Lecture 13. Dissipation of Gravity Waves

gvpvvtv

v

0div

z

x

Governing equations:

equilibrium:

small-amplitude waves:

gzp

v

0

,0

qpp

v

0

- small

small addition to equilibrium pressure

vqtv

v

0divLinearized equations:

air

liquid

For 2D case,

.

,

,0

2

2

2

2

2

2

2

2

zv

xv

zq

tv

zv

xv

xq

tv

zv

xv

zzz

xxx

zx

Seek solution in the form of a plane wave,That is,

kxtie ~

.

,

,0

2

2

zzz

xxx

zx

vvkqvi

vvkikqvi

vikv

zx vik

v 1

xxx vvkik

vk

q 2

kxti

kxtizz

kxtixx

ezqq

ezvv

ezvv

Re

,Re

,Re

Equations for the amplitudes are

Seek solution of the last equation in the form, ze~

Auxiliary (complimentary) equation is 22222 kiki

Roots,

i

kk 24,32,1 ,

Let us denote

i

km 2 The square root of a complex number produces two different complex values. For m, the root with a positive real part will be chosen.

Finally, these three equations can be reduced to one equation for vz,

zzivzzz vkvkvvivki 422 2

mzmzkzkzz DeCeBeAev

Solution,

Here,

Boundary conditions:

1. At the solution is bounded. This gives B=D=0, or, z

mzkzz CeAev

2. At interface (ζ defines the shape of interface): txz ,0kikn 1,0n

is a unit vector normal to the interface

x-projection:

000

xv

zv zx

xzxz

z-projection:

020200

zv

qgzv

pp zzzzzz

(1)

(2)

(1): 01 22 CAikCmAkik

02 222 CkmAk

(2): First, we take into account that

Hence, eq. (2) can be rewritten as

, or,

Or,

iv

dtd

v zzz

02

zv

qiv

g zz

02 22222 mCkAkmCmkCmAkiCAigk +

022

zv

vvkik

vki

vg z

xxxz

02 222 CkmAk

These equations can be rewritten as,

02

022222

22

CkmAk

CkmigkAkiigk

This system have non-trivial (non-zero) solutions only if determinant of the matrix of coefficients equals zero

02

22222

22

kmk

kmigkkiigk

0222 22222 kmigkkkmkiigk

02222 2222

kmigkk

ikkiigk

or

0222 222 kiigkmkkik or

And finally,

mkkkigk 3222 44 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

gk20 This the known dispersion relation for the waves on the surface of an inviscid liquid

kxtitkkxti eee 022~

dissipation

2k - damping coefficient

Shorter waves (with smaller wave lengths, and hence with larger k) propagate slower ( with speed ) and dissipate faster.

The terms of the first order (proportional to ν) are,

22 kigk

2010 42 ki 2

1 2 ki

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,

wave

kg

kc 0

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)

http://www.youtube.com/watch?v=W4mlquoOSOk

http://www.youtube.com/watch?v=9OOZQxmYnmo

http://www.youtube.com/watch?v=3U3FMCGfEa0

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.

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.

21

11RR

p 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:

appp ei

2

internal external

radius

Wetting

The behaviour of liquids on solid surfaces is also of considerable practical importance.

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

solid

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.

liquidgas

θ

The drop may spread freely over the surface, or it remain as a drop with a specific contact angle θ.

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).

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,

yxz ,

2

2

2

2

21

11yxRR

Boundary conditions with account of the surface tension

iikkikk nRR

nn

21,1,2

11

This equation can be also written

ikikiki nRR

nnpp

21,2,121

11

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

i

kikiki xnn

RRpp

,2,1

2121

11

n

is a unit normal vector directed into medium 1.

Boundary condition for an interface between two inviscid liquids:

ii nRR

npp

2121

11

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.

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.