Unsteady motion

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. 

The outline of this paper is as follows. First, a theoretical model of unsteady motions in a combustion chamber with feedback control is constructed. The formulation is based on a generalized wave equation which accommodates all influences of acoustic wave motions and combustion responses. Control actions are achieved by injecting secondary fuel into the chamber, with its instantaneous mass flow rate determined by a robust controller. Physically, the reaction of the injected fuel with the primary combustion flow produces a modulated distribution of external forcing to the oscillatory flowfield, and it can be modeled conveniently by an assembly of point actuators. After a procedure equivalent to the Galerkin method, the governing wave equation reduces to a system of ordinary differential equations with time-delayed inputs for the amplitude of each acoustic mode, serving as the basis for the controller design. 

A wave equation governing the unsteady motions is then derived by decomposition of all dependent variables as sums of the mean and fluctuation parts. Thus... 

The upper bound of the region of stable steady motion is shown in Fig. 6.7 as a function of (CdRcx ) and /. For large /, secondary motion starts at Rcy = 100, i.e., (Cd Rcj ) = 23.4. At lower /, steady motion persists to higher Rcy the boundary shows a maximum at Rcy = 172, (CdRcj ) = 32.6 for/ = 8 X 10 Three kinds of secondary motion have been observed (S8), although the distinctions between them are not sharp. Immediately above the transition to unsteady motion, a disk shows regular oscillations about a diameter the amplitude of oscillation and of the associated horizontal motion increases with... 

In the present chapter, we neglect wall effects and unsteady motion including splitting. These factors are considered in Chapters 9, 11, and 12, respectively. The fluid mechanics of large bubbles and drops are discussed before turning to mass transfer. 

In this chapter we first discuss the equations of motion for particles at low Re. Semiempirical extensions beyond the creeping flow regime are then considered. It is useful to distinguish two general kinds of unsteady motion ... 

As for steady motion, analytic solutions for unsteady motion of rigid and fluid particles are available only in creeping flow. The solution was developed by Basset (B3) the outline given here follows the treatment of Landau and Lifshitz (L4). The governing equation is the unsteady form of Eq. (1-36), i.e., for axisymmetric flow,... 

Equation (11-11) depends on neglect of inertial terms in the Navier-Stokes equation. Neglect of inertia terms is often less serious for unsteady motion than for steady flow since the convective acceleration term is small both for Re 0 (Chapters 3 and 4), and for small amplitude motion or initial motion from rest. The second case explains why the error in Eq. (11-11) can remain small up to high Re, and why an empirical extension to Eq. (11-11) (see below) describes some kinds of high Re motion. Note also that the limited diffusion of vorticity from the particle at high cd or small t implies that the effects of a containing wall are less critical for accelerated motion than for steady flow at low Re. 

Although Eqs. (11-30) to (11-32) give a good description of free acceleration from rest, this does not necessarily mean that they apply to all types of unsteady motion. Fall or rise from rest is a particular case in which the creeping flow assumptions apply initially. The approach is less realistic if Re is initially large (e.g., for a particle released in a flowing fluid). 

Neglect of added mass and history simplifies calculation of unsteady motion considerably. However, for y characteristic of particles in liquids, this introduces substantial errors as illustrated by curve 4 in Fig. 11.7. The accuracy of the simplification improves as y and Re increase, but even for y as high as 10 trajectories calculated neglecting history and added mass substantially underpredict the duration of accelerated motion. Neglect of added mass causes the predicted trajectory to be in error from the start of the motion. Since it is the... 

With these simplifications, W and X can be generated as functions of T, with the particle characterized by a single dimensionless parameter, either Rejs, A d or Rqj. Figure 11.13 shows predictions for a particle released from rest W = X = 0 at T = 0), while Fig. 11.14 gives trajectories for particles projected vertically upwards such that the particle comes to rest at T = 0. Figures 11.13 and 11.14 enable rapid estimations for many problems involving unsteady motion of particles in gases. 

SUMMARY We investigate the unsteady motion of mass reservoir formed by the accretion onto the magnetosphere around rotating neutron stars. The unsteady motion of the reservoir induces secondary accretion to neutron star by R-T instability. The nonperiodic or quasiperiodic phenomena of X-ray bursters seems to be related to this property of mass reservoir on the magnetosphere. We classify the typical dynamical state of the reservoir into three types with the parameters which are accretion rate M and angular velocity of neutron star ft. They are nonsequential oscillation sequential periodic (quasi-periodic) oscillation, and chaotic oscillation states. 

