Entry flow

The method devised by Holland and Thake [ 1 ] for estimating the cooling air (vv, ), as a fraction of mainstream entry flow to a blade row (vvg), i.e. tp = wjw, was described by Horlock et al. [2] and is reproduced in Appendix A Fig. A. 1 shows diagrammatically the notation employed there and the same symbols are defined and used below. 

Flow through a convergent channel, either two-dimensional or axisymmetric, are probably the most investigated geometries (due to their connection with the technical die entry flow problem). Most of the other convergent flows could be derived from the abrupt contraction flow which is depicted schematically in Fig. 25. 

DEFINITION A single entry flow diagram is a block if and only if it can be built up firan the basic blocks in (1) by a finite number of applications of block... 

Obviously a block is a single entry flow diagram and the basic blocks are tree-like. We need only see that the operations of block composition and block iteration preserve the property of being tree-like. If B is fearned by directly connecting node n of block by block composition to entry node of block Bj, clearly n is an ancestor of ej, so that if both and Bj are treelike, clearly B is tree-like. If B is formed by block iteration by directly connecting exit node n of Block B to entry node of, then n remains a descendant of in B, so if is tree-like then B is also tree-like. 

Notice that our proof that a tree-like single entry flow diagram is a block actually gives an algorithm for dividing such a graph G into subblocks. Namely, one finds A, selects n to minimize a(n) and separates out Sn. If is equal to G, then G is formed by block iteration from S which is S minus... 

Now that we have shown that tree-likB single entry flow diagrams are blocks and vice versa and in the process given a construction for expressing any tree-like single entry flow diagram as a block construction of tree subschemes, we are in a position to show how to convert a program to block structured form. 

S. R. Galante and P. L. Frattini, Spatially resolved birefringence studies of planar entry flow, J. Non-Newt. Fluid Mech., 47, 289 (1993). 

The term melt fracture has been applied from the outset [9,13] to refer to various types of visible extrudate distortion. The origin of sharkskin (often called surface melt fracture ) has been shown in Sect. 10 to be related to a local interfacial instability in the die exit region. The alternating quasi-periodic, sometimes bamboo-like, extrudate distortion associated with the flow oscillation is a result of oscillation in extrudate swell under controlled piston speed due to unstable boundary condition, as discussed in Sect. 8. A third type, spiral like, distortion is associated with an entry flow instability. The latter two kinds have often been referred to as gross melt fracture. It is clearly misleading and inaccurate to call these three major types of extrudate distortion melt fracture since they do not arise from a true melt fracture or bulk failure. Unfortunately, for historical reasons, this terminology will stay with us and be used interchangeably with the phase extrudate distortion. ... 

Nonreactive entry flow. Figure 1 shows the streamlines for a 4 1 entry flow. Here a grid of 100 four-node linear elements was used to model the upper symmetric half of a plane capillary, and a fully-developed Poiseuille velocity was imposed on the reservoir entry as a boundary condition. The streamlines are identical with published experimental and numerical results, although the grid used here was not intended to be fine enough to capture the weak recirculation which develops in the stagnant corner of the reservoir. 

Figure 1. Streamlines for nonreactive 4 1 entry flow, Newtonian fluid with imposed Poiseuille flow at inlet.

Figure 2. Contours of constant temperature for convectionless entry flow, with heat generation by viscous dissipation only.

Figure 3. Entry flow centerline temperatures at various Peclet numbers.

From a numerical viev point, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the munerical problem, in relation to the nonlinearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry. 

This paper describes a finite element formulation designed to simulate polymer melt flows in which both conductive and convective heat transfer may be important, and illustrates the numerical model by means of computer experiments using Newtonian extruder drag flow and entry flow as trial problems. Fluid incompressibility is enforced by a penalty treatment of the element pressures, and the thermal convective transport is modeled by conventional Galerkin and optimal upwind treatments. 

Figure 5 shows the contours of shear stress near the throat, normalized on the theoretical value expected at the capillary wall. The shear contours are in agreement with the isochromatic fringes seen in birefringence photography of entry flow (7), which is an example of a possible experimental verification of the flow model. 

Figure 3. Finite element mesh and computed velocity field for 4 1 entry flow.

Boger, D.V. "Circular Entry Flows of Inelastic and Viscoelastic Fluids" to appear in "Advances in Transport Process" Wiley International. 

The thermal entry flow with fully developed velocity profile... 

In the same way as shown in the previous section, the heat transfer coefficient and from that the mean Nusselt number Nume = amed/A (the index e stands for entry flow) can be obtained from the temperature profile. The Nusselt number can be calculated from an empirical equation of the form... 

Laminar flow that is developing hydrodynamically or thermally, (entry flow) constant wall temperature ... 

This chapter focuses on fluid flow, leaving the combination of fluid flow, heat transfer, and diffusion to Chapter 11. Examples of fluid flow include entry flow into a pipe, flow in a microfluidic T-sensor, turbulent flow in a pipe, time-dependent start-up of pipe flow, flow in an orifice, and flow in a serpentine mixer. The examples demonstrate many of the techniques that are useful in the program FEMLAB. 

Next consider entry flow into a pipe, as illustrated in Figure 10.1. The flow is taken as uniform in radius at the entrance (taking the value 1.0 at each radial position), and the velocity has to develop into its fully developed profile. The pipe radius is taken as 0.5. How far downstream do you have to go to get a fully developed solution (Another way to pose this question is How much error do you make if you assume fully developed flow when it is not )... 

Figure 10.2. Mesh, arrow plot, and streamlines for entry flow into a pipe (a) mesh (h) arrow plot ...

Next consider the entry flow of a non-Newtonian fluid, essentially the same problem as... 

Figure 10.13. (a) Velocity profile at exit and (b) centerline velocity for turbulent entry flow in a pipe. 

Step 4 Set the boundary conditions as you did for entry flow slip/symmetry at the centerline and no-slip on the solid wall. However, at the inlet choose the Normal flow/ pressure option and put in the value 2.5 Pa. At the exit choose the Normal flow/pressure option and put in zero. 

This chapter illustrated how to use FEMLAB to solve the Navier-Stokes equations in a variety of situations. Some of these problems are classic, such as entry flow into a pipe and transient start-up of pipe flow. Most examples were for laminar flow in two dimensions, but one model was turbulent flow into a pipe, and another model was for a complicated three-dimensional geometry. To review, the chapter covered the following specific features ... 

Solve for the entry flow of a Newtonian fluid into a pipe using the parameters in the example. Reproduce Figure 10.2 through Figure 10.5 plus Figure 10.7. 

Solve for the entry flow of a power-law fluid into a region between two flat plates. The viscosity is given by... 

The plates are a distance of 0.005 m apart, and the applied pressure gradient is 60 Pa in a length of 0.3 m. The density of the fluid is that of water. (Hint Follow the steps in the example of entry flow in a pipe, except use a power-law formula for viscosity and use two flat plates rather than a pipe.)... 

---