Viscous flows

Viscous flow is characterized by proportionality between the stress and the rate of deformation, that is, between t and dy/dt, and is desaibed by Newton s viscosity law  

In contrast to elastic behavior, this idealized viscous behavior is completely irreversible, both mechanically and thermodynamically. Irreversibility implies that the initial shape of the body is not restored after the shear stress has been relieved. Viscous flow is accompanied by the dissipation of energy, that is, by the conversion of all work into heat. The rate of energy dissipation, that is, the power dissipated per unit volume, is given by 

Physical-Chemical Mechanics of Disperse Systems and Materials 

The viscous flow mechanism, which was first proposed by Frenkel, can be operative in the sintering of viscous materials like glass. If the material follows the behaviour of a Newtonian fluid, the neck growth and shrinkage kinetics are expressed as follows. 

The viscosity of the glass phase decreases with an increase in temperature. When the viscosity is low, the glass phase can flow under the action of a stress. The viscosity is also related to the volume fraction of the glass phase. The relation is shown in Equation 15.39. 

is the effective viscosity that is affected by the volume fraction of the phase, f. The intrinsic viscosity that is independent of the volume fraction is denoted as ri . The strain rate, which is same as the creep rate, is dependent upon the shear stress and the effective viscosity. 

When the pore walls strongly absorb gas molecules, surface diffusion and/or capillary condensation accompanied by (surface) flow occurs. Usually this is the case with gases which condense rather easily at moderate temperature-pressure conditions (in any case being below their critical point) and we are dealing with vapour flow. 

Configurational diffusion is a separate class and occurs when the pore diameter is a factor of 1-5 larger than the molecular diameter. 

When the number of intermolecular collisions is strongly dominant Kn 1), forced flow under a pressure or concentration gradient in a capillary can be 

9 — TRANSPORT AND SEPARATION PROPERTIES OF MEMBRANES WITH GASES AND VAPOURS 

In real porous media, Eq. (9.1) must be modified to account for the number of capillaries per unit volume (porosity) and the complexities of the structure (tortuosity t). This leads to  

Consider the flow over a flat plate as shown in Figs. 5-l and 5-2. Beginning at the leading edge of the plate, a region develops where the influence of viscous forces is felt. These viscous forces are described in terms of a shear stress r between the fluid layers. If this stress is assumed to be proportional to the normal velocity gradient, we have the defining equation for the viscosity, 

The constant of proportionality p. is called the dynamic viscosity. A typical set of units is newton-seconds per square meter however, many sets of units are used for the viscosity, and care must be taken to select the proper group which will be consistent with the formulation at hand. 

The region of flow which develops from the leading edge of the plate in which the effects of viscosity are observed is called the boundary layer. Some arbitrary point is used to designate the y position where the boundary layer ends this point is usually chosen as the y coordinate where the velocity becomes 99 percent-of the free-stream value. 

Initially, the boundary-layer development is laminar, but at some critical distance from the leading edge, depending on the flow field and fluid properties, small disturbances in the flow begin to become amplified, and a transition process takes place until the flow becomes turbulent. The turbulent-flow region may be pictured as a random churning action with chunks of fluid moving to and fro in all directions. The transition from laminar to turbulent flow occurs when 

This particular grouping of terms is called the Reynolds number, and is dimensionless if a consistent set of units is used for all the properties  

A fluid of constant density (p) and viscosity (p.) is contained in a very long horizontal pipe of length L and radius R [14]. The fluid is at rest initially. At f = 0, a pressure gradient 

Then it is convenient to introduce the following dimensionless variables 

Substitution of the dimensionless variables into the differential equation gives 

This is a case in which the differential equation is nonhomogeneous. However, the fact that the system is expected to reach a steady state as t — oo can be used to reduce the differential equation to a homogeneous one. That is, assume a solution of the dimensionless system to be 

the weight function is for the case of a singular Sturm-Liouville problem as discussed in Chapter 4. Then with the aid of integral tables for Bessel functions [3, 21], we get 

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A more recent model for the preexponential factor including viscous flow across the solid-liquid interface is [14]... 

A simple law, known as Darcy s law (1936), states that the volume flow rate per unit area is proportional to the pressure gradient if applied to the case of viscous flow through a porous medium treated as a bundle of capillaries,... 

A third definition of surface mobility is essentially a rheological one it represents the extension to films of the criteria we use for bulk phases and, of course, it is the basis for distinguishing states of films on liquid substrates. Thus as discussed in Chapter IV, solid films should be ordered and should show elastic and yield point behavior liquid films should be coherent and show viscous flow gaseous films should be in rapid equilibrium with all parts of the surface. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Finally we require a case in which mechanism (lii) above dominates momentum transfer. In flow along a cylindrical tube, mechanism (i) is certainly insignificant compared with mechanism (iii) when the tube diameter is large compared with mean free path lengths, and mechanism (ii) can be eliminated completely by limiting attention to the flow of a pure substance. We then have the classical Poiseuille [13] problem, and for a tube of circular cross-section solution of the viscous flow equations gives 2... 

Chapcer 4. GAS MOTION IN A LONG TlfBE AT THE LIMIT OF BULK DIFFUSION AND VISCOUS FLOW... 

This determines the total flux at the li/nic of viscous flow. Equations (5.18 and (5.19) therefore describe the limiting form of the dusty gas model for high pressure or large pore diameters -- the limit of bulk diffusion control and viscous flow,... 

Ac this point It is important to emphasize that, by changing a and p, it is not possible to pass to the limit of viscous flow without simultaneously passing to the limit of bulk diffusion control, and vice versa, since physical estimates of the relative magnitudes of the factors and B... 

