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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

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Height of a meniscus h0h0 θ R a Applying the Young-Laplace equation we obtain The meniscus will be approximately hemispherical with a constant radius of curvature, Hence, is the capillary length. h 0 may be positive and negative, e.g. for mercury θ~140 0 and the meniscus will fall, not rise. For water, α =73*10 -3 N/m, and in 0.1mm radius clean glass capillary, h 0 =15cm.

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Let us calculate the rate at which the meniscus rises to the height h 0. 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, p 1, and at the top of the capillary (just under the meniscus), p 2, is Thus,

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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,

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Finally, Or, introducing, we obtain As, t/τt/τ h/h0h/h0 For water in a glass capillary of 0.1mm radius,

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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,

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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

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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):

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and the 1 st law of thermodynamics (applied for a fluid particle of unit mass, V =1/ ρ ): (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)

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

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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

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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 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. thermal conductivity conservation of energy for an ideal fluid

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

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

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

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16 LHS (1+2): RHS: LHS=RHS (canceling like terms): (7)

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

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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 )

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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

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

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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: -- temperature conductivity

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

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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

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