Stress in fluids

In order to appreciate the effect of forces acting on a fluid it is helpful first to consider the behaviour of a solid subjected to forces. Although the deformation behaviour of a fluid is different from that of a solid, the method of describing forces is the same for both. 

It has been found experimentally that most solid materials exhibit a particularly simple relationship between the shear stress and the shear strain, at least over part of their range of behaviour  

The shear stress is proportional to the shear strain and the constant of proportionality G is known as the shear modulus. It does not matter how rapidly the solid is sheared, the shear stress depends only on the amount by which the solid is sheared. 

This expression is valid even when the displacement x is not proportional toy, in which case the strain is not uniform throughout the sample. 

In contrast to the behaviour of a solid, for a normal fluid the shear stress is independent of the magnitude of the deformation but depends on the rate of change of the deformation. Gases and many liquids exhibit a simple linear relationship between the shear stress r and the rate of shearing  

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A second aim has been to make the book more nearly self-contained and to this end a substantial introductory chapter has been written. In addition to the material provided in the first edition, the principles of continuity, momentum of a flowing fluid, and stresses in fluids are discussed. There is also an elementary treatment of turbulence. 

Hanratty TJ and Campbell JA 1996 Chapter 9, Measurement of wall shear stress. In Fluid Mechanics Measurements, 2nd Edition, Editor Goldstein RJ, Taylor and Francis, Washington, DC, pp. 575-648. 

J. Guven, Laplace pressure as a surface stress in fluid vesicles, arXiv cond-matl06022 9vl, 2006. 

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

When constmction is complete, the pipeline must be tested for leaks and strength before being put into service industry code specifies the test procedures. Water is the test fluid of choice for natural gas pipelines, and hydrostatic testing is often carried out beyond the yield strength in order to reHeve secondary stresses added during constmction or to ensure that all defects are found. Industry code limits on the hoop stress control the test pressures, which are also limited by location classification based on population. Hoop stress is calculated from the formula, S = PD/2t, where S is the hoop stress in kPa (psig) P is the internal pressure in kPa (psig), and D and T are the outside pipe diameter and nominal wall thickness, respectively, in mm (in.). 

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

Supercritical and Freeze Drying. To eliminate surface tension related drying stresses in fine pore materials such as gels, ware can be heated in an autoclave until the Hquid becomes a supercritical fluid, after which drying can be accompHshed by isothermal depressurization to remove the fluid (45,69,72) (see Supercritical fluid). In materials that are heat sensitive, the ware can be frozen and the frozen Hquid can be removed by sublimation (45,69). 

The shear stress in the fluid is much higher near the impeller than it is near the tank wall. The difference is greater in large tanks than in small ones. 

Fig. 5.10 shows an annular element of fluid of radius r and thickness dr subjected to a shear stress in the capillary. When the element of fluid emerges from the die it will recover to the form shown by ABCD. 

Chapter 4 describes in general terms the processing methods which can be used for plastics and wherever possible the quantitative aspects are stressed. In most cases a simple Newtonian model of each of the processes is developed so that the approach taken to the analysis of plastics processing is not concealed by mathematical complexity. Chapter 5 deals with the aspects of the flow behaviour of polymer melts which are relevant to the processing methods. The models are developed for both Newtonian and Non-Newtonian (Power Law) fluids so that the results can be directly compared. 

Meshalin, V. S. 1974. About turbulent stress in impinging jets. In Mechanics of (jases and Fluids. Moscow. 

Viscosity, dynamic Sometimes called absolute viscosity, the shear stress in a fluid divided by the velocity gradient. 

To understand how the dispersed phase is deformed and how morphology is developed in a two-phase system, it is necessary to refer to studies performed specifically on the behavior of a dispersed phase in a liquid medium (the size of the dispersed phase, deformation rate, the viscosities of the matrix and dispersed phase, and their ratio). Many studies have been performed on both Newtonian and non-Newtonian droplet/medium systems [17-20]. These studies have shown that deformation and breakup of the droplet are functions of the viscosity ratio between the dispersity phase and the liquid medium, and the capillary number, which is defined as the ratio of the viscous stress in the fluid, tending to deform the droplet, to the interfacial stress between the phases, tending to prevent deformation ... 

Non-Newtonian fluids vary significantly in their properties that control flow and pressure loss during flow from the properties of Newtonian fluids. The key factors influencing non-Newtonian fluids are their shear thinning or thickening characteristics and time dependency of viscosity on the stress in the fluid. 

The conjoint action of a tensile stress and a specific corrodent on a material results in stress corrosion cracking (SCC) if the conditions are sufficiently severe. The tensile stress can be the residual stress in a fabricated structure, the hoop stress in a pipe containing fluid at pressures above ambient or in a vessel by virtue of the internal hydraulic pressure created by the weight of its contents. Stresses result from thermal expansion effects, the torsional stresses on a pump or agitator shaft and many more causes. 

Smith, T., A Capillary Model for Stress-corrosion Cracking of Metals in Fluid Media, Corros. Sci., 12, 45 (1972)... 

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