Viscosity second coefficient

Dynamic viscosity of the liquid carrier Shear spin viscosity Second coefficient of viscosity Bulk spin viscosity... 

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

The numerical correlations given by the direct BEM simulations are similar to the expressions given earlier. In fact, the first coefficient in the power expansion is close to the one predicted by Einstein [15]. The second coefficient in the power expansion is between the value suggested by Guth and Simha [25] and one suggested by Vand [65, 66], In Figure 10.29, the calculated BEM relative viscosity is collapsed for all cases. 

The coordinate transformation also enables a nondimensional version of the equation for momentum conservation to be written from equations (1-2) and (1-5). For this purpose, Prandtl numbers based on the first and a modified second coefficient of viscosity, respectively, may be defined as Pri = /iooCp, /A and Pf2 = + Koo)Cp,ooMoo- To allow for... 

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. 

Now in some cases a direct study of a polymer under 0 conditions is not feasible or, more frequently, not desired. The need to work under 0 conditions when determining unperturbed chain dimensions might be circumvented if one could rely on theories connecting accurately measurable quantities such as intrinsic viscosity, second virial coefficient etc. obtained in good solvents, with the chain expansion factor. 

Essentially exact values for V2 and have been reported by Wajnryb and Dahler [42] for both stick and slip boundary conditions. These values are recorded in Table V.4, along with estimates obtained by several previous investigators. In Figure 5.16, predictions based on the formula (5.297) are compared with the available experimental data. The solid curve is based on the stick boundary condition values V2 = 2.5 (Einstein) and = 5.9147 (Wajnryb and Dahler). To obtain the dashed curve, which agrees much better with (some of) the experimental data and for which virial coefficients with V2 = 5.0781. 

Mathematical description of the process of polymer melting in the extrusion channel is complex when ultrasound is used. The description requires firstly, consideration of the mass flow of the polymer, knowledge of the flow characteristics of the melt, the temperature and pressure of extrusion, sizes of the channel and frequency of ultrasonic oscillations. Secondly, coefficient of swelling of the extrudate, effective viscosity of polymer, pressure of melt, and frequency of oscillations. 

The temperature dependence of the second coefficient in a virial expansion of the viscosity [Eq. (7)] can now be written in terms of the Gibbs free energy AG per mole of contact points. Doing so, one may write the relative viscosity as (again /O can be substituted for O as discussed in Sec. II.B) [60]... 

Second coefficient of viscosity A (Pas) Shear/bulk spin viscosity tj, A (kgms rd ) Surfactant density ps (kg/m ) ... 

According to the second law of thermodynamics the transport coefficients for F = G (that is, the thermal conductivity, shear and bulk viscosities, and coefficients of diffusion) cannot be negative. To show that this is actually the case, consider the inequality... 

This theory covers the complete density range from dilute gas to solidification. However, the hard sphere model is inappropriate for a real gas at low and intermediate densities where specific effects of intermolecular forces are significant. It is therefore necessary to modify the theory this has been done in different ways. In Section 5.2, a method is described for determination of the second viscosity virial coefficient and the translational part of the thermal conductivity coefficient. Although this approach is not rigorous, it does provide a useful estimate for these coefficients - especially for low reduced temperatures. 

Since the existence of internal degrees of fteedom and their interaction with translational modes of motion has a negligible effect on viscosity, the second viscosity virial coefficient of polyatomic fluids can be formally identified with that of monatomic gases (57" = 5 ). 

Fig. 5.3. The reduced second viscosity virial coefficient as a function of reduced temperature. Curves 1 - Rainwater-Friend theory (S = 1.04 and 9 = 1.25) 2 - Rainwater-Friend theory (5 = 1.02 and 6> = 1.15) 3 - MET-I.

For application to real gases, this theory has been modified (Enskog 1922 Hanley et al. 1972 Hanley Cohen 1976 Vogel etal. 1986 Ross etal. 1986). Although this has no rigorous theoretical basis, it does provide an alternative rqiresentation of the second viscosity virial coefficient and the translational part of the second thermal conductivity virial coefficient, which is particularly useful at reduced temperatures below T = 0.5, the lower limit of the coefficients in Table 5.1. On the basis that a real fluid differs from a hard sphere fluid mainly in the temperature dependence of the collision frequency, the pressure P of the hard-sphere fluid is replaced by the thermal pressure T(dP/dT)p of... 

