Physics basics velocity

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

A design based on the enclosure open-area control velocity does not consider all the other variables listed as affecting dust control. Calculation procedures to predict many of the other variables would be very complicated if not impossible to perform. Physical modeling of the problem and solution w as therefore used as the basic design tool. 

Since the physical properties of a system are interconnected by a series of mechanical and physical laws, it is convenient to regard certain quantities as basic and other quantities as derived. The choice of basic dimensions varies from one system to another although it is usual to take length and time as fundamental. These quantities are denoted by L and T. The dimensions of velocity, which is a rate of increase of distance with time, may be written as LT , and those of acceleration, the rate of increase of velocity, are LT-2. An area has dimensions L2 and a volume has the dimensions L3. 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

The visualization of light as an assembly of photons moving with light velocity dates back to Isaac Newton and was formulated quantitatively by Max Planck and Albert Einstein. Formula [1] below connects basic physical values ... 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

The occurrence of sewer sediments is primarily determined by the physical characteristics of wastewater solids and the hydraulic conditions. Basically, sewers should be designed and operated in a way that does not result in permanent deposits. This ideal performance of a sewer is not generally observed, and sediments may be more or less temporarily accumulated in sewers. In combined sewer networks, sediments may settle under dry-weather conditions when the wastewater velocity and shear stress at the bottom are low and be... 

The strategy in a molecular dynamics simulation is conceptually fairly simple. The first step is to consider a set of molecules. Then it is necessary to choose initial positions of all atoms, such that they do not physically overlap, and that all bonds between the atoms have a reasonable length. Subsequently, it is necessary to specify the initial velocities of all the atoms. The velocities must preferably be consistent with the temperature in the system. Finally, and most importantly, it is necessary to define the force-field parameters. In effect the force field defines the potential energy of each atom. This value is a complicated sum of many contributions that can be computed when the distances of a given atom to all other atoms in the system are known. In the simulation, the spatial evolution as well as the velocity evolution of all molecules is found by solving the classical Newton equations of mechanics. The basic outcome of the simulation comprises the coordinates and velocities of all atoms as a function of the time. Thus, structural information, such as lipid conformations or membrane thickness, is readily available. Thermodynamic information is more expensive to obtain, but in principle this can be extracted from a long simulation trajectory. 

Basic Breakup Modes. Starting from Lenard s investigation of large free-falling drops in still air,12671 drop/droplet breakup has been a subject of extensive theoretical and experimental studies[268] 12851 for a century. Various experimental methods have been developed and used to study droplet breakup, including free fall in towers and stairwells, suspension in vertical wind tunnels keeping droplets stationary, and in shock tubes with supersonic velocities, etc. These theoretical and experimental studies revealed that droplet breakup under the action of aerodynamic forces may occur in various modes, depending on the flow pattern around the droplet, and the physical properties of the gas and liquid involved, i.e., density, viscosity, and interfacial tension. 

The basic assumptions implied in the homogeneous model, which is most frequently applied to single-component two-phase flow at high velocities (with annular and mist flow-patterns) are that (a) the velocities of the two phases are equal (b) if vaporization or condensation occurs, physical equilibrium is approached at all points and (c) a single-phase friction factor can be applied to the mixture if the Reynolds number is properly defined. The first assumption is true only if the bulk of the liquid is present as a dispersed spray. The second assumption (which is also implied in the Lockhart-Martinelli and Chenoweth-Martin models) seems to be reasonably justified from the very limited evidence available. 

The purpose of this chapter is to provide a comprehensive discussion of some simple approaches that can be employed to obtain information on the rate of heat and mass transfer for both laminar and turbulent motion. One approach is based on dimensional scaling and hence ignores the transport equations. Another, while based on the transport equations, does not solve them in the conventional way. Instead, it replaces them by some algebraic expressions, which are obtained by what could be called physical scaling. The constants involved in these expressions are determined by comparison with exact asymptotic solutions. Finally, the turbulent motion is represented as a succession of simple laminar motions. The characteristic length and velocity scales of these laminar motions are determined by dimensional scaling. It is instructive to begin the presentation with an outline of the basic ideas. 

Distribution functions are usually first met in physical chemistiy when the crude treatment of molecular velocities in the kinetic theory of gases (all the molecules taken as having the same mean speed) is replaced by Maxwell s seminal equation showing that the number of molecules having velocities between narrow limits depends very much on what velocities are chosen. This is shown in Fig. 9.1. Thus, this first and basic distribution law of Maxwell, the distribution of velocities, gives an unexpected result (the nonsymmetrical nature of the distribution), which still causes us to think, more than a century after its publication. 