A fundamental account of unsteady motion in a dense suspension requires knowledge of phase interactions and origins of wavy stratified motion [Zhu, 1991 Zhu et al., 1996]. Flow stratification involves a dense phase of solids flowing along the bottom of a horizontal pipe with wave motions, or a layer of the dense phase of solids flowing near the wall of a vertical pipe with wave motions. The former was analyzed, in part, on the basis of the theoretical work of de Crecy (1986) on stratified liquid-vapor flows. 

Unsteady flow also includes such topics as oscillations in connected reservoirs and in U tubes and such phenomena as tidal motion and flood waves. Likewise, the field of machinery regulation by servomechanisms is intimately connected with unsteady motion. However, all these topics are considered to be beyond the scope of the present text. 

Filbey equation (7). For cases of small deformation and deformation gradients, the general linear viscoelastic model can be used for unsteady motion of a viscoelastic fluid. Such a model has a memory function and a relaxation modulus. Bird and co-workers (6, 7) gave details of the available models. 

Let one be told by a person, whose veracity he cannot doubt of, that one of his sons is suddenly kill d, tis evident the passion this event wou d occasion, wou d not settle into pure grief, till he got certain information, which of his sons he had lost. Here there is an evil certain, but the kind of it uncertain Consequently the fear we feel on this occasion is without the least mixture of joy, and arises merely from the fluctuation of the fancy betwixt its objects. And tho each side of the question produces here the same passion, yet that passion cannot settle, but receives from the imagination a tremulous and unsteady motion.41... 

Chisnell RE (1987) The unsteady motion of a drop moving vertically under gravity. J Fluid Mech 176 443-464... 

Ogibalov, P. M. and Mirzadzhanzade, A. Kh., Unsteady Motions of Viscoplastic Media, Izd. Moskov. Univ., Moscow, 1970 [in Russian],... 

Benard experimented with several different liquids and found that volatile liquids (ether at 15°C, alcohol at 50°C) produced rapid, chaotic, permanently unsteady motions. A qualitative description of the currents in evaporating ether was given by B6nard, but the bulk of his report concerns spermaceti. 

By quasi-steady motion here, we mean an unsteady motion for which the time-dependent terms in the equations of motion and boundary conditions are of higher order in R than 0(.R In R). 

Several mechanisms may be proposed to explain the process of spontaneous emulsification, all of which are related to the properties of the interfacial film. The first mechanism is due to interfacial turbulence that may occur as a result of mass transfer or by non-uniform adsorption of the surfactant molecules at the OAV interface. The interface shows unsteady motions - streams of one phase are ejected and penetrate into the second phase. This is illustrated in Figure 4.1(a) which shows the localized reduction in interfacial tension caused by non-uniform adsorption of surfactants or mass transfer of surfactants across the interface (5-7). When the two phases are not in chemical equilibrium, convection currents may be formed which transfers the liquid rich in surfactants towards the areas deficient in surfactants. These convection currents may give rise to... 

Stanyukovich, K, P., Unsteady Motion of a Continuous Media, Pergamon Press, London, 1960, pp. 372-381. 

A molecular rearrangement is expected around the contact line having an effect on the advancing contact lines creating an unsteady motion and a complex structure in this region, while for the receding angle the contact line should move more steadily. 

Gupta, R. K., Unsteady motion of a spheroid in an elastico-viscous liquid, Z. Ang. Math. Phys. 27 273-279 (1976). 

K.P. Staniukovich, Unsteady Motion of Continuous Media (Pergamon, London, 1960), p. 745... 

The two last terms on the right-hand side of (2.2.1) relate to fast, unsteady motion. The added mass term accounts for the fact that when accelerating a particle from rest, the surrounding fluid must also be accelerated. This appears to add mass to the particle. The Basset integral says that the drag will, by rapidly changing motion, depend not only on its instantaneous velocity relative to the fluid, but also on the previous motion since the fluid flow pattern may not have had time to adjust, due to the fluid inertia. These two terms are zero in steady movement. 

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