Conservation of momentum, equations

Several discrete forms of the conservation of momentum equation, (9.3), can be derived, depending on the type of mesh and underlying assumptions. As an example, assume the equation will be solved on staggered spatial and temporal meshes, in two dimensions, in rectangular geometry, and with the velocities located at the nodes. Assume one quarter of the mass from each adjacent element is associated with the staggered element as shown in Fig. 9.11. 

Mixture conservation of mass equation Mixture conservation of momentum equation Mixture conservation of energy equation Slip equation (concerning the difference in velocity)... 

The high temperature gas generated in the combustion chamber of a rocket engine is expanded through a De Laval nozzle to convert a major portion of the enthalpy to mechanical thrust. The thrust F of a rocket engine is defined by the following conservation of momentum equation ... 

The expression for overall conservation of momentum, equation (26), is also identical with the corresponding relation in the kinetic theory, ... 

The first is that, because of the local equilibrium at any 6, the ionic concentrations are related to the potential according to Boltzmann s law. In the second place, the conservation of momentum equation ]4.6.5] can. for a relatively thin double layer, be decomposed into its normal emd tangential parts... 

We shall use the conservation of momentum equation in the form [4.6.6]. The argument for doing so is that the time derivative terms contribute only weakly to the force balance, as already pointed out by DeLacey and White. For high Reynolds number cases this simplification is not allowed, but we shall not consider this. 

If Eq. 4.48 is used along with the conservation of momentum equations, the energy of the scattered photon as a function of the scattering angle 6 can be calculated. The result is (see Evans)... 

The continuum mechanics modeled by JAS3D are based on two fundamental governing equations. The kinematics is based on the conservation of momentum equation, which can be solved either for quasi-static or dynamic conditions (a quasistatic procedure was used for these analyses). The stress-strain relationships are posed in terms of the conventional Cauchy stress. JAS3D includes at least 30 different material models. 

As downstream pressure Pj is decreased and upstream pressure P is kept fixed, the flow increases until the internal pressure of the tube drops somewhat below the external pressure P. Then the tube partially collapses, decreasing in cross-sectional area according to its pressure-area or tube law relationship A(P), shown in Fig. 4.11i>. As P is further decreased, the tube reduces its cross-sectional area while the average velocity of the flow increases. However, their product, the volumetric flow rate F, does not increase as shown in Fig. 4.11c. A simplified understanding of this behavior may be seen from the conservation of momentum equation for the flow, a Bernoulli equation, where the average fluid velocity U is defined as the ratio of flow to cross-sectional area, U = FIA, all of the terms pressure dependent. 

The equation of motion is really the equation for the conservation-of-momentum equation (2.8-3), which we can write as... 

In the early 1800s the Navier-Stokes equations were independently derived by G. G. Stokes and M. Navier [5]. These are a set of equations that describe the motion of a fluid. It consists of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation. The flow condition can be estimated using these equations. 

The Lagrangian conservation equations in two dimensions for slabs, cylinders, and spheres are given below. The sign conventions of the stress deviators are reversed from those of Appendix A to be negative in compression and positive in tension so cr = P — Sxx and mass equation is automatically satisfied. The conservation of momentum equation is... 

The conservation of momentum equation is derived from Newton s second law of motion, which expresses proportionality between applied force and resulting acceleration of a particle. Momentum equation for Newtonian viscous fluid is given by the Navier-Stokes equation... 

Mass balances must be used with the flow pattern (knowm a priori from theory or experimental measurements) to establish species concentrations and fluxes at any point in the fuel cell. When the flow pattern is a priori unknown, conservation-of-momentum equations (also called equations of motion) must be used with mass balance equations to establish the velocity and concentration profiles. Conservation of momentum for gases leads to the following equations (Navier-Stokes equations), in which k represents one of the three orthogonal directions in the coordinate sj stem (x, y, and z) ... 

---