Streamline flow definition

As indicated in Section 3.7.9, this definition of ReMR may be used to determine the limit of stable streamline flow. The transition value (R ur)c is approximately the same as for a Newtonian fluid, but there is some evidence that, for moderately shear-thinning fluids, streamline flow may persist to somewhat higher values. Putting n = 1 in equation 3,140 leads to the standard definition of the Reynolds number. 

For the flow of power-law fluids through packed beds of cubes, cylinders and gravel chips [Machac and Dolejs, 1981 Chhabra and Srinivas, 1991 Sharma and Chhabra, 1992 Sabiri and Comiti, 1995 Tiu et al, 1997], the few available data for streamline flow correlate well with equation (5.56) if the equal voliune sphere diameter, ds, multiplied by sphericity, j/, is employed as the effective diameter, efr = dsf in the definitions of the Reynolds number and friction factor. 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

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... 

The physical meaning of the stream function is that fluid flows along streamlines, which are lines of constant stream function. Since, by definition, flow cannot cross streamlines, the mass flow rate between any two streamlines must be constant. Furthermore the magnitude of the flow rate between two streamlines is determined by the difference in the values of the streamfunction on the two streamlines. 

Along a streamline (i.e., a line of constant ty), = 0. Equation 3.24 requires that mass flux cannot cross a streamline, since along the streamline dm = 0, and by definition, rh is the mass flow crossing the line (area). Equation 3.24 also requires that the mass flow rate between any two streamlines is related simply to the difference of the stream function on the two streamlines... 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

Fluid flow rarely follows ihe commonly accepted idea of streamlines, since the velocities necessary for viscous flow of this nature are almost always lower than those found expedient to employ, Most flows are turbulent in nature. They become turbulent at a definite velocity, the value of which was studied by Reynolds and this value is incorporated in the well-known Reynolds Number. A general thermodynamic equation of energy of a fluid under flow conditions would be as follows ... 

For a laminar flow through a straight microchannel, the fluid particles move in definite paths called streamlines and there are no components of fluid velocity normal to the duct axis. The projection of Eq. 6 along the axial direction (z) gives... 

We include in the term microcirculation those vessels with lumens (internal diameters) that are some modest multiple—say 1 to 10—of the major diameter of the unstressed RBC. This definition includes primarily the arterioles, the capillaries, and the postcapillary venules. The capillaries are of particular interest because they are generally from 6 to 10 /on in diameter, i.e., about the same size as the RBC. In the larger vessels, RBC may tumble and interact with one another and move from streamline to streamline as they course down the vessel. In contrast, in the microcirculation the RBC must travel in single file through true capillaries (Berman and Fuhro, 1969 Berman et al., 1982). Clearly, any attempt to adequately describe the behavior of capillary flow must recognize the particulate nature of the blood. 

In this design, a plate with the required opening is placed abruptly at the end of the die flow channel with a minimum amount of streamlining. This type of die is simple, easy to make, and easy to modify. However, there is a large dead flow region, and degradation is a definite concern with polymers with limited thermal stability. This type of die, therefore, should be used with relatively stable polymers and preferably only for short times. 

A chaotic flow produces either transverse homocHnic or transverse heterocHnic intersections, and/or is able to stretch and fold material in such a way that it produces what is called a horseshoe map, and/or has positive Liapunov exponents. These definitions are not equivalent to each other, and their interrelations have been discussed by Doherty and Ottino [63]. The time-periodic perturbation of homoclinic and heteroclinic orbits can create chaotic flows. In bounded fluid flows, which are encountered in mixing tanks, the homoclinic and heteroclinic orbits are separate streamlines in an unperturbed system. These streamhnes prevent fluid flux from one region of the domain to the other, thereby severely limiting mixing. These separate streamlines generate stable and unstable manifolds upon perturbation, which in turn dictate the mass and energy transports in the system [64-66]. 

However, the same definition would apply to the case of 7t-electron currents in the same molecules, in which the separatrix coincides with the single vortical line through the centre of the molecule. It would be also applicable to diamagnetic atoms, in which the delocalized flow beyond the nucleus consists of concentric circular streamlines about a vortical stagnation axis identifiable with the separatrix [60]. 

First, let us consider the kinematic definition of a streamline. A streamline is a flow trajectory across which fluid motion is absent fluid moves tangentially to it. Thus, its local slope must be equal to the ratio of the vertical to the horizontal velocities,... 

Thus, if the mass conservation requirement div q = 0 holds, and it does for incompressible flows where q is the Darcy velocity, the identity guarantees that we can represent q = curl V, which reduces to our planar in that limit. The use of a three-dimensional T, however, does not render path tracing any easier than dealing with three velocity functions (in two-dimensional problems, a single streamfunction suffices). Thus, we will not pursue any further discussion. But the idea of streamlines or pathlines as tangents locally parallel to the velocity vector is still attractive, and kinematically, we would expect a definition along the lines of dx/dt u, dy/dt v, and dz/dt w, where denotes proportionality. Let us consider an interface located anywhere within a flow, that is, any surface marked by red dye, and describe it by the locus of points f(x,y,z,t) = 0 (4-103)... 

Here, if the fiber of radius rf is perpendicular to the gas flow, then 2b is the width of the region of gas flow (see Figure 6.3.9A) which is cleaned completely of any particles by the single fiber. An essentially identical definition may be employed when the filter bed consists of granular particles of radius rf-, in that case, 2b will be the diameter of a cylindrical tube of contaminated gas, which will be cleaned of dust particles by the spherical collector in the filter bed. The value of b is obtained from the solution to the governing equation (6.3.40) and the gas velocity profile. The streamline corresponding to b is the limiting trajectory. 

---