Testing

The correctness of the code has been checked with the following tests. They are used to ensure each part of the code behaves as intended and the physics is represented correctly:

• The integrator for the line integrals has been evaluated by setting up a closed domain of elements once around the singularity in the Stokeslet/Stresslet and once away from

1The content of a 2013 minisymposium at a SIAM conference on this topic can be found here:

http://web.eecs.utk.edu/˜luszczek/conf/siamcse2013_eigvalsolv/

it and checking the following identities, as presented in [54], withCas the contour of a

control volumeΩ: _Z

Cni(x)Gi j(x,x0)dl=0 ifx0is outside of the control volume, i.e.x06∈Ω:

Z

CTi jk(x,x0)nk(x)dl(x) =0 ifx0is in the control volume, i.e. x0∈Ω:

Z

CTi jk(x,x0)nk(x)dl(x) =δ_{i j}

If the precision of the integrator is insufficient, there will be a residue, i.e. the result of the integral will be not exactly zero or one.

• To validate the stress boundary conditions, a circular domain with a stress boundary condition of the form f⊥=const, fk=0 in an inviscid fluid was set up to confirm that the fluid velocities on the interface become zero.

• By replacing the circle with an ellipse and allowing for the ellipse to evolve, volume conservation has be verified and the relaxation time scale compared to published results, as presented in Ref. [78].

• For mixed boundary conditions, a box representing a gravity driven channel flow with periodic boundary conditions in x-direction and walls with partial slip in y-direction has been modeled. To confirm the slip condition is implemented correctly, this system has to exhibit a parabolic flow profile with an extrapolation length to zero velocity identical to the slip length. Another system that is suitable for testing is the deformation of a droplet in a shear flow, as discussed in Ref. [58].

• A simple free interface flow that has been studied extensively is a droplet spreading on a homogeneous substrate for different wetting conditions[82, 68]. In this system, the spreading velocity of a droplet depending on the baselength can be determined and compared to the analytical scaling given by Tanner[82].

As the main test if free interface flows are simulated physically correct, it was checked if the Cox-Voinov scaling for the shape of a moving fluid interface close to the contact line is obtained. For this, a droplet in a shear geometry is simulated. In this geometry a droplet is located between two substrates with Navier slip condition at the top and the bottom that move in opposing direction with constant velocity.

The free interface forms a straight line in equilibrium as the top and bottom wall were chosen to be neutrally wetting, i.e. with a substrate contact angle of 90^◦. Therefore, the observed interfacial configurations for moving substrates should agree with the predictions for interface configurations derived as perturbations from a moving fluid wedge.

The code can be validated by confirming that the obtained steady profile of the free interface for a homogeneous substrate fulfills the Cox-Voinov relation. The self-similar solution for the shape of a moving contact line, as discussed previously, is:

θ(x)³=θ³₀+9Caln(x/lc) (3.2)

withθ₀as the microscopic contact angle,xthe distance to the contact line, andlc a micro-scopic length scale determining the lower cutoff.

The numerical simulations confirm the scaling for the intermediate length scales above the slip length and an order of magnitude below the system size. Above this length scale, the macroscopic solution determined by recirculation in the bulk flow away from the contact line determines the shape of the fluid interface. The scaling and the deviation for small and high distances can be seen in Fig. 3.6(b), which shows the obtained stationary solution plotted according to the equation 3.2. When studying the non-steady states of the interface before it reaches an equilibrium, the capillary velocityCaused in the equation should not be deter-mined with the driving velocity of the system, as shown in Fig. 3.6(a), but with the current velocity of the contact relative to the substrate, as shown in Fig. 3.6(b). Even before the global steady state is reached, the fluid interface follows the scaling law in a region close to the contact line, when rescaling with the local contact line velocity.

−4 −2 0 2 4 6 8 10

Fig. 3.6: Slope of the fluid interface over distance to the contact line for differ-ent times as it approaches the steady state, rescaled to show logarithmic dependence with the distance from the contact line of the Cox-Voinov scaling for ls=1.0·10⁻⁴,Ca=0.05. (a) rescaled with the capillary number of the system and (b) rescaled with the capillary number deter-mined by the contact line velocity relative to the substrate. The dashed line shows a unit slope.

An interesting aspect is the fact that the microscopic length scale, i.e. the point where the scaling law approachesθ0, in the equation is not exactly the slip length of our system. If this was the case, the extrapolated line from the linear regime in the plot would cross the origin (0,0) in the graph. The most probable explanation for this is that during the derivation of the scaling law it is assumed that the microscopic contact angleθ0gives the slope of the interface at the lower cut-off heightlc. In contrast to that, in our system the microscopic contact angle is enforced on the last element, given by the last data point in Fig. 3.6(b), around one order of magnitude below the slip length. That means the contact angle observed atlswill already deviate slightly from the equilibrium contact angle due to the hydrodynamic deformation taking place in the slip region.

It was also attempted to show the Cox-Voinov scaling for initially curved interfaces. The influence of the curvature on the equilibrium shape of the fluid interface is not negligible

for more than one order of magnitude below the system size and could not be corrected for, therefore the attempt to resolve the scaling in space for that system did not succeed.