1. Lecture 13. Dissipation of Gravity Wavesz x
air
Governing equations:
liquid
equilibrium:
small-amplitude waves:
- small
small addition to equilibrium pressure
Linearized equations:

2. For 2D case,
Seek solution in the form of a plane wave,
That is,
Equations for the amplitudes are

3. Finally, these three equations can be reduced to one equation for vz,
Here,
Seek solution of the last equation in the form,
Auxiliary (complimentary) equation is
Roots,
Let us denote
The square root of a complex number produces two different complex values.
For m, the root with a positive real part will be chosen.
Solution,

4. Boundary conditions:
1. At the solution is bounded. This gives B=D=0, or,
2. At interface (ζ defines the shape of interface):
is a unit vector normal to the interface
x-projection:
(1)
z-projection:
(2)
(1):

5. (2): First, we take into account that
Hence, eq. (2) can be rewritten as
, or,
Or, +
These equations can be rewritten as,
This system have non-trivial (non-zero) solutions only if determinant of the matrix of coefficients equals zero

6. or or
And finally,
This is the general dispersion relation for the gravity waves on the free surface of a viscous liquid.
Let us assume that viscosity is small. The terms that contain viscosity are small. Let us also assume that ω can be split into ω0+ ω1, where ω0>> ω1.
Next, we will analyse different orders of the dispersion relation.
The leading terms (that do not contain ν) are
This the known dispersion relation for the waves on the surface of an inviscid liquid

7. The terms of the first order (proportional to ν) are,
Finally, the frequency of gravity waves is defined by
Only the main terms are written here. Terms, proportional to ν in higher orders, are neglected.
ω is complex. Let us analyse time evolution of the derived solutions,
dissipation
wave
- damping coefficient
Shorter waves (with smaller wave lengths, and hence with larger k) propagate slower ( with speed ) and dissipate faster.

8. Lecture 14. Surface tension
Phenomena involving surface tension effects:
Soap bubbles;
Breakup into drops of a stream of water flowing out of a tap (basis of the ink jet printer or gel encapsulation processes to encase everything from monoclonal antibodies to perfume)
liquid
gas
The molecules at the surface are attracted inward, which is equivalent to the tendency of the surface to contract (shrink). The surface behaves as it were in tension like a stretched membrane.

9. Young-Laplace equation
Owing to surface tension, there will be a tendency to curve the interface, with a pressure difference across the surface with the highest pressure on the concave side.
Young-Laplace equation
α is the surface tension coefficient;
R1 and R2 are the radii of curvature of the surface along any two orthogonal tangents;
Δp is the pressure difference across the curved surface.
For a spherical bubble or drop:
radius
internal
external

10. Wetting
The behaviour of liquids on solid surfaces is also of considerable practical importance.
solid
For a planar interface, the liquid molecules could be attracted more strongly to the solid surface than between the liquid molecules themselves (e.g. water on clean glass).
liquid
Normally when a liquid drop is placed on a solid surface, it will be in contact not only with the surface but often with a gas.
gas
liquid θ
The drop may spread freely over the surface, or it remain as a drop with a specific contact angle θ.

11. Wetted/unwetted surfaces
θ=0: the solid is ‘completely wetted’; θ=π: the solid is ‘completely unwetted’; 0<θ< π/2: ‘wetted’; π/2 <θ< π: ‘unwetted’ (mercury on glass θ~1400).

12. Shape of a liquid interface
We need formulae which determine the radii of curvature, given the shape of the surface. These formulae are obtained in differential geometry but in general case are somewhat complicated. They are considerably simplified when the surface deviates only slightly from a plane. Let be the equation of the surface; we suppose that ζ is small, i.e. that the surface deviates only slightly from the plane z=0. Then,

13. Boundary conditions with account of the surface tension
is a unit normal vector directed into medium 1.
This equation can be also written
This equation is still not completely general, as α may not be constant over the surface (may depend on temperature or impurity concentration). Then, besides the normal force, there is another force tangential to the surface. Adding this force, we obtain the boundary condition

14. Boundary condition for an interface between two inviscid liquids:

15. Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton, he possessed a phenomenal natural mathematical faculty superior to that of any of his contemporaries.
Thomas Young (13 June 1773 – 10 May 1829) was English polymath. Young made notable scientific contributions to the fields of vision, light, solid mechanics, energy, physiology, language, musical harmony, and Egyptology.