Parabolic profile

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

Compared with the parabolic profile for a Newtonian fluid (n = 1), the profile is flatter for a shear-thinning fluid ( < 1) and sharper for a shear-thickening fluid (n > l). The ratio of the centre line (uCl) to mean (k) velocity, calculated from equation 3.133, is ... 

Toothpaste flow is an extreme example of non-Newtonian flow. Problem 8.2 gives a more typical example. Molten polymers have velocity profiles that are flattened compared with the parabolic distribution. Calculations that assume a parabolic profile will be conservative in the sense that they will predict a lower conversion than would be predicted for the actual profile. The changes in velocity profile due to variations in temperature and composition are normally much more important than the fairly subtle effects due to non-Newtonian behavior. 

Figure 4.8. A stack of refracting concave beryllium lenses with parabolic profile (ESRF, France and RWTH Aachen, Germany)... 

Measurements by FIA occur under conditions where laminar flow predominates over turbulent flow (Fig. 3, a and b) and hence a parabolic profile of the concentration of analyte solution inside the carrier stream is developed. The layers of the analyte that are adjacent to the inner surface of the transportation tube flow slowly owing to the friction forces developed between these two different... 

In Figure 26c and 26d some more details are plotted for the PE-modified lipid. On the PE chain there are some features deviating from the parabolic profile near the lipid-water interface. We should realise that in the membrane... 

Via a simple integration of this parabolic profile, one can find a relationship between v and the effective flow rate vf (volume of liquid crossing any section of the channel per unit of time) ... 

An important feature of the EOF is its flat flow profile, compared to the parabolic profile of hydrodynamic flows (Fig. 17.4). The reason for this characteristic is that the EOF originates almost at the wall of the capillary, owing to the extremely small size of the double layer... 

The simulations and theoretical approaches reveal that the density profile is somewhat lost in the presence of polydispersity [193,202], and special features (including a kink in the profile) are observed for a bimodal chain distribution [195,203 ]. The profiles are flatter for the highest grafting densities, due to the influence of high-order terms. Laradji et al. [204] performed an MC simulation with an off-lattice model that uses the Hamiltonian employed in the SCF calculations (i.e., it only considers binary interactions). The parabolic profiles are... 

However, deviations from the parabolic profile become progressively important as the length of the polymers N or the grafting density pa decreases. In a systematic derivation of the mean-field theory for Gaussian brushes [52] it was shown that the mean-field theory is characterized by a single parameter, namely the stretching parameter fi. In the limit p oo, the difference between the classical approximation and the mean-field theory vanishes, and one obtains the parabolic density profile. For finite /3 the full mean-field the-... 

Fig. 3 Self-consistent mean-field results for the density profile (normalized to unity) of a brush for different values of the interaction parameter f). In a the distance from the grafting surface is rescaled by the scahng prediction for the brush height, h, and in b it is rescaled by the unperturbed polymer radius Rq. As increases, the density profiles approach the parabolic profile (shown as dashed lines)...

The classical FEE retention equation (see Equation 12.11) does not apply to ThEEE since relevant physicochemical parameters—affecting both flow profile and analyte concentration distribution in the channel cross section—are temperature dependent and thus not constant in the channel cross-sectional area. Inside the channel, the flow of solvent carrier follows a distorted, parabolic flow profile because of the changing values of the carrier properties along the channel thickness (density, viscosity, and thermal conductivity). Under these conditions, the concentration profile differs from the exponential profile since the velocity profile is strongly distorted with respect to the parabolic profile. By taking into account these effects, the ThEEE retention equation (see Equation 12.11) becomes ... 

Clearly, in the absence of a radial temperature or velocity gradient, no radial mass transfer can exist unless, of course, a reaction occurs at the bed wall. When a system is adiabatic, a radial temperature and concentration gradient cannot exist unless a severe radial velocity variation is encountered (Carberry, 1976). Radial variations in fluid velocity can be due to the nature of flow, e.g. in laminar flow, and in the case of radial variations in void fraction. In general, an average radial velocity independent of radial position can be assumed, except from pathological cases such as in very low Reynolds numbers (laminar flow), where a parabolic profile might be anticipated. 

Figure 8.11—Effecl of diffusion on the efficiency obtained in HPLC and CE. Diffusion increases with the square of tube diameter. This is, thus, more important in HPLC. In CE. the electrolyte is repelled by the wall leading to an almost perfect plane-like flow contrary to the usual parabolic profile obtained under hydrodynamic flow. However, other factors that depend on the difference in conductivity between the electrolyte and solutes can lead to peak deformation.

It is well known that in turbulent pipe flow the parabolic profile present in laminar flow becomes blunter, so that the ratio uma,x/u decreases. A similar effect has been found for the relatively deep flows in open channels at small slopes by Jeffreys (J4), who obtained values of us/u down to 1.06, and by Horton et al. (H19), who measured values as low as 1.1. It can be expected that in the flow of thin films the ratio will decrease in turbulent flow from the value of 1.5, but by a very much smaller amount than observed in the deep flows noted above. 

In many situations, the system is primed with a buffer solution which is displaced by the protein solution of interest (Fig. 5 a). Assuming constant, laminar established flow, the velocity (V) in a rectangular flow channel of width (w), thickness (b), and length (1), where b 4 w has a characteristic parabolic profile, given by 36)... 

In his analysis of the effect of diffusion on an open-tube distillation column Westhaver (1942) came up with the apparent diffusion coefficient 11 a2 /2/ 48D, and since he assumes a parabolic profile it is at first surprising that this should differ by a factor of 11 from Taylor s result. It appears, however, if the more general problem in which the solute can be retained on the wall be considered, that the value of k varies continuously from to is as the fraction of solute held on the wall varies from 1 to 0. This result is implicit in Golay s analysis of the tubular chromatographic column (Golay 1958). He considers the stationary phase of the column as a very thin retentive layer held on the wall and derives an expression for the dispersion by arguments entirely analogous to Taylor s. He has also discussed the effect of diffusion in the retentive layer. 

For a first approximation, when we assume a Poiseuille parabolic profile, the necessary length Lp is two times larger ... 

By an analogous argument, the laminar flow in a circular pipe of radius ro gives the following velocity parabolic profile ... 

The residence time of such a well is again best visible at the exit. The parabolic profile this time is much wider than for the structured case. The maximum relative deviation amounts to 233%, which is 6.5 times larger than for the structured well. This is important because it demonstrates that micro structures are indeed a means to obtain a narrow overall residence time distribution. The error introduced by manufacturing tolerances (estimated 5 pm absolute tolerance in a 320 pm wide channel) is 1.6% in width, a value which does not influence this evaluation. 

As the flow accelerates into the gaps around the cylinder, it possesses a greater relative amount of extension. Ultimately, at distances far downstream from the cylinder, the flow is expected to relax back toward a parabolic profile. In these plots, the symbols represent the measured velocities and the solid curves are the results of a finite element, numerical simulation. The constitutive equation used was a four constant, Phan-Thien-Tanner mod-el[193], which was adjusted to fit steady, simple shear flow shear and first normal stress difference measurements. The fit to the velocity data is very satisfactory. 

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