Motion equations validity

Note that the hydrodynamic boundary layer depends on the diffusion coefficient. Introducing the proportionality constant K° results in an equation valid for any desired hydrodynamic system based on relative fluid motion as proposed in Ref. 10 ... 

In this model, the velocity and concentration fields are described by the equations valid for the steady laminar motion of a semiinfinite fluid along a plate. Therefore, the velocity components in each of the paths satisfy the... 

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

For most systems, where the velocity of motion of the nuclei is slow relative to the electron velocity, this decoupling of electronic and nuclear motion is valid and is referred to as the adiabatic approximation. Equation (II.3) thus defines an electronic eigenstate (rn,Rv), parametric in the nuclear coordinates, and a corresponding eigenvalue Ek(RN) that is taken to represent the potential-energy curve or surface corresponding to state k. 

For the polar vibrations of the two sublattices against each other, according to Huang the following equation of motion is valid... 

The liquid and the vapor motion is described by the Navier-Stocks set of equations, valid for the non-compressible fluid ... 

The alternative VOF models designed to describe stratified and dispersed flows with internal motions in all the phases are based on the whole domain formulation. The basis for this approach is a set of equations valid for the whole calculation domain, in which the governing mass and momentum balances are expressed as [132, 214, 32, 183, 164, 92, 222, 227] ... 

A full asymptotic analysis by Robertson and Acrivos23 has shown that this solution of the creeping-motion equations is a uniformly valid first approximation for Re 1. 

Under the conditions of validity of the two-electronically-adiabatic-state approximation it is possible to change from the i]/al,ad(r q) (n = i, j) electronically adiabatic representation to a diabatic one 1,ad(r q) (n = i, j) for which the VR Xn(R) terms in the corresponding diabatic nuclear motion equations are significantly smaller than in the adiabatic equation or, for favorable conditions, vanish [24-26]. Such an electronically diabatic representation is usually more convenient for scattering calculations involving two electronically adiabatic PESs, but not for those involving a single adiabatic PES. This matter will be further discussed in Sec. III.B.3 for the case in which a conical intersection between the E ad(q) and Ejad(q) PESs occurs. 

For a rotating sphere viscometer, the tangential velocity on the surface of the sphere is ft sin 0. This reveals the angular dependence of 1)0 at any radial position, because if one moves into the fluid at larger r and constant 9, and a separation of variables solution to the (f)-component of the equation of motion is valid, then the sin 9 dependence shouldn t change. Hence,... 

Let us consider more precisely the conditions under which the above adiabatic separation of electronic and nuclear motions is valid. For this purpose, using (3.1) we can write the general solution of the exact wave equation (2.1) as... 

The governing equation used by Kobatake and Fujita [21] is the simple equation for motion of fluids using a simple model which incidentally does not reproduce the conditions of Teorell s experiments. In this model, it is assumed that mass flow inside the membrane is slow enough and in a steady state at every instant. Furthermore, the density of the solution is assumed to be equal to that of water. The theoretical formulation ignores the electro-osmotic effect which is quite important. In addition, the theory cannot be quantitatively checked on account of various difficulties as pointed by Mikulecky and Caplan [12]. Similar problem arises with respect to the formalism developed by Arnow [22]. Mears and Page developed a theory [8] based on the following force-balance equation valid at any instant ... 

Molecular mechanics, in which we solve Newton s equation of motion, only valid for situations where no bonds are broken or formed, i.e., conformational changes... 

One drawback is that, as a result of the time-dependent potential due to the LHA, the energy is not conserved. Approaches to correct for this approximation, which is valid when the Gaussian wavepacket is narrow with respect to the width of the potential, include that of Coalson and Karplus [149], who use a variational principle to derive the equations of motion. This results in replacing the function values and derivatives at the central point, V, V, and V" in Eq. (41), by values averaged over the wavepacket. 

If the magnitudes of the dissipative force, random noise, or the time step are too large, the modified velocity Verlet algorithm will not correctly integrate the equations of motion and thus give incorrect results. The values that are valid depend on the particle sizes being used. A system of reduced units can be defined in which these limits remain constant. 

In literature, some researchers regarded that the continuum mechanic ceases to be valid to describe the lubrication behavior when clearance decreases down to such a limit. Reasons cited for the inadequacy of continuum methods applied to the lubrication confined between two solid walls in relative motion are that the problem is so complex that any theoretical approach is doomed to failure, and that the film is so thin, being inherently of molecular scale, that modeling the material as a continuum ceases to be valid. Due to the molecular orientation, the lubricant has an underlying microstructure. They turned to molecular dynamic simulation for help, from which macroscopic flow equations are drawn. This is also validated through molecular dynamic simulation by Hu et al. [6,7] and Mark et al. [8]. To date, experimental research had "got a little too far forward on its skis however, theoretical approaches have not had such rosy prospects as the experimental ones have. Theoretical modeling of the lubrication features associated with TFL is then urgently necessary. 

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