Unsteady flows

The conservation equations in orthogonal, curvilinear coordinate-systems may easily be derived from equations (l)-(4). Since the complete form is complicated and will not be required in subsequent problems, the 

The flow will be assumed to be in the direction, and the relation between the physical distances (sj, S2, S3) and the coordinates (x, X2, X3) will be taken as 

From equation (5) it is found that the off-diagonal elements of p vanish (Pi2 = P23 = P31 =0) that the diagonal elements of p are given by 

By utilizing the expression for the divergence of a diagonal tensor in orthogonal, curvilinear coordinate systems, one can show that equation (2) reduces to 

Equations (19)-(22) may also be obtained by specializing the conservation equations given in [1] to the case in which d/dx2 = 0, d/dx = 0, the 2 and 3 components of all vectors are zero, the pressure tensor is diagonal, and 01 = 1. In Cartesian coordinate systems 2 = 3 = 1- 

---

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

Matching the flow between the impeller and the diffuser is complex because the flow path changes from a rotating system into a stationary one. This complex, unsteady flow is strongly affected by the jet-wake of the flow leaving the impeller, as seen in Figure 6-29. The three-dimensional boundary layers, the secondary flows in the vaneless region, and the flow separation at the blades also affects the overall flow in the diffuser. 

Domercq, O., and Thomas, R., 1997, Unsteady Flow Investigation in a Transonic Centrifugal Compressor Stage, AIAA Paper No. 97-2877. 

Winterbone, D.E., Nikpour, B., and Alexander, G.L., 1990, Measurement of the Performance of a Radial Inflow Turbine in Conditional Steady and Unsteady Flow, IMechE, Paper No. 0405/015. 

An unsteady flow is one in which the conditions at any point van with time such a flow is also called a transient flow. 

A typical envelope opening has a complicated shape and is often subject to unsteady flow conditions at its inlet and outlet.- There are no simple analytical solutions for the flow through such openings. The most-used equation representing flow characteristics is the so-called power law ... 

Dead-end Pores Dead-end volumes cause dispersion in unsteady flow (concentration profiles ar> ing) because, as a solute-rich front passes the pore, transport oceurs by molecular diffusion into the pore. After the front has passed, this solute will diffuse back out, thus dispersing. 

Iim88] Lim, H.A., Lattice gas automata of fluid dynamics for unsteady flow, Complex Systems 2 (1988) 45-58. 

Important for polymer processing is the fact that when the concentration of a hard filler is increased in the composite, the unsteady flow (in the sense of large-scale distortions) of the extrudate occurs at higher shear rates (stresses) than in the case of the base polymer [200, 201,206]. Moreover, the whirling of the melt flow is even suppressed by small additions of filler [207]. 

This method of solution of problems of unsteady flow is particularly useful because it is applicable when there are discontinuities in the physical properties of the material.(6) The boundary conditions, however, become a little more complicated, but the problem is intrinsically no more difficult. 

Figure 2.40 shows the unsteady flow upstream of the ONE in one of the parallel micro-channels of d = 130 pm at = 228kW/m, m = 0.044 g/s (Hetsroni et al. 2001b). In this part of the micro-channel single-phase water flow was mainly observed. Clusters of water appeared as a jet, penetrating the bulk of the water (Fig. 2.40a). The vapor jet moved in the upstream direction, and the space that it occupied increased (Fig. 2.40b). In Fig. 2.40a,b the flow moved from bottom to top. These pictures were obtained at the same part of the micro-channel but not simultaneously. The time interval between events shown in Fig. 2.40a and Fig. 2.40b is 0.055 s. As a result, the vapor accumulated in the inlet plenum and led to increased inlet temperature and to increased temperature and pressure fluctuations.

Ozawa M, Akagawea K, Sakaguchi T (1989) Flow instabilities in paraUel-channel flow systems of gas-liquid two-phase mixtures. Int 1 Multiphase Flow 15 639-657 Peles YP (1999) VLSI chip cooling by boiling-two-phase flow in micro-channels. Dissertation, Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa Peles YP, Yarin LP, Hetsroni G (2001) Steady and unsteady flow in heated capUlary. Int J Multiphase Flow 22 577-598... 

Peles el al. (2000) elaborated on a quasi-one-dimensional model of two-phase laminar flow in a heated capillary slot due to liquid evaporation from the meniscus. Subsequently this model was used for analysis of steady and unsteady flow in heated micro-channels (Peles et al. 2001 Yarin et al. 2002), as well as the study of the onset of flow instability in heated capillary flow (Hetsroni et al. 2004). 

Fig. 9.14 Unsteady flow in a heated capillary Dl = 0.3, Ja = 1.83, PeL = 598. Reprinted from Peles et al. (2001) with permission...

The quasi-one-dimensional model described in the previous chapter is applied to the study of steady and unsteady flow regimes in heated micro-channels, as well as the boundary of steady flow domains. The effect of a number of dimensionless parameters on the velocity, temperature and pressure distributions within the domains of liquid vapor has been studied. The experimental investigation of the flow in a heated micro-channel is carried out. 

Peles YP, Yarin LP, Hetsroni G (2001) Steady and unsteady flow in a heated capillary. Int J Multiphase Flow 27 577-598... 

Landetman CS (1994) Micro-channel flow boding mechanisms leading to burnout. J Heat Transfer Electron Syst ASME HTD 292 124-136 Levich VG (1962) Physicochemical hydrodynamics. Prentice Had, London Morijama K, Inoue A (1992) The thermohydraudc characteristics of two-phase flow in extremely narrow channels (the frictional pressure drop and heat transfer of boding two-phase flow, analytical model). Heat Transfer Jpn Res 21 838-856 Peles YP, Yarin LP, Hetsroni G (2001) Steady and unsteady flow in a heated capdlary. Int J Multiphase How 27 577-598... 

Unlike at adiabatic conditions, the height of the liquid level in a heated capillary tube depends not only on cr, r, pl and 6, but also on the viscosities and thermal conductivities of the two phases, the wall heat flux and the heat loss at the inlet. The latter affects the rate of liquid evaporation and hydraulic resistance of the capillary tube. The process becomes much more complicated when the flow undergoes small perturbations triggering unsteady flow of both phases. The rising velocity, pressure and temperature fluctuations are the cause for oscillations of the position of the meniscus, its shape and, accordingly, the fluctuations of the capillary pressure. Under constant wall temperature, the velocity and temperature fluctuations promote oscillations of the wall heat flux. 

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 ... 

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

---