Viscous disk models

Evolution of the Solar Nebula. 04.4.2.1 Viscous accretion disk models... 

Figure 4 Midplane temperature as a function of heliocentric radius for a solar nebula with varying mass (inside 10 AU) undergoing mass accretion at a rate of a solar mass in —0.1-1 Myr, compared to various cosmochemical constraints, and the results of a viscous accretion disk model (dashed line) with a mass of 0.24 solar masses (source Boss, 1998).

Given that convective instabihty is no longer considered to be a possible driver of disk evolution, the Ruden and Pollack (1991) models are interesting, but not likely to be applicable to the solar nebula. Unfortunately, httle effort has gone into generating detailed viscous accretion models in the interim the theoretical focus seems... 

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

While the viscous model for the evolution of protoplanetary disks has had some success in matching some of the general properties of protoplanetary disks, such as the observed mass accretion rates and effective temperatures, the exact source of the viscosity remains the subject of ongoing studies. Currently, the most popular candidates for driving the mass transport in protoplanetary disks are the magneto-rotational instability (MRI) and gravitational instability. A third candidate, shear instability, has also been proposed based on laboratory experiments of rotating fluids (Richard Zahn 1999), but questions remain as to whether these results can be extended to the scale of protoplanetary disks. 

A comparison of disk-dispersal timescales, such as that presented in Fig. 1 of Hollenbach et al. (2000), suggests that viscous spreading and photoevaporafion are the major dispersal mechanisms. Models combining these two mechanisms were first developed by Clarke etal. (2001) and later refined by Alexander etal. (2006a,b). The evolution of an accreting and photoevaporating disk can be summarized as follows. In the first 106 7yr viscous evolution proceeds relatively unperturbed by photoevaporafion. Once the viscous accretion inflow rates fall below the photoevaporation rates a gap opens up close to rg and the inner disk rapidly ( 105 yr) drains onto the central star. At this point direct ionization of the disk inner edge (the flux is not anymore attenuated by the inner-disk atmosphere) disperses the... 

The finite hfetime of circumstellar disks, and the fact that young stars are observed to be accreting material, has given rise to a model which views the protoplanetary nebula as a viscous accretion disk in which material was transported radially inwards, ultimately falling onto the Sun (Lynden-Bell and Pringle, 1974). As a consequence of this accretion, the mass of the... 

It is difficult to account for hysteresis in a second-order model because of its nonlinearity with viscous damping (disk movement in the relieving fluid), its lack of repeatability, and its dependence on built-up backpressure (and hence the particular application). However, where additional viscous damping is deliberately added, a linearized analysis will yield useful results. ... 

Problem 4-8. Pressure-Driven Radial Flow Between Parallel Disks. The flow of a viscous fluid radially outward between two circular disks is a useful model problem for certain types of polymer mold-filling operations, as well as lubrication systems. We consider such a system, as sketched here ... 

Fig. 8 All MTM ( ) and pin-on-disk ( ) data at different concentrations of aqueous glycerol mixtures, plotted on a lubrication-tegime map, obtained from the Esfahanian-Hamrock-Dowson equations [28] for a circular contact (ellipticity parameter k = 1). The four different regimes in the dimensionless viscosity (gv) versus elastic (gD parameter plot are iso-viscous rigid (IR), iso-viscous elastic (IE), piezo-viscous rigid (VR), and piezo-viscous elastic (VE). All the values reported in this study lie in the iso-viscous elastic regime. It should be noted that while the equations in [28] apply to rolling contact, the model can also be used for sliding geometries at the low speeds used in our pin-on-disk experiments [35]...

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