We study the force on a 2D cylinder near a wall in two potential flows: the flow that is due to the circulation 2πκ about the cylinder and the uniform streaming flow with velocity V past the cylinder. The pressure is computed with Bernoulli's equation, and the viscous normal stress is calculated with VPF; the shear stress is ignored. The forces on the cylinder are computed by integration of the normal stress over the surface of the cylinder. In both of the two cases, the force perpendicular to the wall (lift) is due to only the pressure and the force parallel to the wall (drag) is due to only the viscous normal stress. Our results show that the drag on a cylinder near a wall is larger than on a cylinder in an unbounded domain. In the flow induced by circulation or in the streaming flow, the lift force is always pushing the cylinder toward the wall. However, when the two flows are combined, the lift force can be pushing the cylinder away from the wall or toward the wall.

The flow that is due to the circulation about the cylinder

Figure 10.1 shows a cylinder with radius a near the wall x = 0.

When a vessel containing liquid is made to vibrate vertically with constant frequency and amplitude, a pattern of standing waves on the gas–liquid surface can appear. For some combinations of frequency and amplitude, waves appear; for other combinations the free surface remains flat. These waves were first studied in the experiments of Faraday (1831), who noticed that the frequency of the liquid vibrations was only half that of the vessel. Nowadays, this would be described as a symmetry-breaking vibration of a type that characterized the motion of a simple pendulum subjected to a vertical oscillation of its purpose.

The first mathematical study of Faraday waves are due to Rayleigh (1883a, 1883b) but the first definitive study is due to Benjamin and Ursell (1954; hereafter BU) who remark that “The present work has been made possible by the development of the theory of Mathieu functions.”

Faraday's problem is a rich source of problems in pattern formation, bifurcation, chaos, and other topics within the framework of fluid mechanics applications in the modern theory of dynamical system. Under the excitation of different parameters governing the Faraday system, different patterns, stripes, squares, hexagons, and time-dependent states can be observed. These features have spawned a large recent literature on Faraday waves. The experiments of Ciliberto and Gollub (1985) and Simonelli and Gollub (1989) on chaos, symmetry, and mode interactions are often cited.

In this chapter we present the form of the Navier–Stokes equations implied by the Helmholtz decomposition in which the relation of the irrotational and rotational velocity fields is made explicit. The idea of self-equilibration of irrotational viscous stresses is introduced. The decomposition is constructed first by selection of the irrotational flow compatible with the flow boundaries and other prescribed conditions. The rotational component of velocity is then the difference between the solution of the Navier–Stokes equations and the selected irrotational flow. To satisfy the boundary conditions, the irrotational field is required, and it depends on the viscosity. Five unknown fields are determined by the decomposed form of the Navier–Stokes equations for an incompressible fluid: the three rotational components of velocity, the pressure, and the harmonic potential. These five fields may be readily identified in analytic solutions available in the literature. It is clear from these exact solutions that potential flow of a viscous fluid is required for satisfying prescribed conditions, such as the no-slip condition at the boundary of a solid or continuity conditions across a two-fluid boundary. The decomposed form of the Navier–Stokes equations may be suitable for boundary layers because the target irrotational flow that is expected to appear in the limit, say at large Reynolds numbers, is an explicit to-be-determined field.

In this chapter we carry out an analysis of the stability of a liquid jet into a gas or another fluid by using VPF. This instability may be driven by KH instability that is due to a velocity difference and a neck-down that is due to capillary instability. KH instabilities are driven by pressures generated by a dynamically active ambient flow, gas or liquid. On the other hand, capillary instability can occur in a vacuum; the ambient can be neglected. KH instability is included by a discontinuity of the velocity at a two-fluid interface. This discontinuity is inconsistent with the no-slip condition for Navier–Stokes studies of viscous fluids, but is consistent with the theory of potential flow of a viscous fluid. We start our study with an analysis of capillary instability.

Capillary instability of a liquid cylinder in another fluid

The study of this problem is especially valuable because it can be solved exactly and was solved by Tomotika (1935). This solution allows one to compute the effects of vorticity generated by the no-slip condition. The ES can be compared with irrotational solutions of the same problem. One effect of viscosity on the irrotational motion may be introduced by evaluation of the viscous normal stress at the liquid–liquid interface on the irrotational motions.

