Navier Stokes Characteristic Boundary Conditions

The steady-state heat equation (Eq. 3.284) is often used as the model equation for an elliptic partial-differential equation. An important property of elliptic equations is that the solution at any point within the domain is influenced by every point on the boundary. Thus boundary conditions must be supplied everywhere on the boundaries of the solution domain. The viscous terms in the Navier-Stokes equations clearly have elliptic characteristics. 

Clearly, in the limit Re y> 1, the leading-order approximation for the solution to this problem is identical to the inviscid flow problem for a solid sphere. Although the no-slip boundary condition has been replaced in the present problem with the zero-shear-stress condition, (10-197), this has no influence on the leading-order inviscid flow approximation because the potential-flow solution can, in any case, only satisfy the kinematic condition u n = 0 at r = 1. Hence the first approximation in the outer part of the domain where the bubble radius is an appropriate characteristic length scale is precisely the same as for the noslip sphere, namely, (10-155) and (10-156). However, this solution does not satisfy the zero-shear-stress condition (10-197) at the bubble surface, and thus it is clear that the inviscid flow equations do not provide a uniformly valid approximation to the Navier-Stokes... 

Another characteristic of turbulent flows is unpredictability, that is the high sensitivity of the solution to very small perturbations that are always present in real physical systems or numerical simulations. This unpredictability, also known as dynamical chaos, is a well known feature of much simpler low-dimensional nonlinear dynamical systems. Although in a strict mathematical sense a unique solution of the Navier-Stokes equation always exists for well-posed initial conditions (at least for large finite times), in practice the details of the forcing and boundary conditions are only known within some approximations and thus the solution in the turbulent regime repre-... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

It is clear from the above discussion that surface properties are extremely important in microscale systems and their importance grows as the characteristic channel dimension decreases. However, there is no straightforward way to take these effects into account, with the models developed to describe this phenomenon being problem specific. Generally they are based on a combination of classical solutions of the Navier-Stokes equations, coupled with ad hoc models of molecular slip flow. Therefore, in the simulation of microchannel flows, it is important to keep in mind that the use of the no-slip boundary condition may not be appropriate and that additional physics may need to be included in the modeling to capture the correct behavior. 

It is well known that the continuum theory in the Navier-Stokes equations only validates when the mean free path of the molecules is smaller than the characteristic length scale of the gas flow. Otherwise, the fluid will no longer be in thermodynamic equilibrium and the linear relationship between the shear stress and rate of shear strain cannot be applied. The commonly used no-slip boundary condition at the fluid-solid interface is not fully valid, and a slip length has to be introduced. 

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