Laminar translator

Bubble Laminar translational flow at high Reynolds numbers rr U = U is the fluid velocity at infinity... 

This is valid for any Newtonian fluid in any (circular) pipe of any size (scale) under given dynamic conditions (e.g., laminar or turbulent). Thus, if the values of jV3 (i.e., the Reynolds number 7VRe) and /V, (e/D) for an experimental model are identical to the values for a full-scale system, it follows that the value of N6 (the friction factor) must also be the same in the two systems. In such a case the model is said to be dynamically similar to the full-scale (field) system, and measurements of the variables in N6 can be translated (scaled) directly from the model to the field system. In other words, the equality between the groups /V3 (7VRc) and N (e/D) in the model and in the field is a necessary condition for the dynamic similarity of the two systems. 

Common geometries used to make viscosity measurements over a range of shear rates are Couette, concentric cylinder, or cup and bob systems. The gap between the two cylinders is usually small so that a constant shear rate can be assumed at all points in the gap. When the liquid is in laminar flow, any small element of the liquid moves along lines of constant velocity known as streamlines. The translational velocity of the element is the same as that of the streamline at its centre. There is of course a velocity difference across the element equal to the shear rate and this shearing action means that there is a rotational or vorticity component to the flow field which is numerically equal to the shear rate/2. The geometry is shown in Figure 1.7. 

To explain the Green function method for the formulation of Dx, D and D, of the fuzzy cylinder [19], we first consider the transverse diffusion process of a test fuzzy cylinder in the solution. As in the case of rodlike polymers [107], we imagine two hypothetical planes which are perpendicular to the axis of the cylinder and touch the bases of the cylinder (see Fig. 15a). The two planes move and rotate as the cylinder moves longitudinally and rotationally. Thus, we can consider the motion of the cylinder to be restricted to transverse diffusion inside the laminar region between the two planes. When some other fuzzy cylinders enter this laminar region, they may hinder the transverse diffusion of the test cylinder. When the test fuzzy cylinder and the portions of such other cylinders are projected onto one of the hypothetical planes, the transverse diffusion process of the test cylinder appears as a two-dimensional translational diffusion of a circle (the projection of the test cylinder) hindered by ribbon-like obstacles (cf. Fig. 15a). 

Limitation of a reaction by translational diffusion in solution is a rather rare case. Much more frequently the limitation of the observed overall reaction rate is by external mass transfer (through a laminar film around a solid macroscopic carrier) (Chapter 5, Section 5.5.1) or internal mass transfer (diffusion of substrate or product through the pores of a solid carrier or a gel network to an enzyme molecule in the interior of the carrier) (Chapter 5, Section 5.5.2). 

Viscous transport is laminar or streamline, if molecules in the (imaginary) planes of fluid in any geometry, flowing along a velocity gradient, do so with the same translational and rotational velocities. Streamlines are occa-... 

The flow cell translates time into distance and the combination of the three and varying the flow rates gave a range of observations from 0 to 30 s. SHG measurements of the static aqueous/dodecane interface were made at each port before and after the flow experiment to calibrate the observations from each port For a laminar (non-turbulent) flow, the two flow rates should be in the inverse ratio of the fluid viscosities this ratio for dodecane on water is 0.65 at 25°C, very close to the observed flow rate ratio of 0.67. The bulk flow rates for each liquid were measured by collecting the volume of liquid flowing in a known time. Since the cell operates under non-turbulent conditions, the velocity of each layer at the interface must be the same, but the average velocities of the two layers are different. Ideally a model of the flow conditions inside the cell would be used to accurately determine the velocity of the interface. Since this was not... 

D. Magde, W.W. Webb, E.L. Elson, Fluorescence correlation spectroscopy. 111. Uniform Translation and Laminar Flow. Biopolymers 17, 361-376 (1978)... 

Theoretical flow equations were derived for the molecular flow region by Knudsen (Kl) as far back as 1909. These equations for molecular flow and Poiseuille s Law for laminar flow, were the basis for vacuum flow computation until the later years of World War II. Normand (Nl) was the first to translate these equations into practical forms for engineering applications. In this reference Normand gives useful empirical rules for applying Knudsen s equations to ducts of rectangular cross-section, non-uniform cross-section, baffles, elbows, etc. 

In a laminar flow at a definite shear rate, different parts of the polymer molecule move at different rates depending on whether they are in the zone of rapid or relatively slow flow, and as a result the polymer molecule is under the action of a couple of forces which makes it rotate in the flow. Rotation and translational movement of polymer molecules causes friction between their chain segments and the solvent molecules. This is manifested in an increase in viscosity of the solution compared to the viscosity of the pure solvent. 

Problem 10-9. Translating Flat Plate. Consider the high-Reynolds-number laminar boundary-layer flow over a semi-infinite flat plate that is moving parallel to its surface at a constant speed (7 in an otherwise quiescent fluid. Obtain the boundary-layer equations and the similarity transformation for f (r ). Is the solution the same as for uniform flow past a semi-infinite stationary plate Why or why not Obtain the solution for f (this must be done numerically). If the plate were truly semi-infinite, would there be a steady solution at any finite time (Hint. If you go far downstream from the leading edge of the flat plate, the problem looks like the Rayleigh problem from Chap. 3). For an arbitrarily chosen time T, what is the regime of validity of the boundary-layer solution ... 

Formula (5.6.4) is valid for an arbitrary laminar flow without closed streamlines for particles and drops of an arbitrary shape. The quantity Sh(l,Pe) corresponds to the asymptotic solution of the linear problem (5.6.1) at Pe > 1. For spherical particles, drops, and bubbles in a translational or linear straining shear flow, the values of Sh(l, Pe) are shown in the fourth column in Table 4.7. 

The term chaotic mixing was introduced by Ottino [40] do describe laminar, distributive mixing with continuous or periodic translation of cavity walls. This flow stretches, folds, and transports a drop more effectively than a steady-state translation. The experiments also showed presence of mixing islands where little mixing took place. 

Many technologies have been developed to monitor or measure fluid flow in microdevices. The underlying mechanisms of these devices are based mainly on thermal or mechanical principles examples of the variables to be measured are temperature, differential pressure, and drag force, which translate to thermal changes, deflection of cantilever beams, and shear strain, respectively. Most of these microflow sensors are manufactured by microelec-tromechanical systems (MEMS) processes without moving parts, and the flow rate measurements are mainly translated from velocity detection. For example, because flows in microchannels are in most cases laminar, the pressure drop Ap along a channel can be expressed as follows ... 

In the case of confined channels, relation (1.43) is just the translation of the usual pressure drop over a distance z for forced laminar flows to the case of capillary flows. 

Magde D, Webb WW, Elson EL (1978) FluOTescence correlation spectroscopy. 3. Uniform translation and laminar-flow. Biopolymers 17(2) 361-376... 

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