Lecture 17: Natural (Free)
convection
http://www.youtube.com/watch?v=qIlcXp-cKHg
1. Assume that gravitational field affects the flow

divv  0


 
v 
p
 v   v  
v  g
t


T
 v   T  T
t
1
2. Statics:

p0

g 0
0
3. We separate the static parts and the non-static contributions:
p  p 0  p ,    0   
4. Assume that
p

 1,
 1
p0
0
5. Linearization of the pressure term
p0  p1    0  p0 p p0 
p0  p
p0  p





 2 





0  
0 1   0
0
0
0
0
p
6. Substituting this into Navier-Stokes equation

p p  p

 
v 
 v   v  


  v  g
t






v 
p    
or
 v   v  
 g v
t


0
0
2
0
0
0
0
0
2
7. In incompressible flow, density variations are due to temperature nonisothermalities
          T 
0
0
0




v 
p 
 v   v  
 T g v
t

0
Finally, the governing equations for thermal convection in
Boussinesq approximation

v 
p



 v  v  
v  T g
t
0
T  
 v   T   T 
t

div v  0
Let us non-dimensionalise these equations.
3
Dimensions of the viscosity and
thermal conductivity coefficients
m2
     
s
1
K
Dimension of the heat
expansion coefficient
  
Gravity acceleration


g  gk
L -- length scale (typical size)

L2

-- time-scale (also called convective time scale)
v   L  

L
-- scale of velocity
p    v       -- scale of pressure
L 
2
2
0
T   
0
-- scale of temperature (Θ is the typical temperature difference)
Non-dimensionalisation


t dim  tnon dim , v dim  v v non dim ,
pdim  v pnon dim , Tdim  Tnon dim
4
Non-dimensionalization of the Navier-Stokes equation:
v  v  v 2 v   v    0 v 2 p   v v  T gk
 t
or
L
L
0
L2





v
gL 
 v   v  p   v  GrT k , Gr 
t

3
2
Heat transfer equation,
 T  v  
v   T    T 

 t
L
L
2
or
T  
1

 v   T  
T , Pr 
t
Pr

Non-dimensionalization of the continuity equation is trivial:

divv  0
5
Non-dimensionalised equations
of thermal convection





v
 v   v  p   v  GrT k
t
T  
1
 v   T  
T 
t
Pr

divv  0
Non-dimensional parameters:
gL  -- Grashof number (defines the strength of the forcing
Gr 

(buoyancy) term; characterises intensity of the convective flow)
3
2
Pr 


-- Prandtl number (characterises the fluid properties; defines
the relative importance of thermal conduction and convection
as two mechanisms of heat transfer, Pr>>1: convection
6
dominates, Pr<<1: convection can be disregarded)
Czochralski process
When the silicon is fully melted
(~15000C), a small seed crystal mounted
on the end of a rotating shaft is slowly
lowered until it just dips below the
surface of the molten silicon. The seed
crystal's rod is slowly pulled upwards.
Silicon
ingot
The Czochralski
process is a one of the
methods of crystal growth
from melt, used to obtain
single crystals of
semiconductors (e.g.
silicon, germanium and
gallium arsenide), metals
(e.g. palladium, platinum,
silver, gold), salts, and
synthetic gemstones.
Main difficulty: strong convective flows in the melt. From one hand,
convection makes the melt more uniform, which is positive. From another
hand, strong flows near the crystallization plane introduce crystal defects.
Convection is controlled by rotation/vibration of ingot and crucible, by
7
magnetic field, …
Solutal convection
Consider incompressible isothermal fluid flow with an admixture.
Equation of mass conservation for an impurity:
C 
(this equation is valid for not very large C)
 v   C  DC
t
Diffusion
coefficient
Variations in density can be caused by admixture:           C
0
0




v 
p 
 v   v  
v  Cg
t
0
C 
 v   C  DC
t

divv  0
P.S. If flow is non-isothermal, the thermosolutal (or double-diffusive)
convective flow is induced.
0
Equations of
solutal
convection:
8
Lecture 1Flows driven by
surface tension gradients
Spatial variations in surface tension result in added tangential stresses
at an interface, giving rise to fluid motions in the underlying bulk liquids.
The motion induced by tangential gradients of surface tension is termed
the Marangoni effect.
Examples:
the camphor ball ‘dances’ on a water surface
(http://www.youtube.com/watch?v=Pe88T45VdR8),
the calming effect of ‘oil troubled waters’
(http://www.youtube.com/watch?v=00PPPt7EJqo).
9
This motion is induced at an interface, where, besides
the normal force,

there is another force tangential to the surface, f t   . Adding this
force, we obtain the following boundary condition

1
1 



p

p



n




n



2
1,ik
2 ,ik
k
 1
 i
R
R
x i
 1
2 


n is the unit normal vector directed into medium 1.
   T 
   C 
   q 
The surface tension coefficient is function of temperature; the
thermocapillary flows are induced.
function of concentration; the difusocapillary flows.
function of electric charge; the electrocapillary flows.
10
Thermocapillary motion in a thin
liquid layer
The thermocapillary motion generated in an open rectangular shallow pan
with a very thin layer at the bottom.
The difference in side wall
temperatures results in a
temperature gradient along

z
T2
the surface. For liquids,
g
T1
(cold)

