Axisymmetric stream function

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

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 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. 

A useful theorem due to Payne and Pell (P3) enables the drag on an axisymmetric body to be calculated directly from the stream function ij/ for steady... 

For axisymmetric flow the species continuity equation, Eq. (1-38), written in terms of the dimensionless concentration (j) and stream function (see Chapter 1) is... 

For steady-state (no time variation) two-dimensional flows, the notion of a streamfunction has great utility. The stream function is derived so as to satisfy the continuity equation exactly. In cylindrical coordinates, there are two two-dimensional situations that are worthwhile to investigate the r-z plane, called axisymmetric coordinates, and the r-0 plane, called polar coordinates. 

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 ... 

Explain the relationship between the stream-function values at the comers and the mass flow rate crossing the line (actually a surface for the axisymmetric situation) that connects the comer points. 

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

The axisymmetric inviscid stagnation flow is described in terms of a stream function having the form... 

The derivation of the two-dimensional planar equations is analogous to the approach for axisymmetric coordinates. The stream function in the planar situation is... 

As with the axisymmetric stagnation-flow case, deriving the tubular stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.59, and 3.60). The approach depends on essentially the same assumptions as the axial stagnation flows described earlier, albeit with the similarity requiring no variation in the axial coordinate. The velocity field is presumed to be described in terms of a stream function that has the form... 

Consider a sphere of radius a moving along a straight line in an infinite stagnant fluid medium at a velocity V. The cylindrical coordinates are selected such that the z-axis coincides with the path of the sphere, and the origin is an arbitrarily fixed point on the path. Assuming that the fluid is incompressible and its motion is axisymmetric, the stream function if may be defined such that... 

We now transform the governing equations in cylindrical coordinates into polar coordinates. Since the motion is axisymmetric, the transformation from (r, z) to (R, 6), as shown in Fig. 3.1, is analogous to the transformation from Cartesian coordinates (x, y) to cylindrical coordinates (r, 0) in a two-dimensional domain. The stream function is related to the velocity components in polar coordinates by... 

Consider two-dimensional flow over an axisymmetric body. Write the governing equations in terms of a suitably defined stream function and vorticity. 

The motion of a sphere moving through a stagnant incompressible fluid is equivalent to the uniform fluid flow about a fixed sphere. For axisymmetrical potential flow the velocity components of the fluid can be obtained by use of the Stokes stream function as explained by [26, 170] ... 

C. REPRESENTATION OF TWO-DIMENSIONAL AND AXISYMMETRIC FLOWS IN TERMS OF THE STREAM FUNCTION... 

Figure 7-13. The streamlines for axisymmetric flow in the vicinity of a solid sphere with uniaxial extensional flow at infinity. When the direction of motion is reversed at infinity, the undisturbed flow is known as biaxial extensional flow. The stream-function values are calculated from Eq. (7-185). Contour values are plotted in equal increments equal to 0.5.

Drops and bubbles. Axisymmetric shear flow past a drop was studied in [474,475], We denote the dynamic viscosities of the fluid outside and inside the drop by p and p.2- Far from the drop, the stream function satisfies (2.5.3) just as in the case of a solid particle. Therefore, we must retain only the terms with n = 3 in the general solution (2.1.5). We find the unknown constants from the boundary conditions (2.2.6)-(2.2.10) and obtain... 

Since the problem is axisymmetric, we introduce the stream function as... 

We use a local orthogonal curvilinear system of dimensionless coordinates f, ), ip, where 77 varies along and is normal to the surface of the particle. In the axisymmetric case, the azimuth coordinate tp varies from 0 to 27r in the plane case, it is supposed that 0 <

constant value = s. The dimensionless fluid velocity components can be expressed via the dimensionless stream function rp as follows ... 

Suppose that in the spherical (or cylindrical) coordinates, the surface of a particle (drop, bubble) is described by the equation r = R(9), where r is the dimensionless (referred to the characteristic length) radial coordinate and 9 is the angular coordinate. Then the velocity field near the interface is determined by the dimensionless stream function rp = [r-R(9)]mf(9), and the value F(k, k+1) in (4.6.22) is calculated by the following formulas [166] in the axisymmetric case, 0 < 8 < rr and... 

