Curvilinear velocity

The type of motion considered here is shown in Fig. 3. We follow one labeled chain in the layer. The chain reptates on the self-similar grid. It is subdivided into subunits (i) with a spatial size Zi equal to the distance to the wall, and a number of monomers gi - gizi) Instead of computing directly the diffusion constant Du, we use as an intermediate the mobility p = D /kT of the chain (where kT is the thermal energy). When the chain reptates, the subunit (i) has a certain curvilinear velocity Ud It is important to realize that Ud is different for different subunits what is conserved is the tube current J , i.e., the number of monomers, crossing one given point on the tube, per second. 

If the number of segments at the internal coil surface at r, is dn and these are convected away, tangentially, at a rate u, given by Eq. (18), an increase in the curvilinear velocity, that supplies segments to the layer surface at r, must occur which is given by... 

P.G. de Gennes (College de France) I understand your estimate for the curvilinear velocity vd but not the later estimate of the... 

Curvilinear velocity Effective force Disengagement time Reduced end-to-end projection Distribution function of blob orientations Critical chain size for mobility saturation Orientation time of a chain Pulse time... 

The energy dissipated during this process is proportional to the square of the curvilinear velocity, and can be written as,... 

Fig. 35. Velocity distribution along different streamlines in abrupt contraction flow (s is the curvilinear distance along the streamline from the entrance, as shown in Fig. 32).

If the enzyme concentration is fixed at a value well below the substrate concentration, and the concentration of substrate is then titrated, one finds that the initial velocity of the reaction varies as illustrated in Figure 2.7. At the lower substrate concentrations, the initial velocity tracks linearly with substrate concentration. At intermediate values of [5], the initial velocity appear to be a curvilinear function of [5], while at higher substrate concentrations, the initial velocity appears to reach a maximum level, as if the active site of all the enzyme molecules are saturated with... 

If the inhibition is found to be rapidly reversible, we must next determine if the approach to equilibrium for the enzyme-inhibitor complex is also rapid. As described in Chapter 4, some inhibitors bind slowly to their target enzymes, on a time scale that is long in comparision to the time scale of the reaction velocity measurement. The effect of such slow binding inhibition is to convert the linear progress curve seen in the absence of inhibitor to a curvilinear function (Figure 5.10). When nonlinear progress curves are observed in the presence of inhibitor, the analysis of... 

The components of the velocity vector (v, v2, V3) align with the curvilinear-coordinate directions ( 1, 2, 3). 

When particles experience a mean curvilinear motion and also have Brownian agitation, they are deposited on obstacles by both mechanisms. For very small particles of radii less than 0.1 /xm, Brownian motion dominates particle collection on surfaces. For larger particles, inertial forces dominate. An example of the difference in collection efficiency for spherical collectors of different size is shown in Fig. 3 for different particle diameters and aerosol flow velocity. 

Equation 6.1 represents a system of three differential equations for the coordinates x, y, and z (or for some curvilinear coordinates qlt q2> q3) expressed as functions of time t. Solution of these equations defines a trajectory of the particle for certain initial conditions of position and velocity. Several examples will be examined. 

The magnitude of the accelerating force that acts on a particle in curvilinear motion depends on the particle inertia. The greater the inertia of the particle, the greater will be the displacement. Inertia depends on particle mass and velocity. Heavy particles will be displaced more from the streamlines in which they are traveling than light ones, and increases in velocity will increase displacement for a particle of given mass. 

The process by which something moves from c ie position to another is referred to as motion that is, a chang-ii position involvii Hme, velocity and acceleration. Motions can be classified as linear or translational (motion aloi a stra ht line), rotational (motion about some axis), or curvilinear (a combination of linear and rotational). A detailed description of all aspects of motion is called kinematics and is a hjndamental part of mechanics. 

In circular motion velocity is always parallel to the direction of motion and perpendicular to the radius of motion. The acceleration required to change the velocity s direction, called centripetal acceleration, is always perpendicular to the velocity and toward the center of motion. To change the velocity s magnitude an acceleration is required in the direction of the velocity. Hence, acceleration is required to change both magnitude and direction of velocity and are in different directions. This is applicable to curvilinear motion in general. 

Illustrative Cases. Three cases are illustrated in Figure 9, marked by the circles labeled A, B, and C. Case A refers to classical experiments by Swift and Friedlander (27) on the coagulation of monodisperse latex particles (diameter = 0.871 pm) in shear flow and in the absence of repulsive chemical interactions. Considering a velocity gradient of 20 s 1, HA is 0.0535, log HA is — 1.27, and dfdj is 1.0 for these experimental conditions. The circle labeled A in Figure 9 marks these conditions and indicates that the hydrodynamic corrections to Smoluchowsla s model predict a reduction of about 40% in the aggregation rate by fluid shear. The experimental measurements by Swift and Friedlander showed a reduction of 64%. This observed reduction from Smoluchowski s rectilinear model was therefore primarily physical or hydro-dynamic and consistent with the curvilinear model. 

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