Poisson flow

The motion of polydispersed particulate phase is modeled making use of a stochastic approach. A group of representative model particles is distinguished. Motion of these particles is simulated directly taking into account the influence of the mean stream of gas and pulsations of parameters in gas phase. Properties of the gas flow — the mean kinetic energy and the rate of pulsations decay — make it possible to simulate the stochastic motion of the particles under the assumption of the Poisson flow of events. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

The prevailing theory of heat, popularized by Sinieon-Denis Poisson, Antoine Lavoisier and others, was a theory of heat as a substance, caloric. Different materials were said to contain different quantities of caloric. Fourier had been interested in the phenomenon of heat from as early as 1802. Fourier s approach was pragmatic he studied only the flow of heat and did not trouble himself with the vexing question of what the heat actually was. 

The Poisson process represents only one possible way of assigning joint distribution functions to the increments of counting functions however, in many problems, one can argue that the Poisson process is the most reasonable choice that can be made. For example, let us consider the stream of electrons flowing from cathode to plate in a vacuum tube, and let us further assume that the plate current is low enough so that the electrons do not interact with one another in the... 

The area of a peak is the integration of the peak height (concentration) with respect to time (volume flow of mobile phase) and thus is proportional to the total mass of solute eluted. Measurement of peak area accommodates peak asymmetry and even peak tailing without compromising the simple relationship between peak area and mass. Consequently, peak area measurements give more accurate results under conditions where the chromatography is not perfect and the peak profiles not truly Gaussian or Poisson. 

Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

The validity of the Poisson distribution for silver nucleation is demonstrated in Fig. 5.48B. The assumption for this kind of treatment is that the nucleus formation is irreversible and that the event is binary consisting of a discontinuous process (nucleus formation) and a continuous process (flow of... 

At the instant of contact between a sphere and a flat specimen there is no strain in the specimen, but the sphere then becomes flattened by the surface tractions which creates forces of reaction which produce strain in the specimen as well as the sphere. The strain consists of both hydrostatic compression and shear. The maximum shear strain is at a point along the axis of contact, lying a distance equal to about half of the radius of the area of contact (both solids having the same elastic properties with Poisson s ratio = 1/3). When this maximum shear strain reaches a critical value, plastic flow begins, or twinning occurs, or a phase transformation begins. Note that the critical value may be very small (e.g., in pure simple metals it is zero) or it may be quite large (e.g., in diamond). 

The charge within the metal is confined to an extremely thin layer < 1 A) at the surface. There is no charge separation within the metal since if there were there would be an associated potential difference and this would cause a current to flow. Potential V and charge density p (coulomb [C] m 3) are related by the Poisson equation ... 

The divergence of (4.21) yields a Poisson equation for p. However, the residual stress tensor r6 is unknown because it involves unresolved SGS terms (i.e., UfiJfi). Closure of the residual stress tensor is thus a major challenge in LES modeling of turbulent flows. 

Get for yourself [advises Poisson] a pound of pure mercury freshly dug from a mine. Next, take some Roman Vitriol and some calcinated, ordinary table-salt. Grind and thoroughly mix these together. Put these last two materials into a large glazed earthenware crock. Place this upon a slow fire and keep it there until the matter begins to melt and to flow. 

From this equation it is clear that there is an instantaneous relationship between the velocity field and the pressure field that is described by a Poisson equation. It does not depend directly on the viscosity nor involve any viscous transport terms. Note, again, that this result is only for incompressible flows. 

It is seen that the relationship between streamfunction and vorticity is described by a Poisson equation. Depending on the particular coordinates, the operator on the right-hand side may reduce to a Laplacian. In this case of axisymmetric flow, the operator is not a Lapla-cian. 

Although the principles of heat flow have lieen understood and treated mathematically since the early 19th century (Fourier, LaPlace. Poisson, Peclel. Lord Kelvin. Riemann. and many otherst. it was not until nearly... 

Electrokinetic phenomena can be understood with the help of two equations The known Poisson equation and the Navier3-Stokes4 equation. The Navier-Stokes equation describes the movement of a Newtonian liquid, i.e., a liquid whose viscosity does not change when it flows and when it is sheared. In order to make the equation plausible we consider an infinitesimal quantity of the liquid having a volume dV = dx dy dz and a mass dm. If we want to write Newtons equation of motion for this volume element we have to consider three forces ... 

As discussed earlier, with a nozzle of 70 pm and a stream moving at 10 m/s, our system is committed to a vibration frequency of about 30,000 cycles per second (30 kHz) in order to get drops to form. If we prefer a margin of safety, we may want to charge and sort three drops at a time in that case, we will want a particle in no more than every third drop. This means that our total particle flow rate can be no faster than 10,000 particles per second. Because cells are not spaced absolutely evenly in the flow stream (they obey a Poisson distribution), most sorting operators like to have particles separated by about 10-15 empty drops. With a 70 pm nozzle and a stream velocity of 10 m/s, this restricts our total particle flow rate to about 2000-3000 particles per second. For sorting cells of very low frequency within a mixed population, this may involve unacceptably long sorting times. 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

In this section, we will proceed to develop a finite element formulation for the two-dimensional Poisson s equation using a linear displacement, constant strain triangle. Poisson s equation has many applications in polymer processing, such as injection and compression mold filling, die flow, potential problems, heat transfer, etc. The general form of Poisson s equation in two-dimensions is... 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

The boundary conditions are defined in the same way as with the flow analysis network. The nodes whose control volumes are empty or partially filled are assigned a zero pressure, and the gate nodes are either assigned an injection pressure or an injection volume flow rate. Just as is the case with flow analysis network, a mass balance about each nodal control volume will lead to a linear set of algebraic equations, identical to the set finite element formulation of Poisson s or Laplace s equation. The mass balance (volume balance for incompressible fluids) is given by... 

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