Wall slip with Newtonian fluid

Now, for convenience, we assume that the barrel surface is stationary and that the upper and lower plates representing the screws move in the opposite direction, as shown in Fig. 6.57, but for flow rate calculations, it is the material retained on the barrel rather than that dragged by the screw that leaves the extruder. We assume laminar, isothermal, steady, fully developed flow without slip on the walls of an incompressible Newtonian fluid. We distinguish two flow regions marked in Fig. 6.57 as Zone I and Zone II. In the former, the flow is between two parallel plates with one plate moving at constant velocity relative to... 

A more convenient, but entirely equivalent, problem for analysis is to consider the position and shape of the interface to be fixed, with the boundary translating at a velocity U. If we calculate the velocity and pressure fields for an incompressible, Newtonian fluid, assuming no-slip at the solid wall and the kinematic plus no-slip conditions at the interface, we find that the tangential stress component on the boundary exhibits a nonintegrable singularity as the distance to the contact point goes to zero, i.e.,... 

In spite of this, it is perhaps useful to briefly consider the conditions at solid boundaries and fluid interfaces for complex/non-Newtonian fluids. One reason for doing this is that it provides additional emphasis to the idea from the proceeding paragraphs that there will be conditions when the commonly applied no-slip condition breaks down. It should be stated, at the outset, that the question of slip or no-slip is still a matter of current research interest for complex fluids. Nevertheless, the occurrence of sbp is generally accepted to be much more common for complex/non-Newtonian fluids than for Newtonian/small molecule liquids. In the latter case, we have seen that slip generally involves either extreme shear stresses or solid walls that exhibit extremely weak attractive interactions with the hquids, and the issue is primarily one of basic scientific interest. Polymer melts, on the other hand, commonly exhibit apparent manifestations of slip that play a critical role in the success or failure of certain types of commercial processing applications.43... 

It is clear from the above that extreme care must be exercised in the characterization and rheological eva-luation of concentrated emulsions. Few, if any, com-mercial viscometers are designed to give reliable results for nonNew-tonian fluids. Not only are modifications of the hardware often called for, but also the software of automated instra-ments is generally incapable of dealing with yield-stress fluids, end effects, and wall slip. For example, to correct for end effects, it will not do to use a calibration or instrument factor for any but Newtonian fluids. Unfortunately, there are no shortcuts in this field ... 

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. 

The effects of the Reynolds number on the extrusion of Newtonian fluid from square and rectangular dies has been considered. As with planar and axisymmetric jets, extrudates from three-dimensional dies swell at low Reynolds numbers but contract at high ones. Depending on its aspect ratio, limiting die swell from the rectangle varies that of the square (0.7255) and that of two-dimensional planar case (0.8333). Wall slip reduces die swell and in the cases of perfect slip, completely eliminates it. 

The system of equations to be solved is siuroimded with double bars in Table 1.1. It needs to be associated with boundary conditions. For a Newtonian fluid, such as water or air, with solid walls consisting of common materials, the boimdary condition is a no-slip condition for the fluid at the wall. The velocity of the fluid becomes zero on a fixed solid wall, or it equals the velocity of the wall if the latter is in motion. 

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. 

Physically, the cause of the pressure drop for internal flow is the viscous interaction with the wall, where there is a no-slip condition (Vwaii = Ffluid). The one-dimensional shear force at the wall for a Newtonian fluid is proportional to the fluid strain and is related to fluid viscosity ... 

Using finite element techniqnes, a mathematical model was developed for the two-dimensional analysis of non-isothermal and transient flow and mixing of a generalised Newtonian fluid with an inert filler. The model could incorporate no-slip, partial-slip or perfect-slip wall conditions using a universally applicable numerical technique. The model was used to simulate the convection of carbon black with flowing rubber in the dispersive section of a tangential rotor (Banbury) mixer. The Carreau equation was used to model the rheological behaviour of the fluid in this example. 31 refs. 

Flow in a Tube. Laminar flow with a flat velocity profile and slip at the walls can occur when a viscous fluid is strongly heated at the walls or is highly non-Newtonian. It is sometimes called toothpaste flow. If you have ever used Stripe toothpaste, you will recognize that toothpaste flow is quite different than piston flow. Although Vflr) = u and z(7) = 1, there is little or no mixing in the radial direction, and what mixing there is occurs by diffusion. In this situation, the centerline is the critical location with respect to stability, and the stability criterion is... 

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. 

Solution This flow is z-axisymmetric. We, thus, select a cylindrical coordinate system, and make the following simplifying assumptions Newtonian and incompressible fluid with constant thermophysical properties no slip at the wall of the orifice die steady-state fully developed laminar flow adiabatic boundaries and negligible of heat conduction. 

One of the fundamental assumptions in fluid mechanical formulations of Newtonian flow past solids is the continuity of the tangential component of velocity across a boundary known as the "no-slip" boundary condition (BC) [6]. Continuum mechanics with the no-slip BC predicts a linear velocity profile. However, recent experiments which probe molecular scales [7] and MD simulations [8-10] indicate that the BC is different at the molecular level. The flow boundary condition near a surface can be determined from the velocity profile. In molecular simulations, the velocity profile is calculated in a simitar way to the calculation of the density profile. The region between the walls is divided into a sufficient number of thin slices. The time averaged density for each slice is calculated during a simulation. Similarly, the time averaged x component of the velocity for all particles in each slice is determined. The effect of wall-fluid interaction, shear rate, and wall separation on velocity profiles, and thus flow boimdary condition will be examined in the following. 

In summary, we have so far seen that there are two types of boundary conditions that apply at any solid surface or fluid interface the kinematic condition, (2-117), deriving from mass conservation and the dynamic boundary condition, normally in the form of (2-122), but sometimes also in the form of a Navier-slip condition, (2-124) or (2-125). When the boundary surface is a solid wall, then u is known and the conditions (2-117) and (2-122) provide a sufficient number of boundary conditions, along with conditions at other boundaries, to completely determine a solution to the equations of motion and continuity when the fluid can be treated as Newtonian. 

In this work, heat and fluid flow in some common micro geometries is analyzed analytically. At first, forced convection is examined for three different geometries microtube, microchannel between two parallel plates and microannulus between two concentric cylinders. Constant wall heat flux boundary condition is assumed. Then mixed convection in a vertical parallel-plate microchannel with symmetric wall heat fluxes is investigated. Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. In this study, the usual continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. 

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