Viscous medium equations

Now suppose that the harmonic oscillator represented in Fig. 1 is immersed in a viscous medium. Equation (32) will then be modified to include a damping force which is usually assumed to be proportional to the velocity, -hx. Thus,... 

An additional length, intermediate in size between /0 and Zk, which often arises in formulations of equations for average quantities in turbulent flows is the Taylor length (A), which is representative of the dimension over which strain occurs in a particular viscous medium. The strain can be written as (U /l0). As before, the length that can be constructed between the strain and the viscous forces is... 

For calculating the time-dependent properties of biopolymers, the equations of motion of the molecule in a viscous medium (i.e., water) under the influence of thermal motion must be solved. This can be done numerically by the method of Brownian dynamics (BD) [83]. Allison and co-workers [61,62,84] and later others [85-88] have employed BD calculations to simulate the dynamics of linear and superhelical DNA BD models for the chromatin chain will be discussed below. 

Consider a system consisting of a particle diffusing in a viscous medium and under the influence of external forces. Such a system may be described2 by a Smoluchowski equation of the form... 

In the derivation of Stokes law, the assumption of a perfectly viscous medium means that no inertial forces are considered. This was done to linearize the Navier-Stokes equation. If these inertial effects are included in a first-order approximation, it is possible to extend the applicability of Stokes law up to a Reynolds number of about 5. Then the resisting force can be expressed as... 

The simplest type of flow of a medium that yields itself to an analytical description within the framework of the precise hydrodynamical equations of viscous liquids (Navier-Stocks equations) is the Couette flow. This flow occurs under the impact of tangential stresses generated in a viscous liquid by a solid surface moving in it. The magnitude of the force that has to be applied to this surface to securse its movement in the viscous medium characterizes the tangential stresses and the velocity of its movement — the shear velocity. 

Notice that the equation was evaluated by considering a particulate which moves through a fluid being pushed by the force resulting from impacts of many molecules of the (viscous) medium. In the same time, it experiences hydrodynamic resistance (friction). The dynamical viscosity of gases rather weakly depends on p and... 

It is important to notice the similarity between Eqs. 1.1,1.6, and 1.20. The heat conduction equation, Eq. 1.1, describes the transport of energy the diffusion law, Eq. 1.6, describes the transport of mass and the viscous shear equation, Eq. 1.20, describes the transport of momentum across fluid layers. We note also that the kinematic viscosity v, the thermal diffusivity a, and the diffusion coefficient D all have the same dimensions L2/f. As shown in Table 1.10, a dimensionless number can be formed from the ratio of any two of these quantities, which will give relative speeds at which momentum, energy, and mass diffuse through the medium. 

In a ccnirifugalfield, a particle sedimenting through a viscous medium also reaches a terminal velocity u. The ccnirilugal acceleration is analytical radius. In this case, the Stokes equation has the form ... 

In Section B, we included the impact of viscous medium by adding the viscous force (27.53) to Equation (27.13). In that treatment, the viscous force was considered as competitive with other forces arising from Hamiltonian (27.1). The consequence of that approach was the outcome which showed the impact of viscosity being so strong that the soliton solution (27.74) decays almost instantaneously into its asymptotic form which is localized bell-shaped mode given by expression (27.81). However, there is an alternative approach where viscous force has features of small perturbation. We refer them as big and small viscosities [30]. 

Theoretical considerations and existing experimental data indicate that under certain conditions, the rate of photoisomerization strongly depends on the microviscosity around the isomerized molecule and upon the effect of steric hindrance. In a viscous medium, the apparent rate constant of trans-cis photoisomerization ki o is controlled by the reorganization rate of the process in the medium (Equations 4.2 and 4.3). This method was used for the measurement of fluidity of biological membranes and microviscosity of a specific site of a protein. On this theoretical basis, fluorescence-photochrome immunoassay (FPHIA)] were used. 

If an N-unit chain is immersed in a viscous medium, then the movement of any monomer i can be described by the modified Langevin equation ... 

We hence start with a cubic network formed by beads which have an identical friction constant, f, and which are connected to each other by means of elastic Hookean springs with elasticity constant K, see Fig. 3. The network is embedded into an effective viscous medium and is a regular structure in the sense of connectivity only. Every site of the cubic network is denoted by a three-dimensional index 12 = (a,/3,y). The Langevin equation of motion, Eq. 2, can be rewritten here as ... 

Stokes-Einstein Equation To drag a particle suspended in a viscous medium at a constant velocity v, a constant force of F = v must be applied to the particle (Fig. 3.12). The coefficient is called the friction coefficient. Einstein showed that the diffusion coefficient D of the particle in a quiescent solution at temperature T is related to by... 

Then the total Newtonian equation of motion of the ball has the exciting force, F, the acceleration and deceleration of the vibration, the drag of the viscous medium and the Hooke s Law restoring force of the spring. This gives for the force and the restraints acting in opposite directions ... 

The fluctuations of the velocity and coordinates of an atom cause it to move in a diffusive fashion. This will take place if the length of the path the atom travels during a time t I/7 is much smaller than the size of its region of interaction with the field. Such a diffusive motion of the atom is well described by the Fokker-Planck equation (Minogin 1980). Diffusive redistribution of atomic velocities has been observed in an atomic beam propagating in a counterrunning light wave (Balykin et al. 1981). In the three-dimensional case, the diffusive motion of the atom takes place in the space of both velocities and coordinates. It is similar to the motion of a particle in a viscous medium and has therefore been termed optical molasses (Chu et al. 1985). 

The dynamics of an isolated Kuhn segment chain in its bead-and-spring form is considered in a viscous medium without hydrodynamic backflow or excluded-volume effects. The treatment is based on the Langevin equation generalized for Brownian particles with internal degrees of freedom. A first, crude formalism of this sort was reported by Kargin and Slonimskii [45]. In-... 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

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