III. Applications 137
11.4. Staggered slip boundary condition
Chapter 11. The capillary rise problem
11.4.1. Capillary rise using the staggered slip boundary condition
We reconsider the capillary rise problem forΩ=1 to examine the effect of the staggered slip boundary condition on a dynamic wetting simulation. The numerical results forL=R/5 are reported in Fig. 11.9. In particular, a mesh study for the choiceω =1 shows that in this case the limiting value for the oscillation amplitude is approached frombelow; see Fig. 11.9(a). Hence in contrast to the standard Navier slip condition (ω=0), the viscous dissi-pation appears to be overestimated on coarse meshes forω=1. This is a remarkable qualitative change in the behavior of the numerical solution.
In order to quantify the numerical error for different choices of the parameterω, we investigate the discrete maxi-mum error in the dynamic rise height defined as
E∞=max
i |Hnum(ti)−Href(ti)|, (11.27) where the continuous reference functionHrefis obtained from a linear interpolation of the numerical values for a well-resolved simulation with 192 cells per radius with the standard Navier slip condition (ω=0). A convergence study forω∈ {0,13,12,23,1}is given in Fig. 11.9(b). In general, at least a first-order convergence of the solution is observed in the considered range of mesh resolutions. Following Fig. 11.9(b), the accuracy can be increased with the staggered slip condition.
The numerical results for the reduced slip length L=R/50 are reported in Fig. 11.10. Like in the previous example, the reference solution is obtained from a simulation withω=0 on a very fine mesh with 192 cells per radius. As before, the numerical solutions forω=1 approximates the oscillation amplitude from below. However, in this case, the maximum error E∞is similar forω=1 andω =0; see Fig. 11.10(a) and Fig. 11.10(b). Yet the results in Fig. 11.10(c) show that the accuracy is increased by an order of magnitude for the choiceω=1/2. The error is even smaller forω=2/3 on coarse meshes with 8 and 16 cells per radius but a saturation of the error appears for 32 and 64 cells per radius; see Fig 11.10(d). In summary, the numerical results show that the staggered slip condition improves the accuracy for the capillary rise significantly if the slip length is not well-resolved by the mesh.
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11.4. Staggered slip boundary condition
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Rise Height [cm]
time [s]
4 cells/radius
8 cells/radius 16 cells/radius
32 cells/radius 64 cells/radius Reference
2.1 2.2 2.3 2.4
0.19 0.2 0.21
(a) Results forω=1. (b) Convergence in the maximum norm.
Figure 11.9.: Capillary rise with the staggered slip boundary condition, results forL=R/5.
0.8 1 1.2 1.4 1.6 1.8 2 2.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Rise Height [cm]
time [s]
4 cells/radius
8 cells/radius 16 cells/radius
32 cells/radius 64 cells/radius Reference
1.9 2 2.1
0.19 0.2 0.21
(a)ω=0
0.8 1 1.2 1.4 1.6 1.8 2 2.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Rise Height [cm]
time [s]
4 cells/radius
8 cells/radius 16 cells/radius
32 cells/radius 64 cells/radius Reference
1.6 1.7 1.8 1.9 2
0.19 0.2 0.21
(b)ω=1
0.8 1 1.2 1.4 1.6 1.8 2 2.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Rise Height [cm]
time [s]
4 cells/radius 8 cells/radius
16 cells/radius 32 cells/radius
64 cells/radius Reference
1.9 2
0.19 0.2 0.21
(c)ω=1/2 (d) Convergence in the maximum norm.
Figure 11.10.: Capillary rise with the staggered slip boundary condition, results forL=R/50.
Chapter 11. The capillary rise problem
11.4.2. Single-phase channel flow with staggered slip
Figure 11.11.: Illustration of the single-phase channel flow problem.
In order to gain more insights into the impact of the staggered slip boundary condition, we come back to the single-phase channel flow problem in two dimensions described in Example 1.4 (see p. 12). Here we consider the flow through a channel of heightH (see Fig. 11.11) which is driven by a constant pressure gradient∇p= (−G,0) =const. The incompressible Navier Stokes equations are solved subject to the impermeability and Navier slip conditions at the top and bottom wall of the channel, i.e.
vy=0, vx−L∂yvx=0 for y=0,H.
