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