A more detailed study of fuel cloud dispersion, though one lacking direct exptl verification, was made by Rosenblatt et al (Ref 23). The purpose of their study was to develop and use physically based numerical simulation models to examine the cloud dispersion and cloud detonation with fuel mass densities and particle size distributions as well as the induced air pressures and velocities as the principal parameters of interest. A finite difference 2-D Eulerian code was used. We quote The basic numerical code used for the FAE analysis was DICE, a 2-D implicit Eulerian finite difference technique which treats fluid-particle mixtures. DICE treats par-... 

What problems face the theory of combustion The theory of combustion must be transformed into a chapter of physical chemistry. Basic questions must be answered will a compound of a given composition be combustible, what will be the rate of combustion of an explosive mixture, what peculiarities and shapes of flames should we expect We shall not be satisfied with an answer based on analogy with other known cases of combustion. The phenomena must be reduced to their original causes. Such original causes for combustion are chemical reaction, heat transfer, transport of matter by diffusion, and gas motion. A direct calculation of flame velocity using data on elementary chemical reaction events and thermal constants was first carried out for the reaction of hydrogen with bromine in 1942. The problem of the possibility of combustion (the concentration limit) was reduced for the first time to thermal calculations for mixtures of carbon monoxide with air. Peculiar forms of propagation near boundaries which arise when normal combustion is precluded or unstable were explained in terms of the physical characteristics of mixtures. 

Basic courses in physics teach us that power is work per unit time, and work is a measure of energy which may be defined as force times distance. Therefore, power is in physical units of force times distance per unit time, or force times velocity. It therefore should come as no surprise that traveling power waves are defined for strings as... 

Filtration is a physical separation whereby particles are removed from the fluid and retained by the filters. Three basic collection mechanisms involving fibers are inertial impaction, interception, and diffusion. In collection by inertial impaction, the particles with large inertia deviate from the gas streamlines around the fiber collector and collide with the fiber collector. In collection by interception, the particles with small inertia nearly follow the streamline around the fiber collector and are partially or completely immersed in the boundary layer region. Subsequently, the particle velocity decreases and the particles graze the barrier and stop on the surface of the collector. Collection by diffusion is very important for fine particles. In this collection mechanism, particles with a zig-zag Brownian motion in the immediate vicinity of the collector are collected on the surface of the collector. The efficiency of collection by diffusion increases with decreasing size of particles and suspension flow rate. There are also several other collection mechanisms such as gravitational sedimentation, induced electrostatic precipitation, and van der Waals deposition their contributions in filtration may also be important in some processes. 

Equations 9.3-22 and 9.3-26 are the basic equations of the melting model. We note that the solid-bed profile in both cases is a function of one dimensionless group ijj, which in physical terms expresses the ratio of the local rate of melting per unit solid-melt interface JX /X to the local solid mass flux into the interface Vszps, where ps is the local mean solid bed density. The solid-bed velocity at the beginning of melting is obtained from the mass-flow rate... 

A discussion of vapor-phase fundamentals begins with the basic gas laws, which apply to any vapor-phase deposition technique. These techniques employ gases at low pressure (less than 1 atm) and therefore are well described by basic laws such as the ideal gas law and the kinetic gas theory, which are presented in undergraduate physical chemistry. For the purposes of vapor deposition, the critical gas parameters include (1) concentration, (2) velocity distribution, (3) flux, and (4) mean free path. The concentration of gas particles in a low-pressure gas, less than 1 atm, is given by the ideal gas law,... 

In order to determine the distributions of pressure, velocity, and temperature the principles of conservation of mass, conservation of momentum (Newton s Law) and conservation of energy (first law of Thermodynamics) are applied. These conservation principles represent empirical models of the behavior of the physical world. They do not, of course, always apply, e.g., there can be a conversion of mass into energy in some circumstances, but they are adequate for the analysis of the vast majority of engineering problems. These conservation principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p viscosity, p conductivity, k and specific heat, cp. Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to be known. (Non-Newtonian fluids in which p depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems in which the variation of these properties across the flow field can be ignored, i.e., in which the fluid properties can be assumed to be constant in obtaining fire solution. Such solutions are termed constant... 

Since the basic assumption of boundary layer theory is that (8IL) is small, it follows from this that (v/uj) is also small. Thus, in the boundary layer the lateral velocity component, v, is very much smaller than the longitudinal component, u, which, of course, is what is physically to be expected. 

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