Lecture 15: Capillary motion Capillary motion is any flow governed by forces associated with surface...
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Transcript of Lecture 15: Capillary motion Capillary motion is any flow governed by forces associated with surface...
Lecture 15: Capillary motion
Capillary motion is any flow governed by forces associated with surface tension.
Examples: paper towels, sponges, wicking fabrics. Their pores act as small capillaries, absorbing a comparatively large amount of liquid.
Capillary flow in a brick
Water absorption by paper towel
Height of a meniscus
h0 θ
R
a
cos21
aRR
aghpp liqatm
cos20
Applying the Young-Laplace equation we obtain
The meniscus will be approximately hemispherical with a constant radius of curvature,
Hence,
cos2cos
22
0 aagh c
gc 2 is the capillary
length.
h0 may be positive and negative, e.g. for mercury θ~1400 and the meniscus will fall, not rise. For water, α=73*10-3N/m, and in 0.1mm radius clean glass capillary, h0=15cm.
Let us calculate the rate at which the meniscus rises to the height h0.
Assume that the velocity profile is given by the Poiseuille profile,
82;
4
4
0
22 AardrvQra
Av
a
dtdha
hppAa
aQ
v 88
221
2
2
gha
pp cos221
ha
ghadt
dh
8
cos2 2
The average velocity is
Here is the instantaneous distance of the meniscus above the pool level.
0hh
The pressure difference at the pool level, p1, and at the top of the capillary (just under the meniscus) , p2, is
Thus,
thhhh
gah
ag
hhga
dd8
cos2
d8
022
hhh
hh
hhhhh
hhhh
d1dd
0
0
0
00
0
cthhhhga
002 ln8
chhga
002 ln08
Or, separating the variables,
For integration, it is also continent to rearrange the terms in the rhs
Integration gives
The constant of integration c can be determined from initial condition, at . Hence,0h 0t
00
020
0
002 ln
8ln
8hh
hhh
gah
hhh
hhga
t
208
gah
0hh hh
ht
0
0ln
t
hh exp10
Finally,
Or, introducing , we obtain
00
0lnhh
hhh
t
As ,
t/τ
h/h0
For water in a glass capillary of 0.1mm radius, s12
For this solution, we assumed the steady Poiseuille flow profile. This assumption is not true until a fully developed profile is attained, which implies that our solution is valid only for times
For water in a capillary tube of 0.1mm radius,
2at
sa 22
10~
Lecture 16: Non-isothermal flow
• Conservation of energy in ideal fluid• The general equation of heat transfer• General governing equations for a single-
phase fluid• Governing equations for non-isothermal
incompressible flow
Conservation of energy in ideal fluid
e
v 2
2
-- total energy of unit volume of fluid
kinetic energy
internal energy,
e is the internal energy per unit mass
pvv
tv
vt
0div
Let us analyse how the energy varies with time: .
For derivations, we will use the continuity and Euler’s equation (Navier-Stokes equation for an inviscid fluid):
ev
t
2
2
vp
ev
tS
Tpvv
v
vev
tp
tS
Tpvv
vev
t
div
div
22
22222
222
ddddd
2
1 pSTpSTe
and the 1st law of thermodynamics (applied for a fluid particle of unit mass, V=1/ρ):
vev
tep
vvv
vev
te
tv
v
te
te
tvv
te
vt
div
div
2
2
222
2
2
222
22
22 vv
vvvvvvvv kkikki
(differentiation of a product)
(use of continuity equation)
(use of Euler’s equation)
Next, we will use the following vector identity (to re-write the first term):
1:
2:
Equation (1) takes the following form:
(use of continuity equation)
0 SvtS
tS d
d
If a fluid particle moves reversibly (without loss or dissipation of energy), then
SvtS
Thv
v
vhv
tS
TSvThvv
vev
t
2
2222
222
div
div
p
eh
SThpp
STpp
eh dddd
ddd
dd
2
We will also use the enthalpy per unit mass (V=1/ρ) defined as
3:
Equation (2) will now read
Sn
Sn
Sn
VV
dSpvSev
v
Shv
vVhv
vVev
t
d
dddivd
2
2222
222
hv
v2
2
Finally,
conservation of energy for an ideal fluid
-- energy flux
In integral form,
hv
vev
t 22
22
div
using Gauss’s theorem
energy transported by the mass of fluid
work done by the pressure forces
12
The conservation of energy still holds for a real fluid, but the energy flux must include
(a) the flux due to processes of internal friction (viscous heating),
(b) the flux due to thermal conduction (molecular transfer of energy from hot to cold regions; does not involve macroscopic motion).
