Boundary Conditions at Solid Walls and Fluid Interfaces

French physicist, P. G. DeGennes, were responsible, in part for his achieving the Nobel prize in physics in 1991), there are still many open issues in modeling the microdynamics of entangled polymers, and we are only now approaching constitutive models that can be used for fluid mechanics predictions.37 

It is likely, in the interim, while we await models from the molecular modeling perspective for the more difficult complex fluids, that the most success in predicting fluid mechanics results for non-Newtonian fluids will come from a hybrid approach combining some elements of both continuum mechanics and molecular modeling to produce relatively simple empirical models. There is a great deal of current research focused on all aspects of constitutive model development on numerical analysis of flow solutions based on these models and on experimental studies of many flows. There are a number of books and references available, but this is a complicated field that really requires a textbook/class of its own. At this point, it is time to return from our little sojourn into the land of complex fluids and come back to the principle subject of Newtonian fluids. 

BOUNDARY CONDITIONS AT SOLID WALLS AND FLUID INTERFACES 

We are concerned in this book with the motion and transfer of heat in incompressible, Newtonian fluids. For this case, the equations of motion, continuity, and thermal energy, 

The obvious question is this What conditions should be imposed Without a molecular or microscopic theory for guidance, there is no deductive route to answer this question. The application of boundary conditions then occupies a position in continuum mechanics that is analogous to the derivation of constitutive equations in the sense that only a limited number of these conditions can be obtained from fundamental principles. The rest represent an educated guess based to a large extent on indirect comparisons with experimental data. In recent years, insights developed from molecular dynamics simulations of relatively simple 

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L. Boundary Conditions at Solid Walls and Fluid Interfaces... 

L. BOUNDARY CONDITIONS AT SOLID WALLS AND FLUID INTERFACES... 

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... 

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. 

The above remarks point out the interest of direct measurements of the boundary condition (BC) for the fluid velocity at a fluid solid interface. To obtain reliable information on the flow velocity BC of a fluid, with a spatial resolution from the wall down to molecular sizes, is a particularly difficult challenge. Conventional velocimetry techniques (even laser velocimetry) are far from such a resolution. We have developed a near field laser velocimetry technique which allows to increase significantly the spatial resolution compared to more conventional velocimetry techniques. This technique has been used to characterize the friction between a polymer melt and a solid wall and to understand how surface modifications weakening the interactions between a solid and a given simple fluid affected the fluid -- wall friction. 

In some foods, a thin layer of low-viscosity fluid forms at the solid-fluid interface that in turn contributes to lower viscosity values. The boundary condition that at the solid-fluid interface the fluid velocity is that of the wall is not satisfied. This phenomenon is known as slip effect. Mooney (1931) outlined the procedures for the quantitative determination of slip coefficients in capillary flow and in a Couette system. The development for the concentric cylinder system will be outlined here for the case of the bob rotating and details of the derivation can be found in Mooney (1931). 

To solve the equations for the pressure and velocity of the fluid, one must specify boundary conditions. Usually one assumes that the fluid sticks to a solid wall, so that v(r, t) = 0 when r is on the solid surface. (This no-slip boundary condition may not be completely accurate at microscopic length scales.) The other boundary condition that is often important is that flows at infinity are unperturbed by the boundaries. Finally, at surfaces or interfaces, there is continuity of the normal and tangential forces. The force on a surface is related to the normal component of the stress tensor, Eq. (1.143) the ith component of the force per unit area, fs = dFs/dS, obeys... 

In the pure convection problem, heat transfer through the wall is characterized by an appropriate thermal boundary condition directly or indirectly specified at the wall-fluid interface. In a pure convection problem, the solution of the temperature problem for the solid wall is not needed the velocity and temperature are determined only in the fluid region. However, the heat transfer through the sohd walls of the microchannel by conduction may have significant normal and/or peripheral as well as axial components, or the wall may be of ncmuniform thickness. In these cases the temperature problem for the solid wall needs to be analyzed simultaneously with that for the fluid in order to calculate the real wall-fluid interface heat flux distribution. In this case the wall-fluid heat transfer is referred to as conjugate heat tranter. 

Mass transport across isodensity lines should become particularly important when the lubrication approximation breaks down. This should happen near the contact line in the case when two alternative fluid densities near the solid wall are possible. If. say. the boundary densities are Psv 1 and / 5/ = 1 - a, a 1, the three-phase contact line can be viewed as a sharp transition between 0(1) positive and negative values of the nominal thickness h, such that < 1 on either side. This can be treated as a shock of Eq. (91) or (93). The Hugoniot condition, which should ensure zero net flux through the shock, is the equality of chemical potentials on both sides. Unfortunately, this condition cannot be formulated precisely, since the sharp-interface limit of the surface tension term is inapplicable in the shock region. Moreover, our test computations of the profile of the dense layer using Eq. (94) with different boundary conditions imposed on the shock at h = ho showed that the spreading velocity is very sensitive to the conditions on the shock. 

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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