Stream function spherical coordinates

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. 

Therefore, a stream function T may be introduced in the meridian plane of the cyclone, i.e., the r, 9) plane in the spherical coordinate system ... 

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... 

Relative movement between the droplet and the surrounding fluid can induce circulatory motion inside the droplet and affect the mass and heat transfer. These circulatory velocities depend on the Reynolds number, and start to occur for Reynolds numbers higher than 20. For Re < 1, the stream function

spherical coordinates with the origin on the sphere center is... 

To solve the corresponding hydrodynamic problem, it is convenient to use the spherical coordinates and introduce the stream function according to (2.1.3). Condition (2.5.2) acquires the form... 

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... 

The problem can be solved effectively by converting the convection-diffusion equation into the well studied heat conduction equation by introducing the stream fimction P as a new variable. In terms of the stream function the velocity components in spherical coordinates z and 0 are,... 

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 ... 

The sign convention is arbitrary, provided that one of the two velocity components has a negative sign. These relations between Vr and and the stream function, given by (8-103) and (8-105), conserve overall mass for an incompressible fluid. When p constant, the simplified equation of continuity in spherical coordinates. 

The angular dependence of the stream function represents one of the Legendre polynomials that is unaffected by the operator for creeping viscous flow in spherical coordinates. In other words,... 

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. 

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... 

In this section, minimal attention has been paid to the methods for solving the stream function equations since the basic purpose has been to illustrate the method for obtaining the correction It should be noted that the solutions in the form of stream functions for many regular geometric shapes are available and can be used to calculate the first correction In cylindrical and spherical coordinates, the general solutions have been provided by Haberman and Sayre (1958) and their method can be applied to many of the separable coordinate systems (Happel and Brenner, 1983). 

For flow past a sphere the stream function ij/ can be used in the Navier-Stokes equation in spherical coordinates to obtain the equation for the stream function and the velocity distribution and the pressure distribution over the sphere. Then by integration over the whole sphere, the form drag, caused by the pressure distribution, and the skin friction or viscous drag, caused by the shear stress at the surface, can be summed to give the total drag. 

Begin with equations (34)-(35) and using the usual spherical coordinate system x = (r, 0, ) fixed to the bubble, with corresponding velocity field u = (ur,u, U0). Due to axisymmetry (no (j) dependence), the continuity equation (35) can be satisfied by introducing a stream-function ip(r,9,t) such that (see [2]) ... 

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