Stationary flow regimes

Rearranging the thermal balance equation (9.41) and using expressions (9.46-9.50) we arrive at 

The temperature distribution within the liquid and vapor domains of a heated micro-channel is plotted in Pig. 9.3. The liquid entering the channel absorbs heat 

Equation (9.62) contains seven dimensionless parameters accounting for the hydrodynamic and thermal effects. The regimes in which stable flows in a heated capillary are possible correspond to the following interval of the length Zp 0 Zp 1. The latter allows us to use Eq. (9.62) to define the domains of the existence of stable and unstable flow regimes. In the multi-dimensional parametric space (Z, i9p Pep Ja Ts ki. c pL.g) the limiting values of these parameters corre- 

Spend to a surface subdividing the parametric space into domains, which correspond to various flow regimes. 

The solution of Eq. (9.62) may be represented as 35 spatial (or 21 planar) diagrams which establish correlations between any three (or two, for the planar diagram) parameters when all the other parameters are fixed. These diagrams allow us to outline the domains of stable flow regimes and to determine the values of the parameters corresponding to the change between the stable and unstable flows. 

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Chapter 9 consists of the following in Sect. 9.2 the physical model of two-phase flow with evaporating meniscus is described. The calculation of the parameters distribution along the micro-channel is presented in Sect. 9.3. The stationary flow regimes are considered in Sect. 9.4. The data from the experimental facility and results related to two-phase flow in a heated capillary are described in Sect. 9.5. 

Non-stationary flow regime of evaporating fuel spray... 

We can see that a) even a small (in comparison with the loads at experimental deformation conditions) preliminary shearing influence on oligomeric liquid reduces the time of transition to the stationary flow regime (where q = qs, qs is not dependent on Xt) irreversibly and significantly (by decimal exponents) b) within the studied range of Yp, the qs absolute value does not depend on Yp, however, it increases significantly if the liquid takes a short relaxation after preliminary deformation. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

Fig. 11.7 The dependence (PeL) 1 domain of stationary steady regimes of flow, 2 domain of unsteady states. PeL = PeLtr point of transition from the stable to unstable flow regime. Reprinted from Hetsroni et al. (2004) with permission...

The study of laminar vortical structures has been well documented in the Taylor-Couette (TC) flow for an extensive range of gap sizes [33]. Of particular interest are the experimentally determined boundaries in the map of flow regimes [34], and specifically the flow regime where the propagating helical vortex and stationary toroidal vortex modes become stable simultaneously and interact in the axisym-... 

Fig. 4.4.5 Gradual blurring (staring on locations marked by arrow) of MRI spin-tagging spin-echo images of Taylor—Couette—Poiseuille flow as the axial flow is increased (from left to right). The images correspond to longitudinal sections of the flow and the axial flow is upwards. The dashed line marks the location of one of the stationary helical vortices which characterize the SHV mode. This flow regime corresponds to the transition from the SHV (steady) to partial PTV (unsteady) regimes as Re increases, as shown in Figure 4.4.2.

The transition from laminar to turbulent flow on a rotating sphere occurs approximately at Re = 1.5 4.0 x 104. Experimental work by Kohama and Kobayashi [39] revealed that at a suitable rotational speed, the laminar, transitional, and turbulent flow conditions can simultaneously exist on the spherical surface. The regime near the pole of rotation is laminar whereas that near the equator is turbulent. Between the laminar and turbulent flow regimes is a transition regime, where spiral vortices stationary relative to the surface have been observed. The direction of these spiral vortices is about 4 14° from the negative direction of the azimuthal angle,. The phenomenon is similar to the flow transition on a rotating disk [19]. 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

FIGURE 1.36 General scheme for the process of CPC and mobile phase flow regime for the descending mode of CPC (insert bottom left) used for the CPC separation of dichlorprop with [mono-ll-octadecylthio-h ls-10,ll-dihydroquinidinyl)]-l,4-phthalazine as chiral selector. Elution profiles for dichlorprop after injection of 366 mg racemate, amolar ratio r = 1 of loaded dichlorprop to total selector present in the rotor, and arotor speed of 1100 rpm. Stationary phase, 10 mM selector in methyl tert-butylether mobile phase, 100 mM sodium phosphate buffer (pH 8) flow rate 3 mLmin temperature, 25°C. (Reproduced from E. Gavioh et ah. Anal. Chem., 76 5837 (2004). With permission.)... 

