Inertia free flow

In the case of an inertia-free flow (at low Reynolds numbers) of a quasi-Newtonian power-law fluid with rheological index n close to 1 past a gas bubble, the drag coefficient can be calculated by the formula [190]... 

A well known result from the theory of inertia free flows is that the effect of the tube wall on the particle motion through a capillary is to slow the particle down relative to the fluid in the neighborhood of the wall. The mean particle velocity given by Eq. (5.7.2) must therefore be too large. From their moment analysis, Brenner 5c Gaydos found for small values of A that to terms of lowest order in A... 

Assuming inertia free flow with no pressure gradient, the above equation simplifies to... 

On a reciprocating conveyor, material is carried forward in a horizontal direction by frictional contact with the trough. Inertia causes the material to be left in that position as the trough is quickly returned to the initial position. These conveyors are useful for handling granular free-flowing materials with a minimum of attrition. 

Froude w inertia force flow with a free surface,... 

In the discussion of forced flow we neglected the influence of free flow and in reverse the effect of forced flow was neglected in our handling of free flow. However, frequently a free flow will overlap a forced flow as a result of density gradients. As we have already seen in 3.9.1, eq. (3.308), the decisive quantity for this is Gr/Re2. If it is of the order 1, the buoyancy and inertia forces are equal, whilst for Gr/Re2 -C 1 the forced, and for Gr/Re2 1, the free flow predominates. 

Forced and free flow can, depending on the direction of the inertia and buoyancy forces, either mutually stimulate or dampen each other. In a forced flow overlapping a free flow, the heat and mass transfer can either be improved or inhibited. As an example of this we will look at a heated plate, Fig. 3.51. A free flow in the upwards direction develops, which can be strengthened Fig. 3.51a, or weakened, Fig. 3.51b, by a forced flow generated by a blower. Experiments have shown that the heat transfer coefficient can be calculated well by using equations of the form... 

Numerous available numerical solutions the Navier-Stokes equations, as well as experimental data (see a review in [94]), provide a detailed analysis of the flow pattern for increasing Reynolds numbers. For 0.5 < Re < 10, there is no flow separation, although the fore-and-aft symmetry typical of inertia-free Stokes flow past a sphere is more and more distorted. Finally, at Re = 10, flow separation occurs at the rear of the particle. 

The spherical form of a drop or a bubble in Stokes flow follows from the fact that the flow is inertia-free. However, even for the case in which the inertia forces dominate viscous forces and the Reynolds number cannot be considered small, the drop remains undeformed if the inertia forces are small compared with the capillary forces. The ratio of inertial to capillary forces is measured by the Weber number We = p U a/a, where cr is the surface tension at the drop boundary. For small We, a deformable drop will conserve the spherical form. 

Figure 10.3 The influence of the finite dimension of particles in inertia-free flotation on their trajectory in the vicinity of a floating bubble. The liquid flow lines corresponding to target distances b(a,) and are indicated by dashed lines. The continuous lines are characteristic of the deviation of the trajectory of particles from the liquid flow lines under the influence of short-range hydrodynamic interaction...

The diffusion layer thickness is seen to grow as the cube root of the streamwise distance rather than the square root, as in unbounded flow past a surface. Moreover, the growth is independent of the kinematic viscosity and therefore of the Schmidt number, although it does depend on viscosity through Both of these conditions result from the inertia free character of the channel flow itself that is, viscous forces predominate, and the mass density p does not enter the problem. 

To the extent that dispersion in an inertia free porous medium flow arises from a nonuniform velocity distribution, its physical basis is the same as that of Taylor dispersion within a capillary. Data on solute dispersions in such flows show the long-time behavior to be Gaussian, as in capillaries. The Taylor dispersion equation for circular capillaries (Eq. 4.6.30) has therefore been applied empirically as a model equation to characterize the dispersion process in chromatographic separations in packed beds and porous media, with the mean velocity identified with the interstitial velocity. In so doing it is implicitly assumed that the mean interstitial velocity and flow pattern is independent of the flow rate, a condition that would, for example, not prevail when inertial effects become important. 

A porous medium is modeled as made up of uniformly distributed straight circular capillaries of the same diameter. The flow through each capillary is an inertia free Poiseuille flow. By comparing the Poiseuille pressure drop and the Darcy pressure drop formulas, deduce an expression for the permeability. Discuss the difference between the result obtained and the Kozeny-Carman permeability. 