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... 

Axisymmetric Stream Function in Spherical Coordinates. It is necessary to understand the stream function in sufficient depth because additional boundary conditions are required to solve linear fourth-order PDFs relative to the typical second-order differential equations that are characteristic of most fluid dynamics problems. Consider the following two-dimensional axisymmetric flow problem in which there is no dependence on the azimnthal angle 4> in spherical coordinates ... 

Whereas the stream function for planar flow in rectangnlar coordinates has units of volnmetric flow rate per nnit depth, ir for axisymmetric flow in spherical coordinates has nnits of volnmetric flow rate ... 

Shortcut Methods for Axisymmetric Creeping Flow in Spherical Coordinates. All the previous results can be obtained rather quickly with assistance from information in Happel and Brenner (1965, pp. 133-138). For example, the general solution for the stream function for creeping viscous flow is... 

Potential Flow around a Gas Bubble Via the Stream Function. The same axisymmetric flow problem in spherical coordinates is solved in terms of the stream function All potential flow solutions yield an intricate network of equipotentials and streamlines that intersect at right angles. For two-dimensional ideal flow around a bubble, the velocity profile in the preceding section was calculated from the gradient of the scalar velocity potential to ensure no vorticity ... 

Hence, two-dimensional axisymmetric potential flow in spherical coordinates is described by = 0 for the scalar velocity potential and = 0 for the stream function. Recall that two-dimensional axisymmetric creeping viscous flow in spherical coordinates is described by E E ir) = 0. This implies that potential flow solutions represent a subset of creeping viscous flow solutions for two-dimensional axisymmetric problems in spherical coordinates. Also, recall from the boundary condition far from submerged objects that sin 0 is the appropriate Legendre polynomial for the E operator in spherical coordinates. The methodology presented on pages 186 through 188 is employed to postulate the functional form for xlr. 

In summary, Laplace s equation must be satisfied by the scalar velocity potential and the stream function for all two-dimensional planar flows that lack an axis of symmetry. The Laplacian operator is replaced by the operator to calculate the stream function for two-dimensional axisymmetric flows. For potential flow transverse to a long cylinder, vector algebra is required to determine the functional form of the stream function far from the submerged object. This is accomplished from a consideration of Vr and vg via equation (8-255) ... 

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... 

Calculate the stream function for axisymmetric fully developed creeping viscous flow of an incompressible Newtonian fluid in the annular region between two concentric tubes. This problem is analogous to axial flow on the shell side of a double-pipe heat exchanger. It is not necessary to solve algebraically for all the integration constants. However, you must include all the boundary conditions that allow one to determine a unique solution for i/f. Express your answer for the stream function in terms of ... 

Axisymmetric irrotational (i.e., potential) flow of an incompressible ideal fluid past a stationary gas bubble exhibits no vorticity. Hence, V x v = 0. This problem can be solved using the stream fnnction approach rather than the scalar velocity potential method. Develop the appropriate equation that governs the solution to the stream function f for two-dimensional axisymmetric potential flow in spherical coordinates. Which Legendre polynomial describes the angular dependence of the stream function ... 

The tangential velocity component in spherical coordinates is expressed in terms of the stream function I for two-dimensional axisymmetric flow as... 

The earliest analysis of the motion of two rigid spheres began with Jeffery (1915). He considered the motion of two rigid spheres that rotated about their centers. Later, Stimson and Jeffery (1926) solved the axisymmetric problem of two spheres translating at an equal velocity along their lines of center. This exact solution serves as a reference as to the accuracy of other approximate treatments. The solution is based on determining Stokes stream function for the motion of the fluid and from this the forces necessary to maintain the motion of the spheres. The force for equal spheres is given as... 

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... 

Figure 10-11. A schematic representation for streaming flow past an arbitrary axisymmetric body. The body geometry is specified by the function r(x). which measures the distance from the symmetry axis to the body surface as a function of the position x.

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