Slow viscous liquid flow

As noted previously, mixing in highly viscous liquids is slow both at the molecular scale, on account of the low values of diffusivity, as well as at the macroscopic scale, due to poor bulk flow. Whereas in low viscosity liquids momentum can be transferred from a rotating impeller through a relatively large body of fluid, in highly viscous liquids only... 

In the molten state polymers are viscoelastic that is they exhibit properties that are a combination of viscous and elastic components. The viscoelastic properties of molten polymers are non-Newtonian, i.e., their measured properties change as a function of the rate at which they are probed. (We discussed the non-Newtonian behavior of molten polymers in Chapter 6.) Thus, if we wait long enough, a lump of molten polyethylene will spread out under its own weight, i.e., it behaves as a viscous liquid under conditions of slow flow. However, if we take the same lump of molten polymer and throw it against a solid surface it will bounce, i.e., it behaves as an elastic solid under conditions of high speed deformation. As a molten polymer cools, the thermal agitation of its molecules decreases, which reduces its free volume. The net result is an increase in its viscosity, while the elastic component of its behavior becomes more prominent. At some temperature it ceases to behave primarily as a viscous liquid and takes on the properties of a rubbery amorphous solid. There is no well defined demarcation between a polymer in its molten and rubbery amorphous states. 

At this stage, one has to take into account the volume compressibility of the material, since upon feed-up the hold-on time of material under pressure is determined by compressibility and slow viscous flow. If the pressure of injection PQ is sufficiently high, then at this stage a liquid may be considered to be Newtonian with viscosity q ,. Keeping this in mind, we may state that the calculation given below will be applicable to various plastisols (of types I and II) with the only difference that for plastisol I q = const, while for plastisol II q = q. For the sake of simplicity, the analysis will be performed for the case of a flat mould filled through a slit runner (Fig. 10 a). 

The intense heat dissipated by viscous flow near the walls of a tubular reactor leads to an increase in local temperature and acceleration of the chemical reaction, which also promotes an increase in temperature the local situation then propagates to the axis of the tubular reactor. This effect, which was discovered theoretically, may occur in practice in the flow of a highly viscous liquid with relatively weak dependence of viscosity on degree of conversion. However, it is questionable whether this approach could be applied to the flow of ethylene in a tubular reactor as was proposed in the original publication.199 In turbulent flow of a monomer, the near-wall zone is not physically distinct in a hydrodynamic sense, while for a laminar flow the growth of viscosity leads to a directly opposite tendency - a slowing-down of the flow near the walls. In addition, the nature of the viscosity-versus-conversion dependence rj(P) also influences the results of theoretical calculations. For example, although this factor was included in the calculations in Ref.,200 it did not affect the flow patterns because of the rather weak q(P) dependence for the system that was analyzed. 

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

M. L. Booy, A Noniterative Numerical Solutions of Poisson s and Laplace s Equations with Applications to Slow Viscous Flow , Trans. ASME Series D, 88, 725-733 (1966) also, Isothermal Flow of Viscous Liquids in Corotaing Twin Screw Devices, Polym. Eng. Sci., 20, 1220 (1980). 

Preparation of Antimony Trifluorodichloride (SbF3Cl2). This is made in the steel reaction vessel, described on p. 59. A known quantity of antimony fluoride is placed in the vessel the vessel is evacuated, the needle valve is closed, and the whole is weighed. Connection is established to a chlorine cylinder, and the needle valve is opened to permit qhlorine to fill the vessel. Part of it is absorbed rapidly by the salt, with evolution of heat. Soon the reaction slows down as indicated by the rate of pressure fall when the needle valve is dosed. Weighing indicates the amount of chlorine present in the vessel. When the absorption practically ceases, the valve is closed, and the connection with the chlorine tank is removed. The reaction vessel is alternately heated gently, then allowed to cool in order to permit SbFsCl2, which is a viscous liquid, to flow and expose fresh surfaces of crystalline antimony trifluoride. The operation is ended after the absorption of the desired quantity of chlorine. 

