Quasi-one-dimensional equations

Below the system of quasi-one-dimensional equations considered in the previous chapter used to determine the position of meniscus in a heated micro-channel and estimate the effect of capillary, inertia and gravity forces on the velocity, temperature and pressure distributions within domains are filled with pure liquid or vapor. The possible regimes of flow corresponding to steady or unsteady motion of the liquid determine the physical properties of fluid and intensity of heat transfer. 

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is ... 

Keywords Capillary instability of liquid jets Curvature Elongational rheology Free liquid jets Linear stability theory Nonlinear theory Quasi-one-dimensional equations Reynolds number Rheologically complex liquids (pseudoplastic, dilatant, and viscoelastic polymeric liquids) Satellite drops Small perturbations Spatial instability Surface tension Swirl Temporal instability Thermocapillarity Viscosity... 

A more involved version of the quasi-one-dimensional equations of the dynamics of thin liquid jets was proposed in [37, 38] where radial inertia in the jet cross-section was accounted for. The final version of these equations for a Newtonian viscous jet with a straight axis derived in [37-39] has the form... 

Summarizing, in the linear stability theory of capillary breakup of thin free liquid jets, the quasi-one-dimensional approach allows for a simple and straightforward derivation of the results almost exactly coinciding with those obtained in the framework of a rather tedious analysis of the three-dimensional equations of fluid mechanics. This serves as an important argument for further applications of the quasi-one-dimensional equations to more complex problems, which do not allow or almost do not allow exact solutions, in particular, to the nonlinear stages of the capillary breakup of straight thin liquid jets in air (considered below in this chapter). 

On the other hand, capillary breakup of sufficiently viscous liquid jets is a longwave phenomenon, and its description in the framework of the quasi-one-dimensional equations of the dynamics of liquid jets is sufficiently accurate. The effect of the viscosity on the capillary breakup of highly-viscous liquid jets was studied numerically by Yarin [29]. The initial perturbation of the jet surface was imposed as a harmonic... 

At the late stage of capillary breakup near the jet cross-section where the breakup will eventually occur, liquid flow completely forgets the initial conditions. It is dominated by the local flow conditions and becomes self-similar. The numerical description of the latest stages of capillary breakup is unreliable near the cross-sections where the cross-sectional radius tends to zero. A theoretical description of such self-similar final jet pinching is given in [79-84], assuming either inertia or viscosity dominated flows in the tiny threads and, in particular, using quasi-one-dimensional equations. 

Y arin et al. [29, 111] gave a theory of the capillary breakup of thin jets of dilute polymer solutions and formation of the bead-OTi-the-string structure (some additional later results can be foimd in [90]). The basic quasi-one-dimensional equations of capillary jets (1.49) and (1.50) are supplemented with an appropriate viscoelastic model for the longitudinal stress. Yarin et al. [29, 111] used the Hinch rheological constitutive model, which yields the following expression... 

Keywords Bending instability of liquid jets Buckling of liquid jets Electrified liquid jets Electrospinning Elongational rheology Newtonian and rheologically complex liquids Quasi-one-dimensional equations of the dynamics of liquid jets Small and finite perturbations Viscoelastic polymeric liquids... 

General Quasi-One-Dimensional Equations of Dynamics of Free Liquid Jets... 

These expressions close the system of the general quasi-one-dimensional equations of free liquid jets moving in air with arbitrary speeds. 

The present model takes into account how capillary, friction and gravity forces affect the flow development. The parameters which influence the flow mechanism are evaluated. In the frame of the quasi-one-dimensional model the theoretical description of the phenomena is based on the assumption of uniform parameter distribution over the cross-section of the liquid and vapor flows. With this approximation, the mass, thermal and momentum equations for the average parameters are used. These equations allow one to determine the velocity, pressure and temperature distributions along the capillary axis, the shape of the interface surface for various geometrical and regime parameters, as well as the influence of physical properties of the liquid and vapor, micro-channel size, initial temperature of the cooling liquid, wall heat flux and gravity on the flow and heat transfer characteristics. 