Two common types of one-dimensional flow regimes examined in interfacial studies Poiseuille and Couette flow [37]. Poiseuille flow is a pressure-driven process commonly used to model flow through pipes. It involves the flow of an incompressible fluid between two infinite stationary plates, where the pressure gradient, Sp/Sx, is constant. At steady state, ignoring gravitational effects, we have... 

As is well known, when the gravity and the drag force acting on the particle are numerically the same but in opposite directions, the relative velocity of the particle with respect to the gas flow will be kept constant such a relative velocity is called terminal velocity , denoted as u, = (u - p) and is numerically equal to the sedimentation velocity of the particle in a stationary gas. Setting dupldt = 0 and using Eqs. (2.33) to (2.35) and the data listed in Table 2.1, the terminal velocities for various flow regimes can be directly obtained from Eq. (2.32) as follows ... 

Concentric-cylinder viscometers are in widespread use. Figure 3d represents a partial section through such an instrument in which liquid is contained and sheared between the stationary inner and rotating outer cylinders. Either may be driven, but the flow regime... 

Problem 8-5. The annular region between two concentric rigid spheres of radii a and A.a (with k > 1) is filled with Newtonian fluid of viscosity /i and density p. The outer sphere is held stationary whereas the inner sphere is made to rotate with angular velocity fl. Assume that inertia is negligible so that the fluid is in the Stokes flow regime. 

Several wind models of analytical nature exist. They differ in their level of physical sophistication and in their way to parametrize the wind characteristics. In all cases, the wind is assumed to be spherically symmetric, which appears to be a reasonable first approximation even in two-dimensional simulations, at least late enough after core bounce. In addition, the wind is generally treated as a stationary flow, meaning no explicit time dependence of any physical quantity at a given radial position. Newtonian and post-Newtonian descriptions of a spherically symmetric stationary neutrino-driven (supersonic) wind or (subsonic) breeze emerging from the surface of a PNS have been developed. The reader is referred to [24] for the presentation of a Newtonian, adiabatic and steady-state model for the wind and breeze regimes, and for a general-relativistic steady-state wind solution. 

As an incompressible fluid of infinite extent approaches and flows past either a spherical solid pellet or a gas bubble, a mobile component undergoes inteiphase mass transfer via convection and diffusion from the sphere to the fluid phase. The overall objective is to calculate the mass transfer coefficient and the Sherwood number at any point along the interface (i.e., the local transfer coefficients), as well as surface-averaged transfer coefficients. The results are applicable in the laminar flow regime (1) when the sphere is stationary and the fluid moves,... 

Effect of Flow Regime on the Dimensionless Mass Transfer Correlation. For creeping flow of an incompressible Newtonian fluid around a stationary solid sphere, the tangential velocity gradient at the interface [i.e., g 9) = sin6>] is independent of (he Reynolds number. This is reasonable because contributions from accumulation and convective momentum transport on the left side of the equation of motion are neglected to obtain creeping flow solutions in the limit where Re 0. Under these conditions. 

Answer For boundary layer mass transfer across solid-liquid interfaces, = I and y =. In the creeping flow regime, z = - This problem is analogous to one where the solid sphere is stationary and a hquid flows past the submerged object at low Reynolds numbers. 

Unlike porous pellets, it is mathematically feasible to account for chemical reaction on the well-defined catalytic surfaces that bound the flow regime in regular polygon duct reactors. A qualitative description of the boundary conditions is based on a steady-state mass balance over a differential surface element. Since convective transport vanishes on the stationary catalytic surface, the following contributions from diffusion and chemical reaction are equated, with units of moI/(areatime) ... 

For flow around submerged objects, a = 1 in the creeping flow regime, and a = 0 for turbulent flow. Since the hydrodynamic drag force exerted by fluid B on solid particle A acts in the opposite direction of va when the fluid is stationary, and the gravitational force acts downward, these two forces are balanced ... 

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