Abstract Macromolecular coils are deformed in flow, while optically anisotropic parts (and segments) of the macromolecules are oriented by flow, so that polymers and their solutions become optically anisotropic. This is true for a macromolecule whether it is in a viscous liquid or is surrounded by other chains. The optical anisotropy of a system appears to be directly connected with the mean orientation of segments and, thus, it provides the most direct observation of the relaxation of the segments, both in dilute and in concentrated solutions of polymers. The results of the theory for dilute solutions provide an instrument for the investigation of the structure and properties of a macromolecule. In application to very concentrated solutions, the optical anisotropy provides the important means for the investigation of slow relaxation processes. The evidence can be decisive for understanding the mechanism of the relaxation. 

Rapid fluid flow cannot be achieved with active metal brazes because of the need to form solid wettable reaction product layers for their liquid fronts to advance. Equations (10.1) to (10.2) relating liquid flow rates to the opposed effects of surface energy imbalances and of viscous drag are not relevant. Actual penetration rates are so slow, usually of the order of 1 pm.s, that the usual practice is to place the active metal braze alloy within the joints rather than expecting it to fill them, and, as explained already, gap width is not the dominant consideration when designing ceramic-metal joints. 

Small Reynolds Number Flow, Re < 1. The slow viscous motion without interfacial mass transfer is described by the Hadamard (66)-Rybcynski (67) solution. For infinite liquid viscosity the result specializes to that of the Stokes flow over a rigid sphere. An approximate transient analysis to establish the internal motion has been performed (68), Some simplified heat and mass transfer analyses (69, 70) using the Hadamard-Rybcynski solution to describe the flow field also exist. These results are usually obtained through numerical integration since analytical solutions are usually difficult to obtain. 

Langlois WE (1964) Slow Viscous Flow. Macmillan, New York Laux H (1998) Modeling of dilute and dense dispersed fluid-particle flow. Dr Ing Thesis, Norwegian University of Science and Technology, Trondheim, Norway Lawler MT, Lu P-C (1971) The role of lift in radial migration of particles in a pipe flow. In Zandi 1 (ed) Advances in Solid-Liquid Flow in Pipes and its Apphcations. Pergamon Press, Oxford, Chap 3, pp. 39-57 Lee SL (1987) Particle drag in a dilute turbulent two-phase suspension flow. Int J Multiphase Flow 13(2) 247-256... 

A simplifying approximation often made in fluid mechanics, where the terms arising due to the inertia of fluid elements is neglected. This is justified if the Reynolds number is small, a situation that arises, for example, in the slow flow of viscous liquids such as when pouring honey over toast. 

Thermoset polymers are those whose precursors are heated to an appropriate temperature for a short time, so that they will flow as a viscous liquid a slow, chemical cross-linking reaction then causes the liquid to solidify to form an infusible mass. The precursor materials may be of low molecular weight some mixed precursors will flow and cross-link at room temperature. 

Das, S.P., 1966. Slow steady flow of a viscous liquid in an annulus with uniform arbitrary injection and... 

Bouncing putty is based on a polydimethylsiloxane polymer modified with boric acid, additives, fillers, and plasticizers to give a material that on shock behaves like an elastic material but flows like a viscous liquid on slow application of pressure. 

The Newtonian constitutive equation is the simplest equation we can use for viscous liquids. It (and the inviscid fluid, which has negligible viscosity) is the basis of all of fluid mechanics. When faced with a new liquid flow problem, we should try the Newtonian model first. Any other will be more difficult. In general, the Newtonian constitutive equation accurately describes the rheological behavior of low molecular weight liquids and even high polymers at very slow rates of deformation. However, as we saw in the introduction to this chapter (Figures 2.1.2 and 2.1.3) viscosity can be a strong function of the rate of deformation for polymeric liquids, emulsions, and concentrated suspensions. 

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