PoiseuiUe flow

The geometry of the flow is supposed infinite i.e. very large in one direction), such as in the case of a flow around an obstacle, or in a long die. The flow at infinity is assumed to be known (uniform, PoiseuiUe flow,. ..). For computational purposes, one introduces artificial boundaries, at a finite, hopefully not too large, distance. The problem is to define which conditions to impose on the artificial boundaries in order to obtain a solution of the truncated problem, which is as close as possible to the solution of the original problem. 

It is worth noticing that Tlapa and Bernstein [71] have proven that the Squire theorem holds true for the PoiseuiUe flow of m upper-convected Maxwell fluid. It means that any instability, which may be present for three dimensional disturbances, is also present for two dimensional ones at a lower value of the Reynolds number. This property is not true, in general, for non-Newtonian fluids [72]. 

Segre G and Silberberg A, Behavior of microscopic rigid spheres in PoiseuiUe flow. Part 1 and 2, Fluid Mech. 1962 14 115-157. 

Knudsen diffusion [20,30,32,37 ] depending on gas pressure and mean free path in the gas phase applies to pores between 10 A and 500 A in size however, there are examples in the literature where it was observed for much larger pores [41 ]. In this region, the mean free path of molecules in gas phase A is much larger than the pore diameter d. It is common to use the so-called Knudsen number K = X d o characterize the regime of permeation through pores. When 1, viscous (PoiseuiUe) flow is realized. The condition for Knudsen diffusion is 1. An intermediate regime is realized when A), 1. 

The smaller the temperature difference Tna T, the smaller is the smectic order parameter, that is the amplitude of the density wave. Consequently, the permeation coefficient in SmA should decrease upon approaching the SmA-N transition. Indeed, in experiment, very close to Tna the PoiseuiUe flow is observed, as in the nematic phase, but already at Tna T > 0.3 K the plug flow occurs with apparent viscosity two orders of magnitude larger than q. 

It is well known that the essence of field-flow fractionation (FFF) is in the interaction between the distribution of the sample particles in the transversal field and the non-uniformity of the longitudinal flow profile. The classical FFF is realized in the channel with the flow driven by the pressme drop. The flow, in this case, is called PoiseuiUe flow and its profile is parabolic. 

For large values of k, the first two terms in the square brackets are substantially non-zero only in the Debye layer vicinity of the walls, whereas everywhere else the EOF profile is dominated by the last two linear terms in the square brackets. Therefore, the asymmetric EOF profile can be close to trapezoidal or close to triangular depending on the exact values of the zeta-potentials of the walls. If the signs of the zeta-potentials of the walls are different, then the liquid moves in one direction near one wall and in the opposite direction near another wall (this case can be interesting for the preseparation of the particles having different densities). The last term in Eq. 2 corresponds to the pressure-driven PoiseuiUe flow. 

Comparison of R and x values for the flow profiles presented in Fig. 1 for the case of the FFF parameter A -C 1 gives R = 6 and x = 24A for classical FFF with PoiseuiUe flow and R = 2 and x = 8A for FFF with a triangular EOF (Ci = 0). The most interesting result corresponds to the case of FFF with a combined triangular EOF and counterdirected PoiseuUle flow (with... 

Barrat and Bocquet [7] carried out the molecular dynamics simulation of Couette and Poiseuille flows. In Couette flow, the upper wall is moved with a constant velocity and in PoiseuiUe flow an external force drives the flow. Sample results from molecular dynamics simulation are reproduced in Fig. 6. The application of no-slip boundary condition leads to the expected linear and parabolic velocity profile respectively for Couette and Poiseuille flow. However, the velocity profile obtained from molecular dynamics simulation shows a sudden change of velocity in the near-waU region indicating slip flow. The velocity profile for Couette flow away from the solid surface is linear with slope different from that of the no-slip case. The velocity for the slip flow case is higher than that observed for the no-sUp case for Poiseuille flow. For both Couette and PoiseuiUe flow, the partial slip boundary condition at the wall predict similar bulk flow as that observed by molecular dynamics simulation. Some discrepancy in the velocity profile is observed in the near-wall region. 

Telescopic Flow n A picturesque name for laminar flow in a circular tube, derived from visualizing successively smaller cylindrical shells of liquid, from the tube wall toward the center, each moving faster than the next outer one, sliding like the tubes of a sectional telescope. See Laminar Flow and PoiseuiUe Flow. 

Capillary flow PoiseuiUe flow pressure drop Newtonian fluids, wide range of viscosity Simple... 

PoiseuiUe flow exists inside the fiber bore (for pressure drop calculation purposes) ... 

Figure 2.10 A sketch of the geometry for calculating the viscous energy dissipation in a PoiseuiUe flow...

The relaxation of PoiseuiUe flow to obtain an expression for the rate of viscous dissipation of energy, — is presented in the previous section. In this section, let us analyze the steady-state PoiseuiUe flow. Here, the pressure P dSi) to the left-hand side on dS is higher than the pressure P dS2) to the right-hand side on dS2 given as... 

Note that the left-hand side of this equation disappears due to steady-state translation invariance behavior of the PoiseuiUe flow. The first term in the right-hand side of equation (2.100) contributes to the mechanical energy and the second term contributes to the viscous dissipation. 

Figure 4.5 Example of Taylor dispersion in a microchannel with a steady PoiseuiUe flow (a) initial flat concentration of the solute, (b) stretching of the solute to a paraboloid-shaped plug neglecting diffusion, and (c) solute plug with diffusion indicated by vertical arrows...

The equation can also be solved by superimposing an ED flow u ir) and a standard PoiseuiUe flow Up r), with opposite sign for Ap. Note that the superposition procedure works because of the linearity of the N-S equation due to negligible inertial term (V V) V ... 

Segre, G., Silberberg, A. (1961), Radial PoiseuiUe flows of suspensions . Nature, 189, 209. 

We finally return to the solution of Design Problem V (Fig. 6.31). The flow in the die is helical in nature that is, it consists of an axial PoiseuiUe flow and a drag Couette rotational flow due to the rotation of the mandrel. The analysis of the striation thickness of each layer will be based on simple geometrical and kinematical arguments, and it will be shown that the two approaches give the same results. For Newtonian fluids, the axial and angular flow fields are independent and given in Tables 2.7 and Example 6.7, respectively. 

Fig. 6 Velocity field of a fluid near a square cylinder in a PoiseuiUe flow at Reynolds number Re = Vniasf-/v = 30. The channel width is eight times larger than the cylinder size L. A pair of stationary vortices is seen behind the obstacle, as expected for Re < 60. From [81]...

K. Travis, B. D. Todd and D. J. Evans. PoiseuiUe flow of molecular fluids. PhysicaA Stat. Theor. Phys. 240 (1-2), 1997, 315-327. 

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