Inertial momentum

Reynolds number flows /vRe N -°Vp /vRe — pV2 pV/D AQp izDp PV2 Tw/8 Pipe flow rw =wall stress (inertial momentum flux)/ (viscous momentum flux) Pipe/internal flows (Equivalent forms for external flows)... 

For turbulent flow, with roughly uniform distribution, assuming a constant friction factor, the combined effect of friction and inertial (momentum) pressure recovery is given by... 

The rotational speed and inertial momentum of the rotating mass represents a substantial amount of stored rotational energy, preventing sudden changes in voltage output when the input is momentarily interrupted. 

The numerator is the flux of inertial momentum carried by the fluid along the tube in the axial direction. The denominator is proportional to the viscous shear stress in the tube, which is equivalent to the flnx of viscous momentum normal to the flow direction, i.e. in the radial direction. Thus, the Reynolds number is a ratio of the momentum flux due to inertia (in the flow direction) to the momentum flux due to viscous stresses (in the radial direction). 

Most schemes that have been proposed to propel starships involve plasmas. Schemes differ both in the selection of matter for propulsion and the way it is energi2ed for ejection. Some proposals involve onboard storage of mass to be ejected, as in modem rockets, and others consider acquisition of matter from space or the picking up of pellets, and their momentum, which are accelerated from within the solar system (184,185). Energy acquisition from earth-based lasers also has been considered, but most interstellar propulsion ideas involve nuclear fusion energy both magnetic, ie, mirror and toroidal, and inertial, ie, laser and ion-beam, fusion schemes have been considered (186—190). 

Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, totaf energy, and mechanical energy may be found in Whitaker (ibid.). Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. 

Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, Vp = pg. Letting z be directed vertically upward, so that g, = —g where g is the gravitational acceleration (9.806 mVs), the pressure field is given by... 

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

Inertial forces are developed when the velocity of a fluid changes direction or magnitude. In turbulent flow, inertia forces are larger than viscous forces. Fluid in motion tends to continue in motion until it meets a sohd surface or other fluid moving in a different direction. Forces are developed during the momentum transfer that takes place. The forces ac ting on the impeller blades fluctuate in a random manner related to the scale and intensity of turbulence at the impeller. 

The only force opposing the downward flow of the heated air or upward flow of the cooled air is a buoyancy force. In their analysis, Helander and Jakowatz also suggested accounting for inertial forces due to the entrainment of room air. However, this suggestion is not in an agreement with a principle of momentum conservation used in most of the existing models for isothermal jets. 

The current Archimedes number for the resulting jet grows along the jet as it does in any nonisothermal jet. However, the consequent momentum additions by directing jets increases the inertial forces in rhe resulting jet and thus at a certain cros.s-section the current Archimedes number falls. The number of directing jets after which Ar reaches the peak can be calculated using... 

Gases, vapors, and fumes usually do not exhibit significant inertial effects. In addition, some fine dusts, 5 to 10 micrometers or less in diameter, will not exhibit significant inertial effects. These contaminants will be transported with the surrounding air motion such as thermal air current, motion of machinery, movement of operators, and/or other room air currents. In such cases, the exterior hood needs to generate an airflow pattern and capture velocity sufficient to control the motion of the contaminants. However, as the airflow pattern created around a suction opening is not effective over a large distance, it is very difficult to control contaminants emitted from a source located at a di,stance from the exhaust outlet. In such a case, a low-momentum airflow is supplied across the contaminant source and toward the exhaust hood. The... 

Mach s principle, as formulated by Wheeler [wheel64a], states that the inertial properties of an object are determined by the energy-momentum distribution throughout all of space. 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

The Reynolds number Re = vl/v, where v and l are the characteristic velocity and length for the problem, respectively, gauges the relative importance of inertial and viscous forces in the system. Insight into the nature of the Reynolds number for a spherical particle with radius l in a flow with velocity v may be obtained by expressing it in terms of the Stokes time, t5 = i/v, and the kinematic time, xv = l2/v. We have Re = xv/xs. The Stokes time measures the time it takes a particle to move a distance equal to its radius while the kinematic time measures the time it takes momentum to diffuse over... 

The collection efficiency of particles at a stage of an impactor is based on curvilinear motion and assumes Reynolds numbers for flow greater than 500 but less than 3000. Figure 8A illustrates the principle of inertial sampling in which particles with high momentum travel in the initial direction of flow of an airstream impacting on an obstructing surface and those with low momentum adjust to the new direction of flow and pass around the obstruction. The efficiency of this phenomenon can be described as follows ... 

Figure 8.4 Inertial separators increase the efficiency of separation by giving the particles downward momentum.

Inertial collectors. Inertial collectors were also discussed in Chapter 8 and illustrated in Figure 8.4. The particles are given a downward momentum to assist tihe settling. Only particles in excess of 50 im can be reasonably... 

The forces that exist within a fluid at any point may arise from various sources. These include gravity, or the weight of the fluid, an external driving force such as a pump or compressor, and the internal resistance to relative motion between fluid elements or inertial effects resulting from variation in local velocity and the mass of the fluid (e.g., the transport or rate of change of momentum). 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

With regard to the drag on a sphere moving in a Bingham plastic medium, the drag coefficient (CD) must be a function of the Reynolds number as well as of either the Hedstrom number or the Bingham number (7V Si = /Vne//VRe = t0d/fi V). One approach is to reconsider the Reynolds number from the perspective of the ratio of inertial to viscous momentum flux. For a Newtonian fluid in a tube, this is equivalent to... 

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