Boundary conditions Navier slip condition

Here n is the unit normal to the boundary, u and T are the (continuum) velocity and stress, and P is an empirical parameter known as the slip coefficient. The Navier-slip condition says, simply, that there is a degree of slip at a solid boundary that depends on the magnitude of the tangential stress. We note, however, that it is generally accepted that the slip coefficient is usually very small, and then the no-slip condition (2 123) appears as an excellent approximation to (2-124) for all except regions of very high tangential stress. 

Although the Navier-slip condition has been largely ignored for many years in favor of the corresponding no-slip boundary condition, there has been a growing interest in the Navier-slip condition in recent years, even for Newtonian fluids, driven by both new experimental observations and by certain theoretical problems that arise from application of the no-slip condition. The best known of the theoretical problems arises when a contact line separating the two immiscible fluids moves with velocity - U along a solid surface at which the no-slip condition is assumed to apply, as sketched in Fig. 2-11. 

The Navier-Stokes equation is one of the basic governing equations for study of fluid flow related to various disciplines of engineering and sciences. It is a partial differential equation whose integration leads to the appearance of some constants. These constants need to be evaluated for exact solutions of the flow field, which are obtained by imposing suitable boundary conditions. These boundary conditions have been proposed based on physical observation or theoretical analysis. One of the important boundary conditions is the no-slip condition, which states that the velocity of the fluid at the boundary is the same as that of the boundary. Accordingly, the velocity of the fluid adjacent to the wall is zero if the boundary surface is stationary and it is equal to the velocity of the surface if the surface is moving. This boundary condition is successful... 

Clearly then, the continuum approach as outlined above is faulty. Furthermore, since our erroneous result depends only on the non-slip boundary condition for V together with the Navier-Stokes equation (4.3), one or... 

E. Bansch, B. Hdhn. Numerical treatment of the Navier-Stokes equations with slip-boundary condition. Preprint 9-98, Mathematische Fakultat Freiburg. SIAM J Sci Comput (submitted). 

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure boundary condition and two velocity boundary conditions (for each velocity component) to completely specify the solution. The no slip condition, which requires that the fluid velocity equal the velocity of any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather... 

P 7] The topic has only been treated theoretically so far [28], A mathematical model was set up slip boundary conditions were used and the Navier-Stokes equation was solved to obtain two-dimensional electroosmotic flows for various distributions of the C, potential. The flow field was determined analytically using a Fourier series to allow one tracking of passive tracer particles for flow visualization. It was chosen to study the asymptotic behavior of the series components to overcome the limits of Fourier series with regard to slow convergence. In this way, with only a few terms highly accurate solutions are yielded. Then, alternation between two flow fields is used to induce chaotic advection. This is achieved by periodic alteration of the electrodes potentials. 

P 21] The mixing of gaseous methanol and oxygen was simulated. The equations applied for the calculation were based on the Navier-Stokes (pressure and velocity) and the species convection-diffusion equation [57]. As the diffusivity value for the binary gas mixture 2.8 x 10 m2 s 1 was taken. The flow was laminar in all cases adiabatic conditions were applied at the domain boundaries. Compressibility and slip effects were taken into account The inlet temperature was set to 400 K. The total number of cells was —17 000 in all cases. 

Since the Navier s slip hypothesis of the last century, most experiments have failed to obtain positive evidence for a slip boundary condition on macroscopic scales in low molar mass liquids. However, Navier s notion of slip turns out to be extremely useful and convenient for the latest description of flow anomalies of highly entangled polymer melts including linear polyethylenes (LPE). The ability of a melt/solid interface to possess two profoundly different states as shown by Fig. 4a,b clearly reveals the potential role of interfacial slip in governing various melt flow phenomena in high pressure extrusion. Before reviewing recent experimental studies that have elucidated the molecular origins of different flow... 

In addition to the limitations of the continuum approaches in being able to accurately represent transport processes under strongly nonequilibrium conditions, the formulation of physically meaningful boundary conditions may also be problematic. For the Euler equations, the boundary conditions at the vehicle surface must be adiabatic for energy and no slip for momentum. Use of the Navier-Stokes equations allows stipulation of isothermal temperature and slip velocity conditions. However, under strongly nonequilibrium conditions, these boundary conditions will fail to reproduce the physical behavior accurately. The situation for the Burnett equations is even worse since the required boundary conditions must include second order effects. 

Another study, [17], used helium as their working fluid and carried out the experiments in 51.25 X 1.33 micrometer microchannels. They showed that, as long as the Knudsen number is in the slip flow range, the Navier-Stokes equations are still applicable and the discontinuities at the boundaries need to be represented by the appropriate boundary conditions. They obtained the following formula for the mass flow rate including the slip effects... 

More reeently, [26] has eonfirmed the need to include the second order slip condition at higher Kn number values. Their work was both theoretical and experimental using nitrogen and helium in a silicon channels. They used the second order slip approximation to obtain the equation for the volumetric flow rate and related it to the ratio of inlet to outlet pressure. It was shown that when using the Navier-Stokes equation, the boundary conditions must be modified to include second order slip terms as the Knudsen number increases. They also studied in depth the accommodation coefficient Fv and verified the need for further study. It was shown that as the Knudsen number increases, the momentum accommodation value deviates further and further from unity for instance Kn -0.5 yields Fv 0.8 for helium. The values found for nitrogen were quite similar. The measurements agreed with past studies such as [11] for lower Kn. 

Whilst the Navier-Stokes equation (3.98) is of second order, Euler s equation (3.103) only contains first order terms. As its order is one lower than the Navier-Stokes equation, after integration one less boundary condition can be satisfied. As a result of this, the no-slip condition, zero velocity at the wall, cannot be satisfied. Rather a finite velocity at the wall is obtained, as the absence of friction was presumed, whilst in real flows the velocity is zero at the wall. 

The problem considered here differs from the canonical problem by the presence of a source term (i.e. the force) on the right-hand side of the complete Navier—Stokes equations (3.29). This force vanishes outside the EPR, for z (h, 1 - h), is opposite to the local flow direction, and is proportional to some power of its velocity (here, we consider the linear or quadratic law). The boundary condition at the entrance x = 0 is evident, U = 1, V = 0 (homogeneous velocity distribution). There are non-slip conditions on the walls z = 0 and z = 1. The further formulation of the problem is somewhat different for linear and quadratic EPRs. 

For many purposes, we will find that antiplane shear problems in which there is only one nonzero component of the displacement field are the most mathematically transparent. In the context of dislocations, this leads us to first undertake an analysis of the straight screw dislocation in which the slip direction is parallel to the dislocation line itself. In particular, we consider a dislocation along the X3-direction (i.e. = (001)) characterized by a displacement field Usixi, X2). The Burgers vector is of the form b = (0, 0, b). Our present aim is to deduce the equilibrium fields associated with such a dislocation which we seek by recourse to the Navier equations. For the situation of interest here, the Navier equations given in eqn (2.55) simplify to the Laplace equation (V ms = 0) in the unknown three-component of displacement. Our statement of equilibrium is supplemented by the boundary condition that for xi > 0, the jump in the displacement field be equal to the Burgers vector (i.e. Usixi, O" ") — M3(xi, 0 ) = b). Our notation usixi, 0+) means that the field M3 is to be evaluated just above the slip plane (i.e. X2 = e). 

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. 

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... 

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