Stream function definition

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),... 

In two-dimensional, incompressible, steady flows, there is a relatively simple relationship between the vorticity and the stream function. Consider the axisymmetric flow as might occur in a channel, Fig. 3.12. Beginning with the axisymmetric stream function as discussed in Section 3.1.2, substitute the stream-function definition into the definition of the circumferential vorticity u>q ... 

Notice several important aspects about the transformed equations. Of course, xjr is now an independent variable, with the relationship between r and x/r coming from the stream-function definition... 

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. 

With the definition of stream function in hand, consider its relationship to mass-flow rate. The mass-flow rate (kg/s) crossing any differential area is given by... 

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... 

The boundary conditions provide a tight coupling between the vorticity and stream-function fields. Also velocities still appear in the convective terms. Given the stream-function field, velocity is evaluated from the definition of stream function. That is, velocity is computed from stream-function derivatives. 

From the definition of the axisymmetric stream function (Section 3.1.2), it can be seen that... 

With the definition of the stream function in Eq. (9.20), the x-velocity component may be represented by... 

By definition (4.27), the stream function / satisfies the first equation in equation (4.26). The boundary conditions for / are... 

From the definition of the axisymmetric stream function and elementary continuity considerations, the volume flow rate between any two stream surfaces is simply IttA I. It follows that the mass flux of particles intercepted by the spherical collector is... 

Our result for the collector efficiency, using the stream function from above and the definition is... 

The distance E yi l e limiting trajectory from the axis 6 = 0 as r—as shown in Fig. 8.3.2, follows directly from the definition of the stream function and the fact that lp = E yiaU. Note that the corresponding limiting trajectory distance for the spherical collector is ph. ... 

Here, we have again used the solution for the stream function near a cylinder Eq. (8.3.24) with h > Up. Substitution of Eq. (8.4.23) with dp = 0, on evaluating the definite integral, yields... 

The entire discussion of the stream function has been for two-dimensional flow. The definition of a satisfactory stream function for three-dimensional flow is more difficult. However, if the flow is symmetric about some axis, e.g, uniform flow around some body of revolution, then it is possible to define a different stream function which is convenient for that problem. This three-dimensional stream function is called Stokes stream function [6] to distinguish it from the Lagrange stream function, which is discussed in this chapter. 

With the problem as stated there are three types of flow, a pressure driven flow, a plate driven flow and a mixture of the two. A pressure driven flow is taken to be a flow dominated by Poiseuille flow in the channel ODEF but which is not a pure Poiseuille flow. Conversely a plate driven flow is a flow dominated by Couette flow in the channel. Using these definitions, results are presented for both a plate driven and a pressure driven flow. In addition the pressure generated on the moving plate is also presented for a fixed flow rate Q and for various speeds U. The results for a flow rate of 3.0 and a plate speed of 1.0 are shown in Figure (5). The upper diagram shows lines of constant stream function, calculated at equal intervals between 0 and Q (3.0 in this example). The velocity profiles beneath are calculated at the stations (a) to (e) in the x-direction and to a normalised height of 1,0 (the film thickness) in the y-direction. 

S-6.2.2 Streamlines. In 2D simulations, a quantity called the stream function, lJf, is defined in terms of the density and gradients of the x- and y-components of the velocity, U and V. In terms of cylindrical coordinates, which are most appropriate for axisymmetric stirred tank models, the definition takes the form... 

Good heat transfer on the outside of the reactor tube is essential but not sufficient because the heat transfer is limited at low flow rates at the inside film coefficient in the reacting stream. The same holds between catalyst particles and the streaming fluid, as in the case between the fluid and inside tube wall. This is why these reactors frequently exhibit ignition-extinction phenomena and non-reproducibility of results. Laboratory research workers untrained in the field of reactor thermal stability usually observe that the rate is not a continuous function of the temperature, as the Arrhenius relationship predicts, but that a definite minimum temperature is required to start the reaction. This is not a property of the reaction but a characteristic of the given system consisting of a reaction and a particular reactor. 

By definition the exit age distribution function E is such that the fraction of the exit stream with residence times between t and t + St is given by Ed/. 

The general formulation for a dynamic-programming problem, presented in a simplified form, is shown in Fig. 11-11. On the basis of the definitions of terms given in Fig. 11-lla, each of the variables, x1+1, xt, and dt, may be replaced by vectors because there may be several components or streams involved in the input and output, and several decision variables may be involved. The profit or return Pt is a scalar which gives a measure of contribution of stage i to the objective function. 

The time it takes a molecule to pass through a reactor is called its residence time 6. Two properties of 6 are important the time elapsed since the molecule entered the reactor (its age) and the remaining time it will spend in the reactor (its residual lifetime). We are concerned mainly with the sum of these times, which is 6, but it is important to note that micromixing can occur only between molecules that have the same residual lifetime molecules cannot mix at some point in the reactor and then unmix at a later point in order to have different residual lifetimes. A convenient definition of residence-time distribution function is the fraction J ) of the effluent stream that has a residence time less than 0. None of the fluid can have passed through the reactor in zero time, so / = 0 at 0 = 0. Similarly, none of the fluid can remain in the reactor indefinitely, so that Japproaches 1 as 0 approaches infinity. A plot of J 6) vs 0 has the characteristics shown in Fig. 6-2a. 

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