Velocity field development, axial

Figure 2. Radial-axial velocity field and temperature contours for a rotating-disk reactor at an operating condition where a buoyancy-driven recirculation vortex has developed. The disk temperature is HOOK, the Reynolds number is 1000, Gr/Re / = 6.2, fo/f = 1.28, and L/f = 2.16. The disk radius is 4.9 cm, the spin rate is 495 rpm. The maximum axial velocity is 55.3 cm/sec. The gas is helium.

Figure 1. Development of the axial velocity field obtained by numerical solution of the parabolic differential equations...

Fig. 10.20 Axial location of the two planes perpendicular to the counterrotating screws, where velocity fields were calculated. Plane (I) is at the middle of the side, and plane (II) at the middle of the calender gaps. [Reprinted by permission from T. Katziguara, Y. Nagashima, Y. Nakano, and K. Funatsu, Numerical Study of Twin Screw Extruders by 3-D Flow Analysis - Development of Analysis Technique and Evaluation of Mixing Performance for Full Flight Screws, Polym. Eng. Sci., 36, 2142 (1996).]...

We consider fully developed incompressible laminar flow, considering slip at the walls, inside a circular micro-tube or a parallel plates micro-channel subjected to a pressure gradient dp/dz that varies in an arbitrary functional form with the time variable. The velocity field is represented by u(r,t), which varies with the transversal coordinate, r, and time, t. The related time-dependent axial momentum equation (z-direction) is then written in dimensionless form as ... 

Dynamic interfacial effects due to capillary advancement in the presence of electroosmotic flows in hydrophobic circular microchannels have recently been investigated by Yang et al. [8]. In a general sense, their theoretical development is based on the prototype equation of motion of the form of Eq. 1, with an additional term appearing in the right-hand side to model the electroosmotic body force. Not only that, the quantification of the viscous drag force is also adapted to accommodate the influences of electroosmotic slip. To address these issues carefully, one may first derive an expression for electroosmotic flow velocity in presence of an axial electric field strength of Ej in the solution as... 

Some time after the flow has started, the time variation of the temperature approaches zero if the botmdary conditions are independent of time. Assuming a steady-state axial pressure distribution and a fully developed velocity field inside the microchannel, the energy equation becomes ... 

To study laminar flow forced convection in microcharmels tmder the assumptions of steady-state axial pressure distribution and fully developed velocity field, the balance of the thermal energy can be written as follows [1] (see Convective Heat Transfer in Microcharmels ) ... 

Figure 3.3 Nonidealities in a PFR boundary layer development. The nonuniformities in the velocity fields cause mixing problems, giving rise to axial and/or radial dispersion effects.

With these governing equations and boundary conditions in place, and if the input pressure distribution that drives the flow field is known, it is possible to develop a formal solution for the axial velocity. For an oscillatory flow, the input pressure would normally be expected to be of a sinusoidal form P(x, r, t) = const e . Following Reference 18, the method of characteristics shows the pressure distribution throughout the tube to be P(x, r, t) = A(x, where c is the wave speed in the... 

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