Effect of the Velocity Gradient

As elaborated above, the Ericksen-Leslie theory takes into account only the first rank effect of the velocity gradient of liquid crystalline polymers. If the amplitude of the velocity gradient is high enough, these theories are no longer valid and the steady viscosity of liquid crystalline polymers... 

The procedure of Mason and Evans has the electrical analog shown in Figure 2.2, where voltages correspond to pressure gradients and currents to fluxes. As the argument stands there is no real justification for this procedure indeed, it seems improbable that the two mechanisms for diffusive momentum transfer will combine additively, without any interactive modification of their separate values. It is equally difficult to see why the effect of viscous velocity gradients can be accounted for simply by adding... 

In each of these cases, it is correctly assumed that the upthrust acting on the particles is determined by the density of the suspension rather than that of the fluid. The use of an effective viscosity, however, is valid only for a large particle settling in a fine suspension. For the sedimentation of uniform particles the increased drag is attributable to a steepening of the velocity gradients rather than to a change in viscosity. 

The positive value of N2 thus reflects an expansion of the distribution of configurations in the flow direction, relative to the distribution in the direction of the velocity gradient. Negative N2 values imply an effective contraction of the distribution along the direction of the velocity gradient relative to the 3 direction. 

In the previous sections we have shown that the inclusion of the director of the underlying nematic order in the description of a smectic A like system leads to some important new features. In general, the behavior of the director under external fields differs from the behavior of the layer normal. In this chapter we have only discussed the effect of a velocity gradient, but the effects presented here seem to be of a more general nature and can also be applied to other fields. The key results of our theoretical treatment are a tilt of the director, which is proportional to the shear rate, and an undulation instability which sets in above a threshold value of the tilt angle (or equivalently the shear rate). 

The dependence of the velocity gradients on the co-ordinates leads to possible migration of macromolecules in a flow (Aubert and Tirell 1980 Brunn 1984) - the effect, which is not discussed in this monograph. 

In a deformed system, the average form of the macromolecular coil can be approximated by an ellipsoid. The effective volume of the macromolecular coil depends on the velocity gradients. The expansion of the effective volume as a series in powers of the velocity gradients does not contain the first-order term, so vu =0. This means that, at low velocity gradients, the coil does not change its volume (one says the coil is orientated by flow). At larger velocity gradients, the volume of the coil is increased. 

A similar general expression also can be used under nonsink conditions. Here it is useful to describe the velocity of transport (i.e., flux J) expressing the effect of the concentration gradient (AC) and the length of the dimension along the line (Ax), as well as to introduce the diffusion coefficient as the proportionality constant (the Pick law). 

The resulting induced flow may be laminar (usually at small temperature differences and/or viscous fluids) or turbulent, or often in a transition or mixed laminar and turbulent regime. Correlations may be based on analytical or experimental studies, and numerical methods are now available. A major limitation to analytical solutions is that constant viscosity is generally assumed, whereas the variation of viscosity with temperature is likely to have a major effect upon the velocity gradient and the dominant flow regime near the heat-transfer surface. 

Two of the various types of laminar flow, and the effects such flows have on drops, are depicted in Figure 11.7. It also gives the definitions of the velocity gradient. 

The effective thickness of the liquid film rigidly attached to the oscillating surface is equal to I - and is less than the thickness of the inhomogeneous layer, I. The increase of the permeability leads to the enhancement of the velocity gradient in the layer, which results in a decrease of the shift due to mass loading, and an increase of the width caused by the energy dissipation. When the layer thickness is the shortest length of the problem, K 5, L < and < S, the frequency shift is also proportional to the density of the liquid and does not depend on viscosity ... 

The effects of the velocity profile are mitigated by molecular diffusion in the radial direction (i.e., the cross-sectional direction). Diffusion in the axial direction is negligible. Reactor performance is better than in a laminar flow reactor with no diffusion. If the radial diffusion is high enough, concentration gradients in the reactor cross section are eliminated, and reactor performance approaches that of piston flow. 

The extension of the rear stagnant cap takes place at increasing surfactant concentration, adsorption and retardation of the surface which also results in a decrease of inertia forces. Any variation of surface concentration and its gradient at any point of the surface results in a change of the velocity gradient. This is a local effect. 

We shall next consider the rigorous theory of Poiseuille flow, i.e., the steady laminar flow, caused by a pressure gradient, of an incompressible fluid through a tube of circular cross-section.< > We shall suppose that the tube is of infinite length so that end effects can be ignored. Let us choose a cylindrical polar coordinate system r(pz with z along the axis of the tube. In the steady state the only component of the velocity gradient is = dv/dr. It is natural to expect that the director is everywhere in the rz plane... 

Based Diffusometry, Fig. 3 Illustration of the effect of a velocity gradient on the cross-correlation... 

The heat transfer rate from a flush-mounted shear stress tensor depends on the near-wall flow, i.e., the magnitude of the velocity gradient. For a laminar two-dimensional thermal boundary layer developing over the heated sensor with an approaching linear velocity profile (Fig. 2) and negligible free convectimi effect, the heat loss from the thermal element can be derived from the thermal boundary layer equation as... 

In air d would be of the order of several mm. The boundary layer is an important concept in fluid mechanics because it allows a fluid to be separated into two regions (a) the boundary layer, which contains the whole of the velocity gradient and all viscous effects (b) the bulk fluid in which viscous forces are small compared with other forces. Flow in the boundary layer may be laminar or turbulent however, very close to the wall viscous forces always dominate in a thin region called the viscous sub-layer. 

Phenomenologically, the viscous stress is the stress which vanishes instantaneously when the flow is stopped. On the other hand the elastic stress does not vanish until the system is in equilibrium. The elastic stress is dominant in concentrated polymer solutions, while viscous stress often dominates in the suspensions of larger particles for which the Brownian motion is not effective. Whichever stress dominates, the rheological properties can be quite complex since both and are functions of the configuration of the beads and therefore depend on the previous values of the velocity gradient. Note that the viscous stress only appears in the system with rigid constraints.t... 

The pretransitional effect can be also observed by viscoelasticity. Firstly we derive the constitutive equation from the kinetic equation. Again, eqn (10.39) in the presence of the velocity gradient k is rewritten to give a kinetic equation for (see eqn (8.149)) ... 

---