Our calculations show an asymptotic shape of the wedge, because the surface tension tends to zero. The azimuthal component of the mean curvature improves the definition of the finger neck and smoothes the interface there.
List of Tables
Chapter 1
Introduction and Background
Removing Stiffness of Surface Tension in Two and Three Dimensions
In the first part of the thesis, we propose the use of "curvature" as a new dynamic variable when calculating free two- and three-dimensional interfaces with curvature adjustment. To demonstrate the robustness of the method, we apply our method to a number of interesting applications.
Numerical Study of Two-Dimensional Hele-Shaw Flow with Suction and Three-Dimensional Axisymmetric Flow
However, Nie and Tian do not address the confining behavior of the interface as surface tension tends to zero. Surface tension poses additional difficulties in the calculation of the dynamically evolving fluid interface.
Chapter 2
Removing the Stiffness of Surface
Tension in Computing Two-Dimensional Interfaces and Three-Dimensional
Filaments
The KI-K2-W-L Formulation for Three-Dimensional Fil- aments
The formulation is more subtle for three-dimensional filaments, since there are two normal vectors (e.g. the normal and the bi-normal vectors). W measures the rate of rotation of the NI-N2 plane around T, and may (e.g. the Kirchhoff bar model) or not (e.g. motion due to curvature) have a physical meaning.
Application to the Kirchhoff Rod Model
As in the two-dimensional case, Country w can be updated using an explicit integration method. We can effectively invert the implicit discretization for the diffusion terms in the 11:1 and 11:2 equations.
A pplication to Nearly Parallel Vortex Filaments
Some Implementation Issues
- Reconstruction of the Interface from Curvature
The reason is that in the case of curved motion, .. we are only dealing with the shape of the curve r which is determined only by the tangential vector. Thus the twist w is important in the evolution of the rod and actually depends on ).3.
Numerical Results
- Motion By Curvature in Two Dimensions
- Motion by Curvature for Three-Dimensional Filaments
In our numerical calculations, we use curve length and curvature as our dynamic variables. We find that using the same number of points, the largest possible time steps that yield stable discretizations are of the same order for both methods. The motion of the three-dimensional filament due to curvature is slightly different from that of its two-dimensional counterpart.
Thus we are able to calculate cP. The curvature of the unit circle is 1. Thus 1);1 and 1);2 are completely determined by Eq. Second, the magnitude of the contact force must be controlled to prevent excessively large forces from destabilizing our solution. Finally, we will test our method on the motion of anti-parallel vortex filaments.
Appendix: Evolution Equations for rLl, rL2 and w for Three-Dimensional Filaments
Chapter 3
Removing the Stiffness of Surface Tension in Computing
Three-Dimensional Surfaces
The Equations of Motion of Three-Dimensional Water Wave Problems
Since the flow in each region is non-rotating, we introduce velocity potentials ¢l and ¢2 given by Ul = V' ¢l and U2 = V' ¢2, where UI/U2 is the velocity above/below the surface. The surface can be thought of as a distribution of intensity density dipoles; in the following, the potential in the fluid domain outside the surface is defined as. Using the limit for the double layer potential, we obtain the Fredholm integral equation of the second kind for p at the surface.
By differentiating (3.16) with respect to X and then integrating by parts, we obtain the velocity on the surface to be. The main work is spent on evaluating the surface velocity and velocity potential which requires O(N4) calculations where N is the number of particles used to discretize the surface in each dimension. More precisely, we replace the boundary integral for the surface velocity by its leading contribution at high modes.
Some Implementation Issues
- Tangential Velocities
- Reconstruction of the Surface from Curvature
- Preconditioned Conjugate Gradient Method
Using the same idea, we can also derive an iteration method for constructing a rectangular coordinate system. In our numerical experiments, only a few iterations are needed for convergence with an iteration error of 10-11. To reconstruct the surface from the curvature, we use the same idea as the two-dimensional cases.
This difficulty is solved by allowing the coefficients to be exactly 0 after the reconstruction process. Note that the linear operator on the left side of (3.73) is symmetric and positive definite. We use the predetermined conjugate gradient method. Empirically, it only takes two to three iterations to converge with an iterative error of 10-11.
Numerical Results
- Motion By Mean Curvature for Three-Dimensional Surfaces We begin our numerical experiments with motion by mean curvature for three-dimensional
- Gravity Waves
We see that the wave drops around t = 0.6 but has recovered with a lower maximum amplitude at t = 0.8. T = 0.01 case, we see that the peak of the wave is more pointy, which is expected since a larger surface tension tends to smooth the surface more. We also set the surface tension to be zero and observe almost no change in the surface wave profiles for the case T = 1 X 10-4.
The movement begins with wave valleys rising and wave crests falling (See Fig. 3.24). Unfortunately, due to the limited computer memory, we do not have the necessary resolution here to further calculate the wave motion to determine if the motion becomes singular. Then the wave rises in the opposite corners and forms rays in these places.
