Linear viscous flow

We consider both cases of simplified Couette flow, Fig. 8.3, and Poiseuille flow, Fig. 8.4, see [3] 

For Poiseuille ffow the middle block is driven by constant force fV = Fi — F2 with / the force density and V the volume of the system. Newton s equation is 

In the case of Coutte ffow there is no pressure-volume work 

But in the case of Poiseuille ffow the pressure-volume work done on the system is 

In the moving frame of velocity vs the kinetic energy changes dK are 

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C. Pozrikidis. Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University, 1992. 

In the same way as our dashpot represents the whole range of linear viscous flow, the spring represents the whole range of linear elastic behavior. If you are having a bad day and haven t wrapped your mind around the idea of a picture representation of a type of mechanical response, imagine that you have an unlimited supply of springs (or dashpots), each with a different value of the modulus (or viscosity), so that you can model any-... 

The dominant characteristic of fluids, on the other hand, is not their elasticity, but rather their viscosity. The equation of motion for simple linear viscous flow is Newton s law... 

Figure 5.5 Linear viscous flow between rotating coaxial cylinders.

Pozrikidis C (1992) Boundary-integral and singularity methods for linearized viscous flow. Cambridge University Press, Cambridge... 

Simmons, J.H, (1998) What is so exciting about non-linear viscous flow in glass, molecular dynamics simulations of brittle fracture and semiconductor-glass quantum composites. J. Noncryst. Solids, 239 1-15. 

The spring element is considered to behave according to Hooke s law in a linear elastic manner while the mechanical behavior of the dashpot equates linear viscous flow of a Newtonian fluid. Following the models of, for example. Lord Kelvin and Woldemar Voigt... 

The Maxwell model allows the approximation of elastic and creep strain under static stress as a function of time by assigning elastic compliance leading to instant elastic strain to the spring part of the model. The flow due to creep is represented by the linear slope related to the single dashpot element with linear viscous flow properties. 

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. 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

Viscous flow is a sort of creep. Like diffusion creep, its rate increases linearly with stress and exponentially with temperature, with... 

Fig. 23.7. A modulus diagram for PMMA. It shows the glassy regime, the gloss-rubber transition, the rubbery regime and the regime of viscous flow. The diagram is typical of linear-amorphous polymers.

Ic. Cross-Linking of Polymer Chains.—Formation of chemical bonds between linear polymer molecules, commonly referred to as cross-linking, also may lead to the formation of infinite networks. Vulcanization of rubber is the most prominent example of a process of this sort. Through the action of sulfur, accelerators, and other ingredients present in the vulcanization recipe, sulfide cross-linkages are created by a mechanism not fully understood (see Chap. XI). Vulcanized rubbers, being typical network structures, are insoluble in all solvents which do not disrupt the chemical structure, and they do not undergo appreciable plastic, or viscous, flow. 

In order to go further into the experimental check we constructed Arrhenius plots of the fluorescence quantum yield of BMPC in a few solvents (methanol, ethanol, propanol, hexanol and methylene chloride), all of which showed good linearity. The activation energies and A/kp ratios, calculated from the slopes and intercepts of those plots, are collected in Table 1. The smooth increase of both parameters in the alcohol series is mainly associated with the increase of solvent viscosity. On the other hand, decrease of the solvent dielectric constant from 32.7 (methanol) to 8.9 (dichloromethane) causes a small but significant increase of the activation energy also, this increase is probably somewhat compensated by the decrease of the viscous-flow... 

Non-linear viscoelastic flow phenomena are one of the most characteristic features of polymeric liquids. A matter of very emphasised interest is the first normal stress difference. It is a well-accepted fact that the first normal stress difference Nj is similar to G, a measure of the amount of energy which can be stored reversibly in a viscoelastic fluid, whereas t12 is considered as the portion that is dissipated as viscous flow [49-51]. For concentrated solutions Lodge s theory [52] of an elastic network also predicts normal stresses, which should be associated with the entanglement density. 

Knudsen writes the conductivity coefficient as a linear combination of the viscous flow coefficient, Fv, and the free molecular flow coefficient denoted Fm,... 

The chamber-to-condenser pathway usually involves dimensions on the order of 10 cm or greater even for laboratory freeze dryers. Here, the ratio of radius to mean free path is on the order of 0.01 or less, and flow should be essentially viscous flow. Conductance should be linear in pressure, with a zero pressure intercept near zero. Qualitative agreement with theory is observed (data not shown). 

In Figure 2 we presented the permeability coefficient K of oxygen as a function of the mean gas pressure experimentally obtained for a sample of porous material from acetylene black modified with 35% PTFE. The experimental linear dependence is obtained. The intercept with the abscissa corresponds to the Knudsen term DiK. The value obtained is 2,89.1 O 2 cm2/s. The slope of the straight line is small, so that the ratio K,/ Dik at mean gas pressure 1 atm. is small ( 0.1) which means that the gas flow is predominantly achieved by Knudsen diffusion and the viscous flow is quite negligible. At normal conditions (1 atm, 25°C) the mean free path of the air molecules (X a 100 nm) is greater than the mean pore radii in the hydrophobic material (r 20 nm), so that the condition (X r) for the Knudsen-diffusion mechanism of gas transport is fulfilled. 

For viscous dominated flows, it can be assumed that the gas inertia and the gas gravitational forces are negligible. By dropping the gas inertia and gravity time from the gas momentum equation and simplifying the dimensionless drag coefficient to the linear viscous term, the set of dimensionless equations does not include gas-to-solid density ratio as a parameter. 

Figures 5 and h show how the shape of the creep curve is modified by changes in the constants of the model. The values of the constants are given in Table I. Curve I is the same as shown in Figure 4, curve II shows onlv a small amount of viscous creep, and in curve 111, viscous flow is a prominent part of the total creep. The same data were used in Figures 5 and 6, but notice the dramatic, change in the shapes of the curves when a linear time scale is replaced by a logarithmic time scale. In the model, most of the recoverable creep occurs "Within about one decade of the retardation time.

Creep behavior is similar to viscous flow. The behavior in Equation 14.17 shows that compliance and strain are linearly related and inversely related to stress. This linear behavior is typical for most amorphous polymers for small strains over short periods of time. Further, the overall effect of a number of such imposed stresses is additive. Non-creep-related recovery... 

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