Linearity, consequences

As Figure 25 8 shows the glucose units of cellulose are turned with respect to each other The overall shape of the chain however is close to linear Consequently neigh boring chains can pack together m bundles where networks of hydrogen bonds stabilize the structure and impart strength to cellulose fibers... 

Note that nonhnear constraints can be treated in this manner through linearization. Consequently, adjustments to the measurements are required. The result from the reconciliation process is this set of adjusted measurements,... 

Discrepancy of TFL from EHL results from the elasticity of lubricants. The viscosity-to-elasticity ratio is introduced to account for the effects of the elasticity. If the viscosity plays a dominant role, it keeps constant, but if the elasticity prevails, the angle of the director varies linearly consequently the viscosity varies. 

It appears that there is one parameter ft. When ft tends to infinity, bq becomes small relative to all other pertinent magnitudes, so that one approaches the Fokker-Planck limit. However, at the same time Aeqq becomes small compared to lq, so that one simultaneously approaches the limit in which the fluctuations may be treated as linear. Consequently,... 

In a suspended carbon nanotube, in addition to the purely Coulomb energy, we also have the nanomechanical corrections. Generally, these corrections make the relations between V and Vg, which describe the boundaries of Coulomb blockade regions, non-linear. Consequently, the Coulomb diamonds in suspended nanotubes are not diamonds any more, but instead have a curvilinear shape (with the exception of the case Cr = Cr = 0). Their size is also not the same and decreases with Vg - Thus, the mechanical degrees of freedom affect the Coulomb blockade diamonds. However, since these effects originate from the nanomechanical term which is typically a small correction, its influence on Coulomb diamonds is small as well. For the E-nanotube, these effects do not exceed several percents for typical gate voltages. 

This analysis has limitations. The thermal conductivity detector has a low sensitivity for H2 yet high concentrations cannot be allowed because response is not linear. Consequently, the flow of helium through the electrolysis cell must be adjusted to Keep the H2 concentration in the sensitive range. 

It should be mentioned, too, that during a phase transition of a pure substance (e.g., melting), the temperature of the sample remains constant, but the furnace temperature still increases linearly. Consequently, the heat flow rate to the sample increases linearly, too, until all substance is transformed then it decreases exponentially to the level it had before the transition took place. 

Figure 14.10 shows capillary viscometry data at 150°C for the lUPAC A LDPE considered earlier in Figure 14.7. It is apparent that these data blend nicely with the data generated using a cone-and-plate viscometer. At shear rates greater than about 1 sec , the viscosity-versus-shear rate behavior is linear. Consequently, one can say that the shear stress x (where subscripts have been dropped for convenience) is given by...

---