Hydrodynamics viscosity equation

An alternative point of view assumes that each repeat unit of the polymer chain offers hydrodynamic resistance to the flow such that f-the friction factor per repeat unit-is applicable to each of the n units. This situation is called the free-draining coil. The free-draining coil is the model upon which the Debye viscosity equation is based in Chap. 2. Accordingly, we use Eq. (2.53) to give the contribution of a single polymer chain to the rate of energy dissipation ... 

Rearranging Equation 12-60 tells you that the product of the molecular weight and the intrinsic viscosity of a polymer is proportional to the hydrodynamic volume (Equation 12-61) ... 

Waldeck, on the other hand, conclude from their studies on 4,4 -dimethyl-stilbene in n-alkane solvents that the 1-dimensional hydrodynamic Kramers equation is not appropriate. In their view a 2-dimensional co-ordinate system with a reaction trajectory related to solvent viscosity is more appropriate. Further contributions to this subject are certain. 

Equation (6-94) has been found to be valid for a number of filled systems up to a value of f of about 0.3, whereas (6-95) and (6-96c) can be used at somewhat higher concentrations. These equations were first used to describe the viscosity of liquids with suspended solid particles. In fact equation (6-94) was derived using basic hydrodynamic principles. Equations of this type have been "borrowed" to be used for the elasticity of filled elastomers, based on the analogy between steady viscous flow and elastic deformation as described in equations (3-4) and (2-14), respectively. Certainly an additional justification... 

Readers are referred to the original source (41) for details of the method. In essence, the CV is used alone to determine a viscosity distribution of the polymer in terms of corresponding molecular weight times intrinsic viscosity w,[ri], andM[ ], values. The latter product is related directly to hydrodynamic volume [Equation (2)] and is obtain from a universal calibration with standard polymers, using the... 

The fundamental viscosity equation of Einstein is fully confirmed by this investigation, which affords considerable support to all the formulas derived in this field by the aid of hydrodynamic theory. 

In these equations, v is the hydrodynamic viscosity of the solution, w the angular rotation rate. The Prandtl number is usually of the order of 1000 - that is, the diffusion layer at a rde is only about 1/5 the thickness of the hydrodynamic layer. An excellent work on the significance of Pr in electrochemistry is that of Vielstich (1953). 

In this case the relaxation forces are the only relevant contributions and the hydro-dynamic interactions do not play a significant role in the range of concentration of 0-2 M except for low viscosity solvents [30]. Neglecting hydrodynamic interactions, equation (5.13) gives, to the first order approximation ... 

From various studies" " it is becoming clear that in spite of a heat flux, the overriding parameter is the temperature at the interface between the metal electrode and the solution, which has an effect on diffusion coefficients and viscosity. If the variations of these parameters with temperature are known, then / l (and ) can be calculated from the hydrodynamic equations. 

While mathematically attractive, this force law is of limited interest physically it represents only the interaction between permanent quadrupoles, and even this with neglect of angles of orientation. However, although the details of the dependence of viscosity upon temperature are affected by the force law used, the general form of the hydrodynamic equation in the Navier-Stokes approximation is not affected. 

Yet as long ago as 1966 the problem of calibration in GPC was solved. In that year, Benoit and his co-workers recognised that GPC separates on the basis of the hydrodynamic volume of the polymer molecules in solution. The intrinsic viscosity [rj] is related to the hydrodynamic volume, V, by the equation ... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

If the preceding analysis of hydrodynamic effects of the polymer molecule is valid, K should be a constant independent both of the polymer molecular weight and of the solvent. It may, however, vary somewhat with the temperature inasmuch as the unperturbed molecular extension rl/M may change with temperature, for it will be recalled that rl is modified by hindrances to free rotation the effects of which will, in general, be temperature-dependent. Equations (26), (27), and (10) will be shown to suffice for the general treatment of intrinsic viscosities. 

Equations (18) and (19) show that the diffusion coefficient is inversely proportional to both solvent viscosity and the molar volume of the hydrodynamic... 

Electrostatic repulsion of the anionic carboxylate groups elongates the polymer chain of partially hydrolyzed polyacrylamides increasing the hydrodynamic volume and solution viscosity. The extensional viscosity is responsible for increased resistance to flow at rapid flow rates in high permeability zones (313). The screen factor is primarily a measure of the extensional (elonga-tional) viscosity (314). The solution properties of polyacrylamides have been studied as a function of NaCl concentra-tion and the parameters of the Mark-Houwink-Sakaruda equation calculated... 

The specific viscosity )jsp of a dilute solution of spheres is directly related to their hydrodynamic volume VV Nl denotes Avogadro s number. Typically the intrinsic viscosity [tj] follows a scaling law, the so-called Mark-Houwink-Sakurada equation ... 

There are numerous equations in the literature describing the concentration dependence of the viscosity of dispersions. Some are from curve fitting whilst others are based on a model of the flow. A common theme is to start with a dilute dispersion, for which we may define the viscosity from the hydrodynamic analysis, and then to consider what occurs when more particles are added to replace some of the continuous phase. The best analysis of this situation is due to Dougherty and Krieger18 and the analysis presented here, due to Ball and Richmond,19 is particularly transparent and emphasises the problem of excluded volume. The starting point is the differentiation of Equation (3.42) to give the initial rate of change of viscosity with concentration ... 

Substitution of Equation (3.62) into Equation (3.60) gives the relative zero shear viscosity. When the shear rate makes a significant contribution to the interparticle interactions, the mean minimum separation can be estimated from balancing the radial hydrodynamic force, Fhr, with the electrostatic repulsive force, Fe. The maximum radial forces occur along the principle axes of shear, i.e. at an orientation of the line joining the particle centres to the streamlines of 6 = 45°. This is the orientation shown in Figure 3.19. The hydrodynamic force is calculated from the Stokes drag, 6nr 0au, where u is the particle velocity, which is simply... 

The next problem is that the variation of rG with shear rate is only valid as written in Equation (3.65) at low volume fractions because the solvent viscosity is used to calculate the value of rQ over the shear rate range. If an effective medium treatment is used to make a simple estimate of the effect of many-body hydrodynamic interactions we have ... 

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