Hydrodynamic boundary layer, thickness

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Figure 8. Variation of the hydrodynamic boundary layer thickness (So, equation (26), continuous line), the diffusion layer thickness (<5,-, equation (34), dotted line) and the ensuing local flux (/, equation (32), dashed line) with respect to the distance from the leading edge (y) in the case of laminar flow parallel to an active plane (the surface is a sink for species i). Parameters /), = 10-9nrs, v= 10 3ms, c = lmolm-3, and v — 10-6 m2 s 1. Notice that c5, o (as required for the derivation of the flux equation (32)), and that the flux decreases when <5, increases...

The boundary layer equations may be obtained from the equations provided in Tables 6.1-6.3, with simplification and by an order-of-magnitude study of each term in the equations. It is assumed that the main flow is in the x direction. The terms that are too small are neglected. Consider the momentum and energy equations for the two-dimensional, steady flow of an incompressible fluid with constant properties. The dimensionless equations are given by Eqs. (6.46) to (6.48). The principal assumption made in the boundary layer is that the hydrodynamic boundary layer thickness 8 and the thermal boundaiy layer thickness 8t are small compared to a characteristic dimension L of the body. In mathematical terms,... 

Use these assumptions to find a relation between the hydrodynamic boundary layer thickness 8 and the thermal boundary layer thickness 8,. For gases, Pr is of the order of unity. For liquids, Pr ranges from about 10 to 1000. For liquid metals, Pr ranges from about 0.003 to about 0.03. Deduce the relative thicknesses of 8 and 8, for gases, liquids and liquid metals. 

When a fluid flow of constant velocity, u, impinges parallel to the edge of a plate, the boundary condition is such that the fluid velocity is zero on the surface of the plate. This results in the formation of a hydrodynamic boundary layer in which the flow velocity parallel to the surface varies with distance normal to the surface. The hydrodynamic boundary layer thickness increases with distance, jc, from the upstream edge of the plate as given by equation (10.5) [7] ... 

For sufficiently large electrodes with a small vibration amplitude, aid < 1, a solution of the hydrodynamic problem is possible [58, 59]. As well as the periodic flow pattern, a steady secondary flow is induced as a consequence of the interaction of viscous and inertial effects in the boundary layer [13] as shown in Fig. 10.10. It is this flow which causes the enhancement of mass-transfer. The theory developed by Schlichting [13] and Jameson [58] applies when the time of oscillation, w l is small in comparison with the time taken for a species to diffuse across the hydrodynamic boundary layer (thickness SH= (v/a>)ln diffusion timescale 8h/D), i.e., when v/D t> 1. Re needs to be sufficiently high for the calculation to converge but sufficiently low such that the flow does not become turbulent. Experiment shows that, for large diameter wires (radius, r, — 1 cm), the condition is Re 2000. The solution Sh = 0.746Re1/2 Sc1/3(a/r)1/6, where Sh (the Sherwood number) = kmr/D and km is the mass-transfer coefficient,... 

Substituting for the hydrodynamic-boundary-layer thickness from Eq. (5-21) and using Eq. (5-36) gives... 

To show how one might proceed to analyze a new problem to obtain an important functional relationship from the differential equations, consider the problem of determining the hydrodynamic-boundary-layer thickness for flow over a flat plate. This problem was solved in Chap. 5, but we now wish to make an order-of-magnitude analysis of the differential equations to obtain the functional form of the solution. The momentum equation... 

Let us first consider the simple flat plate with a liquid metal flowing across it. The Prandtl number for liquid metals is very low, of the order of 0.01. so that the thermal-boundary-layer thickness should be substantially larger than the hydrodynamic-boundary-layer-thickness. The situation results from the high values of thermal conductivity for liquid metals and is depicted in Fig. 6-15. Since the ratio of 8/8, is small, the velocity profile has a very blunt shape over most of the thermal boundary layer. As a first approximation, then, we might assume a slug-flow model for calculation of the heat transfer i.e., we take... 

Consider one side of a disk whose diameter is large compared to 6, rotating in a large vessel of liquid. Since liquid rotates with the disk and acquires a radial as well as an angular motion, it must also flow toward the face of the disk. The velocity of the perpendicular flow can be considered independent of distance except very near the disk. The distance at which flow becomes essentially parallel to the disk can be called the hydrodynamic boundary layer thickness. The distance at which a concentration gradient begins was calculated by Levich (7) to be... 

Another factor that does not depend on power and is shared by DI water and SCI cleaning is the acoustic boundary layer thickness, which is a function of frequency and viscosity only. The boundary layer decreases from thousands of micrometers to a fraction of a micrometer when megasonics is applied at any power. The acoustic boundary layer thickness is inversely proportional to the frequency and directly proportional to the viscosity of the fluid. For example, a flow with a velocity of 4 m/s (maximum streaming velocity for the considered equipment) has a hydrodynamic boundary layer thickness of 1500 pm at the center of the wafer. By contrast, the acoustic boundary layer thickness on a wafer in a 850-kHz megasonic cleaning tank is 0.61 pm. The effect of a very... 

Both qualitative and quantitative insight can be garnered from transient X -i, i-t and r -t measurements in quiescent or stirred solutions, while measurements of steady-state behavior are best performed under well-defined hydrodynamic conditions. Typically, a rotating disc electrode (RDE), or a related method, is used to specify and/or modulate the hydrodynamic boundary layer thickness, 8. With an RDE the boundary layer is specified by... 

The effect of the inhibitor coverage on the deposition kinetics can also be mapped by referencing the additive-perturbed deposition kinetics to the additive-free case, in accord with Equation 2.1 [58]. Recent results analyzed in this manner are shown in Figure 2.12, where the coverage is a distinct function of the additive concentration in the bulk electrolyte and the hydrodynamic boundary layer thickness [58], The dotted lines are simulations based on a steady-state model [57] that considers metal deposition to proceed through an adsorbed intermediate M+ads, which competes with coumarin for available surface sites ... 

Collision efficiency was calculated by the method proposed for the first time by Dukhin Derjaguin (1958). To calculate the integral in Eq. (10.25) it is necessary to know the distribution of the radial velocity of particles whose centre are located at a distance equal to their radius from the bubble surface. The latter is presented as superposition of the rate of particle sedimentation on a bubble surface and radial components of liquid velocity calculated for the position of particle centres. Such an approximation is possibly true for moderate Reynolds numbers until the boundary hydrodynamic layer arises. At a particle size commensurable with the hydrodynamic layer thickness, the differential of the radial liquid velocity at a distance equal to the particle diameter is a double liquid velocity which corresponds to the position of the particle centre. Such a situation radically differs from the situation at Reynolds numbers of the order of unity and less when the velocity in the hydrodynamic field of a bubble varies at a distance of the order ab ap. At a distance of the order of the particle diameter it varies by less than about 10%. Just for such conditions the identification of particle velocity and liquid local velocity was proposed and seems to be sufficiently exact. In situations of commensurability of the size of particle and hydrodynamic boundary layer thickness at strongly retarded surface such identification leads to an error and nothing is known about its magnitude. 

FIGURE 18.17 Depletion layer at an electrode surface in the absence (a) and presence (b) of external magnetic field applied perpendicular to the diffusional flux. J is the diffusional flux of the substrate, Cei is the substrate concentrations at the electrode surface, 5 and do are the Nemst diffusion layer thickness and hydrodynamic boundary layer thickness, respectively, and Uq, is the fluid velocity on the outer edge of the hydrodynamic boundary layer. (Adapted with permission from Ref. [118]. Copyright 2004, American Chemical Society.)... 

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