Chapter 4
Numerical Study of Hele-Shaw Flow with Suction
The Governing Equations
Here n is the exterior unit normal to f, T is the surface tension, and ~ is the interface average curvature. The kinematic boundary condition (4.3) states that the normal component of the velocity field is continuous across the interface. The relation (4.4), known as the Laplace-Young boundary condition, gives an account of how the presence of surface tension changes the pressure across the interface.
The interface governing equations can be put into a convenient form by introducing the complex position variable z(a, t) = x(a, t) + iy(a, t) and the complex conjugate velocity W(a, t) = u(a, t) - iv(a, t). 34; is the (unnormalized) vortex sheet strength, which measures the tangential velocity jump across the interface with . Here, A(a, t) is arbitrary and only determines the parameterization (frame) of the interface, but does not affect its dynamics.
The Numerical Method
The stiffness is hidden by the small spatial scales of Uo in the equation for O. Here we use the following fourth-order explicit/implicit method studied by Ascher, Ruuth and Wetton [1]. L is first updated using a fourth-order explicit Adams-Bashforth multistep scheme to obtain Ln+1 before en+1 is calculated via FFT.
It has been shown [3] that Fredholm's integral equation for '"Y has a globally convergent Neumann series. We solve '"Y by fixed-point iteration, accelerated by constructing a fourth-order extrapolated initial guess from previous time steps. It usually takes several iterations to obtain a convergent solution for '"Y when the interface is relatively smooth.
Numerical Results
- An Analysis of Numerical Errors
A time sequence of the interface evolution for this small surface tension is shown in Fig. A comparison with the curvature corresponding to 5 = 5 X 10-5 (Fig.4.6(a)) shows the singular tendency of the interface limiting behavior. The viscosity of the external fluid alone does not prevent the formation of nodules in the interface.
However, the side depressions on the finger neck are developed for less surface tension as fig. The effect of small surface tension is to slightly round the tip of the thin finger. Figure 4_17: Development of the interface for A with different surface tensions as it approaches the sink.
Further Discussion
In addition, at sufficiently small values of surface tension, lateral indentations develop, forming a neck at the top of the finger. The formation of the neck and the bulging of the finger are intriguing phenomena that can be related to the influence of the zero surface tension singularity for very small values of surface tension. Since the neck formation and finger protrusion are observed around tc, we rule out the effects of the daughter singularity for the specific flow we consider here.
The viscosity of the external fluid has a smoothing effect on the flow and a thinner finger develops. The small surface tension only slightly surrounds the tip of the edge before this part of the interface rapidly accelerates toward the sink. It is observed that the angle of the asymptotic wedges decreases with Aw We believe that this angle will collapse to zero (a peak) in the limit as AJ.l -+ O.
Chapter 5
Numerical Study of Axisymmetric Flow with Suction through Porous Media
The Governing Equations
The non-linearity of the flow arises from the boundary conditions at the fluid interface (fluid spot boundary). The kinematic boundary condition (5.3) states that the normal component of the velocity field is continuous across the interface. Since the flow is potential, it can be described by the dynamics of the free interface.
Thus, the motion of the asymmetric flow through porous media with suction is described by Eq. We non-dimensionalize the equations of motion by setting Q = -1 and by setting the initial radius of the interface to 1. We see that the governing equations are very similar to those for the two-dimensional Hele-Shaw flow, except for the additional component of the mean curvature, and the integrands of the boundary integrals.
The Numerical Method
- Evaluation of the Principal-value Integrals
- Time Integration: Removing the Surface-tension Induced Stiff- ness
- Filtering and Numerical Stability
This makes accurate evaluation of principal value integrals for axisymmetric flow even more difficult to achieve. Through curvature, the surface tension introduces higher-order derivatives of the interface position that are nonlinearly and nonlocally related to the flow. The leading-order behavior of U at small scales can be obtained by noting that, for x f- 0, the leading-order terms of the integrals in the velocity [Eq.
This is a plot of the magnitude of the Fourier coefficients of the complex vortex sheet velocity 'U + iv [Eq. Very high spatial resolution is required to keep the high-frequency components of the non-smooth sampling error below an acceptable filter level. To summarize, filtering is applied as follows to calculate the motion of the axisymmetric flow interface:
Numerical Results
We stop the calculation when the distance of the tip of the cone to the sink is 0.0148. At this time, the formation of a corner singularity at the tip of the cone as it reaches the sink is evident. This can be clearly assessed on the graph of the tangent angle e(ex, t) in fig.
However, for the initial data just considered, the interface is at least smooth and eventually singular, exactly at one of the poles where the finger develops. The interfaces in both cases are not drawn at the same time t, but when the fingertip reaches 0.0865. For Hele-Shaw flow, there is an asymptotic form of fingers at the late stage of interface motion [10J. 5.13 interface profile for axial symmetry.
Chapter 6 Conclusions
Bibliography
A numerical study of the effect of surface tension and noise on an expanding Hele-Shaw bubble. Rele-Shaw flows with a free boundary created by injecting liquid into a narrow channel.
