Free momentum equation

In Eq. (67) the classical energy of a free particle, a = mu2, has been substituted, with u its velocity and mv its momentum. Equation (67) is of course the well-known relation of deBroglie. 

The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor flow. This assumption, however, needs to be modified by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor flows in the symmetrical coordinates ( ,2) are given as Eqs. (43) and (47). When the Knudsen number of the vapor flow is between 0.01 and 0.1, the flow is in the slip regime. In this regime, the flow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor flow at the solid surface can be given by... 

Consider a two-dimensional free convective flow. The momentum equation for the jc-direction, which is assumed to be in the vertical direction, is ... 

A consideration of the orders of magnitude of the terms in the momentum equation for boundary layer flow indicates that if u = o(u ), where u is characteristic free-stream velocity, then the buoyancy force term will be important if ... 

For the free-convection system, the integral momentum equation becomes... 

The velocity components in die free stream region of a flat plate are u = V = constant and v = 0. Substituting these into Ihe. t-momentum equations (Eq. 6-28) gives dP/dx = 0. Therefore, for flow over a flat plate, the pressure remains constant over the entire plate (both inside and outside the boundary layer). 

At the inlet to the finite element domain, the flow is parallel so the equations of the lubrication approximation are used to specify the inlet velocity profile. These equations are integrated from -oo to the inlet, generating an equation relating the flow rate to the inlet pressure. The remaining boundary conditions are as shown in Figure 3. The only complexity here is that the fluid traction, n T, at the free surface has to be specified as a boundary condition on the momentum equation. A force balance there gives, in dimensionless form,... 

In order to see that a free flow can only exist if the density gradient and body forces are not parallel to each other, we will discuss a fluid that is initially at rest, Wj = 0. According to the momentum equation (3.48) it holds for this that... 

We will now deal with free flow on a vertical, flat wall whose temperature d0 is constant and larger than the temperature in the semi-infinite space. The coordinate origin lies, in accordance with Fig. 3.43, on the lower edge, the coordinate x runs along the wall, with y normal to it. Steady flow will be presumed. All material properties are constant. The density will only be assumed as temperature dependent in the buoyancy term, responsible for the free flow, in the momentum equation, in all other terms it is assumed to be constant. These assumptions from Oberbeck (1879) and Boussinesq (1903), [3.45], [3.46] are also known as the Boussinesq approximation although it would be more correct to speak of the Oberbeck-Boussinesq approximation. It takes into account that the locally variable density is a prerequisite for free flow. The momentum equation (3.294) in... 

Simple molecular photoionization is the interaction of electromagnetic radiation with a molecule (in its ground state) to generate an ion (in a variety of different possible energy, and symmetry states) and a free electron Equation (17.2). In this process, mass, momentum and energy are conserved, according to Equation (17.3). 

Free-surface flow is a limiting case of flow with interfaces, in which the treatment of one of the phases is simplified. For instance, for some cases of gas-liquid flow, we may consider the pressure pgas in the gas to depend only on time and not on space and the viscous stresses in the gas to be negligible. For such flows the jump condition formulation must be used, since the bulk momentum equation breaks down. The jump conditions become boundary conditions on the border of the liquid domain [245, 183[ ... 

The pressure-based method was introduced by Harlow and Welch [67] and Chorin [30] for the calculation of unsteady incompressible viscous flows (parabolic equations). In Chorines fractional step method, an incomplete form of the momentum equations is integrated at each time step to 3ueld an approximate velocity field, which will in general not be divergence free, then a correction is applied to that velocity field to produce a divergence free velocity field. The correction to the velocity field is an orthogonal projection in the sense that it projects the initial velocity field into the divergence free... 

We are going to explain the procedure of forming wave packets out of plane waves for the free Dirac equation. The free stationary Dirac equation Ho tp = Erp has no square-integrable solutions at all. But it turns out that for E > me there are bounded oscillating solutions (here bounded means that the absolute value [ipix, t) of the solution remains below a certain constant M for all x and t). As for the Schrodinger equation, it is comparatively easy to find these solutions. They are similar to plane waves with a fixed momentum (wavelength), that is, they are of the form... 

SHEAR-STRESS DISTRIBUTION IN A CYLINDRICAL TUBE. Consider the steady flow of a viscous fluid at constant density in fully developed flow through a horizontal tube. Visualize a disk-shaped element of fluid, concentric with the axis of the tube, of radius r and length dL, as shown in Fig. 5.1. Assume the element is isolated as a free body. Let the fluid pressure on the upstream and downstream faces of the disk be p and p + dp, respectively. Since the fluid possesses a viscosity, a shear force opposing flow will exist on the rim of the element. Apply the momentum equation (4.14) between the two faces of the disk. Since the flow is fully developed, j8j, = and Fj, = F , so that E F = 0. The quantities for substitution in Eq. (4.15) are... 

