Shear flow boundary conditions

P. A. Thompson, M. O. Robbins. Shear flow near solids epitaxial order and flow boundary conditions. Phys Rev A 47 6830-6837, 1990. 

In this chapter we review molecular dynamics simulations of thin films confined between two surfaces under shear. Potential models, temperature regulation methods, and simulation techniques are presented. Three properties (friction, shear viscosity, and flow boundary condition) that relate the dynamic response of confined thin films to the imposed shear velocity are presented in detail. 

In this chapter we review the application of MD techniques to the study confined thin films under shear. Three properties of particular interest are friction, shear viscosity, and flow boundary condition. For nanometer-scale material properties such as indentation and adhesion, we refer the reader to a recent review by Harrison and Brenner [49],... 

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 the SFA experiments there is no way to determine whether shear occurs primarily within the film or is localized at the interface. The assumption, made by experimentalists, of a no-slip flow boundary condition is invalid when shear localizes at the interface. It has also not been possible to examine structural changes in shearing films directly. MD simulations offer a way to study these properties. Simulations allow one to study viscosity profiles of fluids across the slab [21], local effective viscosity inside the solid-fluid interface and in the middle part of the film [28], and actual viscosity of confined fluids [29]. Manias et al. [28] found that nearly all the shear thinning takes place inside the adsorbed layer, whereas the response of the whole film is the weighted average of the viscosity in the middle and inside the interface. Furthermore, MD simulations also allow one to examine the structures of thin films during a shear process, resulting in an atomic-scale explanation [12] of the stick-slip phenomena observed in SFA experiments of boundary lubrication [7]. 

Three properties of fluids under shear are discussed in detail flow boundary condition, friction, and shear viscosity. It has been shown that the no-slip boundary condition assumed in fluid mechanical formulations of Newtonian flow past solids can fail at the molecular level. The velocity profiles deviate most from the continuum linear form at small pore separations, low temperatures, high pressures, and high shear rates. Friction is controlled by two factors - interfacial strength and in-plane ordering. 

P. A. Thompson and M. O. Robbins, Phys. Rev. A, 41, 6830 (1990). Shear Flow Near Solids—Epitaxial Order and and Flow Boundary Conditions. 

This completes the solution to 0(Re l/2). It should be noted that the first two terms in (10-246) are, in fact, nothing but the first two terms in the inviscid solution, evaluated in the inner region, namely, (10-213). Thus, to 0(Re x/2), we see that the solution in the complete domain consists of the inviscid solution (10-155) and (10-156), with an 0(Re l/2) viscous correction in the inner boundary-layer region to satisfy the zero-shear-stress boundary condition at the bubble surface. Because the viscous correction in the inner region is only C)( Re l/2), the governing equation for it is linear. Hence, unlike the no-slip boundary layers considered earlier in this chapter, it is possible to obtain an analytic solution for the leading-order departure from the inviscid flow solution. 

Any rheometric technique involves the simultaneous assessment of force, and deformation and/or rate as a function of temperature. Through the appropriate rheometrical equations, such basic measurements are converted into quantities of rheological interest, for instance, shear or extensional stress and rate in isothermal condition. The rheometrical equations are established by considering the test geometry and type of flow involved, with respect to several hypotheses dealing with the nature of the fluid and the boundary conditions the fluid is generally assumed to be homogeneous and incompressible, and ideal boundaries are considered, for instance, no wall slip. 

Here F (r) is any force imposed by an external boundary condition such as a shear flow and p(r) is the electrical charge density at r. 

Kaplan, C. R., S. W. Back, E. S. Oran, and J. L. Ellzey. 1994. Dynamics of strongly radiating unsteady ethylene jet diffusion flame. Combustion Flame 96 1-22. Kennedy, C.A., and M. H. Carpenter. 1994. Several new numerical methods for compressible shear-layer simulations. Applied Numerical Methods 14 397-433. Baum, M., T. Poinsot, and D. Thevenin. 1994. Accurate boundary conditions for multicomponent reactive flows. J. Comput. Phys. 116 247-61. 

Grinstein, F. 1994. Open boundary conditions in the simulation of subsonic turbulent shear flows. J. Compt. Physics 115(1) 43. 

For solid particles a sufficient set of boundary conditions is provided by the no slip condition, the requirement of no flow across the particle surface, and the flow field remote from the particle. For fluid particles, additional boundary conditions are required since Eqs. (1-1) and (1-9) apply simultaneously to both phases. Two additional boundary conditions are provided by Newton s third law which requires that normal and shearing stresses be balanced at the interface separating the two fluids. 

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). 

Couette and Poiseuille flows are in a class of flows called parallel flow, which means that only one velocity component is nonzero. That velocity component, however, can have spatial variation. Couette flow is a simple shearing flow, usually set up by one flat plate moving parallel to another fixed plate. For infinitely long plates, there is only one velocity component, which is in the direction of the plate motion. In steady state, assuming constant viscosity, the velocity is found to vary linearly between the plates, with no-slip boundary conditions requiring that the fluid velocity equals the plate velocity at each plate. There... 

For smooth laminar film flow with an interfacial shear the equations of motion remain as in Section III, B, 2, but the boundary condition du/dy = 0 at y = b must be replaced either by... 

Typically, there are two types of boundaries in reacting flows. The first is a solid surface at which a reaction may be occurring, where the flow velocity is usually set to zero (the no-slip condition) and where either a temperature or a heat flux is specified or a balance between heat generated and lost is made. The second type of boundary is an inflow or outflow boundary. Generally, either the species concentration is specified or the Dankwerts boundary condition is used wherein a flux balance is made across the inflow boundary (64). The gas temperature and gas velocity profile are usually specified at an inflow boundary. At outflow boundaries, choices often become more difficult. If the outflow boundary is far away from the reaction zone, the species concentration gradient and temperature gradient in the direction of flow are often assumed to be zero. In addition, the outflow boundary condition on the momentum balance is usually that normal or shear stresses are also zero (64). 

Summarizing, the model of the screw channel flow is governed by eqns. (8.99), (8.105) and (8.106) with boundary conditions eqns. (8.100), (8.101) and (8.104). The constitutive equation that was used by Griffith is a temperature dependent shear thinning fluid described by... 

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