Viscous shear

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

Drop breakage occurs when surrounding fluid stresses exceed the surface resistance of drops. Drops are first elongated as a result of pressure fluctuations and then spHt into small drops with a possibiUty of additional smaller fragments (Fig. 19). Two types of fluid stresses cause dispersions, viscous shear and turbulence. In considering viscous shear effects, it is assumed that the drop size is smaller than the Kohnogoroff microscale, Tj. 

This postulate imposes an idealization, and is the basis for all subsequent property relations for PVT systems. The PVT system sei ves as a satisfactoiy model in an enormous number of practical applications. In accepting this model one assumes that the effects of fields (e.g., elec tric, magnetic, or gravitational) are negligible and that surface and viscous-shear effects are unimportant. 

With turbulent flow, shear stress also results from the behavior of transient random eddies, including large-scale eddies which decay to small eddies or fluctuations. The scale of the large eddies depends on equipment size. On the other hand, the scale of small eddies, which dissipate energy primarily through viscous shear, is almost independent of agitator and tank size. 

In the viscous regime, chemical reactants become associated with each other through viscous shear stresses. These shear stresses exist at all scales (macro to micro) and until the power is dissipated continuously through the entire spectrum. This gives a different relationship for power dissipation than in the case of turbulent flow. 

The flow in the diffuser is usually assumed to be of a steady nature to obtain the overall geometric configuration of the diffuser. In a channel-type diffuser the viscous shearing forces create a boundary layer with reduced kinetic energy. If the kinetic energy is reduced below a certain limit, the flow in this layer becomes stagnant and then reverses. This flow reversal causes... 

The viscous shear properties at any given shear rate are primarily determined by two factors, the free volume within the molten polymer mass and the amount of entanglement between the molecules. An increase in the former decreases the viscosity whilst an increase in the latter, i.e. the entanglement, increases viscosity. The effects of temperature, pressure, average molecular weight, branching and so on can largely be explained in the these terms. 

The defect question delineates solid behavior from liquid behavior. In liquid deformation, there is no fundamental need for an unusual deformation mechanism to explain the observed shock deformation. There may be superficial, macroscopic similarities between the shock deformation of solids and fluids, but the fundamental deformation questions differ in the two cases. Fluids may, in fact, be subjected to intense transient viscous shear stresses that can cause mechanically induced defects, but first-order behaviors do not require defects to provide a fundamental basis for interpretation of mechanical response data. 

Based on a lot of experimental observations, criteria for the drop stability can be defined as below the U curve, namely We < We.cn, the interfacial stress can equilibrate the shear stress, and the drop will only deform into a stable prolate ellipsoid. Above this curve, the viscous shear stress becomes larger than the interfacial stress. The drop is at first extended and finally breaks up into smaller droplets. 

In turbulent flow with high values of Re, the inertia forces become predominant and the viscous shear stress becomes correspondingly less important. 

In the laminar sub-layer, turbulence has died out and momentum transfer is attributable solely to viscous shear. Because the layer is thin, the velocity gradient is approximately linear and equal to Uj,/Sb where m is the velocity at the outer edge of a laminar sub-layer of thickness <5 (see Chapter ll). 

If the heat flux from friction or viscous shear is properly estimated, the surface temperature, which is of interest in most engineering problems, can be determined through integrating an analytical solution of temperature rise caused by a moving point heat source, without having to solve the energy equation. For two solid bodies with velocity u j and Ui in dry contacts, the temperature rises at the surfaces can be predicted by the formula presented in Ref. [22],... 

In a laminar shear flow, the maximum viscous shearing stress, Xn,ax> on the surface of freely suspended spherical particle, such a unicellular micro-organism, is given as ... 

The equivalent of the right-hand side of Eq. (15), the dissipation, is composed of two additive terms. One is due to viscous shear within the liqnid, and the other is dne to viscoelastic losses in the wetting ridge. We may thus write ... 

