Flow field symmetry

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

In the previous section we discussed wall functions, which are used to reduce the number of cells. However, we must be aware that this is an approximation that, if the flow near the boundary is important, can be rather crude. In many internal flows—where all boundaries are either walls, symmetry planes, inlets, or outlets—the boundary layer may not be that important, as the flow field is often pressure determined. However, when we are predicting heat transfer, it is generally not a good idea to use wall functions, because the convective heat transfer at the walls may be inaccurately predicted. The reason is that convective heat transfer is extremely sensitive to the near-wall flow and temperature field. 

A second choice to be made relates to the size of the flow domain. It may be worthwhile to limit the calculational job by reducing the size of the flow domain, e.g., by identifying an axis or plane of symmetry, or, in a cylindrical vessel with baffles mounted on the wall, due to periodicity in the azimuthal direction. Commercial software accomplishes these choices by means of symmetry cells and cyclic cells, respectively although such choices reduce the size of the simulation, they may eliminate the possibility of finding the real (asymmetric, unstable, or transient) 3-D flow field. The presence of feed pipes or drain or withdrawal pipes may also make the use of symmetry or cyclic cells impossible. Again, this issue only plays a role in RANS-type simulations. 

This is our starting point and the infinite dilution case was analysed by Einstein in the early years of this century.12 This analysis was based on the dilation of the flow field because the liquid has to move around the flowing particle. The particles were assumed to be hard spheres so that they were rigid, uncharged and without attractive forces small compared to any measuring apparatus so that the dilatational perturbation of the flow would be unbounded and would be able to decay to zero (the hydrodynamic disturbances decay slowly with distance, i.e. r 1) and at such dilution that the disturbance around one particle would not interact with the disturbance around another. The flow field is sketched in Figure 3.10. The coordinates are centred on the particle so that the symmetry is clear. The result of the analysis for slow flows (i.e. at low Reynolds number) was ... 

Since the pressure field depends only on the magnitude of the velocity (see Eq. (1-22)) and since the flow field has fore-and-aft symmetry, the modified pressure field forward from the equator of the sphere is the mirror image of that to the rear. This leads to d Alembert s paradox that the net force acting on the sphere is predicted to be zero. This paradox can only be resolved, and nonzero drag obtained, by accounting for the viscosity of the fluid. For in viscid flow, the surface velocity and pressure follow as... 

The system considered in this chapter is a rigid or fluid spherical particle of radius a moving relative to a fluid of infinite extent with a steady velocity U. The Reynolds number is sufficiently low that there is no wake at the rear of the particle. Since the flow is axisymmetric, it is convenient to work in terms of the Stokes stream function ij/ (see Chapter 1). The starting point for the discussion is the creeping flow approximation, which leads to Eq. (1-36). It was noted in Chapter 1 that Eq. (1-36) implies that the flow field is reversible, so that the flow field around a particle with fore-and-aft symmetry is also symmetric. Extensions to the creeping flow solutions which lack fore-and-aft symmetry are considered in Sections II, E and F. 

The flow field is symmetric over the period, with velocities in both directions at different times. Because of the symmetry there is no net flow through the duct, and thus the mean velocity profile is exactly zero. The average root-mean-square velocity, however, does have a radial dependence as shown in Fig. 4.11. The root-mean-square velocity is defined as... 

When an isolated sphere is held stationary in a creeping flow field containing suspended particles (see Figure 2), the equation of continuity in spherical coordinates, allowing azimuthal symmetry, is given by... 

If u is a unit vector parallel to the axis of symmetry of a spheroidal particle, then in the absence of Brownian motion (Pe — 00) and of interparticle interactions, the time rate of change of u in a flow field is... 

In spite of the fact that there are actually quite a large number of axisymmetric problems, however, there are many important and apparently simple-sounding problems that are not axisymmetric. For example, we could obtain a solution for the sedimentation of any axisymmetric body in the direction parallel to its axis of symmetry, but we could not solve for the translational motion in any other direction (e.g., an ellipsoid of revolution that is oriented so that its axis of rotational symmetry is oriented perpendicular to the direction of motion). Another example is the motion of a sphere in a simple linear shear flow. Although the undisturbed flow is 2D and the body is axisymmetric, the resulting flow field is fully 3D. Clearly, it is extremely important to develop a more general solution procedure that can be applied to fully 3D creeping-flow problems. 

Figure J. a. The opposed jets. The polymer solution is sucked into the jets along the symmetry axis, o represents the stagnation point in the center of the flow field, b. The flow field visualized by light scattered at 90° from tracer particles, c, A birefringent line between the jets for a 0.1% solution of atactic polystyrene (a-PS). d. The perturbed flow field during flare formation in a semiddute polyethylene oxide solution.

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

Potential Flow Transverse to a Long Cylinder Via the Scalar Velocity Potential. The same methodology from earlier sections is employed here when a long cylindrical object of radius R is placed within the flow field of an incompressible ideal fluid. The presence of the cylinder induces Vr and vg within its vicinity, but there is no axis of symmetry. The scalar velocity potential for this two-dimensional planar flow problem in cylindrical coordinates must satisfy Laplace s equation in the following form ... 

In equilibrium, the pressure in a system is uniform there is no net force. The existence of pressure gradients in a system implies that the system is not in equilibrium — Le., there Is a nonzero velocity (see Fig. 1.13). One can thus say that pressure gradients induce flow. A particularly simple example of this phenomenon is given by the flow field In the region between two solid plates with no-slip boundary conditions. Here, symmetry dictates that the velocity field has a component only in the x direction, which can vary spatially in the y direction Vy = = 0, and Vx = Vx(y). The pressure... 

At rest, the conformation of a flexible chain in dilute solution looks like a coil with spherical symmetry in the long-term. However, its instantaneous shape is asymmetric [125], which means the chain rotates along a streamline of a flow field with velocity gradient. The hydrodynamic drag force from the friction between the chain segments and the solvent molecules can deform the coil from its equilibrium shape. On the other hand, the conformation of the polymer chain is variable and changing all the time because of thermal fluctuation (Brownian motion of the solvent). So the shape of the chain in the flow field depends on how quickly the solvodynamic force deforms the chain and how slow the whole chain relaxes. This evolves two timescales. 

Centrifugal field Pulse-free pumping No pressure-tight interfaces needed Low-pressure load on lids Standard operation in sample prep Robust liquid handling widely decoupled fi om viscosity and surface tension Intrinsic, buoyancy-based bubble removal Coriolis force manipulates flows Rotational symmetry... 

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