Modeling viscous energy dissipation

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

There are a number of additional physical phenomena, such as wall slip, electric effects and viscous energy dissipation, which may need to be taken into account. Generally applicable models are not available for some of these effects, particularly wall slip. 

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... 

Figure 4.8. Sketch of wavenumber bands in the spectral relaxation (SR) model. The scalar-dissipation wavenumber kd lies one decade below the Batchelor-scale wavenumber kb. All scalar dissipation is assumed to occur in wavenumber band [/cd, oo). Wavenumber band [0, k ) denotes the energy-containing scales. The inertial-convective sub-range falls in wavenumber bands [k, k3 ), while wavenumber bands [/c3, /cD) contain the viscous-convective sub-range.

The second model introduced by Hunter and cowor)ters (20,21) is the elastic floe model. In this case, the structural units (which persist at high shear rates) are assumed to be small floes of particles (called floccules) which are characterised by the ability of the particle structure to trap some of the dispersion medium. In this energy dissipation is considered to arise from two processes, namely the viscous flow of the suspension medium around the floes (which are the basic flow units) and the energy involved in stretching the floes to brealc the floe doublets apart so that the amount of structure in the system is preserved inspite of the floc-floc collision. This model gives the following expression for the yield value. 

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. 

The boundary conditions treated in seismic condition are constraints. The left, right boundary is fixed without any movement in the horizontal direction. The bottom boundary is fixed without any movement in the vertical direction. But, the slope surface and internal unit allows a large deformation. For a seismic event (Dynamic Analysis), the model boundary will reflect the waves which is not conducive to the seismic wave energy dissipation. The bottom of the slope is also set to a viscous boundary. 

The simplest linear viscous model is Newton s model. This is shown by a piston-dashpot element [10]. The dashpot is an energy dissipation element, and it represents a viscous damping force. It relates the translational and rotational velocity of a fluid (oil) between two points, and an applied load, by using a damping constant. 

In order to resolve the above-mentioned problems, perforated wall structures have been introduced especially in small craft harbors. The simplest perforated wall structure may be a curtain-wall breakwater (sometimes called wave screen or skirt breakwater), which consists of a vertical wall extending from the water surface to some distance above the seabed. Recently, Isaacson et al." proposed a slotted curtain-wall breakwater. Another simple perforated wall structure may be an array of vertical piles, which is called a pile breakwater in this chapter. The closely spaced piles induce energy dissipation due to viscous eddies formed by the flow through the gaps. To examine the wave scattering by vertical piles, hydraulic model tests have been used. Efforts toward developing analytical models to calculate the reflection and transmission coefficients have also been made. Recently, Suh et introduced a curtain-wall-pile breakwater, the upper part of which is a ciu tain waU and the lower part consisting of an array of vertical piles. They developed a mathematical model that predicts various hydrodjmamic characteristics of a cmtain-wall-pile breakwater. More recently, Suh and Ji extended the model to a multiple-row breakwater. 

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