Friction functional forms

The position-dependent part of the friction is manifest in the spatial dependence of the coupling function g (s). The usual quantum Kramers problem is recovered when g(s) = s. An implicit assumption in Eq. (36) is that the functional form of the coupling g(s) is the same for all modes k. 

Let the factor multiplying tc1/2/2 be called p. Thus we see that if we assume a functional form for the memory function, then it is possible to determine the parameters in the functional form by using the moment theorems of Eq. (162) and to determine, thereby, the transport coefficients, such as the friction coefficient. Moreover, the time correlation function, i /(t), can also be determined. 

Thus if a functional form is chosen for K ooi), K"n((o) and Ku(t) can be determined from the Kramers-Kronig relations. Moreover, the parameters in the functional form, Af j co), can be related to the moments p2 , in addition to the friction constant H(0), so that these parameters can thereby be determined. 

The first result agrees with what solution chemists expect for the effect of the "microscopic viscosity " The second result tells us that the sensitivity of the friction coefficients on a local viscosity change largely depends on the mode of solvent motions. The shear mode (the viscosity B coefficient) is the most sensitive of the three It is to be noted that these results do not depend on the particular choice of the functional form of the position-dependent viscosity as expected. 

The average friction coefficient and Nusselt number are expressed in functional form as... 

The frictional function which was described as a function of the ratio of the distance associated with the steric repulsion at the interface to the pore radius, however, is still an approximation at most, though it is convenient to use, due to its simplified form. A more appropriate functional form including both steric repulsion and interfacial affinity effects on the restricted motion of the solute molecule in the membrane pore is yet to be developed. A further research effort in this direction is called for. 

Equation (32) indicates how one may achieve the continuum limit of the Hamiltonian form. One notes that if the friction function y(t) appearing in the STGLE is a periodic function with period r then Eq. (32) is just the cosine Fourier expansion of the friction function. The frequencies to, are integer multiples of the fundamental frequency 2tt/t and the coefficients c, are just the appropriate Fourier expansion coefficients. 

The mass-flow and funnel-flow limits in silos are well known and have been used extensively in proper design. The limits for conical hoppers and plane hoppers depend on the hopper half-angle 9, the effective angle of internal friction 5 and the wall friction angle ( ). Once the wall friction angle and effective angle of internal friction have been determined by experimental means, the hopper half angle 0 may be determined. In function form it can be expressed as... 

Intrinsic within this formulation is the assumption that certain geometric and mechanical relationships are known a priori for coronary arteries. In particular, the solution of the problem can be initiated only once the functional forms of the wave speed dependence, c = c(p,x), and the cross-sectional area relationship, S = S(p,x), are specified. Furthermore, the out-flow function, xp, and the friction expression,/, must be expressed as explicit functions of (p,V z,t), together with appropriate initial and boundary conditions. These are all discussed in detail by Rumberger and Nerem (1977). 

Once the functional form for L(t) has been established one can use the fluctuation-dissipation theorem (7.12) to obtain a reasonable model for F(r). The GLE method is capable of discribing the friction fealt by the incoming molecule. Also the thermal respons from the bath to the primary atoms is included. The atoms in the primary zone are given velocities and positions obtained by sampling from a Boltzmann distribution at the surface temperature Tj. 

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