Stress conditions Tangential

The boundary conditions are the same as for steady motion considered in Chapters 1, 3, and 4, i.e., uniform flow remote from the particle, no slip and no normal flow at the particle boundary, and, for fluid particles, continuity of tangential stress at the interface. For a sphere the normal stress condition at the interface is again formally redundant, but indicates whether a fluid particle will remain spherical. 

The tangential-stress condition is satisfied automatically for an inviscid fluid (p = 0) because x = 0 in this case, and there is no viscous contribution to the normal-stress balance. 

The fluid dynamical boundary conditions are similar to those applied in the previous problem, with two notable exceptions. First, u = 0 at z = 0 (i.e., the lower boundary is stationary). Second, the tangential-stress condition is modified to account for the presence of Marangoni stresses that are due to gradients of the interfacial tension at the fluid interface,... 

It remains to determine the four sets of constants, An, Bn, Cn, and Dn, and the function fo that describes the 0(Ca) correction to the shape of the drop. For this, we still have the five independent boundary conditions, (7-207)-(7-210). It can be shown that the conditions (7-207), (7-208), and (7-209) are sufficient to completely determine the unknown coefficients in (7-213) and (7-215). Indeed, for any given (or prescribed) drop shape, the four conditions of no-slip, tangential-stress continuity and the kinematic condition are sufficient along with the far-field condition to completely determine the velocity and pressure fields in the two fluids. The normal-stress condition, (7-210), can then be used to determine the leading-order shape function /0. Specifically, we can use the now known solutions for the leading-order approximations for the velocity components and the pressure to evaluate the left-hand side of (7 210), which then becomes a second-order PDE for the function /(). The important point to note is that we can determine the 0(Ca) contribution to the unknown shape knowing only the 0(1) contributions to the velocities and the pressures. This illustrates a universal feature of the domain perturbation technique for this class of problems. If we solve for the 0(Cam) contributions to the velocity and pressure, we can... 

Because the shape is spherical, the tangential-stress condition takes the form... 

In this regard, it is of interest to contrast the two problems of the streaming motion of a fluid at large Reynolds number past a solid sphere and a spherical bubble. In the case of a solid sphere, the potential-flow solution (10 155)—(10—156) does not satisfy the no-slip condition at the sphere surface, and the necessity for a boundary layer in which viscous forces are important is transparent. For the spherical bubble, on the other hand, the noslip condition is replaced with the condition of zero tangential stress, Tr = 0, and it may not be immediately obvious that a boundary layer is needed. However, in this case, the potential-flow solution does not satisfy the zero-tangential-stress condition (as we shall see shortly), and a boundary-layer in which viscous forces are important still must exist. We shall see that the detailed features of the boundary layer are different from those of a no-shp, sohd body. However, in both cases, the surface of the body acts as a source of vorticity, and this vorticity is confined at high Reynolds number to a thin 0(Re x/2) region near the surface. 

Last, there are the normal- and tangential-stress conditions. The general form of the normal-stress condition [keeping in mind (12 57)] is... 

The tangential-stress condition for a clean, isothermal interface is... 

Huh and Scriven add, to all constraints already taken into account, that the 1/v interface is a material surface, that is not crossed by any matter flux. This adds two conditions, namely, the vanishing of the normal velocity on both sides of the interface. This results in four additional boundary conditions along the 1/v interface continuity of tangential stress, of tangential speed and one condition for the normal speed on each side. Now there are eight conditions in all. 

Tangential stress component at and shear stress neglected Plane stress condition (ae ... 

Let us consider the influence of various combinations of stress conditions, namely, normal fracture, shear, normal compression, and the combined action of direct and tangential stresses of specified ratio, on the strength of adhesive joints, as well as their dependences on materials, adhesion conditions, and polymerization kinetics. Adhesives that were analyzed in Chapter 3 were taken as the objects of investigation. 

