Capillary numerical solutions

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

Figure 9.9 Dimensionless critical droplet size for breakup (capillary number Ca<- = Crj a/r) as a function of viscosity ratio M of the dispersed to the continuous phase for two-dimensional flows in the four-roll mill. The data sets correspond from bottom to top to ff — 1.0(0), 0.8 (A), 0.6 (0), 0.4 (V), and 0.2 ( ), with a defined in Eq. (9-17). The fluids are those described in Fig. 9-7. The solid lines are the predictions of a small-deformation theory, while the dashed lines are for a large-deformation theory. The closed squares are from Rallison s (1981) numerical solutions (see also Rallison and Acrivos 1978). (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

In this work, microscale evaporation heat transfer and capillary phenomena for ultra thin liquid film area are presented. The interface shapes of curved liquid film in rectangular minichannel and in vicinity of liquid-vapor-solid contact line are determined by a numerical solution of simplified models as derived from Navier-Stokes equations. The local heat transfer is analyzed in term of conduction through liquid layer. The data of numerical calculation of local heat transfer in rectangular channel and for rivulet evaporation are presented. The experimental techniques are described which were used to measure the local heat transfer coefficients in rectangular minichannel and thermal contact angle for rivulet evaporation. A satisfactory agreement between the theory and experiments is obtained. 

Pore-size distributions (PSD) are routinely obtained by an algorithm dating back to Barret, Joyner and Halenda [3-4]. Either cylindrical or slit-shaped pores are assumed in these calculations. The BJH method virtually represents numerical solution of an integral equation, which describes adsorption and capillary condensation of adsorbate in pores and utilizes the Kelvin equation. Because the validity of Kelvin equation in micropores can be questioned a new approach based on statistical physics is developing, viz. the density functional theory [5-7]. This approach can supply adsorption isotherms for cylindrical or slit-shaped pores of different sizes in carbonaceous or oxide matrix. The problem then is to sum up these isotherms so that the experimental isotherm is reproduced. Expensive commercial programs are available for this purpose. 

During the 1970s the ECD became firmly established as the most sensitive gas chromatographic detector for some compounds. The kinetic model was described in terms of a numerical solution of the differential equations. This was assisted by the development of the constant current mode of measuring the response and the development of Ni-63 sources for the detector. The purification of the carrier gas and the further development of capillary columns improved the operation of the ECD. In addition, chemical reactions were used to make derivatives with a greater sensitivity in the ECD. Other ion molecule reactions were used to improve the sensitivity of... 

Static capillary phenomena lead to precisely determined geometrical shapes like sessile menisci, pendent menisci, minimal surfaces, which can be used for the physical determination and measurement of the surface tension or the interfacial tensions between fluids. In addition to the simple forms considered herein, more complex forms (e.g., sessile lenticular drops) can be studied. Mathematical resolution of these shapes is a combination of the (numerical) solution of the highly nonlinear Young-Laplace equation together with an appropriate set of boundary conditions. For practical purposes, only axisymmetric forms are readily amenable to mathematical analysis. 

Yin et al. (2006) qnalitatively showed this mechanism by solving relevant flow equations nnmericaUy. Xia et al. (2008) also developed a simplified pore scale model to describe polymer flow. The numerical solutions from Xia et al. have verified the proposed mechanism. Figure 6.22 shows the velocity contours of a Newtonian fluid with Weissenberg number (We) = 0 and a viscoelastic fluid with We = 0.35 in a flow channel with a dead end when the Reynolds number (Re) = 0.001. We can see that the velocity (m/day) of the viscoelastic fluid is higher than that of the Newtonian fluid at the same position of the dead end. This pulling mechanism also works in the case shown in Figure 6.20c, where the residual oil is trapped at the pore throats by capillary force. 

For natural boundary conditions, as a rule, one must use numerical methods or approximate analysis. For example, a numerical solution describing the wave of capillary suction was obtained in [152], and a solution describing centrifugal syneresis was obtained in [490] by using expansions with respect to a small parameter. 

The precision of surface tension measurements using the capillary rise method can be further increased if the deviation of the meniscus shape from the spherical is taken into account. This correction is especially important when capillaries of large radii are used. Corrections for non-spherical meniscus curvature are based on tabulated numerical solutions of the differential Laplace equation [6]. The capillary rise method yields a values with a precision of up to hundredths of mN/m. 

In this study a simulation of such processes for various arrangements of capillaries was performed to attain a better understanding of oxygen supply conditions in the brain cortex. Oxygen tension values were calculated for steady-state conditions by numerical solution of partial differential equations for different capillary-tissue systems. The analysis helps to explain the experimental results obtained from the microstructure of rat... 

