1. 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.
Water absorption by paper towel
Capillary flow in a brick

2. Height of a meniscus
The meniscus will be approximately hemispherical with a constant radius of curvature, h0 θ R a
Applying the Young-Laplace equation we obtain
Hence,
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.

3. Let us calculate the rate at which the meniscus rises to the height h0
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,
The average velocity is
Here is the instantaneous distance of the meniscus above the pool level.
The pressure difference at the pool level, p1, and at the top of the capillary (just under the meniscus) , p2, is
Thus,

4. 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,

5. Finally,
Or, introducing , we obtain
As ,
h/h0
For water in a glass capillary of 0.1mm radius,
t/τ

6. 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,

7. 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

8. Conservation of energy in ideal fluid
-- total energy of unit volume of fluid
kinetic energy
internal energy,
e is the internal energy per unit mass
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):

9. 1:
(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):
and the 1st law of thermodynamics (applied for a fluid particle of unit mass, V=1/ρ):
Equation (1) takes the following form: 2:
(use of continuity equation)

10. We will also use the enthalpy per unit mass (V=1/ρ) defined as
Equation (2) will now read 3:
If a fluid particle moves reversibly (without loss or dissipation of energy), then

11. Finally,
conservation of energy for an ideal fluid
-- energy flux
In integral form,
using Gauss’s theorem
energy transported by the mass of fluid
work done by the pressure forces

12. The general equation of heat transfer
conservation of energy for an ideal fluid
The conservation of energy still holds for a real fluid, but the energy flux must include
the flux due to processes of internal friction (viscous heating),
the flux due to thermal conduction (molecular transfer of energy from hot to cold regions; does not involve macroscopic motion).
For (b), assume that
(i) is related to the spatial variations of temperature field;
(ii) temperature gradients are not large.
Heat flux due to thermal conduction:
thermal conductivity

13. The conservation of energy law for a real fluid
viscous heating
heat conduction
We will re-write this equation by using
(1)
-- continuity equation
e, h and S are the internal energy, enthalpy and entropy per unit mass
(2)
-- Navier-Stokes equation
(3)
-- 1st law of thermodynamics
(4)
-- 1st law of thermodynamics in terms of enthalpy

14. Differentiation of product
(5)
(6)
(7)
1st term in the lhs:
Differentiation of product
(1+5)
(2)
(6)
(4)

15. Differentiation of product
2nd term in the lhs:
Differentiation of product
(3)
(1)

16. LHS=RHS (canceling like terms):
(7)
RHS:
LHS=RHS (canceling like terms):

17. In the lhs,
In the rhs,
Finally,
general equation of heat transfer
heat conducted into considered volume
heat gained by unit volume
energy dissipated into heat by viscosity

18. Governing equations for a general single-phase flow
-- continuity equation
-- Navier-Stokes equation
-- general equation of heat transfer
+ expression for the viscous stress tensor
+ equations of state: p(ρ, T) and S(ρ, T)

19. Incompressible flow
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

20. 1. Typical variations of pressure in a fluid flow,
2. Variations of density,
3. Incompressible flow ≡ slow fluid motion,
4. Hence, we can neglect variations in density field caused by pressure variations
5. Similarly, for variation of entropy.
In general,
but for incompressible flow,
-- specific heat (capacity) under constant pressure

21. For incompressible flow, the general equation of heat transfer takes the following form:
Frequently,
the thermal conductivity coefficient κ can be approximated as being constant;
the effect of viscous heating is negligible.
Then, the general equation of heat transfer simplifies to
-- temperature conductivity

22. Boundary conditions for the temperature field:
1. wall:
given temperature,
given heat flux,
thermally insulated wall,
2. interface between two liquids:
and

23. Governing equations for incompressible non-isothermal fluid flow
-- continuity equation
-- Navier-Stokes equation
-- general equation of heat transfer
Thermal conductivity and viscosity coefficients are assumed to be constant.
+ initial and boundary conditions