (top curve), for D = 1 and W = 1. We see that the numerical solution requires large values of N unless the bed shape is artificially smooth.

69 5.2. Steady-State Thin Film Flow Solving in 1D

(a)N = 400 and δ = 0.025.

(b)N = 800 and δ = 0.025.

(c)N = 800 and δ = 0.5.

Figure 5.2: The bed shape s is shown in blue (bottom curve) and the numerical solution in green (top curve), for D = 1 and W = 5. We see that the numerical solution requires large values of N unless the bed shape is artificially smooth.

Chapter 5. Thin Film Flow System 70

(a)N = 400 and δ = 0.025.

(b)N = 800 and δ = 0.025.

(c)N = 800 and δ = 0.5.

Figure 5.3: The bed shape s is shown in blue (bottom curve) and the numerical solution in green (top curve), D = 1 and W = 10. We see that the numerical solution requires large values of N unless the bed shape is artificially smooth.

71 5.2. Steady-State Thin Film Flow Solving in 1D

Table 5.1: The continuation process with values λ = [1, 0.1, 0.01, .001, 0.0001] and δ = [0.5, 0.4, 0.2, 0.05, 0.025].

λ δ

1 0.5

0.1 0.5

0.01 0.5 0.001 0.5 0.0001 0.5 0.0001 0.4 0.0001 0.2 0.0001 0.05 0.0001 0.025

Table 5.2: Newton iterations for different cases for Kalliadasis’s model with 401 mesh points, D = 1, W = 1, 5 and 10, λ = [1, 0.1, 0.01, 0.001, 0.0001] and δ = [0.5, 0.4, 0.2, 0.05, 0.025] (see Table 5.1).

W=1 W=5 W=10

Iterations Converge Iterations Converge Iterations Converge

9 Yes 9 Yes 9 Yes

6 Yes 6 Yes 6 Yes

5 Yes 5 Yes 5 Yes

5 Yes 5 Yes 103 Yes

4 Yes 6 Yes 104 Yes

4 Yes 104 Yes 102 Yes

6 Yes 104 Yes 103 Yes

31 Yes 104 Yes 108 Diverge

7 Yes 102 Yes 106 Diverge

Chapter 5. Thin Film Flow System 72

Table 5.3: Newton iterations for different cases for Kalliadasis’s model with 801 mesh points, D = 1, W = 1, 5 and 10, λ = [1, 0.1, 0.01, 0.001, 0.0001] and δ = [0.5, 0.4, 0.2, 0.05, 0.025](see Table 5.1).

Table 5.4: Newton iterations for different cases for Kalliadasis’s model with 801 mesh points, W = 1, 5 and 10, D = 1 and δ=[0.5].

λ W=1 W=5 W=10

- Iterations Converge Iterations Converge Iterations Converge

1 10 Yes 10 Yes 9 Yes

Our motivation here is to find the numerical solution for the fully-developed thin film flow model described by Equations (2.31) and (2.32) and to contrast this approach with that of Kalliadasis’s, described in the previous subsections. To make this comparison, we begin by considering a simple case, namely, Sellier’s model to solve the same problem in 1D [53, 115].

This model is described by two coupled second-order ODEs for p(x) and h(x) where:

p− ∂²

73 5.2. Steady-State Thin Film Flow Solving in 1D

We will use the FDM approximation to approximate Equations (5.6) and (5.7). This results in a nonlinear system F (U ) = 0 involving the vector of unknowns U , defined as follows:

U =

We considered in 1D the equivalent version of Sellier’s model which are Equations (5.6) and (5.7) these are scaled slightly differently from the other version in 2D, as we described early in Chapter 2 in Section 2.2.2. We considered this version due to the fact that this model is consis-tent with Kalliadasis’s model and we can make a direct comparison between these two models in 1D. The version of Sellier’s model in 2D that we will carry on with are in Equations (2.31) and (2.32).

We now approximate Equations (5.6) and (5.7) with the FDM for the derivatives in the x direction. In this discretisation, we replace the derivatives by central difference approximations on a uniform grid, where every point xi in the grid has 2 unknown values pi and hi. In other words, every point xi in the grid requires a solution (pi, hi). The resulting discrete problem is as follows:

Chapter 5. Thin Film Flow System 74

These equations together represent a nonlinear system of 2(N − 1) equations with 2(N + 1) unknowns. We set four boundary conditions to complete the system at i = 1 and i = N + 1 as follows,

In the following subsection, we will solve this model with the MATLAB fsolve function.