Mechanical Model Viscous Dissipation

Returning to the problem at hand, the velocity varies with the height z, from 0 at z = 0 to 1.5V at z = Odx. There exists, therefore, a velocity gradient  

We next calculate the entropy source 5 or the energy TS dissipated by viscous phenomena (e epressed per unit length of the line in the t/-direction). Using rf to designate the viscosity, we start with the general expression 

The exact prefactor has been restored in the last equation. The integral in equation (6.4) diverges at both limits. We can take care of this problem by truncating the integral on the high side at x = L (the size of the drop) and 

The numerical value of the dimensionless coefficient I ranges from 15 to 20. Readers should refer to the published literature for a more extensive discussion of the coefficient 

We are now in a position to derive the dependence of F(V) by equating two forms of the dissipation  

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It has been found from MD simulations that friction of SAMs on diamond decreases with the increasing chain length of hydrocarbon molecules, but it remains relatively constant when the number of carbon atoms in the molecule chain exceeds a certain threshold [44], which confirmed the experimental observations. In simulations of sliding friction of L-B films, Glosli and McClelland [45] identified two different mechanisms of energy dissipation, namely, the viscous mechanism, similar to that in viscous liquid under shear, and the plucking mechanism related to the system instability that transfers the mechanical energy into heat, similar to that proposed in the Tomlinson model (see Chapter 9). On the basis of a series work of simulations performed in the similar... 

In case that the decay of impact kinetic energy due to viscous dissipation is the predominant mechanism in droplet flattening, Madejski s full model reduces to ... 

The classic extrusion model gives insight into the screw extrusion mechanism and first-order estimates. For more accurate design equations, it is necessary to eliminate a long series of simplifying assumptions. These, in the order of significance are (a) the shear rate-dependent non-Newtonian viscosity (b) nonisothermal effects from both conduction and viscous dissipation and (c) geometrical factors such as curvature effects. Each of these... 

The enhanced plasticity and toughness of modem adhesives have become an important property in applications where dissipation of energy in the case of impact, the ability to compensate thermal movement, or the reduction of vibration lead to added value and improved service performance. Viscoelastic behavior can theoretically be predicted and analyzed by means of mechanical models including combinations of elastic and viscous elements to simulate the time-dependent viscoelastic stress-strain response to mechanical loads. 

Attempts have often been made to represent the behaviour of solid colloid systems on deformation by mechanical models, which consist of a combination of elastic and viscous elements. The elastic elements arc represented by springs and the viscous by dashpots, the motion of which is retarded by a viscous liquid. Since in actual systems one is also always dealing with a combination of elastic elements (by which potential energy can be stored) and of internal frictional resistances (in which energy is dissipated) the analogy is more than a formal one. One can indeed often make a useful picture of the deformation mechanism by the construction of such models. 

The correspondence between the calculated results based on the model of heat dissipation due to viscous flow and the experimental data in the decrease of the induction period at high shear rates proves that the observed effect is adequately explained by this mechanism. The effects of shearing itself on the kinetics of curing are either absent or of secondary importance. If the experimentally observed decrease in the induction period is more pronounced than predicted by the dissipative model, then it is reasonable to consider additional heat sources, for example, the exothermal effects of a reaction. Heat flux from the surroundings can also influence the kinetics of... 

A convenient physical interpretation may be illustrated by simulating mechanical or electronic models. In the mechanical simulation, a spring represents an elastic or Hookean solid (modulus), while a piston moving in an infinite cylinder filled with a viscous liquid (a dash-pot) represents the Newtonian liquid (viscosity). Thus, the deformation of the solid (spring) is completely recoverable, while that of the liquid (dash-pot) is irrecoverable and is converted to heat. See Figures 4-7, 4-8, 4-9. In conclusion, the elastic energy is conserved and recovered while the viscous energy is dissipated. 

Although the choice of the rheological model varies quite significantly with tp and Pe, some broad albeit approximate trends are also evident in Fig. 27.10. Most importantly, it appears that the poroelastic response is bracketed between Pe = 0.4 and 2. At higher values of Pe one of the viscoplastic models (including purely viscous) applies depending on the value of p. At lower values of Pe bed friction and percolation are the two important dissipation mechanisms. 

Constitutive laws for viscoelastic materials are however more complex than the Hooke s law and consequently the number of parameters to identify increases. Frameworks could also be used to ensure that all parameters of the proposed laws are physically admissible. To do so, it is possible to use the thermodynamic of irreversible processes as the framework. Based on the concepts of continuum mechanics and irreversible thermodynamics, the Clausius-Duhem inequality is obtained for given problems where dissipation mechanisms are of importance, e.g., viscous deformation. Fundamental equations leading to a generic form of the Clausius-Duhem inequality have been well covered by many authors (Bazarov, 1964 Coussy, 2010 Lemaitre and Chaboche, 1990 Mase and Mase, 1999) and thus will be only summarized later in section 3. Based on the generic form of the Qausius-Duhem inequality, models or constitutive laws are further developed considering various assumptions closely related to materials of interest. 

The elastic aftereffect is encountered in solid-like systems with an elastic behavior. The elastic behavior is reversible when the stress is removed, the strain drops gradually to zero, that is, the initial shape of the body is restored, using the energy stored by the elastic element. However, in contrast to true elastic behavior, the elastic aftereffect is thermodynamically irreversible the dissipation of energy takes place in the viscous element. The damping of mechanical oscillations in rubber, caused by harmonic stresses, is the example of a process conforming to the Kelvin model. 

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