Lecture 15: Capillary motionCapillary 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

cos21aRR

aghpp liqatm

cos20

Applying the Young-Laplace equation we obtain

The meniscus will be approximately hemispherical with a constant radius of curvature,

Hence,

cos2cos2

2

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 AardrvQraAva

dtdha

hppAa

aQv

88

221

2

2

gha

pp cos221

hagh

adtdh

8cos2 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

gahag

hhga

dd8cos2d8

022

hhh

hhhhhhh

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 ln8ln8

hh

hhh

gah

hhhhh

gat

208

gah

0hh hh

ht

0

0ln

thh exp10

Finally,

Or, introducing , we obtain

00

0lnhh

hhht

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

ev

2

2

-- total energy of unit volume of fluid

kinetic energy

internal energy,

e is the internal energy per unit mass

pvvtv

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):

evt

2

2

vpevtSTpvvv

vevt

ptSTpvvvev

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):

vevtepvvv

vevte

tvv

te

te

tvv

tev

t

div

div

2

2

222

2

2

222

22

22 vvvvvvvvvv 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 dd

If a fluid particle moves reversibly (without loss or dissipation of energy), then

SvtSThvv

vhvtSTSvThvvvev

t

2

2222

222

div

div

peh

SThppSTppeh ddddddddd

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

dSpvSevv

ShvvVhvvVevt

d

dddivd

2

2222

222

hvv

2

2

Finally,

conservation of energy for an ideal fluid

-- energy flux

In integral form,

hvvevt 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

hvvevt 22

22 div

13

Tvhvvevt

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

1pvvtv

ddddd

2

1 pSTpSTe

peh

pSTppeh dddddd 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 ivvv v v v v

t t t t

22

22 vvvvvvvvvv kkikki

2 2 2 2

2

2 2

2 2

div2 2 2 21div2

div2 2div2 2

v v v v vv vt 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 Tt t t t t t

S p Se v T v h v Tt t

Differentiation of product (3) (1)

16

LHS (1+2):

2

div 2vv h v T

2

2 2

2

2

div2 2

div 2

v et

v v Sh v v h T v S vt

v Sv h T v S vt

RHS:

LHS=RHS (canceling like terms):

divST v S v v Tt

(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 iik

k

vST v S Tt x

general 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 iik

k

vST v S Tt 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 cccpa

,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,

TTSp

pSS

pT

ddd

TTcT

TSS p

p

ddd

-- specific heat (capacity) under constant pressurep

p TSTc

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 xvTTv

tTc

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

flow0vdiv

vpvvtv

-- 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