Creeping flow coordinates

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

To remove momentum fluctuations from the problem, BCAH assume that in the creeping flow limit, in which a system of small mass interacts strongly with a thermally equilibrated solvent, the distribution of values for the momenta for fixed coordinate values stays very near a state of local equilibrium, in which... 

First, consider the case where the flow is parallel to the cylinders. It is assumed that the fluid is moving through the annular space between the cylinder of radius a and the fluid envelope of equivalent radius b, as shown in Fig. 7.14. Assume that the fluid motion is in the creeping flow regime so that inertia terms can be omitted from the Navier-Stokes equations. Thus, in cylindrical coordinates, we have... 

When an isolated sphere is held stationary in a creeping flow field containing suspended particles (see Figure 2), the equation of continuity in spherical coordinates, allowing azimuthal symmetry, is given by... 

E. AXISYMMETRIC CREEPING FLOWS SOLUTION BY MEANS OF EIGENFUNCTION EXPANSIONS IN SPHERICAL COORDINATES (SEPARATION OF VARIABLES)... 

Now, we have expressed the general streamfunction, (7-149), and the disturbance flow contribution in (7-150) and (7-151), in terms of spherical coordinates. However, we have not yet specified a body shape. Thus the linear decrease of the disturbance flow with distance from the body must clearly represent a property of creeping-flows that has nothing to do with specific coordinate systems. Indeed, this is the case, and the velocity field (7-151) plays a very special and fundamental role in creeping-flow theory. It is commonly known as the Stokeslet velocity field and represents the motion induced in a fluid at Re = 0 by a point force at the origin (expressed here in spherical coordinates).17 We shall see later that the Stokeslet solution plays an important role in many aspects of creeping-flow theory. 

Problem 7-20. Sphere in a Parabolic Flow. Use the general eigenfunction expansion for axisymmetric creeping-flows, in spherical coordinates, to determine the velocity and pressure fields for a sohd sphere of radius a that is held fixed at the central axis of symmetry of an unbounded parabolic velocity field,... 

We shall see that the stokeslet solution plays a fundamental role in creeping flow theory. We have already seen in Section E of Chap. 7 that it describes the disturbance velocity far away from a body of any shape that exerts a nonzero force on an unbounded fluid. Indeed, when nondimensionalized and expressed in spherical coordinates, it is identical to the velocity field, (7 151). In the next section we use the stokeslet solution to derive a general integral representation for solutions of the creeping-flow equations. 

A sketch of the physical problem and a definition of the cylindrical polar coordinates (r, creeping-flow limit, the governing equations and boundary conditions are... 

Problem 9-17. Heat Transfer From an Ellipsoid of Revolution at Pe S> 1. In a classic paper, Payne and Pell. J. Fluid Meek 7, 529(1960)] presented a general solution scheme for axisymmetric creeping-flow problems. Among the specific examples that they considered was the uniform, axisymmetric flow past prolate and oblate ellipsoids of revolution (spheroids). This solution was obtained with prolate and oblate ellipsoidal coordinate systems, respectively. 

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... 

Answer For axisymmetric creeping flow in spherical coordinates, the general solution to the low-Reynolds-number equation of change for fluid vorticity (i.e., E ir = 0) is... 

Figure 11-1 Thickness of the mass transfer boundary layer around a solid sphere, primarily in the creeping flow regime. This graph in polar coordinates illustrates 8c 9) divided by the sphere diameter vs. polar angle 9, and the fluid approaches the solid sphere horizontally from the right. No data are plotted at the stagnation point, where 9=0.

Without introducing the stream function or solving this third-order partial differential vector equation for v, it shonld be obvious that the dimensionless velocity vector is a fimction of spatial coordinates and the specific geometry of the flow problem. However, v is not a fimction of the Reynolds number because Re does not appear in (12-9). The generic creeping flow solution is written as... 

To see how we might use the momentum equation, let us return again to the analysis of flow between converging planes, shown in Figure 2.1. We have already seen (Equation 2.5) that rvr = (p(0), which is a consequence of our assumption that = Vs = 0 (purely radial flow). We will assume we have a Newtonian liquid in creeping flow, so we use the equations for cylindrical coordinates in Table 2.4 with P = 0. When we substitute the given form of the velocity into the equations, we find that most terms are identically zero, and the equations simplify to the following ... 

This differential equation is identified often as the equation for creeping flow or Stokes flow. The three components of this equation in the Cartesian coordinate system are easily obtained from Table 6.2.3 by equating the left-hand side of each equation to zero. 

At first glance, three coupled linear third-order PDEs must be solved, as illustrated above. However, each term in the x and y components of the vorticity equation is identically zero because =0 and Vj and Vy are not functions of z. Hence, detailed summation representation of the vorticity equation for creeping viscous flow of an incompressible Newtonian fluid reveals that there is a class of two-dimensional flow problems for which it is only necessary to solve one nontrivial component of this vector equation. If flow occurs in two coordinate directions and there is no dependence of these velocity components on the spatial coordinate in the third direction, then one must solve the nontrivial component of the vorticity equation in the third coordinate direction. 

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,... 

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. 

The two roots are n = — 1, 2, which represent a subset of the four roots for the radial function for two-dimensional axisymmetric creeping viscous flow in spherical coordinates (i.e., n = —1, 1, 2, 4). One of the roots for the potential flow problem (i.e., n = 2) is consistent with the functional form of far from submerged objects. The potential flow solution is... 

Consider creeping viscous flow of an incompressible Newtonian fluid past a stationary gas bubble that is located at the origin of a spherical coordinate system. Do not derive, but write an expression for the tangential velocity component (i.e., vg) and then linearize this function with respect to the normal coordinate r within a Ihin mass transfer boundary layer in the liquid phase adjacent to the gas-liquid interface. Hint Consider the r-9 component of the rate-of-strain tensor ... 

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