Zero wall slip

At the walls of the pipe, that is where s — r, the velocity ux must be zero in order to satisfy the condition of zero wall slip. Substituting the value ux = 0, when s — r. then ... 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

The velocity profile during slip flow in a cylindrical tube is shown in Figure 21. As in conventional fluid flow, the flow velocity in the z direction, u(r), is parabolic, but rather than reach zero at the tube wall, slip occurs, and the velocity at the wall is greater than zero. The velocity does not reach zero until distance h from the wall surface. The derivation of the mass flux equation proceeds along the same lines as the derivation of Poiseuille s law in conventional hydrodynamics, but in slip flow, u(r) = 0 at r = a + h instead of reaching zero at r = a. 

Wall-slip is not an easy phenomenon to detect. Although in principle, the velocity profile should reveal whether or not the fluid velocity is zero at the stationary wall, in reality determining the velocity profile with sufficient resolution near the wall is very difficult. So alternate means, e.g., checking for viscosity variation with appropriate changes in the test geometry, are also widely used in practice. 

A general conclusion that can be made after analyzing the flow behavior of these diverse materials is that wall-slip is affected by pressure. The onset of slip occurs at a certain applied stress (slip velocity must be zero at zero flow) and slip effects are lost above a certain higher stress. It is possible that the onset and the disappearance stress levels for wall-slip could be material specific for a given flow surface conditions. However, little work has been reported using wall-slip as a material characterization parameter. 

It is noted that there is no work associated with the wall stresses since the velocity at the wall is zero (no-slip assumption) [25] [13]. 

Some modifications of the melt flow behavior of thermoplastics that can be observed depending on filler concentration are a yield-like behavior (i.e., in these cases, there is no flow until a finite value of the stress is reached), a reduction in die swell, a decrease of the shear rate value where nonlinear flow takes place, and wall slip or nearwall slip flow behavior [14, 27, 46]. Other reported effects of flllers on the rheology of molten polymers are an increase of both the shear thinning behavior and the zero-shear-rate viscosity with the filler loading and a decrease in the dependence of the filler on viscosity near the glass transition temperature [18, 47-49]. 

Lemon Cake Batter. As another sample we used ready made lemon cake batter. As can be seen from Fig. 5 (left) the flow behaviour differs from the olive oil, hi velocities are more probable than low ones. The probability at velocity zero is almost zero which could be interpreted as a wall slip (/(O) 0 s/mm, cf. Fig. 5, left). The NMR data and the rotational rheometer results for the batter are in reasonable agreement. The increase in viscosity showing up in the classical measurement may possibly be attributed to drying of the sample. The mean velocities calculated from NMR-data (4.4 mm s ) and adjusted at the pump (4.3 mm s" ) are in good agreement. 

S Starch Solution. In Fig. 6 the velocity (cf Fig. 6, left) as well as the viscosity (cf Fig. 6, right) data is shown, exhibiting pronounced wall-slip behaviour ftft) 0 s/mm) as there is essentially no contribution of velocities near zero (cf Fig. 6 left). The calculated average velocity is 2.2 mm/s, evaluation of the NMR-data results in a mean velocity of... 

Fig. 5 Left VPDF of Lemon cake batter showing increased probability at high velocities. The function shows almost zero contribution at w=0, which is a hint to wall slip behaviour. Right Viscosity as a function of shear rate exhibiting shear thinning of the cake batter - the viscosity decreases with increasing shear rate. NMR (o) and rotational rheometer ( ) data.

The slip approach defines the onset of surface defects as related to a slippage of the polymer at the die wall. Accordingly, the zero-wall velocity concept, as used in the constitutive approach, does not apply. Many attempts have been made to experimentally measure the velocity at the wall, however, with mixed success. Still, theories have been developed which support a shear-dependent slippage at a polymer/ solid interface. Slippage may be understood as the buildup of a layer of polymer molecules bonded to the die surface, which undergo a coil-stretch transition under the shear stress. In light of this understanding, the microscopic shp layer development may be in line with the possible formation of a macroscopic surface layer as a consequence of a many... 

For Re > 10000, we have turbulent flow with an almost constant velocity in the main part of the tube and a small laminar sublayer near the wall where the velocity goes down to zero (no-slip condition). For 2300 < Re < 10000, we have a transition regime. Any disturbances, for example, surface roughness or particles of a packed bed, lead to a lower value of the critical Re number. 