As a consequence of this, i enever bulk dlffusional resistance domin ates Knudsen diffusional resistance, so that 1, it follows that fi 1 also, and hence viscous flow dominates Knudsen streaming. Thus when we physically approach the limit of bulk diffusion control, by increasing the pore sizes or the pressure, we must simultaneously approach the limit of viscous flow. This justifies a statement made in Chapter 5. 

Turning finally to the Interpretation of Graham s experimental resul on transpiration, the theory of viscous flow of an ideal gas through a Ion capillary gives... 

The comparison of flow conductivity coefficients obtained from Equation (5.76) with their counterparts, found assuming flat boundary surfaces in a thin-layer flow, provides a quantitative estimate for the error involved in ignoring the cui"vature of the layer. For highly viscous flows, the derived pressure potential equation should be solved in conjunction with an energy equation, obtained using an asymptotic expansion similar to the outlined procedure. This derivation is routine and to avoid repetition is not given here. 

Dual viscous-flow reservoir inlet. An inlet having two reservoirs, used alternately, each having a leak that provides viscous flow. This inlet is used to obtain precise comparisons of isotope ratios in two samples. 

Nonfractionating continuous inlet. An inlet in which gas flows from a gas stream being analyzed to the mass spectrometer ion source without any change in the conditions of flow through the inlet or by the conditions of flow through the ion source. This flow is usually viscous flow, such that the mean free path is very small in comparison with the smallest dimension of a traverse section of the channel. The flow characteristics are determined mainly by collisions between gas molecules, i.e., the viscosity of the gas. The flow can be laminar or turbulent. 

Our approach in this chapter is to alternate between experimental results and theoretical models to acquire familiarity with both the phenomena and the theories proposed to explain them. We shall consider a model for viscous flow due to Eyring which is based on the migration of vacancies or holes in the liquid. A theory developed by Debye will give a first view of the molecular weight dependence of viscosity an equation derived by Bueche will extend that view. Finally, a model for the snakelike wiggling of a polymer chain through an array of other molecules, due to deGennes, Doi, and Edwards, will be taken up. 

Next let us consider the differences in molecular architecture between polymers which exclusively display viscous flow and those which display a purely elastic response. To attribute the entire effect to molecular structure we assume the polymers are compared at the same temperature. Crosslinking between different chains is the structural feature responsible for elastic response in polymer samples. If the crosslinking is totally effective, we can regard the entire sample as one giant molecule, since the entire volume is permeated by a continuous network of chains. This result was anticipated in the discussion of the Bueche theory for chain entanglements in the last chapter, when we observed that viscosity would be infinite with entanglements if there were no slippage between chains. 

The upswing in compliance from the rubbery plateau marks the onset of viscous flow. In this final stage the slope of the lines (the broken lines in Fig. 3.12) is unity, which means that the compliance increases linearly with time. 

Viscotester Viscotron Viscous drag Viscous flow Viscous liquids VI. See Viscosity index. 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

The upper use temperature for annealed ware is below the temperature at which the glass begins to soften and flow (about Pa-s or 10 P). The maximum use temperature of tempered ware is even lower, because of the phenomenon of stress release through viscous flow. Glass used to its extreme limit is vulnerable to thermal shock, and tests should be made before adapting final designs to any use. Table 4 Hsts the normal and extreme temperature limits for annealed and tempered glass. These data ate approximate and assume that the product is not subject to stresses from thermal shock. 

Melt Viscosity. As shown in Tables 2 and 3, the melt viscosity of an acid copolymer increases dramatically as the fraction of neutralization is increased. The relationship for sodium ionomers is shown in Figure 4 (6). Melt viscosities for a series of sodium ionomers derived from an ethylene—3.5 mol % methacrylic acid polymer show that the increase is most pronounced at low shear rates and that the ionomers become increasingly non-Newtonian with increasing neutralization (9). The activation energy for viscous flow has been reported to be somewhat higher in ionomers than in related acidic... 

Rheology. Both PB and PMP melts exhibit strong non-Newtonian behavior thek apparent melt viscosity decreases with an increase in shear stress (27,28). Melt viscosities of both resins depend on temperature (24,27). The activation energy for PB viscous flow is 46 kj /mol (11 kcal/mol) (39), and for PMP, 77 kJ/mol (18.4 kcal/mol) (28). Equipment used for PP processing is usually suitable for PB and PMP processing as well however, adjustments in the processing conditions must be made to account for the differences in melt temperatures and rheology. 

Rheology is the science of the deformation and flow of matter. It is concerned with the response of materials to appHed stress. That response may be irreversible viscous flow, reversible elastic deformation, or a combination of the two. Control of rheology is essential for the manufacture and handling of numerous materials and products, eg, foods, cosmetics, mbber, plastics, paints, inks, and drilling muds. Before control can be achieved, there must be an understanding of rheology and an ability to measure rheological properties. 

The study of flow and elasticity dates to antiquity. Practical rheology existed for centuries before Hooke and Newton proposed the basic laws of elastic response and simple viscous flow, respectively, in the seventeenth century. Further advances in understanding came in the mid-nineteenth century with models for viscous flow in round tubes. The introduction of the first practical rotational viscometer by Couette in 1890 (1,2) was another milestone. 

Mechanical Behavior of Materials. Different kinds of materials respond differently when they undergo basic mechanical tests. This is illustrated in Eigure 15, which shows stress—strain diagrams for purely viscous and purely elastic materials. With the former, the stress is reheved by viscous flow and is independent of strain. With the latter, there is a direct dependence of stress on strain and the ratio of the two is the modulus E (or G). 

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