Dimensionless number Stokes

CCT, critical cracking thickness Boltzmann constant (1.381x10 local permeability [m ] fracture resistance [N m ] average permeability in/of compact [m ] particle shape factor compact thickness [m] initial particle number concentration [m refractive index of particle material refractive index of dispersion material number density of ion i dimensionless number dimensionless number Stokes number Peclet number capillary pressure [N-m ] dynamic pressure [N m ] local liquid pressure in the compact [N-m local solid pressure in the compact [N-m ] superficial fluid velocity [m-s q gas constant [J K ] centre to centre distance [m]... 

The dimensionless numbers are important elements in the performance of model experiments, and they are determined by the normalizing procedure ot the independent variables. If, for example, free convection is considered in a room without ventilation, it is not possible to normalize the velocities by a supply velocity Uq. The normalized velocity can be defined by m u f po //ao where f, is the height of a cold or a hot surface. The Grashof number, Gr, will then appear in the buoyancy term in the Navier-Stokes equation (AT is the temperature difference between the hot and the cold surface) ... 

Before discussing the on.set, and nature, of fluid turbulence, it is convenient to first recast the Navier-Stokes equations into a dimensionless form, a trick first used by Reynolds in his pioneering experimental work in the 1880 s. In this form, the Navier-Stokes equations depend on a single dimensionless number called Reynolds number, and fluid behavior from smooth, or laminar, flow to chaos, or turbulence,... 

Navier-Stokes equations to the swirl reduction experiments and isolate dimensionless numbers that can be used to apply the experiments to various pipes and discharges. 

If the particles are small, and their terminal velocity can be expressed by Stokes Law, u, = dp(ps — pf)g/p, then the dimensionless number is simplified to... 

When dealing with turbulent flows all the relevant dimensionless numbers are evaluated with the available quantities. For example, in DNS, the fluid and particle instantaneous velocities will be employed, whereas in large-eddy simulation (LES) or in Reynolds-average Navier-Stokes-equations (RANS) simulations the filtered or Reynolds-average values will be used. 

When the dimensionless form of the thermal energy equation is compared with the dimensionless Navier-Stokes equation, it is clear that the Peclet number plays a role for heat transfer that is analogous to the Reynolds number for fluid motion. Thus it is natural to seek approximate solutions for asymptotically small values of the Peclet number, analogous to the low-Reynolds-number approximation of Chaps. 7 and 8. 

Fluid flow in small devices acts differently from those in macroscopic scale. The Reynolds number (Re) is the most often mentioned dimensionless number in fluid mechanics. The Re number, defined by pf/L/p, represents the ratio of inertial forces to viscous ones. In most circumstances involved in micro- and nanofluidics, the Re number is at least one order of magnitude smaller than unity, ruling out any turbulence flows in micro-/nanochannels. Inertial force plays an insignificant role in microfluidics, and as systems continue to scale down, it will become even less important. For such small Re number flows, the convective term (pu Vu) of Navier-Stokes equations can be dropped. Without this nonlinear convection, simple micro-/ nanofluidic systems have laminar, deterministic flow patterns. They have parabolic velocity... 

The classification is accomplished by means of the rheological properties and/or by means of dimensionless numbers (e.g. Re-number). For colloidal suspensions, one usually assumes an incompressible, Newtonian liquid phase. The motion of colloidal particles in such liquid is dominated by the viscous properties (i.e. Stokes flow). In spite of this, the whole suspension may flow in a highly turbulent manner, or may show non-Newtonian rheological properties. 

We also met the Stokes number in Qiapter 9, where it was one of the dimensionless numbers used in the scale up of gas cyclones for the separation of particles from gases. There are obvious similarities between the collection of particles in a gas cyclone and collection of particles in the airways of the respiratory system. The Stokes number we met in Chapter 13, describing collision between granules, is not readily comparable with the one used here.)... 

Dimensional analysis of the equations that govern a system enables the identification of the relevant dimensionless numbers for the problem. It also ensures that all parameters have been taken into account, and that they are independent. We illustrate this principle below with the example of Navier-Stokes equations, written in vector form ... 

