Total external force

In a time duration dt, each particle changes its velocity by dv = (F/m) dt, where F is the total external force acting on an individual particle of mass m. The change in xjr can be given by... 

Because we have restricted the discussion to systems for which the total external force on a molecule is zero, we can replace r" by R in the last term ... 

For the particle size considered in this study, the characteristic time scale of acceleration period of the motion is in the order of 10 " s [8] which is much smaller than the time scale of the variation of the field variables. Therefore, the acceleration term can be safely neglected, and the velocity of the particle can be derived by equating the total external force to zero. 

We now focus on macroscopic particles and provide an erqrression for the total external force acting on a particle of mass fWp, density pp, radius Cp, charge Qp, velocity Vp and volume (fWp/pp). We have not included here the Brownian motion force P , nor any force due to thermophoresis, radiation pressure, acoustic force and the electrical force in a nonuniform electrical field given by (3.1.13). Although not generated by an external force field, coulombic types of interactions, London dispersion and electrokinetic forces in the double layer are included in the expression given below ... 

Write down in vectorial form the equation of motion of a particle subjected to a total external force of P. The velocity of the fluid is Vp... 

According to Eq. (18.17) the total force on the particles Ffot = Ftot — F t is equal to the total external force Ftot on the wire reduced by the force, which arises from the slurry pressure p(x). In the case that is small compared to Ftot the particles carry the main load and one obtains F j directly. With the experimentally determined value n = 0.85 the sawing rate is given then by Vg v Ftot F n. The removal rate is mainly proportional to v and Ftot in this case since Fn is almost constant. 

The sum of all internal forces vanishes F is the total external force on the chain. This defines the mobility of the polymer chain Ceb— C- The Einstein relation gives the chain diffusion constant... 

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

CellgeometTy is governed predominantly by the final foam density and the external forces exerted on the cellular stmcture prior to its stabilization in the expanded state. In a foam prepared without such external forces, the cells tend to be spherical or ellipsoidal at gas volumes less than 70—80% of the total volume, and they tend toward the shape of packed regular dodecahedra at greater gas volumes. These shapes have been shown to be consistent with surface chemistry arguments (144,146,147). Photographs of actual foam cells (Fig. 2) show a broad range of variations in shape. 

An elementary solid, such as silver, is regarded as composed of atoms oscillating about fixed centres. The total energy content is therefore partly kinetic and partly potential. Since the solid has a finite compressibility, the atoms may be supposed to be maintained at small distances apart by forces they exert upon one another, and these may be resolved into two sets, one of which opposes a closer approximation of the atoms, and the other tends to draw the latter together. Both are functions of the distance between the atoms, and for a given distance are equal, since the form of the body is altered by external forces alone. 

Conductor-insulator and conductor-vacuum interfaces lack a continuous exchange of free charges, and there is no electrochemical equilibrium. For this reason the work that is performed in transferring charged particles from one phase to the other is not zero. The total work, X, which must be performed by the external forces in transferring (extracting) an electron from a metal (M) into vacuum (0) is called the electron work function (or simply the work function). The work function for all metals is always positive, since otherwise the electrons would leave the conductor spontaneously. 

If there is an external force acting in the same direction on solute molecules, the velocity of these molecules is vz and the resulting flux is cvz. Therefore, the total flux, nz, due to both diffusion and convection is... 

To relate the discrete particle simulations to the KTGFs, it is very useful to analyze the detailed information of the energy evolution in the system. The total energy balance of the system is obtained by calculating all relevant forms of energy as well as the work performed due to the action of external forces. 

This step is so important that some laboratory plants, and usually all pilot and production plants, are equipped with respective mechanisms. The principle is simple The shelves are connected flexibly with inlet and outlet of the brine. The shelves are pressed together, one after the other, by a plate, which is moved by an external force in this way the stoppers are pushed into the closed position. If the pressure necessary for this stopper movement is 1 kg per stopper, the resulting total force for 100 vials per shelf is 100 kg, but if 10 000 vials are loaded per shelf, the total force is 10 tonnes, which has to be applied evenly in order to avoid vials breakage. 

Consider a linear one-box model for concentration C. Estimate the maximum tolerable relative input variation A of a sinusoidal external force (Eq. 21-12) which keeps the relative deviation of C(t) from the mean value C = C smaller than 10%. The total rate constant of the linear system is k = 0.3 yr"1. Assume that the period of the external force is (a) one year, (b) one week. 

MASS (Center of). The center ol mass is (hat point in a collection ol mass-panicles which moves as if the total mass of the collection were concentrated there anil the resultant of all the external forces were acting there. The position sector of such a point is given by... 

After the pressure at the piston returns to zero, the gas is not acted upon by any external forces since the pressure in the cold gas is equal to zero both at x — —oo (in the limit at the boundary of dispersion), and at x — oo (x > X is sufficient) in the cold gas. In the motion, therefore, the total momentum of the gas, f pudx, is conserved. This momentum is equal to the pressure impulse of the piston,... 

The three translations, two rotations, and one vibration provide a total of six independent motions. In the absence of external forces (such as collisions), energy is never... 

Then, in the presence of an external force, the total flux is given by the relation... 

The characteristic nozzle flow assumptions are made, i.e. the flow is laminar, steady, one-dimensional and there are no dissipative or external forces of any kind. The reacting gas is considered to be composed of fx chemical species, each of which is present at a concentration xt (moles per unit mass of mixture). The usual flow variable temperature T, density p, pressure P and velocity u then make a total of jx + 4 variables. The cross-sectional area ratio e is generally specified as a function of the axial distance z and the axial distance z along the flow direction becomes the independent variable. A mass flow rate W per unit reference... 

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