The walls are assumed to be solid and impermeable so that flow velocities are zero. No slip conditions are employed at the boundary walls. These wall boundary conditions are implemented using the near-wall approach of Launder and Spalding [54] for the momentum and scalar transport equations. The method assumes that near the wall, Couette flow prevails and the velocity profile obeys the universal logarithmic law. Therefore, the log-law is generally employed to calculate the velocity component parallel to the wall thus... 

On no-slip walls zero velocity components can be readily imposed as the required boundary conditions (v = v, = 0 on F3 in the domain shown in Figure 3.3). Details of the imposition of slip-wall boundary conditions are explained later in Section 4.2. 

In fluid dynamics it is generally assumed that the velocity of flow at a solid boundary, such as a pipe wall, is zero. This is referred to as the no-slip condition. If the fluid wets the surface, this assumption can be justified in physical terms since the molecules are... 

Both F 0) and F R) vanish for a velocity profile with zero slip at the wall. The mixing-cup average is determined when the integral of F(r) is normalized by Q = 7tR u. There is merit in using the trapezoidal rule to calculate Q by integrating dQ = InrVzdr. Errors tend to cancel when the ratio is taken. 

Note that pressure is treated as a function of z alone. This is consistent with the assumption of negligible Vr- Equation (8.63) is subject to the boundary conditions of radial symmetry, dVJdr = 0 at r = 0, and zero slip at the wall, Fz = 0 SLtr = R. 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

At the solid walls, the boundary conditions state that the velocity is zero (i.e. no slip). Also at the walls, the temperature is either fixed or a zero-gradient condition is applied. At the surface of the spinning disk the gas moves with the disk velocity and it has the disk temperature, which is constant. The inlet fiow is considered a plug fiow of fixed temperature, and the outlet is modeled by a zero gradient condition on all dependent variables, except pressure, which is determined from the solution. 

No slip Is used as the velocity boundary conditions at all walls. Actually there Is a finite normal velocity at the deposition surface, but It Is Insignificant In the case of dilute reactants. The Inlet flow Is assumed to be Polseullle flow while zero stresses are specified at the reactor exit. The boundary conditions for the temperature play a central role in CVD reactor behavior. Here we employ Idealized boundary conditions In the absence of detailed heat transfer modelling of an actual reactor. Two wall conditions will be considered (1) adiabatic side walls, l.e. dT/dn = 0, and (11) fixed side wall temperatures corresponding to cooled reactor walls. For the reactive species, no net normal flux Is specified on nonreacting surfaces. At substrate surface, the flux of the Tth species equals the rate of reaction of 1 In n surface reactions, l.e. 

Apparently, the slip length diverges when 0. In practice, the shear strain in Eq. (5) will approach zero in such a case, thus leaving the velocity jump finite. Several experimental results on cf have been reported [6, 7], most of them indicating values between 0.8 and 1.0, compatible with rough walls. 

In Figure 2.2 DSMC results of Karniadakis and Beskok [2] and results obtained with the linearized Boltzmann equation are compared for channel flow in the transition regime. The velocity profiles at two different Knudsen numbers are shown. Apparently, the two results match very well. The fact that the velocity does not reach a zero value at the channel walls (Y = 0 and Y = 1) indicates the velocity slip due to rarefaction which increases at higher Knudsen numbers. 

Simulations are then performed for gas bubbles emerging from a single nozzle with 0.4 cm I.D. at an average nozzle velocity of lOcm/s. The experimental measurements of inlet gas injection velocity in the nozzle using an FMA3306 gas flow meter reveals an inlet velocity fluctuation of 3-15% of the mean inlet velocity. A fluctuation of 10% is imposed on the gas velocity for the nozzle to represent the fluctuating nature of the inlet gas velocities. The initial velocity of the liquid is set as zero. An inflow condition and an outflow condition are assumed for the bottom wall and the top walls, respectively, with the free-slip boundary condition for the side walls. 

The occurrence of slip invalidates all normal analyses because they assume that the velocity is zero at the wall. Returning to the Rabinowitsch-Mooney analysis, the total volumetric flow rate for laminar flow in a pipe is given by... 

Since the radial domain in the duct ranges between 0 < r < rw, the constant Ci must be zero. Otherwise, the velocity would become unbounded at the centerline. The other constant is determined easily from a no-slip condition at the wall, rw. The solution is... 

The entry-length region is characterized by a diffusive process wherein the flow must adjust to the zero-velocity no-slip condition on the wall. A momentum boundary layer grows out from the wall, with velocities near the wall being retarded relative to the uniform inlet velocity and velocities near the centerline being accelerated to maintain mass continuity. In steady state, this behavior is described by the coupled effects of the mass continuity and axial momentum equations. For a constant-viscosity fluid,... 

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