The seven scales involve three units (time, length, and mass). Four dimensionless numbers should, therefore, be defined to represent the flow. These numbers are determined namrally by evaluating the order of magnitude of the various terms in Navier-Stokes equation. By writing ... 

Navier-Stokes equations can, therefore, be regarded as a functional relationship between these four dimensionless numbers. The solution of Navier-Stokes equations depends on the values of the different dimensionless numbers. Unsteady effects are negligible if y . The gravitational force is negligible if Fr 1. Lastly, dynamic equilibriums are independent from viscosity if Re 1. This formulation means that the flow s length, velocity, pressme, and time scales are linked by a relation. For example, in the case where, simultaneously, y 1, Fr 1 and Re 1, the... 

The approach presented here is the most commonly used when two terms dominate the Navier-Stokes equatioa The flow can then be characterized by the value of a single dimensionless number. This is illustrated in section 3.4 with the classical example of the fall of a ball in a viscous fluid (Stokes experiment). 

To represent a given phenomenon at a differerrt scale, it should be ensured that the physics of the phenomenon is preserved through the change of scale. Depending on the case, the equations of continuum mechanics, Navier-Stokes equations, the transport equations for thermal energy, etc. will provide the appropriate framework to identify, through dimensional analysis, the dimensionless numbers characterizing... 

One thing missing from Eq. (8.1.3) is a dimensionless group that relates directly to the movement or separation of the particles. This can be achieved by introducing the well-known Stokes number , Stk. We can create this new dimensionless number by multiplying together powers of the existing numbers as follows ... 

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

A similar approach to the analysis of hydrocyclones was presented by Svarovsky (1984, 1990). He deduced that the system can be described in terms of three dimensionless groups in addition to various dimensionless geometric parameters. These groups are the Stokes number,... 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

The probability of an inhaled particle being deposited by impaction is a function of the dimensionless Stokes number Stk, which relates particle properties (mass mP, diameter dP, and density, pP) to parameters of the airflow (air velocity vA, viscosity i)A, and airways radius rA) ... 

Typical cascade impactors consist of a series of nozzle plates, each followed by an impaction plate each set of nozzle plate plus impaction plate is termed a stage. The sizing characteristics of an inertial impactor stage are determined by the efficiency with which the stage collects particles of various sizes. Collection efficiency is a function of three dimensionless parameters the inertial parameter (Stokes number, Stk), the ratio of the jet-to-plate spacing to the jet width, and the jet Reynolds number. The most important of these is the inertial parameter, which is defined by Equation 2) as the ratio of the stopping distance to some characteristic dimension of the impaction stage (10), typically the radius of the nozzle or jet (Dj). 

The particle motion along curvilinear pathways and the subsequent deposition rate on nearby bodies are calculated from dimensionless particle force equations. A key parameter that derives from these equations is the Stokes number,... 

Because the Reynolds number is much smaller than 1 and a Newtonian flow behavior is being observed in the first place, the Navier-Stokes equations convert to Stokes equations, and we obtain a system of linear equations for the flow calculations. It therefore follows that there must be a linear relationship both between the flow rate and the pressure and between the flow rate and the power. This is demonstrated in Figs. 8.10 and 8.11 in which the dimensionless conveying and power characteristic are illustrated, respectively. The red lines reflect the Newtonian cases. As expected, a linear relationship is revealed. 

We will first draw the square root of this dimensionless group and then relate it to the well-known Euler, Reynolds and Stokes numbers ... 

It is also possible to derive the Reynolds number by dimensional analysis. This represents a more analytical, but less intuitive, approach to defining the condition of similar fluid flow and is essentially independent of particular shape. In this approach, variables in the Navier-Stokes equation (relative particle-fluid velocity, a characteristic dimension of the particle, fluid density, and fluid viscosity) are combined to yield a dimensionless expression. Thus... 

The quantity 2ut/W is an important dimensionless parameter in impactor studies, known as the Stokes number... 

This dimensionless parameter is used to describe impactor behavior. For impactors with rectangular openings, W is the slit width for circular openings W represents the diameter of the impactor opening. Thus the Stokes number is the ratio of the stop distance to the impactor opening half-width. 

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