Stream function polar

Stream function has value P = q/ikv) at the dividing streamline and so its location can be found by expressing Eq. (10.41) in polar 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... 

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. 

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... 

Solution of Equations in Terms of Stream Function Equation (3.9) in polar coordinates can be satisfied if 0 is in a form... 

Low Reynolds numbers. In [216, 382] the problem on a circular cylinder of radius a in translational flow of viscous incompressible fluid with velocity Ul at low Reynolds numbers was solved by the method of matched asymptotic expansions. The study was carried out on the basis of the Navier-Stokes equations (1.1.4) in the polar coordinates 1Z, 6. Thus, the following expression for the stream function was obtained for IZ/a 1 ... 

The equations of fluid motion inside and outside a circulating drop under viscous flow regime were solved by Hadamard (H2) and Rybczynski (R9) in 1911, and are quoted in hydrodynamics textbooks (L2). The complete derivation is also repeated by Levich (L8). Although Hadamard s stream functions are strictly applicable to the viscous region only, visual observations (GIO, S18) indicated that the function approximates actual flows (E2, H3). Hadamard s stream function inside the drop, as given in polar coordinates with the origin at the center of the drop (K5), is... 

Potential Flow Transverse to a Long Cylinder Via the Stream Function. For two-dimensional planar flow in cylindrical coordinates, the radial and polar velocity components are related to the stream function rfr via the following expressions ... 

Using the line OG as the reference line, 0=0, in cylindrical polar coordinates, with 0 being the origin and angles being measured in radians and positive in the anti-clockwise direction, the boundary conditions on GH can be found. Following the example in Batchelor (9) the form of the stream function, ip, for the moving plate flow is... 

Fig. 2 Meridional circulation stream lines (full lines) as a function of the angle from the polar axis. The position of convection zones is indicated by dotted lines. That identified by A is for a 12000 R main sequence star while that indicated by B is for a 6400 K star.

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