The mathematical problems in this section are specified in the following way, unless stated otherwise. The Navier-Stokes equations govern the motion of the incompressible, Newtonian fluid inside the film [52]. The film is assumed to be symmetric about its vertical centreline (the z-axis). On the free surface of the fluid, several equations must be satisfied the kinematic condition, and the normal and tangential stress conditions [53, 54]. An equation governing the transport of surfactant in the free surface by both advection (fluid... 

For high concentrations of some TCP in PU, the snrfactant develops structure on the film surface and the surface of the film still deforms but doesn t move tangentially. This rigid case [22] is the slow draining limit of the experiment. In that case, the tangential stress condition may be replaced by the condition that the tangential velocity component is zero. We have been able to show mathematically that this case is achieved in the limit of large surface viscosity [59, 60] or 5 —> °o this will be discussed further in the next section. 

Previous work [62, 63] has shown that it is possible to keep the full curvature in the normal and tangential stress conditions and integrate the partial differential equation for the free surface k through the matching region (where the film meets the bath) onto the static meniscus this model relies on those results. While we do not rigorously apply matched asymptotic expansions in this work, the equations contain all of the terms necessary to match the film onto a static meniscus (for the bath surface) and it has been shown that the terms neglected are uniformly small for 8<<1 [65]. 

Tangential stress condition. The stress at any point on the interface in a direction tangential to the interface jumps as we cross from one phase to the other by an amount equal to the force exerted by surface tension gradients (Marangoni forces). 

Next, the tangential stress condition (29) is used to obtain the first evolution equation. The axial velocity correction Wi follows from (86) since the only function of r is Wi, and is found in terms of wq and So- This is then substituted along with the expression (88) for Wo into (29) to yield. The leading order contribution of the tangential stress balance is the consistency condition wor = 0. while the next order yields the first desired evolution equation. The second equation follows from the leading order contributions of the kinematic condition (31). The system to be addressed is ... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

This serai-empirical approach may be compared with a calculation based on the hydrodynamic stress gradient at the equator of a steadily moving drop with a rigid surface, and for Re < 1. The latter condition is easily satisfied for small drops. The tangential stress gradient is given (70, 77) by ... 

Numerical solutions of the flow around and inside fluid spheres are again based on the finite difference forms of Eqs. (5-1) and (5-2) (BIO, H6, L5, L9). The necessity of solving for both internal and external flows introduces complications not present for rigid spheres. The boundary conditions are those described in Chapter 3 for the Hadamard-Rybczynski solution i.e., the internal and external tangential fluid velocities and shear stresses are matched at R = 1 (r = a), while Eq. (5-6) applies as R- co. Most reported results refer to the limits in which k is either very small (BIO, H5, H7, L7) or large (L9). For intermediate /c, solution is more difficult because of the coupling between internal and external flows required by the surface boundary conditions, and only limited results have been published (Al, R7). Details of the numerical techniques themselves are available (L5, R7). 

For the turbulent motion in a tube, the mass transfer coefficient k is proportional to the diffusion coefficient at the power of 2/3. It is easy to realize by inspection that this value of the exponent is a result of the linear dependence of the tangential velocity component on the distance y from the wall. For the turbulent motion in a tube, the shear stress t r0 = const near the wall, whereas for turbulent separated flows, the shear stress is small at the wall near the separation point (becoming zero at this point) and depends on the distance to the wall. Thus, the tangential velocity component has, in the latter case, no longer a linear dependence on y and a different exponent for the diffusion coefficient is expected. For separated flows, it is possible to write under certain conditions that [90]... 

The strengthening effect of a monobloc cylinder due to autoffettage practically is achieved by two effects First the introduction of the compressive (tangential) residual stresses which extend the elastically admissible internal pressure and second the increased available material strength by strain hardening. The maximum admissible pressure for optimum autoffettaged cylinders based on perfectly elastic-plastic materials, completely elastic stress (plus/minus) conditions at the inner bore diameter and the assumption of the GE-hypothesis can be calculated as [11]. 

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