The general equation of heat transfer
v
Tq Heat flux due to thermal conduction:
For (b), assume that
(i) is related to the spatial variations of temperature field;
(ii) temperature gradients are not large.
q
thermal conductivity
conservation of energy for an ideal fluid
hv
vev
t 22
22
div
13
Tvhv
vev
t
22
22
div
viscous heating
heat conduction
The conservation of energy law for a real fluid
We will re-write this equation by using
0
vt
div
1p
vvtv
ddddd
2
1 pSTpSTe
p
eh
p
STpp
ehd
ddd
dd 2
(1)
(2)
(3)
(4)
-- continuity equation
-- Navier-Stokes equation
-- 1st law of thermodynamics
-- 1st law of thermodynamics in terms of enthalpy
e, h and S are the internal energy, enthalpy and entropy per unit mass
14
2
2 2i i
i i
vvvv v v v
t t t t
22
22 vv
vvvvvvvv kkikki
2 2 2 2
2
2 2
2 2
div2 2 2 2
1div
2
div2 2
div2 2
v v v v vv v
t t t t
v pv v v v
v vv v v p v
v vv v v h Tv S v
1st term in the lhs:
Differentiation of product (1+5)
(2)
(5)
(6)
(6)
(4)
(7) AaaAaAAaaAAa iiiiii
divdiv
15
2nd term in the lhs:
div div div
e S pe e e T
t t t t t t
S p Se v T v h v T
t t
Differentiation of product (3) (1)
16
LHS (1+2):
2
div2v
v h v T
2
2 2
2
2
div2 2
div2
ve
t
v v Sh v v h T v S v
t
v Sv h T v S v
t
RHS:
LHS=RHS (canceling like terms):
div
ST v S v v T
t
(7)
17
In the lhs,
k i ikv v
div i k ik k i ik ik i kv v v vIn the rhs,
Finally,
div i
ikk
vST v S T
t xgeneral equation of heat transfer
heat gained by unit volume
energy dissipated into heat by viscosity
heat conducted into considered volume
18
Governing equations for a general single-phase flow
0
vt
div
pvvtv
-- continuity equation
-- Navier-Stokes equation
div i
ikk
vST v S T
t x-- general equation of heat transfer
+ expression for the viscous stress tensor
+ equations of state: p(ρ, T) and S(ρ, T)
19
Incompressible flow
S
pc
2
V
p
T
T c
cc
pa
,22 1
pp TTV
V
11
To define a thermodynamic state of a single-phase system, we need only two independent thermodynamic variables, let us choose pressure and temperature.
Next, we wish to analyse how fluid density can be changed.
-- sound speed
-- thermal expansion coefficient
Tpc
TT
pp
Tp
pT
ddd
ddd
,
2
20
4. Hence, we can neglect variations in density field caused by pressure variations
2vp 1. Typical variations of pressure in a fluid flow,
2. Variations of density,
T
12
cv
Tcv
2
3. Incompressible flow ≡ slow fluid motion,
5. Similarly, for variation of entropy.
In general,
but for incompressible flow,
TTS
ppS
SpT
ddd
TT
cT
TS
S p
p
ddd
-- specific heat (capacity) under constant pressurep
p TS
Tc
21
Frequently,
(i) the thermal conductivity coefficient κ can be approximated as being constant;
(ii) the effect of viscous heating is negligible.
Then, the general equation of heat transfer simplifies to
For incompressible flow, the general equation of heat transfer takes the following form:
k
i
ikp xv
TTvtT
c
div
TTvtT
pc -- temperature
conductivity
22
a) given temperature,b) given heat flux,
c) thermally insulated wall,
Boundary conditions for the temperature field:
wallTT
n
qnT
0nT
1. wall:
2. interface between two liquids:
21 TT andnT
nT
2
21
1
23
Governing equations for incompressible non-isothermal fluid
flow0v
div
vp
vvtv
-- continuity equation
-- Navier-Stokes equation
-- general equation of heat transfer TTvtT
Thermal conductivity and viscosity coefficients are assumed to be constant.
+ initial and boundary conditions