Angle critical tilt

Fig. 6 Plotting the critical values as a function of the bending modulus K shows a convergence of the curves, which is nevertheless not as pronounced as in the case of Fig. 5. The influence of X on the critical tilt angle is significant (A, = 2 in the upper curves and X 1.1 in the lower ones). Again the solid lines show results including the velocity field, the dashed lines correspond to the minimal set of variables, and the dotted lines depict the outcome of the first approach. Note that the wave vectors of the minimal set and of the calculations including the velocity field are indistinguishable within the resolution of the plot...

The angle of repose is a measure for the critical tilt angle . As long as the tilt angle p is smaller than the angle of repose y the material remains stationary on the pan bottom. Only if (3>y does the typical pattern of charge motion develop which results in controlled pellet growth. 

An important subject that one needs to address here is the role of moisture. Typically, in the case of a complete flooding of powder with a wetting liquid, the internal friction and critical tilt angle are significantly lower than in a dry system. However, at a certain average content of the liquid phase, the cohesion between particles and the resistance to shear significantly increase due to the appearance of capillary attractive force (see Chapter 1). 

With disk diameters above 5.25 in., all parameters, eg, water absorption and thermal expansion, become more critical which aggravates the expansion or warp of disks. If in the future disk rotation speeds have to be increased significantly to boost data transfer rates, higher demands will be placed on warp (tilt angle) and modulus to avoid creeping (ie, irreversible elongation in radial direction). A survey of the requirement profile for the substrate material of optical disks is given in Table 5 (182,186,187,189). 

Figure 2.9 Phase diagram for C02, showing solid-gas (S + G, sublimation ), solid-liquid (S + L, fusion ), and liquid-gas (L + G, vaporization ) coexistence lines as PT boundaries of stable solid, liquid, or gaseous phases. The triple point (triangle), critical point (x), and selected 280K isotherm of Fig. 2.8 (circle) are marked for identification. Note that the fusion curve tilts slightly forward (with slope 75 atm K-1) and that the sublimation and vaporization curves meet with slightly discontinuous slopes (angle < 180°) at the triple point. The dotted and dashed half-circle shows two possible paths between a liquid (cross-hair square) and a gas (cross-hair circle) state, one discontinuous (dashed) crossing the coexistence line, the other continuous (dotted) encircling the critical point (see text).

After passing a plateau at a critical film pressure nc the liquid-condensed phase is reached via a phase transition of first order. Here, the amphiphiles exhibit a tilted phase with a decreasing tilt angle (measured against the normal to the subphase). The film is relatively stiff but there is still some water present between the headgroups. 

Let us have a closer look at the differences between the minimal and the extended set of equations and follow these differences along some paths in the parameter space. As mentioned in Sect. 2.3, we can omit some of the physical parameters by using dimensionless parameters. In Figs. 5-9 we show the dependence of the critical values of the tilt angle and wave vector on the dimensionless parameters (as defined... 

Capillary forces in mixed fluid phase conditions are inversely proportional to the curvature of the interface. Therefore, menisci introduce elasticity to the mixed fluid, and mixtures of two Newtonian fluids exhibit global Maxwellian response. For more details see Alvarellos [1], his behavior is experimentally demonstrated with a capillary tube partially filled with a water droplet. The tube is tilted at an angle (3 smaller than the critical angle that causes unstable displacement. Then, a harmonic excitation is applied to the tube in the axial direction. For each frequency, the amplitude of the vibration is increased until the water droplet becomes unstable and flows in the capillary. Data in Figure 3 show a minimum required tube velocity between 40 and 50 Hz. This behavior indicates resonance of the visco-elastic system. The ratio of the relaxation time and characteristic time for pure viscous effect is larger than 11.64. 

It should be taken into account that even while the spin-flop transition possesses the character of a 1-st order phase transition at a precise alignment of the magnetic field along the z-axis, the character of the 1-st order transition is lost when h tilts away from the z-axis by an angle larger than a certain critical value ipc. The critical angle value is deter-... 

At small angles of the field H tilt away from the fourfold crystal axis z (ip C 1), though larger than the critical value ipc 30, a rapid increase of Mz is observed at the spin-flop transition, while Mj passes through a maximum (Fig. 7). It means, that the increase of the magnetization Mz is concurrent with a turn of M to the angle A0 = 7t/2. The shape... 

Lateral overflow, as in water reaching a filled tub. The water that fills the tub blocks the downflow direction of the added water (two-dimensional degrees of hydraulic freedom), and the latter overflows, that is, it flows laterally toward the bases of drainage (Fig. 2.11b). The observation that the water flows laterally indicates its surface (water table) is slightly tilted by the critical angle of flow. The latter is defined by the water viscosity. The water present in the deeper part of the tub does not flow—it is stagnant, bounded in a dead volume situated under... 

For enclosures with smaller aspect ratios (i//L < 12), the next correlation can be used provided that the tilt angle is less than the critical value listed in Table 9-2 [Calton (1978)]... 

For tilt angles greater than the critical value (d < 6 < 90°), the Nussclt number can be obtained by multiplying the Nusselt number for a vertical enclosure by (sin 6) [Ayyaswamy and Catton (1973)],... 

The arm A, which tilts the prism assembly with respect to the observation tube T, is adjusted until the dividing line is brought to the intersection of the cross hairs in the field, thereby giving the critical angle. The value of the refractive index may then be read directly from the graduated scale S, since this is normally calibrated... 

---