Streamline pattern

Nonlaminar flow Ideally, it is a melt flow in a steady, streamlined pattern in and/or out of a tool (die, mold, etc.). Actually, the melt is... 

Figure 2.19 Streamline patterns in a converging-diverging channel for various Reynolds numbers, taken from [107. ...

Figure 2.20 Streamline patterns of secondary flow in quadratic channels for fC=151 (above) and K = 202 (below), taken from [110]. Only the upper half of the channel cross-section is shown.

Figure 2.27 Streamline patterns in a channel with sinusoidal walls (left) and Nusselt number as a function of Reynolds number for the same channel (right), taken from [120]. For comparison, the triangles represent the Nusselt number obtained in parallel-plates geometry.

The simulations of fluid flow and heat transfer in such microstructured geometries were carried out with an FVM solver. Air with an inlet temperature of 100 °C was considered as a fluid, and the channel walls were modeled as isothermal with a temperature of 0 °C. The streamline pattern is characterized by recirculation zones which develop behind the fins at comparatively high Reynolds numbers. The results of the heat transfer simulations are summarized in Figure 2.34, which shows the Nusselt number as a fimction of Reynolds number. For... 

Interestingly, the shape of the wake is similar to that developed behind a hypersonic blunt body where the flow converges to form a narrow recompression neck region several body diameters downstream of the rear stagnation point due to strong lateral pressure gradients. The liquid material, that is continuously stripped off from the droplet surface, is accelerated almost instantaneously to the particle velocity behind the wave front and follows the streamline pattern of the wake, suggesting that the droplet is reduced to a fine micromist. 

Figure 18.9 Velocity and streamline pattern (a) and near-exit region flow details (b)... 

In order to ensure that most of the counterflow is entrained from the ambient medium, PIV pictures are taken of the near field as shown in Fig. 18.9. These figures clearly show that the flow in the vicinity of the nozzle exit is significantly altered by the presence of the suction flow. The velocity field and the associated streamline pattern show a region of reversed flow clearly suggesting the presence of a countercurrent shear layer. It is also clear that most of the reverse flow is the entrained ambient air. 

In unsteady flows, the streamline pattern changes from instant to instant. In "steadyr state flux the streamlines are constant in time and also represent the path lines, the trajectories of the fluid particles (Ref 1, p 18)... 

Figure 6.13 illustrates the streamline patterns and velocity profiles for two rotation rates. The outer flow for the rotating disk is seen to be quite different from the semi-infinite stagnation-flow situation. In the rotating-disk case, the inviscid flow outside the viscous boundary layer has only uniform axial velocity. In the stagnation flow, the axial velocity varies linearly with the distance from the stagnation surface z and the scaled radial velocity v/r is a constant (cf. Fig. 6.6). The rotating-disk solutions reveal that as the rotation rate increases, the axial velocity increases in the outer flow and the boundary-layer thickness decreases as fi1/2 and f2-1/2, respectively.

Figure 1.18 Streamline patterns for various electroosmotic flows when h = 2 (h being the periodicity of arranging the electrodes), (a) U (X) =-U (X) = 1 ...

Figure 1.19 Streamline pattern for the superimposed flow structure, by switching between the flows (a) and (d) given in Figure 1.18, for h = 2 [28] (by courtesy of ACS).

Figure 1.102 Streamline pattern in the slit-shaped interdigital micro mixer at Re = 2160 [37] (by courtesy of AlChE).

At still higher Re (2160), the lamellae become highly intertwined, as evidenced by the streamline pattern (see Figure 1.102) [37]. 

Typical streamline patterns for combined convective flow over a square cylinder at a Reynolds number of 50 and a Prandtl number of 0.7. 

Fig. 1. Schematic of a family of two-dimensional steady incompressible shear flows showing the streamline patterns at the top and the corresponding velocity components at the bottom. By varying X continuously from — 1 to +1, the flow can be varied from pure rotation (without strain) to pure strain (without rotation).

Figure 3.7 Sketches showing ideal streamline patterns created by a rotating and translating cylinder for both clockwise and counter-clockwise rotation. In the real situation, they are slightly more complicated.

The streamline patterns are quite identical for both constitutive equations. However, the vortex is more pronounced for the multimode Phan-Thien Tanner model, whereas the swelling is greater for the generalized Oldroyd-B model... 

The flow of a slurry in a CMP process has been investigated. This wafer-scale model provides the three dimensional flow field of the slurry, the spatial distribution of the local shear rate imposed on the wafer surface, and the streamline patterns which reflect the transport characteristics of the slurry. 

