Flow axial symmetry

In vertical flow, axial symmetry exists and flow patterns tend to be somewhat more stable. However, with slug flow in particular, oscillations in the flow can occur as a result of sudden changes in pressure as liquid slugs are discharged from the end of the pipe. 

Fig. 32. Streamlines in abrupt contraction flow computed at a midtravel distance of the piston at x = — 40 mm (the origin of the abscisse is taken at the orifice entrance). Due to axial symmetry, only half of the flow tube is shown. The dimensionless stream function

The flow behaviour of suspensions of coarse particles is completely different in horizontal and vertical pipes. In horizontal flow, the concentration of particles increases towards the bottom of the pipe, the degree of non-uniformity increasing as the velocity of flow is decreased. In vertical transport, however, axial symmetry is maintained with the solids evenly distributed over the cross-section. The two cases are therefore considered separately. 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

It can be seen in Fig. 10.2 that the flow in the SCISR is in mirror symmetry with respect to the impingement plane, and also approximately in axial symmetry so it is enough to take a half volume of the SCISR, either on the right or the left side, into account in the analysis. 

In order to understand the profile of the pressure fluctuation over the volumetric space inside the reactor, multipoint measurement is carried out in each run with five probes. The measuring points are arranged according to the coordinate system shown in Fig. 11.3, where the x-z plane is the impingement plane, x-y is the horizontal plane, and y-z the vertical plane the values of the coordinates are in mm. The flow inside the SCISR is considered to have an approximately axial symmetry. For convenience, part of the data are interrelated in a pillar coordinate system, and the radial coordinate, r, is determined by... 

The property characterizing the collimation of the beam formed is the spatial distribution 1(9, emitted particles. Due to the axial symmetry of the tube, the angle dependence is only on which is the angle of the emitted particles against the tube axis. The connection between 1(9) and the total rate of flow N is given by... 

To evaluate the heating, a relativistic 1-D Fokker-Planck code was used. The configuration space is 1-D but the momentum space is 2-D, with axial symmetry. This code is coupled to a radiation-hydrodynamic simulation in order to include energy dissipation via ionization processes, hydrodynamic flow, the equation-of-state (EOS), and radiation transport. The loss of kinetic energy from hot electrons is treated through Coulomb and electromagnetic fields. 

For steady flow with axial symmetry, let x be the axis of symmetry, r be the radial coordinate normal to the axis of symmetry, be the component... 

Fig. 17. Molecule with an axial symmetry of optical and hydrodynamic properties in a laminar flow. X flow direction Y direction of velocity gradient Z direction of obs ation 1 axis of symmetry of the molecule...

Because the domain is unbounded inz and there is axial symmetry, the flow is a function of r only. Obtain the solution for ur that vanishes at infinity, and derive the following Eulor-type differential equation for ug ... 

Boundary layer approximation. The Landau problem, which was described above, is an example of an exact solution of the Navier-Stokes equations. Schlichting [427] proposed another approach to the jet-source problem, which gives an approximate solution and is based on the boundary layer theory (see Section 1.7). The main idea of this method is to neglect the gradients of normal stresses in the equations of motion. In the cylindrical coordinates (71, ip, Z), with regard to the axial symmetry (Vv = 0) and in the absence of rotational motion in the flow (d/dip = 0), the system of boundary layer equations has the form... 

Let us consider a solid spherical particle of radius o in a translational Stokes flow with velocity U and dynamic viscosity /i (Figure 2.1). We assume that the fluid has a dynamic viscosity /z. We use the spherical coordinate system. R, 9, ip with origin at the center of the particle and with angle 0 measured from the direction of the incoming flow (that is, from the rear stagnation point on the particle surface). In view of the axial symmetry, only two components of the fluid velocity, namely, Vr and Vg, are nonzero, and all the unknowns are independent of the third coordinate [Pg.58]

The equation as written is valid for two-dimensional, unsteady diffusion boundary layers. The basic features of this result and those that follow are unaltered if the diffusion coefficient is not constant or if the flow is generalized to axial symmetry or three dimensions, although some details will differ. 

The method used to research stability of small perturbations of a cylindrical jet of non-viscous incompressible fluid is similar to the previously discussed method of handling the small perturbation problem. The main difference from the case of a plane surface is the axial symmetry of the problem, which consequently features the characteristic linear size a (radius of the jet). In a coordinate system moving with the jet s velocity, the jet itself is motionless. Let us neglect gravity and take into account only the force of surface tension. Then the pressure along the jet is equal to p = pa+ 2/a (since l i = oo, i 2 = a). Now proceed to linearize the equations and the boundary conditions. Assume the perturbations of the flow to be small and consider the equation of motion. After linearization, i.e. after rejecting the second order terms, one obtains Eq. (17.36) for velocity perturbations and full pressure. Since the flow is a potential one, any perturbation of the velocity potential satisfies the Laplace equation = 0, which in a cylindrical coordinate system (r, 0, z) is written as... 

A central film gate is used so that the flow into the mould cavity has axial symmetry, in preference to a four arm gate, which starts the flow along four radii. The birefringence criterion can only be met by minimising the orientation effects discussed in Chapter 6—molecular orientation in the skin or in the core of the moulding, and residual stresses. 

Moreover, standard LBE is strictly formulated either in two- or three-dimensions. On the other hand, there is a large class of problems related to interfacial flows and instabilities in sprays that can be considered to have an axial symmetry. Such symmetries should be fully exploited for computationally efficient... 

Unlike vertical flow, horizontal flow does not exhibit axial symmetry and the flow pattern is more complex because the gravitational force acts normally to the direction of flow, causing asymmetrical distribution over the pipe cross-... 

Stratifiedflow a clear interface separates the two flows (lamination) and mixing occurs entirely by diffusion Vortex flow, the vortex creates secondary surface area Engulfmentflow the axial symmetry of the flow breaks up (see Section 4.2). 

In order to obtain quantitative results, the flow factor has to be determined this requires knowledge of the stress field in the hopper. Closed-form expressions for ff are not available, except for the simplest case of flow through a straight cylinder. Results from numerical analysis are given by Jenike [8, 9] in graphical form for plane symmetry and axial symmetry. 

The numerical simulations of fluid flows assuming axial symmetry within the main flow. Low feed concentrations are again often assumed and particle trajectories can be studied. 

---