Force-balance equations

The behavior of a bead-spring chain immersed in a flowing solvent could be envisioned as the following under the influence of hydrodynamic drag forces (fH), each bead tends to move differently and to distort the equilibrium distance. It is pulled back, however, by the entropic need of the molecule to retain its coiled shape, represented by the restoring forces (fs) and materialized by the spring in the model. The random bombardment of the solvent molecules on the polymer beads is taken into account by time smoothed Brownian forces (fB). Finally inertial forces (f1) are introduced into the forces balance equation by the bead mass (m) times the acceleration ( ) of one bead relative to the others ... 

For the simplest case of a linear dumbbell in a homogeneous velocity gradient of strain rate s, the force balance equation is the following ... 

These authors have assumed the bubble to be expanding at the orifice, and have used the force balance equation at the time of detachment. The various forces considered by these authors are buoyancy, force due to the addition of mass (P2), excess pressure force, surface tension force, drag force, and force due to the inertia of the liquid. 

The argument holds equally well if the buoyancy force is attributable to the density of the suspension as opposed to that of the liquid. The force balance equation (5.61) then becomes ... 

The average shear strength at the interface, t., whether bonded, debonded or if the surrounding matrix material is yielded, whichever occurs first, can be approximately estimated from a simple force balance equation for a constant interface shear stress (Kelly and Tyson, 1965) ... 

However, some theoretical treatment considers only the special case of friction sliding of a single fiber along a mechanically bonded interface, particularly for some ceramic matrix composites, where the Coulomb friction law applies. See for example Zhou and Mai (1995) and Shetty (1988). Assuming a constant friction at the fiber-matrix interface and neglecting the Poisson effects, Shetty (1988) reported a simple force balance equation for the frictional shear strength, Tfr... 

E is applied parallel to the surface. Each layer of liquid will rapidly attain a uniform velocity relative and parallel to the surface, with electrical and viscous forces balanced. Equating the electrical and viscous forces on a liquid layer of unit area, thickness dx and distance x from the surface, and having a bulk charge density p,... 

With such an understanding on system complexity in mind, the DBS model is composed of two simple force balance equations, respectively, for small or large bubble classes, and one mass conservation equation as well as the stability condition serving as a variational criterion and a closure for conservative equations. For a given operating condition of the global system, six structure parameters for small and large bubble classes (their respective diameters dg, dL, volume fraction... 

To solve for S(z) and R(z), one needs to state the force balance equations for the blown film. The simplest form of these equations, disregarding inertial and gravity forces, are the classic thin-film equations. The forces per unit length in the film in the c [ and < 3 directions are FL/2nR = 8nn and FH/ = Siz33, respectively thus from the thin film equation, we get... 

Steady-State Momentum (Force) Balance Equation... 

By determining the path of a single particle when it is acted upon by a variety of forces, it is possible to predict particle position and behavior. This can be done by solving force balance equations which then give acceleration, velocity, and position of the particle. 

Example 6.2 Write a force balance equation for a particle which is acted upon by gravity in an electric field. 

Foitzik, I., 289 Force, 75,179-180 Force balance equations, 76 Force vector, 180 Form drag, 71... 

Batchelor (1988) developed a force balance equation for the solid phase in fluidized beds using the ensemble average method. With the help of the resulting equation and the theory of linear stability the following criterion was obtained by Ham et al. (1990) ... 

The total force exerted to the microbalance may be given by the following force balance equation ... 

Semiempirical expressions for Tfm have been developed by Lefroy and Davidson from their theoretical analysis discussed in Section IV,B and V,A. From measurements of spout diameter made in beds of kale seeds, polyethylene, and peas at maximum spoutable depths, they determined the value of the left-hand side of their force-balance equation q. (66) ] and found it to be about 0.36 in all cases. Therefore... 

The derivation of the constitutive equation for the dumbbell model can also be extended to allow multiple beads and springs. From the force-balance equations for such a model, one obtains (Bird et al. 1987b Larson 1988)... 

This Eq. (27) may be viewed as a force balance equation, the total force F on the LHS being balanced by a kinetic (K) piece FK plus a contribution which, in three-dimensions, involves the Laplacian (L) of the electron density p(x), and the corresponding force we shall write as Fl. Thus Eq. (27) has the form... 

If the external potential energy —Ze2/r in an atom having nuclear charge Ze) is written as Vext, then their generalization of Eq. (28) is a vector force balance equation ... 

The first achievement of the study of Holas and March [99] is to establish the differential form of the above virial theorem [their Eq. (2.15)]. Again, as in the zero field case treated above, this differential virial theorem is interpreted as a force-balance equation. The well-known Lorentz force of electromagnetism then appears quite naturally in this equation. 

The vapor flood velocity is obtained by substituting the above force expressions in the force balance equation and solving for U/. 

Consider a forced resonator obeying the force-balance equation d x t)... 

In general, the approaches of the trajectory analysis and the build-up models are based on the force-balance equation. Among the forces (external and interparticle) involved in the system, magnetic and hydrodynamic forces, the most significant ones, also compete with one another. The performance of magnetic separation is, therefore, examined in terms of magnetic velocity (Vjj ) and superficial velocity ( Vq) (Watson, 1973),... 

Equation 20 is our momentum, or force balance equation which we may now apply to semiconductors. To do so, we will need constitutive equations for the pa, subject to condition Equation 25, and for the stress tensors Ta. 

Equation 103, we have shown that Equation 103 is really a momentum, or force balance, equation. 

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