Fig. 45. Streamline pattern in a wedge-shaped region (Reynolds number 1,7 x 10 ) [83], Reproduced with permission.

Fluid flow may be steady or unsteady, uniform or nonuniform, and it can also be laminar or turbulent, as well as one-, two-, or three-dimensional, and rotational or irrotational. One-dimensional flow of incompressible fluid in food systems occurs when the direction and magnitude of the velocity at all points are identical. In this case, flow analysis is based on the single dimension taken along the central streamline of the flow, and velocities and accelerations normal to the streamline are negligible. In such cases, average values of velocity, pressure, and elevation are considered to represent the flow as a whole. Two-dimensional flow occurs when the fluid particles of food systems move in planes or parallel planes and the streamline patterns are identical in each plane. For an ideal fluid there is no shear stress and no torque additionally, no rotational motion of fluid particles about their own mass centers exists. 

In the case where a gas flows over the front portion of a curved surface or a convex curve surface, the flow outside the boundary layer accelerates. The boundary layer over most of the front portion remains fairly thin and has a uniform streamline pattern over this portion. According to the Bernoulli equation [48], accelerated flow results in a pressure decrease in the vicinity of the front portion area. In this zone, the pressure gradient is positive, i.e. the direction of pressure gradient is the same as the flow direction. The positive pressure gradient is helpful to push the flow within the boundary layer forwards. The variations of velocity and pressure are expressed as... 

The separated boundary layer and wake displace the outside streamline pattern, which causes the pressure distribution to be significantly altered. Boundary layer separation causes a force on the body called drag force. The drag coefficient is defined as the ratio of total profile drag force divided by the flow pressure and projected area of an object and is expressed as [49]... 

Among references that discuss closed-streamline patterns and eddies in low-Reynolds-number flows, the reader may wish to refer to Ref. 13, Chap. 7, and D. J. Jeffrey and J. D. Sherwood, Streamline patterns and eddies in low-Reynolds-number flow, J. Fluid Mech. 96, 315-34 (1980) A. M. J. Davis and M. B. O Neill, The development of viscous wakes in a Stokes flow when a particle is near a large obstacle, Chem. Eng. Sci. 32, 899-906 (1977) A. M. J. Davis and M. B. O Neill, Separation in a slow linear shear flow past a cylinder and a plane, J. Fluid Mech., 81, 551-64 (1977). 

In rarefied systems of particles, drops, or bubbles, the particle-particle interaction can be neglected in the first approximation then one deals with the behavior of a single particle moving in fluid. In this case, the streamline pattern depends on the particle shape, the flow type (translational or shear), and a number of other geometric factors. 

In chemical technology one often meets the problem of a steady-state motion of a spherical particle, drop, or bubble with velocity U in a stagnant fluid. Since the Stokes equations are linear, the solution of this problem can be obtained from formulas (2.2.12) and (2.2.13) by adding the terms Vr = -U cos6 and V = U[ sin 6, which describe a translational flow with velocity U, in the direction opposite to the incoming flow. Although the dynamic characteristics of flow remain the same, the streamline pattern looks different in the reference frame fixed to the stagnant fluid. In particular, the streamlines inside the sphere are not closed. 

An arbitrary shear Stokes flow past a fixed cylinder is described by the stream function (2.7.9). We restrict our discussion to the case 0 2 fi < 1, in which there are four stagnation points on the surface of the cylinder. Qualitative streamline patterns for a purely straining flow (at CIe 0) and a purely shear flow (at CIe = 1) are shown in Figure 2.10. 

Freely rotating cylinder. Now let us consider convective mass transfer to the surface of a circular cylinder freely suspended in an arbitrary linear shear Stokes flow (Re -> 0). In view of the no-slip condition, the cylinder rotates at a constant angular velocity equal to the angular velocity of the flow at infinity. The fluid velocity distribution is described by formulas (2.7.11). The streamline pattern qualitatively differs from that for the case of a fixed cylinder. For 0 0, there are no stagnation points on the surface of the cylinder and there exist two qualitatively different types of flow. For 0 < Ifigl < 1, there are both closed and open streamlines in the flow, the region filled with closed streamlines is adjacent to the surface of the cylinder, and streamlines far from the cylinder are open (Figure 2.11). For Ifl l > 1, all streamlines are open. 

The inner problem of convective mass and heat transfer is essentially different from the similar outer problem, primarily, by the streamline pattern. This leads to a corresponding qualitative distinction between the dynamics of processes of transient mass transfer inside and outside a drop. In the outer problem considered in Section 4.12, all streamlines are open. The lines near the flow axis carry the... 

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