Shear stresses, definition

Actually, several possibilities exist formulating the wall friction force. The natural boundary layer shear stress definition to use is the one deduced from the fundamental equilibrium boundary layer analysis. The wall shear stress is thus defined as —Om =... 

This leads to a definition of apparent viscosity as the ratio of shear stress to apparent shear rate... 

Now R0 (the shear stress in the fluid at the surface) is equal and opposite to R, the shear stress acting on the surface, —q jQs is by definition the heat transfer coefficient at the surface (h), and (—NA)y=o/ CAjl - CAw) is the mass transfer coefficient ho). Then dividing both sides of equation 12.100 by pu, and of equation 12.101 by u, to make them dimensionless ... 

It is important to distinguish between the momentum flux and the shear stress because of the difference in sign. Some references define viscosity (i.e., Newton s law of viscosity) by Eq. (1-8), whereas others use Eq. (1-9) (which we shall follow). It should be evident that these definitions are equvialent,... 

In the case of non-Newtonian flow, it is necessary to use an appropriate apparent viscosity. Although the apparent viscosity (ia is defined by equation 1.71 in the same way as for a Newtonian fluid, it no longer has the same fundamental significance and other, equally valid, definitions of apparent viscosities may be made. In flow in a pipe, where the shear stress varies with radial location, the value of fxa varies. As pointed out in Example 3.1, it is the conditions near the pipe wall that are most important. The value of /j.a evaluated at the wall is given by... 

The science that deals with the deformation and flow of matter is called rheology. An important rheological concept is the shear force, sometimes called the shear stress, or the force that causes a layer of a fluid material to flow over a layer of stationary material. The rate at which a layer of a fluid material flows over a layer of stationary material is called the shear rate. A fluid flowing through a tube, for example, would be the fluid material, while the tube wall would be the stationary material. An important rheological measurement that is closely related to the resistance to flow is called viscosity. The technical definition of viscosity is the ratio of shear stress to shear rate ... 

Here t is the resulting shear stress, 6 is the phase shift often represented as tan(d), and (O is the frequency. The term 6 is often referred to as the loss angle. The in-phase elastic portion of the stress is To(cosd)sin(wt), and the out-of-phase viscous portion of the stress is To(sind)cos(complex modulus and viscosity, which can be used to extend the range of the data using the cone and plate rheometer [6] ... 

Substituting the shear stresses and velocities and then integrating provides the rate of work for the channel as Eq. 7.88 for barrel rotation. As before for screw rotation, dissipation is a positive definite quantity, and thus the absolute value of the pressure gradient is used. The dissipation between the flight lands and barrel wall is the same for barrel and screw rotation. 

Solving for the shear stress and using the definition of viscosity one obtains ... 

See Definition 1.2 for the general definition of cr. (7ii, i = 1, 2, 3 are denoted normal stresses. cti2 is called the shear stress. 

By definition the term non-Newtonian encompasses all materials which do not obey the direct proportionality between shear stress and shear rate depicted by Eq. (1). 

In Chapter 2 considerable effort is devoted to establishing the relationship between the stress tensor and the strain-rate tensor. The normal and shear stresses that act on the surfaces of a fluid particle are found to depend on the velocity field in a definite, but relatively complex, manner (Eqs. 2.140 and 2.180). Therefore, when these expressions for the forces are substituted into the momentum equation, Eq. 3.53, an equation emerges that has velocities (and pressure) as the dependent variables. This is a very important result. If the forces were not explicit functions of the velocity field, then more dependent variables would likely be needed and a larger, more complex system of equations would emerge. In terms of the velocity field, the Navier-Stokes equations are stated as... 

From this equation it is apparent that the wall shear stress depends linearly on r, since dV /dz is a constant. The stress can be put into a nondimensional form through the definition of a friction factor... 

Eqs. (1.16) are, of course, very well-known. G is known as the storage modulus in shear. It describes that part of the shear stress, which is in phase with the deformation. G" is the loss modulus in shear. It describes that part of shear stress which is 90 degrees out of phase. 6 is the loss angle. These definitions will be used occasionally in this review. 

