Two-dimensional flows

The difference between the one-dimensional and the two-dimensional analysis will increase with increasing helix angle and reducing power law index. From a practical point of view, the use of a two-dimensional analysis becomes important when large helix angles and strongly non-Newtonian fluids are analyzed. The equation of motion in the down-channel direction is the same as used before see Eq. 7.194. A similar expression has to be used for the cross-channel direction. The shear stress profiles can be written as  

The magnitude of the total shear stress is obtained from  

The direction of shear stress x and veiocity v is determined by Xy and Xyx- At this point, there are three unknowns the cross-channei pressure gradient g, the crosschannel shear stress at the screw surface x o, and the down-channel shear stress at the screw surface x . At the time of writing, no analytical solutions to this problem are known. In fact, an analytical solution does not seem possible. Therefore, some numerical scheme has to be used in order to determine the unknowns. Because this problem is of considerable importance to the proper analysis of melt conveying, it will be discussed in some detail. 

Some initial values of g, x , and x can be selected, for instance, by calculating the values for the Newtonian case. From those initial values, the corresponding velocity profiles and the flow rates in the cross-channel direction can be determined. The velocity profile in the x and z directions can be determined from  

The net flow rate in the cross-channel direction should be zero if it is assumed the leakage over the flight is negligible. The cross-channel flow rate is  

---

Two-dimensional flow is in the x and y directions, while three-dimensional flow is in the x, y, and 2 directions. 

The Stream Function Stream functions are defined for two-dimensional flow and for three-dimensional axial symmetric flow. The stream function can be used to plot the streamlines of the flow and find the velocity. For two-dimensional Bow the velocity components can be calculated in Cartesian coordinates by... 

Outside the jet and away from the boundaries of the workbench the flow will behave as if it is inviscid and hence potential flow is appropriate. Further, in the central region of the workbench we expect the airflow to be approximately two-dimensional, which has been confirmed by the above experimental investigations. In practice it is expected that the worker will be releasing contaminant in this region and hence the assumption of two-dimensional flow" appears to be sound. Under these assumptions the nondimensional stream function F satisfies Laplace s equation, i.e.. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

The general features of two-dimensional flow with evaporating liquid-vapor meniscus in a capillary slot were studied by Khrustalev and Faghri (1996). Following this work we present the main results mentioned in their research. The model of flow in a narrow slot is presented in Fig. 10.16. Within a capillary slot two characteristic regions can be selected, where two-dimensional or quasi-one-dimensional flow occurs. Two-dimensional flow is realized in the major part of the liquid domain, whereas the quasi-one-dimensional flow is observed in the micro-film region, located near the wall. 

Now consider the case of two-dimensional flow subjected to the lubrication geometry assumptions that result from analyzing the order of magnitude for the velocities in thin film flow ... 

In order to calculate the shear stress in the chamber several parameter have to be fulfilled To achieve a two-dimensional flow the ratio of width to height of the flow channel is 5 1 [43]. To maintain a laminar flow the Reynolds number, given as... 

Duct flows, like steady two-dimensional flows, are poor mixers. This class of flows is defined by the velocity field... 

Avalosse, Th., and Crochet, M. J., Finite element simulation of mixing 1. Two-dimensional flow in periodic geometry. AlChE J. 43, 577-587 (1997a). 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

The oldest and most widely used method of estimating water age is the calculation of travel times using Darcy s law combined with an expression of continuity. If a field of steady-state, groundwater flow is subdivided into a two-dimensional flow net (figure 1), then Darcy s law can be written as ... 

This method starts with knowledge of the governing equation. The governing equation for steady two-dimensional flow with no pressure gradient is... 

Consider the two-dimensional flow shown in Figure A.l. If the velocity gradient dvjdy is positive it tends to cause the element to rotate in the clockwise direction. Similarly, if dvjdx is positive it tends to cause rotation in the anti-clockwise direction. Thus, the quantity dVy/dx — dvjdy gives the net rate of rotation in the anti-clockwise direction as viewed. It is the clockwise direction about a line parallel to the 2-coordinate as viewed in the positive 2-direction. This quantity is the 2-component of the fluid s vorticity a> ... 

Figure 8.13 Two-dimensional flows in a screw channel with a 6/14 = 1 and operating in extrusion mode. The arrows show the recirculation flows. The shaded area in the lower right corner is expanded in Fig. 8.14 to show the Moffat eddy...

Shock Wave Propagation in a Two-Dimensional Flow Field... 

Two-dimensional flow past infinite cylinders is not treated in detail since such bodies do not meet our definition of a particle (see Chapter 1). 

Mass transfer rates in steady two-dimensional flow normal to the axis of a long cylinder have been computed numerically over a range of Re (D3, M8, W6). 

Reynolds (Rl) suggested that the natures of turbulent momentum and thermal transport were similar. K rm n (K2) extended this analysis and defined eddy conductivity in the following way for steady, uniform, two-dimensional flow ... 

The conservation principle applied to unsteady, nonuniform, laminar, two-dimensional flow results in the following expression if the Fick diffusion coefficient is considered to be isotropic ... 

Note that, as is easily observed from b.v.p.(6.4.45)-(6.4.55), with a concentration gradient present no unidirectional developed Poiseuille-type channel flow is compatible with the boundary conditions (6.4.51). A fairly complicated two-dimensional flow pattern is thus generally expected even away from the edges of the channel. The appropriate rigorous flow calculation is still to be done. Here we shall content ourselves with the following crude order of magnitude estimate of the contribution of the above-mentioned circulation to the solute transport through the channel. 

For steady-state (no time variation) two-dimensional flows, the notion of a streamfunction has great utility. The stream function is derived so as to satisfy the continuity equation exactly. In cylindrical coordinates, there are two two-dimensional situations that are worthwhile to investigate the r-z plane, called axisymmetric coordinates, and the r-0 plane, called polar coordinates. 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

A further reduction of the vorticity equation is possible by restricting attention to two-dimensional flows. Here, since the vorticity vector is orthogonal to the velocity vector, the term (u> V) V vanishes. To retain the two-dimensional flow, the body force f must remain two-dimensional. 

Consider the two-dimensional flow in the r-6 plane of a cylinder, where the vorticity is then purely in the z direction. That is,... 

Consider the behavior of the Navier-Stokes equations for the two-dimensional flow in a conical channel as illustrated by Fig. 3.17. Begin with the constant-viscosity Navier-Stokes equations written in the general vector form as... 

Consider the two-dimensional flow in a channel formed by parallel plates, through which fluid may enter or leave the channel (Fig. 5.13). The similarity analysis of this situation is facilitated by assuming the form of the cross-channel velocity. With an assumed crossstream velocity, the axial-momentum equation can be reduced to an ordinary differential equation for a scaled axial velocity. 

For the boundary-layer equations, where two-dimensional flow is retained, the continuity equation must retain both terms as order-one terms. Otherwise, a purely onedimensional flow would result. Certainly there are situations where one-dimensional flow... 

---