There are two trends in the existing hydrodynamic theories of free films, which differ in the way the (colloidal) interaction forces are taken into account. We mention first the method of Felderhof, who developed a systematic and consistent electrohydrodynamic theory for a nonviscous liquid. His theory was extended to include viscous behavior by Sche and Fijnaut. In these theories the interaction forces are included in the momentum equations. The other theoretical approach considers hydro-dynamic equations without the interaction forces. The influence of these interactions are considered only in the normal stress boundaiy conditions. 

The Strouhal number is a measure of the unsteadiness of the motion. The Froude number is important in free surface flows, for example. The last parameter appearing in the dimensionless momentum equation is... 

In addition to the momentum equation, it is also necessary to satisfy the equation of continuity, which for the incompressible fluid considered is V u = 0. The disturbance velocity is also irrotational hence, in addition, V X u = 0 (Lighthill 1978). Therefore the disturbance velocity field is derivable from a gradient of a velocity potential that is, u = V with the potential satisfying Laplace s equation V — 0. This may seem surprising at first, but Laplace s equation can describe a wave motion when boundary conditions are satisfied at a free surface. 

Projection-type methods were also developed to solve incompressible flows [12]. A projection scheme consists of two steps first the momentum equations are solved without the pressure term for one time-step this gives the velocities. The second step consists of projecting the intermediate velocity back onto the space of the divergence free field (for more details see Ferziger and Peric [12]). 

The top face of the control volume can be considered the free surface and therefore the shear stress is zero (plane A-A in Figure 4.9). The normal stress at this surface is due to the forces exerted by the freeboard gas. if there is no significant saltation of particles on the free surface, then the normal stress at this plane is also zero and equating the net momentum to the net force yields the integral momentum equation... 

The blown film process was briefly described in Section 1.2.6. The process is shown schematically in Figure 10.7. There are many similarities between the blown film and the fiber spinline because of the free surface and the very small transverse dimension relative to the distance between melt extrusion and solidification, and thin sheet equations analogous to the thin filament equations are typically used, although the hoop stress must now be taken into account. The equations for a Newtonian fluid were first published by Pearson and Petrie in 1970, and their approach has been used by nearly all investigators since. There are two steady-state momentum equations because variations in both thickness and width in the stretching direction are important. The mechanics of the solid region above the ill-defined freezeline are... 

Originally, the concept of fluid boundary layer was presented by Prandtl [123]. Prandtl s idea was that for flow next to a solid boundary a thin fluid layer (i.e., a boundary layer) develops in which friction is very important, but outside this layer the fluid behaves very much like a ffictionless fluid. To define a demarcation line between these two flow regions the thickness of the boundary layer, 6, is arbitrarily taken as the distance away from the surface where the velocity reaches 99 % of the free stream velocity (e.g., [55], p. 192 [107], p. 12 [114], p. 545). To proceed giving a thorough description of the equations used for turbulent flows, we need some results from the semi-empirical turbulent boundary layer flow analysis. For a generalized shear flow in the vicinity of a flat horizontal solid wall, the boundary layer flow can be described in Cartesian coordinates. The stress, —Ojty.eff, associated with direction y normal to the wall is apparently dominant, thus the stream-wise Reynolds averaged momentum equation yields ... 

In this section the basis elements of the volume of fluid (VOF) method are described. In general the VOF model is composed of a set of continuity and momentum equations, as well as a transport equation for the evolution of a phase indicator function which is used to determine the location and orientation of the interface. We distinguish between the jump condition—and the whole field formulations of the method, in which both forms are based on a macroscopic view defining the interface as a 2D surface. The jump condition form is especially convenient for free-surface flow simulations, whereas the whole field formulation is commonly used for interfacial flow calculations in which the internal flow of all the phases are of interest. 

In this relation, the operator 6/dXi outside the parentheses on the LHS is the divergence operator inherited from the continuity equation, while 5jf/5xi is the pressure gradient from the momentum equation. The operator 5p/5t is the time advancement of the continuity equation. If the pressure p" satisfies the discrete Poisson equation, the velocity field at time step n + 1 will fulfill continuity (and be divergence free for constant density flows). 

The free surface flow was calculated by using the control volume finite element method (CVFEM). Since the depth of the microchannel is small compared with the length and width, we used the Hele-Shaw approximation for the momentum equation, which is given as follows ... 

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