Forces and Balance Laws

Forces and Balance Laws 2.3.1 Forces Within Continua Stress Tensors... 

The equilibrium orbit theory is based on the concept of the equilibrium radius, originally proposed by Dries sen" and Criner. According to this concept, particles of a given size attain an equilibrium radial orbit position in the cyclone where their terminal settling velocity is equal to the radial velocity of the liquid. Particles are therefore elutriated by the inward radial flow according to the balance of the centrifugal and drag forces, and Stokes law is usually assumed. 

Conservation of Momentum. If the mass of a body or system of bodies remains constant, then Newton s second law can be interpreted as a balance between force and the time rate of change of momentum, momentum being a vector quantity defined as the product of the velocity of a body and its mass. 

A macroscopic momentum balance for a flow system must include all equivalent forms of momentum. In addition to the rate of momentum con-vected into and out of the system by the entering and leaving streams, the sum of all the forces that act on the system (the system being defined as a specified volume of fluid) must be included. This follows from Newton s second law, which provides an equivalence between force and the rate of momentum. The resulting macroscopic conservation of momentum thus becomes... 

Developments to Date. It often has been stated that the basic policy objective of efforts to harmonize the U.S. and European laws is the achievement of consistent and effective protection of health and the environment. However, economic considerations — in particular, the avoidance (or minimization) of non-tariff trade barriers — constitute the principal force behind virtually all of these multilateral efforts. The trade in chemicals and chemical products constitutes a significant part of the overall trade between Western industrialized nations. Specifically, the U.S. enjoys a favorable balance in its chemicals trade, and this is particularly significant given the current recession. Thus, any unnecessary barriers to this trade may impose substantial burdens upon certain segments of the American chemical industry, and may constitute violations of the international General Agreement on Tariffs and Trade (GATT). 

One can apply Stokes law to atmospheric particles to calculate how fast they will settle out of the air when subjected to gravity alone. Thus the terminal settling velocity occurs when the frictional and gravitational forces are balanced, that is,... 

From Newton s law for the gravitation force, Archimedes principle for the buoyant force, and the definition of the drag force, the force balance on a drop becomes... 

General physical laws often state that quantities like mass, energy, and momentum are conserved. In computational mechanics, the most important of these balance laws pertains to linear momentum (when reckoned per unit volume, linear momentum may be expressed as the material density p times velocity v). The balance equation for linear momentum may be considered as a generalization of Newton s second law, which states that mass times acceleration equals total force. As we saw in the previous section, stresses in a material produce tractions, which may be considered as internal forces. In addition, external forces such as gravity may contribute to the total force. These are commonly reckoned per unit mass and are usually referred to as body forces to distinguish them from tractions, which may be considered as surface forces. For a one-dimensional motion, balance of linear momentum requires that (37,38)... 

For each balance law, the values of -0, J and 4> defines the transported quantity, the diffusion flux and the source term, respectively, v denotes the velocity vector, T the total stress tensor, gc the net external body force per unit of mass, e the internal energy per unit of mass, q the heat flux, s the entropy per unit mass, h the enthalpy per unit mass, u>s the mass fraction of species s, and T the temperature. 

The second force component, the dielectric boundary pressure, results from the tendency of a high dielectric medium to displace a low dielectric constant medium if an electrical field is present. This force is always directed along the gradient of the dielectric constant, which means that it constitutes a pure pressure at the solute-solvent interface. The dielectric boundary pressure is the force component that balances the reaction field component of the qE force. For example, in the case of an isolated charge inside a low dielectric cavity, the dielectric boundary pressure provides the equal and opposite force to the qE force urging the charge toward the solvent. It is therefore evident that the dielectric boundary force is quantitatively as important as the reaction field force and that its neglect will lead to a violation of Newton s third law of motion. 

The sedimentation of particles—that is, their downward motion due to gravitational settling—follows Stokes s law. Turbulent diffusion represents an opposing force, and sedimentation equilibrium is established when both forces cancel. Assuming a constant production rate of particles at the earth surface for each size group and a balance of upward and downward fluxes in the atmosphere leads to the following equation ... 

The strategy for solving fluid dynamics problems begins by putting a control volume within the fluid that matches the symmetry of the macroscopic boundaries, and balancing the forces that act on the system. The system is defined as the fluid that is contained within the control volume V, which is completely surrounded by surface S. Since a force is synonymous with the time rate of change of momentum as prescribed by Newton s laws of motion, the terms in the force balance are best viewed as momentum rate processes. The force balance for an open system is stated without proof as l = 2- -3H-4- -5, where... 

Thus, in order to solve the hydrodynamic problem of liquid motion in view of the change of 2 at the interface, we should first And out the distribution of substance concentration, temperature and electric charge over the surface. These distributions, in turn, are influenced by the distribution of hydrodynamic parameters. Therefore the solution of this problem requires utilization of conservation laws - the equations of mass, momentum, energy, and electric charge conservation with the appropriate boundary conditions that represent the balance of forces at the interface the equality of tangential forces and the jump in normal forces which equals the capillary pressure. In the case of Boussinesq model, it is necessary to know the surface viscosity of the layer. From now on, we are going to neglect the surface viscosity. 

Balance of momentum in the objective form (3.84), (3.85) may be interpreted as the general action and reaction law sum of all forces is zero (in total body force (3.86) the force i - v caused by distant, remote stars is included). 

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