The latter can be predicted from the universal expression (5.19) for the reduced second viscosity virial coefficient by using... 

As a result of both theoretical and experimental studies of the second viscosity virial coefficient, it has been possible to develop a generalized representation of its temperature dependence based on the Lennard-Jones (12-6) potential (Rainwater Friend 1987 Bich Vogel 1991). A particular advantage of this s proach is that it is possible to estimate the coefficient BrjiT) for a gas for which no experimental viscosity data as a function of density exist, given a knowledge of the Lennard-Jones (12-6) potential parameters as derived by an analysis of dilute-gas viscosity data. Such an estimation of Bfi has been performed for ethane by use of the reconunended parameters / b = 251.1 K and cr = 0.4325 nm (Hendl etal. 1994). 

This procedure based on the Rainwater-Friend theory is valid only for reduced temperatures T > 0.7 (T = 175 K for ethane). This lower limit will not be exceeded by the viscosity representation of ethane in the vapor phase, since the range of validity of its zero-density contribution has a lower limit of T = 2(X) K. Nevertheless, experimental data in the liquid phase are available at much lower temperatures extending to T = 100 K. In order to use a single overall viscosity correlation it must be ensured that the initial-density contribution extrapolates satisfactorily to low temperatures. For this purpose, for temperatures below T = 0.7, the second viscosity virial coefficient has been estimated by use of the modified Enskog theory (see Chapter 5), which relates B,f to the second and third pressure virial coefficients. Although this method enables Brj to be evaluated, it is cumbersome for practical applications. Therefore, the calculated B, values using both methods have been fitted to the functional form... 

Fig. 14.20. Second viscosity virial coefficient as a function of temperature.

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

Since viscometer drainage times are typically on the order of a few hundred seconds, intrinsic viscosity experiments provide a rapid method for evaluating the molecular weight of a polymer. A limitation of the method is that the Mark-Houwink coefficients must be established for the particular system under consideration by calibration with samples of known molecular weight. The speed with which intrinsic viscosity determinations can be made offsets the need for prior calibration, especially when a particular polymer is going to be characterized routinely by this method. 

Extensive tables and equations are given in ref. 1 for viscosity, surface tension, thermal conductivity, molar density, vapor pressure, and second virial coefficient as functions of temperature. 

SAN resins show considerable resistance to solvents and are insoluble in carbon tetrachloride, ethyl alcohol, gasoline, and hydrocarbon solvents. They are swelled by solvents such as ben2ene, ether, and toluene. Polar solvents such as acetone, chloroform, dioxane, methyl ethyl ketone, and pyridine will dissolve SAN (14). The interactions of various solvents and SAN copolymers containing up to 52% acrylonitrile have been studied along with their thermodynamic parameters, ie, the second virial coefficient, free-energy parameter, expansion factor, and intrinsic viscosity (15). 

Viscosity is equal to the slope of the flow curve, Tf = dr/dj. The quantity r/y is the viscosity Tj for a Newtonian Hquid and the apparent viscosity Tj for a non-Newtonian Hquid. The kinematic viscosity is the viscosity coefficient divided by the density, ly = tj/p. The fluidity is the reciprocal of the viscosity, (j) = 1/rj. The common units for viscosity, dyne seconds per square centimeter ((dyn-s)/cm ) or grams per centimeter second ((g/(cm-s)), called poise, which is usually expressed as centipoise (cP), have been replaced by the SI units of pascal seconds, ie, Pa-s and mPa-s, where 1 mPa-s = 1 cP. In the same manner the shear stress units of dynes per square centimeter, dyn/cmhave been replaced by Pascals, where 10 dyn/cm = 1 Pa, and newtons per square meter, where 1 N/m = 1 Pa. Shear rate is AH/AX, or length /time/length, so that values are given as per second (s ) in both systems. The SI units for kinematic viscosity are square centimeters per second, cm /s, ie, Stokes (St), and square millimeters per second, mm /s, ie, centistokes (cSt). Information is available for the official Society of Rheology nomenclature and units for a wide range of rheological parameters (11). 

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