Equation of Motion for a Particle

The Lagrangian equation of motion of a particle rotating at the radial position r in a centrifugal field with circumferential velocity v is (Eq. 2.2.4 resolved in the radial direction) 

If the motion is steady, the left-hand side is zero. The steady motion of the particle is therefore determined by Stk, and pp does not need to be included explicitly in the analysis, as mentioned in the discussion following Eq. (8.1.4). Also the density ratio Ap/p does not occur in the equations when the added mass and Basset terms are neglected. 

Inspection of the equations of motion of the gas and particle phases has thus confirmed the results of classical dimensional analysis, simplified the results of the analysis further, and has, we trust, increased our understanding of the physical significance of the dimensionless groups. 

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If the total drag force is proportional to the velocity squared, that is to y1, then the equation of motion for a particle moving downwards under the influence of gravity may be written as ... 

Dynamics is the study of matter in motion. The starting point for classical dynamics is Newton s equation of motion for a particle of mass m at position r = r(t), the force F leads to motion described by... 

The quantity y 1 is sometimes called the velocity relaxation time it can be considered to be the time taken for the particle to forget its initial velocity. The Eangevin equation of motion for a particle is therefore... 

The trajectory of a particle moving in a gas can be estimated by integrating the equation of motion for a particle over a time period given by increments of the ratio of the radial distance traveled divided by the particle velocity, that is, r/q. Interpreting the equation of motion, of course, requires knowledge of the flow field of the suspending gas one can assume that the particle velocity equals the fluid velocity at some distance r far from the collecting body. 

Evaluation of the average in equation (43) necessitates investigation of the oscillatory velocity fields. If equation (24) is employed for the gas velocity and a similar representation, = Re V ", is introduced for the particle velocity, then by use of the equation of motion for a particle. 

At this point, let us recall that the equations of motion for a particle i characterized by the generalized coordinate and conjugate momentum tVj are derived from a Lagrangian described in Eq. [48] as... 

In Eq. (A.17), as before X(f) is the vector of thermodynamic forces while M is a symmetric matrix of phenomenological parameters introduced by Machlup and Onsager [4]. We adopt Eq. (A.17) inasmuch as it is the simplest equation of motion that is consistent with the Machlup-Onsager Eq. (A.24). Notice that Eq. (A.17) is similar in form to Newton s equation of motion for a particle system. Thus, we denote the matrix of phenomenological parameters by M in order to emphasize the analogy to particle masses. The analogy, however, is not perfect because M may be nondiagonal [4]. 

The third contribution to the force on the particle is due to random fluctuations caused by interactions with solvent molecules. We will write this force as R(f). The Langevin equation of motion for a particle i can therefore be written" ... 

Up to this point, we have considered the drag force on a particle moving at a steady velocity through a quiescent fluid. Recall that this case is equivalent to the flow of a fluid at velocity oo past the stationary particle. The motion of the particle, however, arises in the first place because of the action of some external force on the particle such as gravity. The drag force arises as soon as there is a difference between the velocity of the particle and that of the fluid. The basis of the description of the behavior of a particle in a fluid is an equation of motion. To derive the equation of motion for a particle of mass nip, let us begin with a force balance on the particle, which we write in vector form as... 

If the reorganization energy is large compared with the coupling, the mass of the quasiparticle is large. In an ordinary ET problem, this treatment becomes a bit clumsy. Chemists measure reaction rates or the mean time for ET. They are less interested in an equation of motion for a particle that stays at the same site most of the time. 

Figure 4 shows the variation of impact force, velocity, compression, and energy transfer rate as a function of time. These values could be obtained by numerical solutions of the above equations, but are given directly by the solution of the differential equation of motion for a "particle with a linear restoring force," e.g., by the method of power series as given by Slater and Frank[7]. 

Few-body problems can be handled by conventional integrators, such as Runge-Kutta or Adams-Moulton methods. Here one calculates the position and velocity for each particle and then the precise two-body interaction for that body with every other particle in the system. Both methods are predictor-corrector procedures in which the next step is computed and corrected iteratively. Leapfrog methods, which use the velocity from one step and the positions from the previous step to compute the new positions, are also computationally efficient and stable. The basic problem is to solve the equations of motion for a particle at position Fj,... 

Corrsin S, Lumley 1 On the equation of motion for a particle in turbulent fluid, AppI Sci Res 6 114-116, 1956. 

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