As discussed in Example 1.4, this problem can be solved explicitly with the solution v(x,y) = (vx(y),0) with vx(y) =GH2
2η y
H−y H
2
+ L H
. (11.28)
Note, that the solution 11.28 is physically stable only if the Reynolds number Re=ρvmaxH
η
is small enough such that the flow is laminar. Using (11.28), the expected Reynolds number can be evaluated as Re=ρH3G
8η2
1+4L H
.
We choose the following fluid parameters ρ=102kg
m3, g=0.1m
s2, η=10−2Pa·s.
The channel height and the pressure gradient are chosen to be
H=10−2m, G=ρg=10 kg m2s2. With these definitions, we have
vx(y) = 5 100
y H−y
H 2
+L H
m
s, hvxi= 1 120
1+6L
H m
s, vmaxx = 5
400
1+4L H
m
s, Re=10 8
1+4L
H
.
(11.29)
Hence, the Reynolds number is small (Re≤6.25 forL≤H) and the flow is laminar.
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11.4. Staggered slip boundary condition
-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02
10-2 10-1 100
Signed Relative Error Mean Velocity
Δx/L
ω = 0 ω = 0.01 ω = 0.1 ω = 0.5 ω = 1.0
(a)L/H=0.25
-0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04
100 101 102
Signed Relative Error Mean Velocity
Δx/L
ω = 0 ω = 0.01 ω = 0.1 ω = 0.5
(b)L/H=2.5·10−3
Figure 11.12.: Signed relative error for the mean velocityhvxi.
FS3D simulation results: The numerical simulations are carried out on uniform meshes with 8, 16, 32, 64 and 128 cells along the diameter of the channel. The simulation is run until steady state is reached. Then we consider the convergence of the averaged velocity (which is proportional to the mass transport rate in the channel)hvxifor the standard discrete Navier slip condition (11.24) and compare to the staggered slip condition (11.26) for different values of the parameterω. The analytical results (11.29) serve as a reference. The signed relative error in the mean velocity, i.e.
hvxinum− hvxi
|hvxi|
is reported in Fig. 11.12 for different choices of the parameterωand a resolved (L/H=0.25) as well as well as an under-resolved slip length (L/H=2.5·10−3). It is found that the numerical method approximates the exact value from below forω∈ {0.1,0.5,1}. In this case, the viscous dissipation is increasingly over-predicted as the mesh becomes coarser. On the other hand, the exact solution is approximated from above for the Navier slip condition (ω=0) andω=0.01. However, a close inspection shows that the convergence forω=0.01 is not monotone (see Fig. 11.13).
10-6 10-5 10-4 10-3 10-2 10-1 100
10-2 10-1 100
Relative Error Mean Velocity
Δx/L ω = 0
ω = 0.01 ω = 0.1
ω = 0.5 ω = 1.0
first-order second-order
(a)L/H=0.25
10-5 10-4 10-3 10-2 10-1 100
100 101 102
Relative Error Mean Velocity
Δx/L ω = 0
ω = 0.01 ω = 0.1
ω = 0.5 first-order second-order
(b)L/H=2.5·10−3
Figure 11.13.: Convergence of the mean velocityhvxi.
In order to assess the convergence of the method quantitatively, the relative error inhvxi, i.e.
|hvxinum− hvxi|
|hvxi|,
Chapter 11. The capillary rise problem
is plotted as a function of the mesh size in units of the slip length; see Fig. 11.13. It is found that the standard Navier slip condition (ω=0) delivers second-order accuracy for the mean velocity. The order of accuracy drops to one for the staggered slip condition withω∈ {0.1,0.5,1}. A local minimum appears forω=0.01 since the convergence is not monotone in this case.
11.4.3. Summary
In summary, the staggered slip boundary condition (11.26) improves the convergence for the capillary rise problem significantly, in particular if the slip length is under-resolved; see Fig. 11.10. Depending on the choice ofω, the correct oscillation amplitude is either approximated from above or from below. On the other hand, the order of convergence for the single-phase channel flow problem drops from two to one when the staggered slip boundary condition is applied instead of the standard Navier slip condition. In the latter case, the average velocity in the channel is approximated from below except for very small values ofω.
It appears that the staggered slip condition increases the dissipation within the numerical method. This way, the staggered slip method can achieve better results than the standard slip condition if the viscous dissipation in the contact line region is not well-resolved. On the other hand, the viscous dissipation is over-estimated for the single-phase channel flow. In conclusion, the staggered slip condition is a possible approach to achieve more re-alistic results for the contact line dynamics on coarse meshes. However, further research efforts are necessary to reach a more quantitative understanding of the underlying mechanisms.
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