(hot)
0
h2
T
h(x)
h1
l
h
x
l
α
α2
α1
 1
The flow in nearly lateral.
Any flow nonuniformities at
the side walls are small and
the flow is essentially 2D.
11
x
Except at the side walls, which are far removed from the bulk flow, the
vertical velocity component is very much smaller than the horizontal
component and any effects of the free surface curvature may be
neglected.

v  uz ,0,0
R1  R2  
We assume that the liquid layer is thin enough to neglect inertial
effects, implying Re<<1 (‘shallow water’ approximation).
Governing equations:
x-projection of the momentum
equation,
z-projection,
p
 2u
 2
x
z
p
  g
z
The continuity equation (in integral
form) for the fully developed flow is
(1)
(2)
h x 
 u z d z  0
0
(3)
12
Boundary conditions:
1. Bottom (z=0): u  0
2. free surface (z=h):
Or, for our flow,
(4)
 patm

 p ni   ik nk 
x i
u d


z dx
(5)
p  patm
(6)

n  0,0,1
Integration of (1) with the use of (4) and (5) gives
d
p
1 p 2
 h z 
z
x 
2 x
 dx
u  
(7)
Integration of (2) with the use of (6) gives
p  patm  g h  z 
(8)
13
(8) specifies the relation between the pressure gradient and variation of
free surface height in the x-direction as
p
dh
 g
x
dx
(9)
By using (7) and (9), condition (3) results in
Or,
  1 
g
3
h
2
 h12 
The velocity profile:
d 2
dh
 gh
dx 3
dx
Here, we assumed that
α=α1 and h=h1 at x=0.
umax
z  3 z  d
u 
 1
2  2 h  dx
u max
h d

4 dx
14
The requirement Re<<1 may be expressed as
Re 
u max h1

h12 d

 1
4 dx
4  2
h 
d dx
or
2
1
The variation of α with T for liquids is close to linear. For water,

mN
 0.15
T
m K
2
m
  106
s
and
Thus
with
mN
This would d
 15 2
give a value dx
m
dT
K
 100
dx
m
  103
kg
m3
Hence,
h1  1mm
We did not consider the gravity-driven convection.
The gravity between the gravity and capillary forces is defined by the
Bond number:
2
 h1 
gh12
Bo    
1
 c 
Here,

 
g
2
c
-- capillary length
15
For water/air interface,   73mN . And, for our configuration, Bo  0.3
1
m
This means that, for the configuration examined, the flows driven by
the surface tension effect (Marangoni convection) dominate (in
comparison with the gravity-driven convective flows).
The surface tension gradients can be important in such very thin
layers of mm size or less, or in a reduced gravity environment. For
example, a crystal grown from its melt under reduced gravity is
governed by convection driven by thermally induced surface tension
gradients rather than buoyancy forces.
16
Ludwig Prandtl (4 February 1875, Freising, Upper Bavaria – 15 August
1953, Gottingen) was a German scientist. He was a pioneer in the
development of rigorous systematic mathematical analyses which he
used to underlay the science of aerodynamics, which have come to form
the basis of the applied science of aeronautical engineering. In the 1920s
he developed the mathematical basis for the fundamental principles of
subsonic aerodynamics in particular.
17
Franz Grashof (July 11,
1826 in Dusseldorf October 26, 1893 in
Karsruhe) was a German
engineer. He was a
professor of Applied
Mechanics at the
Technische Hochschule
Karlsruhe. He developed
some early steam-flow
formulas but made no
significant contribution to
free convection.
Joseph Valentin
Boussinesq (13 March
1842, Saint-Andre-deSangonis, France –Died
19 February 1929, Paris,
France) was a French
mathematician and
physicist who made
significant contributions to
the theory of
hydrodynamics, vibration,
light, and heat.
Carlo Giuseppe Matteo
Marangoni (29 April 1840,
Pavia Italy – 14 April
1925, Florence, Italy) was
an Italian physicist.
He held a position of High
School Physics Teacher at
the Liceo Dante
(Florence) for 45 years,
until retirement in 1916.
He mainly studied surface
phenomena in liquids and
also contributed to
18
meteorology.
Jan Czochralski (23 October 1885 – 22 April 1953) was
a Polish chemist who invented the Czochralski process, which is
used for growing single crystals and in the production of
semiconductor wafers. He was educated at Charlottenburg
Polytechnic in Berlin, where he specialized in metal chemistry.
Czochralski discovered the Czochralski method in 1916, when
he accidentally dipped his pen into a crucible of molten tin. He
immediately pulled his pen out to discover that a thin thread of
solidified metal was hanging from the nib. The nib was replaced
by a capillary, and Czochralski verified that the crystallized metal
was a single crystal. Czochralski's experiments produced single
crystals a millimeter in diameter and up to 150 centimeters long.
In 1950, the method was used to grow single germanium
crystals, leading to its use in semiconductor production.
19