Conservation equation linear momentum

Equation (5) expresses the conservation of linear momentum that defines the position of the center of mass of the molecule, while Eq. (6) is an approximate statement of the conservation of angular momentum of the system These conditions, which are usually attributed to Eckart lead to the relation... 

In a similar manner the homogeneity in space leads to the law of conservation of linear momentum [52] [43]. In this case L does not depend explicitly on qi, i.e., the coordinate qi is said to be cyclic. It can then be seen exploring the Lagrange s equations (2.14) that the quantity dLjdqi is constant in time. By use of the Lagrangian definition (2.6), the relationship can be written in terms of more familiar quantities ... 

It is, perhaps, well to pause for a moment to take stock of our developments to this point. We have successfully derived DEs that must be satisfied by any velocity field that is consistent with conservation of mass and Newton s second law of mechanics (or conservation of linear momentum). However, a closer look at the results, (2-5) or (2-20) and (2 32), reveals the fact that we have far more unknowns than we have relationships between them. Let us consider the simplest situation in which the fluid is isothermal and approximated as incompressible. In this case, the density is a constant property of the material, which we may assume to be known, and the continuity equation, (2-20), provides one relationship among the three unknown scalar components of the velocity u. When Newton s second law is added, we do generate three additional equations involving the components of u, but only at the cost of nine additional unknowns at each point the nine independent components of T. It is clear that more equations are needed. 

We can describe the conservation of linear momentum by noting the analogy between the time-dependent Schrodinger equation, (equation A1 4.108). and (equation A 1.4.99). For an isolated molecule, //does not depend explicitly on t and we can repeat the arguments expressed in (equation Al.4.981. (equation Al.4.991. (equation Al.4.1001. (equation A 1.4.101) and (equation Al.4.102) with X replaced by t and P replaced by -// to show that... 

In Sects. 6, 7, and 8 we have derived equations for the conservation of linear momentum, energy and the mass of molecular species. We now define the gradients of the related variables as k = (Vv) and a = V In T, and b, = V In n,. In general each of these gradients is a function of both position r,. and time t. It is usually adequate to assume that the higher derivatives of these variables are sufiiciently small that, over distances comparable to molecular dimensions, they may be neglected, and thus we may use the following truncated Taylor series for velocity, temperature, and concentration ... 

The largest transverse pressure difference is observed for a director orientation in the xz-plane (0=0). The transverse pressure can be derived from the equation for the conservation of linear momentum... 

Equation (2.39) specifies the velocity u of particle i with respect to the center of mass. The definition of the c.m. system, or the condition of conservation of linear momentum, tells us that wiui -f i2U2 = 0. In other words, irrespective of the laboratory conditions, in the c.m. system the lighter particle is faster moving. Similarly note that, as one would expect, the relative velocity v of die two particles has the same value in the laboratory and in the center-of-mass system ... 

The simplest constitutive equations provide the following three equations for the conservation of linear momentum ... 

Solution of Newton s equations of Fj g as a force completes the description of the particle dynamics. It is clear, however, that small heavy particles cannot be moved by the interface (see electron microscopy data). The maximum radius has been estimated from the conservation of linear momentum. To capture a particle of mass (m), the interface has to transfer to it a linear momentum (mv). If we assume that the particle does not move (or it moves much more slowly than the interface, which is valid for massive particles), then the total linear momentum transferred to the particle is... 

From the fact that f/conmuites with the operators Pj) h is possible to show that the linear momentum of a molecule in free space must be conserved. First we note that the time-dependent wavefiinction V(t) of a molecule fulfills the time-dependent Schrodinger equation... 

The equation of momentum conservation, along with the linear transport law due to Newton, which relates the dissipative stress tensor to the rate of strain tensor = 1 (y. 4, and which introduces two... 

Conservation of linear and angular momentum. If the potential function U depends only on particle separation (as is usual) and there is no external field applied, then Newton s equation of motion conserves the total linear momentum of the system, P,... 

Conservation of linear and angular momenta. After equilibrium is reached, the total linear momentum P [Eq. (9)] and total angular momentum L [Eq. (10)] also become constants of motion for Newton s equation and should be conserved. In advanced simulation schemes, where velocities are constantly manipulated, momentum conservation can no longer be used for gauging the stability of the simulation. 

The conservation equations developed by Ericksen [37] for nematic liquid crystals (of mass, linear momentum, and angular momentum, respectively) are ... 

Equation 5.1.13 shows how heat release acts a volume source. Assuming that the combustion takes place in a uniform medium at rest (Mach < 0), and writing for small perturbations, a = a + a a = p, p, v), the linearized conservation equations for mass and momentum can be used to eliminate the density in 5.1.13 to obtain a wave equation for the pressure in the presence of local heat release ... 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

We have also tried to assess the effects of integrating Hamilton s equations numerically. This is a rather difficult task since the exact solutions to these equations are not known. However, we can use the observed conservation of total energy and linear momentum as an indication that the equations are being integrated properly. For the Stockmayer and modified Stockmayer simulations the total energy and linear momentum were conserved to 0.05 and 0.0006%, respectively, over the 600 integration steps of the production phase of these calculations. 

Fluent is a commercially available CFD code which utilises the finite volume formulation to carry out coupled or segregated calculations (with reference to the conservation of mass, momentum and energy equations). It is ideally suited for incompressible to mildly compressible flows. The conservation of mass, momentum and energy in fluid flows are expressed in terms of non-linear partial differential equations which defy solution by analytical means. The solution of these equations has been made possible by the advent of powerful workstations, opening avenues towards the calculation of complicated flow fields with relative ease. 

From the first two equations one gets conservation of the linear momentum along z ... 

Now Eq. 2.5-2 is a vectorial equation that has three components, reflecting the fact that linear momentum is independently conserved in the three spatial directions. For a rectangular coordinate system, Eq. 2.5-2 becomes ... 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

The jet is governed by four steady-state equations representing the conservation of mass and electric charges, the linear momentum balance, and Coulomb s law for the E field [9]. Mass conservation requires that... 

For each continuous phase k present in a multiphase system consisting of N phases, in principle the set of conservation equations formulated in the previous section can be applied. If one or more of the N phases consists of solid particles, the Newtonian conservation laws for linear and angular momentum should be used instead. The resulting formulation of a multiphase system will be termed the local instant formulation. Through the specification of the proper initial and boundary conditions and appropriate constitutive laws for the viscous stress tensor, the hydrodynamics of a multiphase system can in principle be obtained from the solution of the governing equations. 

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