Slip velocity asymptotic

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

Consider now the boundary layer flow over the surface with a droplet EPR of the height h over it Fig. 3.8 illustrates the physical situation. Both the carying and carried media are governed by the system of equations (3.74) determining the two-dimensional fields U(x,z), V(x,z) and u(x,z) in the boundary-layer approach (with r(x,z) = P vt1U )-The non-slip velocity on the surface z = 0, and the asymptotics lim U = along... 

Asymptotic analysis of ESC valid for extreme nonequilibrium conditions yields the following expression for the ESC-related NEO slip velocity [7] ... 

Dynamic filtration in Newtonian fluids. While borehole annular flows are rarely Newtonian (e.g., fluids such as water or air, where viscous stress is linearly proportional to the rate of strain), many drilling fluids are thin and briny, and at times, simply water. Thus, for analysis purposes, the study of Newtonian flows is more than academic. Furthermore, the mathematical simplicity that it offers sheds some insight into the parameters that influence the value of equilibrium cake thickness in the presence of erosive annular flow. Whether our annular flow is Newtonian or power law, concentric or eccentric, it is important to consider two underlying asymptotic fluid-dynamical models. The first applies during small times when borehole fluid enters the formation radially as filtrate, decelerating with time, while the second deals with large times, when invasion rates are so slow that we essentially have classical no-slip velocity boundary conditions. We will first eonsider the small time limit, assuming that the drill pipe does not rotate. 

We developed a unified flow model that can accurately predict the volumetric flowrate, velocity profile, and pressure distribution in the entire Knudsen regime for rectangular ducts. The new model is based on the hypothesis that the velocity distribution remains parabolic in the transition flow regime, which is supported by the asymptotic analysis of the Burnett equations [1]. The general velocity slip boundary condition and the rarefaction correction factor are the two primary components of this unified model. 

---