Slip at the wall

Both F 0) and F R) vanish for a velocity profile with zero slip at the wall. The mixing-cup average is determined when the integral of F(r) is normalized by Q = 7tR u. There is merit in using the trapezoidal rule to calculate Q by integrating dQ = InrVzdr. Errors tend to cancel when the ratio is taken. 

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... 

Note that pressure is treated as a function of z alone. This is consistent with the assumption of negligible Vr- Equation (8.63) is subject to the boundary conditions of radial symmetry, dVJdr = 0 at r = 0, and zero slip at the wall, Fz = 0 SLtr = R. 

It is possible to derive an expression for the pressnre profile in the x direction using a simple model. We assnme that the flow is steady, laminar, and isothermal the flnid is incompressible and Newtonian there is no slip at the walls gravity forces are neglected, and the polymer melt is uniformly distribnted on the rolls. With these assnmptions, there is only one component to the velocity, v dy), so the equations of continuity and motion, respectively, reduce to... 

Some fluids, as discussed in Section VI, may appear to slip at the wall. This factor may be accounted for quantitatively by methods given by Mooney (M15), but corrections for this factor are necessary before the proposed correlating methods may be applied. 

Exclusive of thixotropic and rheopectic fluids, and of the few materials (see Section VI) which may slip at the wall. 

A rather complete survey of the entire field of viscometry, including the mathematical relationships applicable to various types of instruments, has been made by Philippoff (P4). The problem of slip at the walls of rotational viscometers has been discussed by Mooney (M15) and Reiner (R4). Mori and Ototake (M17) presented the equations for calculation of the physical constants of Bingham-plastic materials from the relationship between an applied force and the rate of elongation of a rod of such a fluid. ... 

Discuss a suitable set of boundary and initial conditions that would be needed to solve the system. Assume no slip at the walls. 

Figure 1 schematically depicts the buildup of a steady velocity profile for a fluid contained between two plates, where one plate is held stationery and the other set in motion. The lack of slip condition requires that the fluid velocity be zero at the stationery wall and the same as the solid at the moving wall. The lack of slip at the walls is the basic premise of rheometry and the interpretation of rheometrical data is usually based on the assumption of no slip at the boundary. 

The slip of viscoelastic polymeric materials (and flow instabilities) was reviewed in detail by Denn (6). Apparent slip at the wall was observed with highly entangled linear polymers, but not with branched polymers or linear polymers with insufficient numbers of... 

Slip at the wall is closely related to extrudate instabilities, but in normal flow situations within machines, in virtually all but exceptional cases, the no-slip condition is assumed for solving flow problems. 

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. 

Assuming no slip at the wall of the capillary, we note that the first term on the right-hand side of Eq. E3.1-5 is zero and it becomes... 

The Rabinowitsch Equation for Fluids Exhibiting Slip at the Wall Derive the Rabinowitsch equation for the case where the fluid has a slip velocity at the wall Vw. [See L. L. Blyler, Jr., and A. C. Hart, Polym. Eng. ScL, 10, 183 (1970).]... 

We next derive the exact velocity profile, assuming steady, laminar, isothermal, and fully developed flow without slip at the walls, of an incompressible Newtonian fluid. The equation of continuity reduces to... 

Cylinders Consider a Power Law model fluid placed between two long concentric cylinders of radii R, and R0. At a certain time the inner cylinder is set in motion at constant angular velocity O rads/s. Assuming steady isothermal laminar flow without slip at the walls, neglecting gravitational and centrifugal forces, the velocity profile is... 

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. 

A force balance over the pipe assuming no slip at the walls gives — APjtr2 = Rw2rcrL, and... 

The boundary conditions are that there is no slip at the wall, that is, u = 0 at r = R, and the velocity is finite within the tube. Since there is symmetry about the tube axis, the boundary condition du/dr = 0 at r = 0 is allowable, and it leads to the same result. 

The first of these conditions follows from the no-slip at the wall requirement while the second simply follows from the fact that the velocity must be continuous at the outer edge of the boundary layer. The third condition follows from the requirement that the boundary layer profile blend smoothly into the freestream velocity distribution in which the viscous stresses are zero. 

Keywords. Polymers, Interfaces, Grafted chains, Adsorbed chains, Polymer brushes, Adhesion, Friction, Chains pull out, Slip at the wall... 

Polymer chains anchored on solid surfaces play a key role on the flow behavior of polymer melts. An important practical example is that of constant speed extrusion processes where various flow instabilities (called sharkskin , periodic deformation or melt fracture) have been observed to develop above given shear stress thresholds. The origin of these anomalies has long remained poorly understood [123-138]. It is now well admitted that these anomalies are related to the appearance of flow with slip at the wall. It is reasonable to think that the onset of wall slip is related to the strength of the interactions between the solid surface and the melt, and thus should be sensitive to the presence of polymer chains attached to the surface. 

The basic law of viscosity was formulated before an understanding or acceptance of the atomic and molecular structure of matter although just like Hooke s law for the elastic properties of solids the basic equation can be derived from a simple model, where a flnid is assumed to consist of hypothetical spherical molecules. Also like Hooke s law, this theory predicts linear behavior at low rates of strain and deviations at high strain rates. But we digress. The concept of viscosity was first introduced by Newton, who considered what we now call laminar flow and the frictional forces exerted between layers within a fluid. If we have a fluid placed between a stationary wall and a moving wall and we assume there is no slip at the walls (believe it or not, a very good assumption), then the velocity profile illustrated in Figure... 

N. El Kissi and J.-M. Piau, The diiferent capillary flow regimes of entangled poly-dimethylsiloxane polymers macroscopic slip at the wall, hysteresis and cork flow, J. Non-Newtonian Fluid Mech., 37 (1990) 55-94. 

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