Convection oscillatory

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

Y.-S. Lee, C.-H. Chun. Experiments on the oscillatory convection of low Prandtl number hquid in Czochralski configuration for crystal growth with cusp magnetic field. J Cryst Growth 180 411, 1997. 

It can easily be shown that for the upwind scheme all coefficients a appearing in Eq. (37) are positive [81]. Thus, no unphysical oscillatory solutions are foimd and stability problems with iterative equation solvers are usually avoided. The disadvantage of the upwind scheme is its low approximation order. The convective fluxes at the cell faces are only approximated up to corrections of order h, which leaves room for large errors on course grids. 

Solve the convection equation of high order (3rd order) essentially non-oscillatory (ENO) upwind scheme (Sussman et al., 1994) is used to calculate the convective term V V

velocity field P". The time advancement is accomplished using the second-order total variation diminishing (TVD) Runge-Kutta method (Chen and Fan, 2004). 

Before leaving this discussion, it is important to note that other forms of Peclet numbers are also possible and may be more appropriate depending on the type of convective influence studied. For example, in the case of oscillatory flows (as in oscillatory viscometers), it is more useful to define the Peclet number as (Rfa/D), where co is the frequency of oscillation. Regardless of the particular definition, the general significance of the Peclet number remains the same, i.e., it compares the effect of convection relative to diffusion. 

So far we have considered an infinite value of the gas-to-particle heat and mass transfer coefficients. One may encounter, however, an imperfect access of heat and mass by convection to the outer geometrical surface of a catalyst. Stated in other terms, the surface conditions differ from those in the bulk flow because external temperature and concentration gradients are established. In consequence, the multiple steady-state phenomena as well as oscillatory activity depend also on the Sherwood and Nusselt numbers. The magnitudes of the Nusselt and Sherwood numbers for some strongly exothermic reactions are reported in Table III (77). We may infer from this table that the range of Sh/Nu is roughly Sh/Nu (1.0, 104). 

Lin, Tsing Fa. Chang, Tsai Shou, and Chen, Yu Feng, Development of Oscillatory Asymmetric Recirculating Flow in Transient Laminar Opposing Mixed Convection in a Symmetrically Heated Vertical Channel , J. Heat Transfer Vol. 115, pp. 342-352, 1993. 

Lai, F.C. and Kulacki, F. A.. "Oscillatory Mixed Convection in Horizontal Porous Layers Locally Heated from Below . Int. J. Heat and Mass Transfer. Vol. 34. pp. 887-890. 1991. 

The ULTRA (Universal Limiter for Tight Resolution and Accuracy) approach of Leonard and Mokhtari [108, 111] was designed to guarantee monotonicity for high-resolution non-oscillatory multidimensional steady-state highspeed convective modeling. 

Leonard BP, Mokhtari S (1990) Beyond first-order upwinding The ULTRASHARP alternative for non-oscillatory steady-state simulation of convection. International Journal of Numerical Methods in Engineering 30 729-766... 

Stemling and Scriven wrote the interfacial boundary conditions on nonsteady flows with free boundary and they analyzed the conditions for hydrodynamic instability when some surface-active solute transfer occurs across the interface. In particular, they predicted that oscillatory instability demands suitable conditions cmcially dependent on the ratio of viscous and other (heat or mass) transport coefficients at adjacent phases. This was the starting point of numerous theoretical and experimental studies on interfacial hydrodynamics (see Reference 4, and references therein). Instability of the interfacial motion is decided by the value of the Marangoni number, Ma, defined as the ratio of the interfacial convective mass flux and the total mass flux from the bulk phases evaluated at the interface. When diffusion is the limiting step to the solute interfacial transfer, it is given by... 

This equation shows that vorticity is a maximum at the moving wall (which acts a source of vorticity) and decays exponentially on a length scale 0(l/Re) by the action of viscous forces. Thus for Rey 1 the flow is effectively irrotational. In this respect the problem is atypical as we normally expect vorticity to decay on a diffusive length scale of 0( / J Re) (cf. oscillatory Couette flow). The reason for this difference is that in the present probiem we have a balance between convective inertia and viscous forces as opposed to a balance between transient inertia and viscous forces. 

In regard to nonsteady tube flows. Mason et al. have observed both inward and outward radial migration of rigid, neutrally buoyant spheres in oscillatory (S9b, G9b) and pulsatile (Tl) flows in circular tubes at frequencies up to 3 cps, at which frequencies inertial effects are likely to be important. We refer here to inertial effects arising from the local acceleration terms in the Navier-Stokes equations, rather than from the convective acceleration terms. In the oscillatory case the spheres (a/R 0.10) attained equilibrium positions at about P = 0.85. Important Reynolds numbers here are those based upon mean tube velocity for one-half cycle and upon frequency. Nonneutrally buoyant spheres in oscillatory flow migrate permanently to the tube axis, irrespective of whether they are denser or lighter than the fluid (K4a). 

Schwabe, D. Scharmann, A. Microgravity Experiments on the Transition from Laminar to Oscillatory Thermocapillary Convection in Floating Zones COSPAR, Advances in Space Research Pergamon Press, to be published in 1984. 

The studies outlined here are the most important ones that established the fundamental of principles of thermal instability in nematics. A number of theoretical and experimental investigations on these and other geometries have since been reported. A particularly interesting study is that of Lekkerkerker who predicted that a homeotropic nematic heated from below (which, it will be recalled, is stable against stationary convection) should become unstable with respect to oscillatory convection. The phenomenon was demonstrated experimentally by Guyon et... 

With their combination of complex kinetics and thermal, convective and viscosity effects, polymerizing systems would seem to be fertile ground for generating oscillatory behavior. Despite the desire of most operators of industrial plants to avoid nonstationary behavior, this is indeed the case. Oscillations in temperature and extent of conversion have been reported in industrial-scale copolymerization (57). 

Figure 9.6 The mechanism of double-diffusive convection in the diffusive regime. The parcel of hot, salty solution that enters the cold, fresh region above loses heat, becoming more dense than the surrounding region. It sinks back to the hot region and regains heat, leading to an oscillatory motion.

---