However, as blood flows along the capillary, oxygen is extracted by the tissue and the oxygen partial pressure in the blood declines rapidly. The oxygen partial pressure in the tissue also declines rapidly and a point is reached where zero-order kinetics no longer applies. The complete form of the Michaelis-Menten equation must be used, requiring a numerical solution of equation (1-118). McGuire and Secomb (2001) showed numerically that under the set of conditions considered in this example, as much as 37% of the tissue was hypoxic. 

Equation (6.5.12), subject to the boundary conditions indicated, can be solved analytically only under limiting conditions on the Debye length described below. In general, it must be solved numerically. Figure 6.5.2 gives numerical solutions obtained by Gross Osterle (1968) for a constant surface potential 2-79 for various values of the Debye length ratio A It can be seen that for A " <0.1 the potential is zero over most of the capillary cross section, whereas for A " > 10 the potential is nearly constant over the cross section. 

This method of successive approximation is called Piccard s iteration, and need not be confined to iterations using the normal stress balance. The kinematic boundary condition, which is the one that deals with the velocities normal to the interfaces (Equation 7.14), has been used as well (Bhattacharji and Savic, 1965). In fact, in numerical solutions, such iterations have to be carried out imtil numerical convergence is reached. It has been observed (Silliman and Scriven, 1980) that iterations using normal stress boundary conditions converge faster if the capillary number is low and those using kinematic boundary conditions converge... 

The opposite limit is that of a very thick annular region, i.e. a >> 1. Physically this is the problem of the capillary instability of a fluid thread immersed in an infinite fluid as first studied by Tomotika [77] Numerical solutions of the dispersion relation indicate that the squeezing mode disappears as a increases (this is expected since the squeezing mode has a band of unstable waves 0 < A < 1/a), and the stretching mode reaches a limit which is independent of 7 - this is because the outer interface is at infinity and has no effect on the stretching mode, to leading order. 

In the following, the results under two different ap- pared. Figures 3a and b show the electrokinetic wall effects proaches, the complete numerical solution of Eq. (12) and for a given particle size dp = 5 p,m under different charge the analytical solution based on Eq. (27), will be com- conditions of the capillary wall. The wall zeta potentials... 

Typical numerical solutions of Eq. (51), following Zhu-mud et al. [8], are presented in Fig. 3. A maximum occurs in the characteristic displacement curves only when the surface tension relaxation is not as fast as compared to the capillary filling. More simplistic scaling estimates in the capillary rise phenomenon can readily be obtained by noting that within certain limits, the capillary rise represents a quasi-steady process, in which the amount of surfactant adsorbed to the solid/liquid interface per unit time is equal to that transported to the liquid/vapour interface by diffusion, which implies... 

The most common approach to the numerical solution of the three-phase non-isothermal flow problems is based on the selection of pressure of one of the phases, temperature and saturations as the unknowns [1-4]. However, the choice of a phase pressure as the primary variable assumes certain difficulties encountered in the numerical solution of three-phase non-isothermal flow problems, which takes into account capillary effects. Some of them, in relation to the isothermal case, are described in [5-7]. These difficulties are mainly related to the unbounded increase in the derivative of capillary pressure functions when saturations approach corresponding residual values. 

Thus, it is clear that the finite difference numerical solutions offered by some authors are not really necessary because problems without capillary pressure can be solved analytically. Actually, such computational solutions are more damaging than useful because the artificial viscosity and numerical diffusion introduced by truncation and round-off error smear certain singularities (or, infinities) that appear as exact consequences of Equation 21-17. Such numerical diffusion, we emphasize, appears as a result of finite difference and finite element schemes only, and can be completely avoided using the more labor-intensive method of characteristics. For a review of these ideas, refer to Chapter 13. As we will show later, capillary pressure effects become important when singularities appear modeling these correctly is crucial to correct strength and shock position prediction. 

McEwen, C.R., A Numerical Solution of the Linear Displacement Equation with Capillary Pressure, Petroleum Transactions, AIME, Vol. 216, 1959, pp. 412-415. 

Other examples of the effect of viscous dissipation on capillary data are given by Cox and Macosko (1974b) and Warren (1988). These workers and Winter (1975, 1977) indicate how to correct data affected by shear heating to true viscosity values. This requires numerical solution of the momentum and energy equations, a capability available in many standard fluid mechanics software packages. However, note that for typical capillary dies the stress level at which viscous dissipation becomes important is near the region for polymer melt fracture, tu, lO Pa. As already pointed out, it is not possible to get true viscosity data aft - the onset of melt h cture. 

Figure 6.11 Dimensionless potential distribution across a cylindrical capillary obtained by the numerical solution of the potential distribution equation...

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