The velocity field

The generalized Navier-Stokes equation for the velocity field v follows from momentum balance 

It is instructive to consider the three simple geometries for plane parallel shear flow, the Miesovicz geometries [73], corresponding to the orientations of the director relative to the flow axis and the shear gradient. Choosing v = u(z)x one has the effective shear viscosities  

The value of the non-vanishing component of the torque F on the director is also given. Another positive effective viscosity is 7o = ai + 014 + a + All shear viscosities are typical of order 10 kg m s.  

4 Unfortunately the definitions of r/i and ri2 are not universally accepted. Our definition is adopted by Blinov [57] whereas the definitions in the review articles [67] and [73] are in the reverse order. 

Note that the flow-alignment parameter A = -72/71 is a reversible quantity. For A 1 the director can align in the flow plane at a fixed angle p with respect to the velocity (x-axis), where = (A-1)/(A + 1) = a3/ 2. For A 1 there is tumbling [73], 

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Let Ap, Au and AT denote the deviations of the mass density, p, the velocity field, u, and the temperature, T, fiom their fiill equilibrium values. The fluctuating, linearized Navier-Stokes equations are... 

The third application is to velocity field fluctuations. For an equilibrium fluid the velocity field is, on average, zero everywhere but it does fluctuate. The correlations trim out to be... 

If a flow satisfies the condition of zero vorticity, that is, the velocity field v is such that V X V = 0, then there exists a function v such that v = Vv. In that case, one can describe the fluid mechanical system with the following Lagrangean density... 

Maxwell considered the motion of a gas in the neighborhood of a plane solid wall, in che presence of a temperature gradient. In particular, when Che velocity field is one dimensional and everywhere parallel to the wall, and the temperature gradient is parallel to the velocity field, he found that... 

In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

The right-hand side of Equation (3,87) is set to zero considering that DA//Dt, DFIDt and the divergence of the velocity field in incompressible fluids are all equal to zero. Therefore, after integration Equation (3.87) yields... 

The results of computations of T o for an isolated fiber are dhistrated in Figs. 17-62 and 17-63. The target efficiency T t of an individual fiber in a filter differs from T o for two main reasons (Pich, op. cit.) (1) the average gas velocity is higher in the filter, and (2) the velocity field around the individual fibers is influenced by the proximity of neighboring fibers. The interference effect is difficult to determine on a purely theoretical basis and is usually evaluated experimentally. Chen (op. cit.) expressed the effecd with an empirical equation ... 

The analytical solution for an infinitely flanged slot can be obtained by assuming that the inlet is composed of elemental point sinks.The velocity field of an infinitely flanged slot can be obtained by assuming the velocity to be uniformly distributed across the opening. The contribution to the velocity potential at point (x, y) due to the elemental line sink of length and located at (0, Q is given by... 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

Semi-Theoretical and Empirical Velocity Fields Since the use of formulas to calculate the velocities outside an arbitrary opening could be very tedious, only some examples of these formulae are given. These calculations are best done on computers and there are some dedicated programs to calculate and visualize the flow fields outside exhaust openings. There could sometimes be problems when calculating the velocity field outside an opening close to... 

IJnflanged Circular, Rectangular, and Slot Openings For unflanged openings no explicit equations exist for the flow fields. However, it is possible to calculate the velocity field outside a specific BEO using computers. These... 

He indicates that the buoyancy is negligible for the velocity field for an Archimedes number less than a critical value equal to Ar, = 0.15. 

C) and Cgm denote the mean concentration in the occupied zone, concentration at a given point P, the mean concentration in the room, and the concentration at the outlet, respectively. To numerically simulate these parameters, the velocity field is first computed. Then a contaminant source is introduced at a cell (or cells) of a region to be studied, and the transport equation for contaminant C is solved. The transport equation for C is... 

The previous methods are mainly used to measure duct flow. When measuring flows on supply or exhaust terminals, different methods are used. The measurement on exhaust terminals is simple to carry out, as the velocity field near the terminal is relatively constant, with no steep gradients or swirls. In the case of a grill, traversing across the terminal surface using a suitable velocity instrument is a good alternative. A suitable instrument for most cases is the vane anemometer. 

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

In the preceding categories of flow, the velocity field is deterministic since it can be calculated (at least in principle) from the constitutive equation of the fluid and the experimental boundary conditions. Turbulent flow, on the other hand, is distinctively unpredictable, as was pointed out a century ago by Osborne Reynolds. 

By making use of the spatial information, the velocity field of an extended, structured object can be obtained unambiguously without errors caused by uncertainty in the position of a feature within the slit. 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

Heat conduction through metal, in the x direction, is negligible because the metal temperature gradient, in this direction, is small. With these assumptions, the velocity field is given by. 

On comparing the two flames, it is evident that the flow structure of the lean limit methane flame fundamentally differs from that of the limit propane one. In the flame coordinate system, the velocity field shows a stagnation zone in the central region of the methane flame bubble, just behind the flame front. In this region, the combustion products move upward with the flame and are not replaced by the new ones produced in the reaction zone. For methane, at the lean limit an accumulation of particle image velocimetry (PIV) seeding particles can be seen within the stagnation core, in... 

Velocity vectors of the gas flow measured using laser Doppler anemometry inside a closed chamber during the formation of a tulip flame. Images of the flame are also shown, though the velocity measurements required many repeated runs, hence, the image is only representative. The chamber has square cross sections of 38.1mm on the side. The traces in the velocity fields are the flame locations based on velocity data dropout. The vorticity generated as the flame changes shape appears clearly in the velocity vectors. 

Temporal sequence of OH-LIF measurements captures a localized extinction event in a turbulent nonpremixed CH4/H2/N2 jet flame (Re 20,000) as a vortex perturbs the reaction zone. The time between frames is 125 ps. The velocity field from PIV measurements is superimposed on the second frame and has the mean vertical velocity of 9m/s subtracted. (From Hult, J. et al.. Paper No. 26-2, in 10th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, 2000. With permission.)... 

Radioactive particle tracking (RPT) can be used to map the velocity field by tracking the position of a single radioactive tracer particle in a reactor. The particle which may consist of a polypropylene shell contains a radionuclide that emits y-rays. 

Even in the case of standard reactors such as stirred tanks and bubble columns, lack of knowledge in this area limits our ability to use particle stress as a selection criterion. The reasons for this lack of knowledge are, on the one hand, that the velocity fields in the reactors, which would allow a certain prediction, can only be obtained by sophisticated measurements and measurement techniques, and on the other hand, the stress on particles becomes evident only as an integral result of a long term process. 

In the velocity field of the determining eddies, which is characterized by the turbulent fluctuation velocity the particles experience a dynamic stress according to the Reynolds stress Eq. (2) ... 

The strategies discussed in the previous chapter are generally applicable to convection-diffusion equations such as Eq. (32). If the function O is a component of the velocity field, the incompressible Navier-Stokes equation, a non-linear partial differential equation, is obtained. This stands in contrast to O representing a temperature or concentration field. In these cases the velocity field is assumed as given, and only a linear partial differential equation has to be solved. The non-linear nature of the Navier-Stokes equation introduces some additional problems, for which special solution strategies exist. Corresponding numerical techniques are the subject of this section. 

The situation is different for incompressible flow. In that case, no equation of motion for the pressure field exists and via the mass conservation equation Eq. (17) a dynamic constraint on the velocity field is defined. The pressure field entering the incompressible Navier-Stokes equation can be regarded as a parameter field to be adjusted such that the divergence of the velocity field vanishes. 

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