Instead of Je Lodge (46) derived another quantity called constrained shear recovery sIn this case a shear recovery is considered, where the liquid is constrained by boundary planes which are rigid and do not change their mutual distance during recovery of the liquid. Subscript oo means that the recovery is measured after an infinite time, reckoned from the moment that the shear stress is made zero. According to Lodge, a quite different type of recovery occurs, when the mentioned restrictions are released. This fact has already been noted in the first paragraph of this section. In the definition of Je, however, the mentioned restrictions are tacitly made. 

In Fig. 5.4 results are plotted, as obtained by Tsvetkov, Garmonova and Stankevich (166) on solutions of linear oligomers of polyoxy-propylene glycol in cyclohexanol at 20° C (full circles). For comparison the ratio of birefringence and shear stress as obtained in the limit of zero shear stress on the bulk oligomers, are inserted (open squares). [Cf. the definition of [ ]/[ ] by eq. (2.33)]. As these bulk materials form viscous... 

POISE (P). A unit of dynamic viscosity. The unit is expressed in dyne second per square centimeter The centipoise (cP) is more commonly used The formal definition of viscosity arises from the concept put forward by Newton that under conditions of parallel flow, the shearing stress is proportional to the velocity giadieut. If lire force acting on each of two planes of aiea A parallel to each oilier, moving parallel lo each other with a relative velocity V, and separated by a perpendicular distance X, be denoted by F. the shearing stress is F/A and the velocity gradient, which will be linear for a true liquid, is V/X. Thus, Ft A = q V/X, where the constant if is the viscosity coefficient or dynamic viscosity of the liquid. The poise is the CGS unit of dynamic viscosity. 

Data from viscometers are often presented as a linear plot of shear stress versus shear rate, sometimes called a rheogram (Figure HI.1.2). This type of plot allows the viewer to see directly if there is Newtonian behavior because the plot will take the form of a straight line through the origin. A non-Newtonian response is, by definition, nonlinear and may or may not pass through the origin. If the sample has an apparent yield stress, then the line or curve will... 

An elastic solid has a definite shape. When an external force is applied, the elastic solid instantaneously changes its shape, but it will return instantaneously to its original shape after removal of the force. For ideal elastic solids, Hooke s Law implies that the shear stress (o force per area) is directly proportional to the shear strain (7 Figure H3.2.1A) ... 

All the preceding particulate handling steps are affected by the unique properties of all particulates, including polymeric particulates while they may behave in a fluidlike fashion when they are dry, fluidized and above 100 pm, they also exhibit solidlike behavior, because of the solid-solid interparticle and particle-vessel friction coefficients. The simplest and most common example of the hermaphroditic solid/ fluidlike nature of particulates is the pouring of particulates out of a container (fluidlike behavior) onto a flat surface, whereupon they assume a stable-mount, solidlike behavior, shown in Fig. 4.2. This particulate mount supports shear stresses without flowing and, thus by definition, it is a solid. The angle of repose, shown below, reflects the static equilibrium between unconfined loose particulates. 

Viscosity and Plasticity—Viscosity and plasticity are closely related. Viscosity may be defined as the force required to move a unit-area of plane surface with unit-speed relative to another parallel plane surface, from which it is separated by a layer of the liquid of unit-thickness. Other definitions have been applied to viscosity, an equivalent one being the ratio of shearing stress to rate of shear. When a mud or slurry is moved in a pipe in more or less plastic condition the viscosity is not the same for all rates of shear, as in the case of ordinary fluids. A material may be called plastic if the apparent viscosity varies with the rate of shear. The physical behavior of muds and slurries is markedly affected by viscosity. However, consistency of muds and slurries is not necessarily the same as viscosity but is dependent upon a number of factors, many of which are not yet clearly understood. The viscosity of a plastic material cannot be measured in the manner used for liquids. The usual instrument consists of a cup in which the plastic material is placed and rotated at constant speed, causing the deflection of a torsional pendulum whose bob is immersed in the liquid. The Stormer viscosimeter, for example, consists of a fixed outer cylinder and an inner cylinder which is revolved by means of a weight or weights. 

If r and q are defined as the total shear stress and heat transfer rate respectively, at any point in the flow, then, using the definitions given in Chapter 5, it follows that ... 

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