Second, sensors are often intended for a single use, or for usage over periods of one week or less, and enzymes are capable of excellent performance over these time scales, provided that they are maintained in a nfild environment at moderate temperature and with minimal physical stress. Stabilization of enzymes on conducting surfaces over longer periods of time presents a considerable challenge, since enzymes may be subject to denaturation or inactivation. In addition, the need to feed reactants to the biofuel cell means that convection and therefore viscous shear are often present in working fuel cells. Application of shear to a soft material such as a protein-based film can lead to accelerated degradation due to shear stress [Binyamin and Heller, 1999]. However, enzymes on surfaces have been demonstrated to be stable for several months (see below). 

The Weber-Reynolds number (Re/We) is defined as the ratio of surf ace tension of a bubble to viscous shear on the bubble surface due to bubble motion ... 

As the drop or bubble gets larger, however, it will become distorted owing to the unbalanced forces around it. The viscous shear stresses tend to elongate the shape, whereas the pressure distribution tends to flatten it out in the direction normal to the flow. Thus the shape tends to progress from... 

In equation 1.94, (Tyx)v is the viscous shear stress due to the mean velocity gradient dvjdy and pv yv x is the extra shear stress due to the velocity fluctuations v x and v y. These extra stress components arising from the velocity fluctuations are known as Reynolds stresses. (Note that if the positive sign convention for stresses were used, the sign of the Reynolds stress would be negative in equation 1.94.)... 

From equation 1.41, the total shear stress varies linearly from a maximum fw at the wall to zero at the centre of the pipe. As the wall is approached, the turbulent component of the shear stress tends to zero, that is the whole of the shear stress is due to the viscous component at the wall. The turbulent contribution increases rapidly with distance from the wall and is the dominant component at all locations except in the wall region. Both components of the mean shear stress necessarily decline to zero at the centre-line. (The mean velocity gradient is zero at the centre so the mean viscous shear stress must be zero, but in addition the velocity fluctuations are uncorrelated so the turbulent component must be zero.)... 

The changing character of the flow in the different regions of the turbulent boundary layer explains certain aspects of the friction factor chart. If the absolute roughness of the pipe wall is smaller than the thickness of the viscous sublayer, flow disturbances caused by the roughness will be damped out by viscosity. The wall is subject to a viscous shear stress. Under these conditions, the line on the friction factor chart... 

Recall also from Section 4.0 that the viscous shear rate, )> , can be related to the viscous shear stress through the viscosity, p, according to Newton s Law of Viscosity, Eq. (4.3) ... 

As discussed in Section 2.6, vorticity is a measure of the angular rotation rate of a fluid. Generally speaking, vorticity is produced by forces that cause rotation of the flow. Most often, those forces are caused by viscous shearing action. As viscous fluid flows over solid walls, for example, the shearing forces caused by a no-slip condition at the wall is an important source of vorticity. The following analysis shows how vorticity is transported throughout a flow field by convective and viscous phenomena. 

The viscous shearing at the stagnation surface is a source of vorticity that is transported into the flow. One way to characterize the boundary layer is in terms of its vorticity distribution. By definition, the circumferential component of the vorticity vector is given as... 

Fiber motion — Jeffery orbits. The motion of ellipsoids in uniform, viscous shear flow of a Newtonian fluid was analyzed by Jeffery [32, 33] in 1922. For a prolate spheroid of aspect ratio a (defined as the ratio between the major axis and the minor axis) in simple shear flow, u°° = (zj), the angular motion of the spheroid is described... 

In this equation is the deviator and a is the spherical part of the stress tensor <7, eij is the strain deviator and e the volumetric part of the strain tensor ij, K = (2M + 3A) /3 is bulk modulus with M and A corresponding to the familiar Lame coefficients in the theory of elasticity, while r) and n can be termed the viscous shear and bulk moduli. 

According to the lubrication approximation, we can quite accurately assume that locally the flow takes place between two parallel plates at H x,z) apart in relative motion. The assumptions on which the theory of lubrication rests are as follows (a) the flow is laminar, (b) the flow is steady in time, (c) the flow is isothermal, (d) the fluid is incompressible, (e) the fluid is Newtonian, (f) there is no slip at the wall, (g) the inertial forces due to fluid acceleration are negligible compared to the viscous shear forces, and (h) any motion of fluid in a direction normal to the surfaces can be neglected in comparison with motion parallel to them. 

---