Volumetric flow rate, axisymmetric

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... 

Problem 3-37. Taylor Dispersion with Streamwise Variations of Mean Velocity. We consider steady, pressure-driven axisymmetric flow in the radial direction between two parallel disks that are separated by a distance 2h. We assume that the volumetric flow rate in the radial direction is fixed at a value Q and that the Reynolds number is small enough that the Navier-Stokes equations are dominated by the viscous and pressure-gradient terms. Finally, the flow is ID in the sense that u = [nr(r, z), 0, 0]. In this problem, we consider flow-induced dispersion of a dilute solute. We follow the precedent set by the classical analysis of Taylor for axial dispersion of a solute in flow through a tube by considering only the concentration profile averaged across the gap, ( ) = f h dz. 

Steady-state numerical simulations of fluid flow and cupric ion transport within an electrochemical fountain plating system are presented. Specifically, the diffusion-limit is determined directly from the computed flux of cupric ions to the wafer under the assumption of complete surface consumption. This maximum flux, in turn, determines the maximum ionic current that can be passed through the electrolyte to the wafer, which is called the limiting current. The goal of the present study is to predict variations in the limiting current density for different electrolyte volumetric flow rates and wafer (cathode) rotation rates. The efficacy of different computational models, including one-dimensional, two-dimensional axisymmetric, and three-dimensional approximations, are assessed via comparisons of numerical predictions with experimental data. 

If one end of a vector is pinned on the symmetry axis at r = 0 and the other end lies somewhere on the lateral surface of the tube at r = R, then this vector maps out a circular cross section of n when it is rotated by Itt radians around the symmetry axis. The volumetric flow rate through this circle is jzR v ), where (u ) is the average fluid velocity through the tube. The axisymmetric stream function at r = is defined by... 

B.I. Consider the axisymmetric tank shown in Figure 4.21. We want to estimate the time that it takes for the tank to drain if filled with water (density 1000 kg/m, viscosity 0.001 Pa s). Let A(t) be the height of water in the tank, h(0) = 2m. Vp(t). The mean velocity in the drain pipe and the volumetric flow rate out of the tank is... 

For premixed fuel-air systems, results are reported in various terms that can be related to a critical equivalence ratio at which the onset of some yellow flame luminosity is observed. Premixed combustion studies have been performed primarily with Bunsen-type flames [52, 53], flat flames [54], and stirred reactors [55, 56], The earliest work [57, 58] on diffusion flames dealt mainly with axisymmetric coflow (coannular) systems in which the smoke height or the volumetric or mass flow rate of the fuel at this height was used as the correlating parameter. The smoke height is considered to be a measure of the fuel s particulate formation and growth rates but is controlled by the soot particle bumup. The specific references to this early work and that mentioned in subsequent paragraphs can be found in Ref. [50],... 

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