Chapter 8 consists of the following in Sect. 8.2 the physical model of the process is described. The governing equations and conditions of the interface surface are considered in Sects. 8.3 and 8.4. In Sect. 8.5 we present the equations transformations. In Sect. 8.6 we display equations for the average parameters. The quasi-one-dimensional model is described in Sect. 8.7. Parameter distribution in characteristic zones of the heated capillary is considered in Sect. 8.8. The results of a parametrical study on flow in a heated capillary are presented in Sect. 8.9. 

Significant simplification of the governing equations may be achieved by using a quasi-one-dimensional model for the flow. Assume that (1) the ratio of meniscus depth to its radius is sufficiently small, (2) the velocity, temperature and pressure distributions in the cross-section are close to uniform, and (3) all parameters depend on the longitudinal coordinate. Differentiating Eqs. (8.32-8.35) and (8.37) we reduce the problem to the following dimensionless equations ... 

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. 

The subject of the present chapter is the analysis of possible states of capillary flow with distinct evaporative meniscus. The system of quasi-one-dimensional mass, momentum and energy equations are applied to classify the operating parameters corresponding to various types of flow. The domains of steady and unsteady states are also outlined. 

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... 

The system of quasi-one-dimensional non-stationary equations derived by transformation of the Navier-Stokes equations can be successfully used for studying the dynamics of two-phase flow in a heated capillary with distinct interface. 

An adequate quantitative description of such a situation requires a two- or even three-dimensional approach. Today, a great variety of numerical models are available that allow us to solve such models almost routinely. However, from a didactic point of view numerical models are less suitable as illustrative examples than equations that can still be solved analytically. Therefore, an alternative approach is chosen. In order to keep the flow field quasi-one-dimensional, the single well is replaced by a dense array of wells located along the river at a fixed distance xw (Fig. 25.2c). Ultimately, the set of wells can be looked at as a line sink. This is certainly not the usual method to exploit aquifers Nonetheless, from a qualitative point of view a single well has properties very similar to the line sink. 

The plug-flow problem may be formulated with a variable cross-sectional area and heterogeneous chemistry on the channel walls. Although the cross-sectional area varies, we make a quasi-one-dimensional assumption in which the flow can still be represented with only one velocity component u. It is implicitly assumed that the area variation is sufficiently small and smooth that the one-dimensional approximation is valid. Otherwise a two- or three-dimensional analysis is needed. Including the surface chemistry causes the system of equations to change from an ordinary-differential equation system to a differential-algebraic equation system. 

In the simplest case under consideration (Newtonian, isothermal, and inertialess motion in the absence of structuring) at h(t) 8 the flow may be assumed to be quasi-one-dimensional, and the distorsions of velocity and pressure profiles in the vicinity of a front and the gate into a forming cavity may be neglected. The basic equation will then have the form ... 

In order to analyze the origin of this difference between the energy spectra of quasi one- and two-dimensional quantum dots in the small confinement regime, the internal space for two electrons is considered as in the quasi-one-dimensional cases. Using a harmonic approximation to the Gaussian confining potential, and neglecting the dependence on the z coordinate, the Hamiltonian of Equation (1) for two electrons takes the form... 

The total current (I) flowing through such a cell is composed of displacement (7C) and conduction contributions (/R), i.e., we neglect inductive effects and can concentrate on capacitive and resistive elements. For mechanistic considerations we consider current densities, which in the quasi one-dimensional (laterally homogeneous) case are connected with the current via i = Ha (a = area), while in the general case this connection reads i = dl/da (a area vector, d/da. gradient operator with respect to the a-coordinates).4 The continuity equation of the charge density (P)... 

Steady-state, quasi-one-dimensional conservation equations... 

The equations governing the steady state, quasi-one-dimensional flow of a reacting gas with negligible transport properties can easily be obtained from equations (l-19)-(l-22). When transport by diffusion is negligible 0 and Dtj 0 for ij = 1,..., N the diffusion velocities, of course, vanish [FJ 0 for / = 1,..., N, see equation (1-14)]. If, in addition, transport by heat conduction is negligible (A 0) and = 0, then the heat flux q vanishes [see equation (1-15)]. Finally, in inviscid flow 0 and K 0), equations (1-16)-(1-18) show that all diagonal elements of the pressure tensor reduce to the hydrostatic pressure, pu = pjj — P33 = P-The steady-state forms of equations (1-20), (l-21a), and (1-22) then become... 

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