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

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.

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,

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,

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

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,

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 -- 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):

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)

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

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

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

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

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

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

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

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

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)

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

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

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

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

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