Stokes stream function

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. 

Stokes stream function for continuous phase relative to particle sphericity, — AJA... 

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] ... 

From Eq. (8.3.24) for the Stokes stream function near a cylinder, the undisturbed velocity field is easily shown to be resolvable into two flows. One is a planar stagnation-type flow shown in Fig. 8.4.IB that is associated with the velocity component at infinity along the line of centers of the cylinder and particle the other is a shear flow normal to the line of centers shown in Fig. 8.4.IG. The respective expressions valid for the cylinder radius a > x + where a is the particle radius, are... 

F2) 0.89(Ape)J " <1 — >10= Laminar flow Thin concentration boundary layer Boundary layer theory Stokes stream function... 

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. 

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... 

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... 

Gupalo and Ryazantsev (GIO) followed the analysis of Acrivos and Taylor (A2) with the Proudman-Pearson stream function rather than Stokes flow. For Sc > 10, the two predictions for Sh agree within 1%, while for Sc = 1 they differ by at most 8% for Pe < 1. The results of Gupalo and Ryazantsev, although valid to higher Re, are still restricted to Pe 0, so that this extension is of little practical value. 

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),... 

Our intent here is not to suggest a solution method but rather to use the stream-function-vorticity formulation to comment further on the mathematical characteristics of the Navier-Stokes equations. In this form the hyperbolic behavior of the pressure has been lost from the system. For low-speed flow the pressure gradients are so small that they do not measurably affect the net pressure from a thermodynamic point of view. Therefore the pressure of the system can simply be provided as a fixed parameter that enters the equation of state. Thus pressure influences density, still accommodating variations in temperature and composition. Since the pressure or the pressure gradients simply do not appear anywhere else in the system, pressure-wave behavior has been effectively filtered out of the system. Consequently acoustic behavior or high-speed flow cannot be modeled using this approach. 

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... 

S.J. Liao. Higher order stream function-vorticity formulation of 2D steady-state Navier-Stokes equation. International Journal for Numerical Methods in Fluids, 15 595-612, 1992. 

Because they do not contain the pressure as a variable. Eqs. (2.76) and (2.77) have been used quite extensively in solving problems for which the boundary layer equations (see later) cannot be used. For this purpose, instead of solving the Navier-Stokes and energy, simultaneously with the continuity equation, it is convenient to introduce the stream function, ip, which is defined such that... 

A direct simulation of the flow field was also attempted in Sengupta et al. (2002), where the following stream function- vorticity formulation of Navier- Stokes equation was used. 

The two-dimensional Navier-Stokes equation is solved in stream function-vorticity formulation, as reported variously in Sengupta et al. (2001, 2003), Sengupta Dipankar (2005). Brinckman Walker (2001) also simulated the burst sequence of turbulent boundary layer excited by streamwise vortices (in X- direction) using the same formulation for which a stream function was defined in the y — z) -plane only. To resolve various small scale events inside the shear layer, the vorticity transport equation (VTE) and the stream function equation (SFE) are solved in the transformed — rj) —... 

Under the long wavelength and quasistationary approximations and with the use of the linearized forms of the hydrodynamic and thermodynamic boundary conditions, first, we solve the Orr-Sommerfeld equation for the amplitude of perturbed part of the stream function from the Navier-Stokes equations. Second, we solve the equation for the amplitude of perturbed part of the temperature in the liquid film. The dispersion relation for the fluctuation of the solid-liquid interface is determined by the use of these solutions. From the real and imaginary part of this dispersion relation, we obtain the amplification rate cr and the phase velocity =-(7jk as follows ... 

The stream function and radial velocity distribution function for a low-Reynold.s-number flow around a sphere are given by the following expressions due to Stokes ... 

Here, we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition,... 

Table 1.1 presents equations for the stream function, obtained from the Navier-Stokes equations (1.1.1), (1.1.2) in various coordinate systems. 

Stream function equations equivalent to the Navier-Stokes equations... 

Let a be the outer radius of the compound drop, and let ae be the radius of the core (0 < e < 1). The exact solution of the problem on the flow past a compound drop in a translational Stokes flow with velocity U can be found in [416], where the stream functions in the phases are given. The drag force is also... 

It follows that in the general solution (2.1.5) of the Stokes equations one must retain only the terms with n = 3. The unknown constants A3, B3, C3, D3 are determined by the boundary no-slip conditions (2.2.1). As a result, we obtain the stream function [474,475]... 

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 ... 

Following [270], we first consider steady-state diffusion to the surface of a solid spherical particle in a translational Stokes flow (Re - 0) at high Peclet numbers. In the dimensionless variables, the mathematical statement of the corresponding problem for the concentration distribution is given by Eq. (4.4.3) with the boundary conditions (4.4.4) and (4.4.5), where the stream function is determined by (4.4.2). 

In the Stokes approximation, the stream function for the flow (4.9.1) is equal to the sum of the stream functions of the constituent flows. 

An arbitrary shear Stokes flow past a fixed cylinder is described by the stream function (2.7.9). We restrict our discussion to the case 0 2 fi < 1, in which there are four stagnation points on the surface of the cylinder. Qualitative streamline patterns for a purely straining flow (at CIe 0) and a purely shear flow (at CIe = 1) are shown in Figure 2.10. 

Let us again first consider collection by a spherical collector assuming the flow to be an inertia free, Stokes flow. The stream function corresponding to the velocity field, defined by Eqs. (8.3.3), is... 

H3) obtained functional approximations for the velocity components by assuming a trial stream function in the Navier-Stokes equations and evaluating the undetermined coefficients from the boundary conditions using the method of residuals. Their relationship can be presented in the form of... 

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