Wall slip, liquid flow

Figure 13 plots an example of the processed PIV frame. The turbulent velocity field and its boundaries, solid wall, and liquid-free surface are simultaneously shown in Figure 13. The turbulence structures such as the coherent vortical structure near the bottom wall and its modification after release from the no-slip boundary condition near the free surface of the open-channel flow, and the evolvement of the free-surface wave can be seen in Figure 13. This simultaneous measurement technique for free-surface level and velocity field of the liquid phase using PIV has been successfully applied to the investigation of wave-turbulence interaction of a low-speed plane liquid wall-jet flow (Li et al., 2005d), and the characteristics of a swirling flow of viscoelastic fluid with deformed free surface in a cylindrical container driven by the constantly rotating bottom wall (Li et al., 2006c).

Magnetic resonance imaging permitted direct observation of the liquid hold-up in monolith channels in a noninvasive manner. As shown in Fig. 8.14, the film thickness - and therefore the wetting of the channel wall and the liquid hold-up -increase nonlinearly with the flow rate. This is in agreement with a hydrodynamic model, based on the Navier-Stokes equations for laminar flow and full-slip assumption at the gas-liquid interface. Even at superficial velocities of 4 cm s-1, the liquid occupies not more than 15 % of the free channel cross-sectional area. This relates to about 10 % of the total reactor volume. Van Baten, Ellenberger and Krishna [21] measured the liquid hold-up of katapak-S . Due to the capillary forces, the liquid almost completely fills the volume between the catalyst particles in the tea bags (about 20 % of the total reactor volume) even at liquid flow rates of 0.2 cm s-1 (Fig. 8.15). The formation of films and rivulets in the open channels of the structure cause the further slight increase of the hold-up. 

A suspension is a dispersion of particles within a solvent (usually a low-molar-mass liquid). Thermodynamics (Brownian motion and collisions) favours the clumping of small particles, and this can be increased by flow. However, particles over 1 pm tend to settle under gravity, unless stability measures have been considered (matching the density of the particle to that of the medium, increasing the Brownian/gravitational force ratio, electrostatic stabilization, steric stabilization). Other complications can occur in the dynamics of suspensions, such as particle migration across streamlines, particle inertial effects and wall slip (Larson, 1999). 

In many PFDs a thin layer of liquid forms at solid boundaries, and this in turn results in slip at the boundaries (22, 22). Wall slip was also observed for polymer solutions in capillary flow (22) and for molten polymers in capillaries made from different materials of construction (42.). Yoshimura and Prud homme (41)... 

The volume fraction of the gas-liquid mixture in the vessel which is occupied by the gas is called the gas holdup, tpG. If the superficial gas velocity in the vessel is vG, then vJ< G is the true gas velocity relative to the vessel walls. If the liquid flows upward, cocurrently with the gas, at a velocity relative to the vessel walls vL/( 1 - tpG), the relative velocity of gas and liquid, or slip velocity, is... 

Fig. 17a-c. Profiles of flow velocities of a curing liquid in the presence of a wall slip (a, b, c are the different conditions of flow, dashed line is the Hquid-sotid boundary)... 

While experimental evidence indicates that fluid flow in microdevices differs from flow in macroscale, existing experimental results are often inconsistent and contradictory because of the difficulties associated with such experiments and the lack of a guiding rational theory. Koo and Kleinstreuer [6] summarized experimental observations of liquid microchannel flows and computational results concerning chamiel entrance, wall slip, non-Newtonian fluid, surface roughness, and other effects. Those contradictory results suggest the need for applying molecular-based models to help establish a theoretical frame for the fluid mechanics in microscale and nanoscale. 

For a Newtonian liquid flow, this assumption is justified if the microchannel has a hydraulic diameter larger than 1 pm. In fact, for liquids the typical mean free path X of molecules under ambient conditions is 0.1-1 nm. Since the fluid velocity tends to evidence a slip at the microchannel walls for Knudsen numbers, defined as Kn = XjD, larger than 0.001, the no-slip botmdary condition has to be abandoned only when the hydraulic diameter of the microchannel becomes less than 1 pm. 

Mathematical modeling of the flow through SSE considers that the screw and the barrel are unwound. The screw is stationary and the barrel moves over it at the correct gap height and the pitch angle. The initial models assumed (i) steady state, (ii) constant melt density and thermal conductivity, (iii) conductive heat only perpendicular to the barrel surface, (iv) laminar flow of Newtonian liquid without a wall slip, (v) no pressure gradient in the melt film, and (vi) temperature effect on viscosity was neglected. Later models introduced non-Newtonian and non-isothermal flows. Present computer programs make it possible to simulate the flow in three dimensions, 3D [39]. 

In order to characterize a liquid flow through microchannels under an inposed pressure gradient Eq. (11) can be used together with the appropriate boundary condition that, for Newtonian liquids, is the no-slip boundary condition at the walls ... 

Consistent with this model, foams exhibit plug flow when forced through a channel or pipe. In the center of the channel the foam flows as a solid plug, with a constant velocity. All the shear flow occurs near the walls, where the yield stress has been exceeded and the foam behaves like a viscous liquid. At the wall, foams can exhibit wall slip such that bubbles adjacent to the wall have nonzero velocity. The amount of wall slip present has a significant influence on the overall flow rate obtained for a given pressure gradient. 

Perhaps the best picture of a viscoplastic fluid is that of a very viscous, even solidlike, material at low stresses. Over a narrow stress range, which can often be modeled as a single yield stress, its viscosity drops dramatically. This is shown clearly in Figure 2.5.5b, where viscosity drops over five decades as shear stress increases from 1 to 3 Pa. (The drop is even more dramatic in Figure 10.7.2.) Above this yield stress the fluid flows like a relatively low viscosity, even Newtonian, liquid. Because of the different behaviors exhibited by these fluids, the model (Bingham, Casson, etc.) and the range of shear rates used to calculate the parameters must be chosen carefully. In Section 10.7 we will discuss microstructural bases for r. It is also important to note that experimental problems like wall slip are particularly prevelant with viscoplastic materials. Aspects of slip are discussed in Section 5.3. 

While the Navier-Stokes equation is a fundamental, general law, the boundary conditions are not at all dear. In fluid mechanics, one usually relies on the assumption that when liquid flows over a solid surface, the liquid molecules adjacent to the solid are stationary relative to the solid and that the viscosity is equal to the bulk viscosity. We applied this no-slip boundary condition in Eq. (6.18). Although this might be a good assumption for macroscopic systems, it is questionable at molecular dimensions. Measurements with the SFA [644—647] and computer simulations [648-650] showed that the viscosity of simple liquids can increase many orders of magnitude or even undergo a liquid to solid transition when confined between solid walls separated by only few molecular diameters water seems to be an exception [651, 652]. Several experiments indicated that isolated solid surfaces also induce a layering in an adjacent liquid and that the mechanical properties of the first molecular layers are different from the bulk properties [653-655]. An increase in the viscosity can be characterized by the position of the plane of shear. Simple liquids often show a shear plane that is typically 3-6 molecular diameters away from the solid-liquid interface [629, 644, 656-658]. 

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