It is desired to estimate the electrophoretic velocity U of a long, nonconducting, charged cylindrical particle of length L and radius a and with a low surface potential I, as a result of the application of an electric field parallel to the symmetry axis. The Debye length is arbitrary but finite, and the flow is a low Reynolds number, inertia free one. 

Following Spielman and the aims of this book, our discussion is confined to the capture of particles in liquid suspension from low-speed laminar flows, where the particles are generally small compared with the collector. The two principal transport mechanisms are (a) Brownian diffusion for submicrometer-size particles, and (b) interception of micrometer-size, nondiffusing, inertia free particles with the collector as a consequence of geometrical collision due to particles following fluid streamlines. Inertial impaction, which can be important for gas-borne particles, is usually unimportant for particles in liquids, because the particle—fluid density difference is smaller and the higher viscosity of liquids resists movement relative to the fluid (Spielman 1977). In this section we shall... 

The diffusional flow rate to a cylindrical collector, whose axis is normal to the flow direction, has also been determined by a procedure similar to that outlined for the spherical collector. Although no steady, uniformly valid, inertia free, Stokes solution exists for an unbounded medium, a solution valid near the... 

Let us again first consider collection by a spherical collector assuming the flow to be an inertia free, Stokes flow. The stream function corresponding to the velocity field, defined by Eqs. (8.3.3), is... 

A similar calculation can in principle be carried out for a cylindrical collector. However, a difficulty arises in that there is no solution of the inertia free, Stokes equation for an infinite cylinder in an otherwise unbounded flow. This was already observed in Section 5.1. Nevertheless we can illustrate the low Reynolds number behavior by using the so-called Stokes-Oseen solution for uniform flow of velocity U past an infinite circular cylinder of radius a whose symmetry axis is perpendicular to the flow. Oseen s method accounts in an... 

The factor 2.00 is not a complete integer but is derived from the natural logarithm of 7.4. Note that is a parameter that characterizes the flow model and is a function of the Reynolds number Re = laUlv. As we discuss in Section 8.5, for an assemblage of cylinders where inertia free solutions can be obtained, A yi can be shown to be a function of the volume fraction of the cylinder assemblage. The solution given by Eq. (8.3.24) is appropriate for Reynolds numbers based on the cylinder diameter of around 1. 

The flow is considered to be inertia free and to obey the Stokes equation, with the solid spherical particle taken to move within the cell with superficial velocity U. This is equivalent to using a coordinate system moving with velocity U. The appropriate boundary conditions are thus... 

Happel (1959) obtained corresponding closed-form inertia free solutions within his model for assemblages of cylinders in those cases where the flow is parallel to the axis of the cylinder and where the flow is at right angles to the cylinder axis. The stream function has the same form as in Eq. (8.3.24), except that is a logarithmic function of the solid volume fraction, independent of the Reynolds number since the flow is inertia free (Spielman 1977). 

The concept of an apparent viscosity introduced for a Couette flow has been applied empirically to a variety of incompressible inertia free steady shear flows through the generalized relation... 

Neglecting gravitational forces and supposing the flow to be an inertia free capillary flow, with no pressure gradient Eq. (6.5.1) simply reduces to a balance between viscous and electrical forces ... 

The energy dissipative and inertia-free resistance R impedes not only the current flow -energy-conserving inert elements, capacitance C, and inductance L are also included in the complex impedance by their frequency-dependent reactive impedances Xc(m) and Xl(co). 

Usually this type of anemometer does not provide information on the flow direction. Vice versa, the. sensors are made as independent of the flow direction as possible—omnidirectional. This is an advantage for free-space ventilation measurements, as the flow direction varies constantly and a direction-sensitive anemometer would be difficult to use. Naturally, no sensor is fully omnidirectional, but satisfactory constructions are available. Due to the high sensor thermal inertia, this type of anemometer is unsuitable for high-frequency flow fluctuation measurement. They can be used to monitor low-frequency turbulence up to a given cut-off frequency, which depends on the dynamic properties of the instrument. 

Photodiodes produce an electric field as a result of pn transitions. On illumination a photocurrent flows that is strictly proportional to the radiation intensity. Photodiodes are sensitive and free from inertia. They are, thus, suitable for rapid measurement [1, 59] they have, therefore, been employed for the construction of diode array detectors. 

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