Transcript SESM3004 Fluid Mechanics

Lecture 13. Dissipation of Gravity
Waves
z
Governing equations:

div v  0






v



   v   v   p  v  g
 t

air
x
liquid

equilibrium: v  0,
p0   gz
small-amplitude waves: v - small
p  p0  q
Linearized equations:
small addition to equilibrium pressure

div v  0


v

 q  v
t
For 2D case, 
v
v
 x  z  0,
z
 x
  2v x  2v x 
q
 v x
,

   2 

2 
x
z 
 x
 t
 v z
  2v z  2v z 
q
  t   z    x 2  z 2 .



Seek solution in the form of a plane wave,
That is,
v x  Rev x z  e i t kx  ,
~ e i t kx 
v z  Rev z z  e i t kx  ,
q  Req z  e i t kx  
Equations for the amplitudes are
 ikvx  v z  0,

2
iv x  ikq    k v x  v x ,
iv  q     k 2v  v  .

z
z
z
1
v z
ik


q  v x   k 2v x  v x 
k
ik
vx 
Finally, these three equations can be reduced to one equation for vz,
Here,   
ik 2v z  iv z  v ziv  2k 2v z  k 4v z 

Seek solution of the last equation in the form, ~ e z
Auxiliary (complimentary) equation is
ik  i    k
2
Roots,
 1, 2  k , 3, 4   k 2 
Let us denote
m  k2 
i

2
2

2 2
i

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,
v z  Ae kz  Be kz  Ce mz  De mz
Boundary conditions:
1. At z   the solution is bounded. This gives B=D=0, or,
v z  Ae kz  Ce mz
2. At interface z
 iknk  0
  x , t  (ζ defines the shape of interface):

n  0,1 is a unit vector normal to the interface
x-projection:
 xz  0   xz
v
v
0 x  z 0
z
x
z-projection:
 zz  0   p   zz  0   p  2
(1):
(1)
v z
v
 0  g  q  2 z  0
z
z
1 2
k A  m 2C   ikA  C   0
ik
2k 2 A  m 2  k 2 C  0
(2)
(2): First, we take into account that
vz
z 

d
 v z  i
dt
Hence, eq. (2) can be rewritten as
g
Or,
+
vz
v
v 
v

 q  2 z  0 , or, g z  v x   k 2v x  v x   2 z  0
i
z
i k
ik
z
 igk2 A  C   i 2 Ak  Cm    k 2  m 2 mC  2k 2 kA  mC   0
2k 2 A  m 2  k 2 C  0
These equations can be rewritten as,
 igk  i 2  2k 2 A   igk  2km C  0
 2
2
2
2k A  m  k C  0
This system have non-trivial (non-zero) solutions only if
determinant of the matrix of coefficients equals zero
 igk  i 2  2k 2
2k 2
 igk  2km
0
2
2
m k
 igk  i
2
or
 igk  i
2
 2k 2 m 2  k 2   2k 2  igk  2km  0
i 

 2k 2  2k 2    2k 2  igk  2km   0
 

or
2k 2   i 2k k  m    igk  i 2  2k 2   0
And finally,
  gk  4ik  4 k k  m 
2
2
2
3
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
02  gk
This the known dispersion relation for the waves on the
surface of an inviscid liquid
The terms of the first order (proportional to ν) are,
201  4i0k 2
1  2ik 2
Finally, the frequency of gravity waves is defined by
  gk  2ik 2
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,
~e
i t kx 
e
2k 2t
 e i 0t kx 
dissipation wave
  k 2 - damping coefficient
Shorter waves (with smaller wave lengths, and hence with larger k)
propagate slower ( with speed c  0  g ) and dissipate faster.
k
k
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
gas
liquid
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.
 1
1 
p     
 R1 R 2 
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:
p  pi  pe 
internal
2
a
external
radius
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.
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 z   x , y  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,
  2  2 
1
1

  2  2 
R1 R 2
 x y 
Boundary conditions with
account of the surface tension
n k 2,ik  n k 1,ik
1
1 
    n i
 R1 R 2 

n is a unit normal vector
directed into medium 1.
This equation can be also written
 p1  p2 n i   1,ik
1
1 
  2,ik nk     n i
 R1 R 2 
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

1
1 

 p1  p2    R  R n i   1,ik   2,ik nk  x
 1
2 
i

Boundary condition for an interface between two inviscid
liquids:
1
1 
 p1  p2 n i     n i
 R1 R 2 
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.