Potential flows of incompressible fluids with constant properties are irrotational solutions of the Navier–Stokes equations that satisfy Laplace's equation. How do these solutions enter into the general problem of viscous fluid mechanics? Under certain conditions, the Helmholtz decomposition says that solutions of the Navier–Stokes equations can be decomposed into a rotational part and an irrotational part satisfying Laplace's equation. The irrotational part is required for satisfying the boundary conditions; in general, the boundary conditions cannot be satisfied by the rotational velocity, and they cannot be satisfied by the irrotational velocity; the rotational and irrotational velocities are both required and they are tightly coupled at the boundary. For example, the no-slip condition for Stokes flow over a sphere cannot be satisfied by the rotational velocity; harmonic functions that satisfy Laplace's equation subject to a Robin boundary condition in which the irrotational normal and tangential velocities enter in equal proportions are required.

The literature that focuses on the computation of layers of vorticity in flows that are elsewhere irrotational describes boundary-layer solutions in the Helmholtz decomposed forms. These kinds of solutions require small viscosity and, in the case of gas–liquid flows, are said to give rise to weak viscous damping. It is true that viscous effects arising from these layers are weak, but the main effects of viscosity in so many of these flows are purely irrotational, and they are not weak.

Viscous potential flow

We obtained the effects of viscosity on irrotational motions of spherical cap bubbles, Taylor bubbles in round tubes, and RT and KH instabilities described in previous chapters by evaluating the viscous normal stress on potential flow. In gas–liquid flows, the viscous normal stress does not vanish and it can be evaluated on the potential. It can be said that, in the case of gas–liquid flow, the appropriate formulation of the irrotational problem is the same as the conventional one for inviscid fluids with the caveat that the viscous normal stress is included in the normal stress balance. This formulation of VPF is not at all subtle; it is the natural and obvious way to express the equations of balance when the flow is irrotational and the fluid viscous.

In this chapter we use the acronym VPF, viscous potential flow, to stand for the irrotational theory in which the viscous normal stresses are evaluated on the potential.

In gas–liquid flows we may assume that the shear stress in the gas is negligible so that no condition need be enforced on the tangential velocity at the free surface, but the shear stress must be zero. The condition that the shear stress be zero at each point on the free surface is dropped in irrotational approximations.

Problems of potential flow in irregular domains bounded by rigid solids and satisfying perhaps conditions at infinity require numerical methods. Computers and software are now so powerful that it can be easier to compute a solution than to find the exact one in a reference book. There are many techniques that may be used to solve Laplace's equation with prescribed boundary conditions. These techniques are readily available even in “search” on the web.

The numerical simulation of the deformation of interfaces between two immiscible fluids or in gas–liquid flows is currently an active topic of research and many options are available for researchers. Level-set methods associated with the names of S. Osher, R. Fedkiw, and J. Sethian, volume-of-fluid methods associated with the name of S. Zaleski, and front-tracking methods associated with the name of G. Trygvasson, are high among the most popular methods. Readers can find references in the comprehensive reviews by Yeung (1982), Tsai and Yue (1996), and Scardovelli and Zaleski (1999), or in “search” on Google.

Perturbation methods

The problem of numerical simulation of the shape of free surfaces in potential flows of inviscid fluids has been considered by various authors. Perturbation methods for nonlinear irrotational waves on an inviscid fluid were introduced by Stokes (1847). He expanded the solution in powers of the amplitude.

The rise velocity of long gas bubbles (Taylor bubbles) in round tubes is modeled by an ovary ellipsoidal cap bubble rising in an irrotational flow of a viscous liquid. The analysis leads to an expression for the rise velocity that depends on the aspect ratio of the model ellipsoid and the Reynolds and Eötvös numbers. The aspect ratio of the best ellipsoid is selected to give the same rise velocity as the Taylor bubble at given values of the Eötvös and Reynolds numbers. The analysis leads to a prediction of the shape of the ovary ellipsoid that rises with same velocity as that of the Taylor bubble.

Introduction

The correlations given by Viana et al. (2003) convert all the published data on the normalized rise velocity Fr = U/(gD)1/2 into analytic expressions for the Froude velocity versus buoyancy Reynolds number, ReG = [D3g (ρl – ρg)ρl]1/2/μ for fixed ranges of the Eötvös number, Eö = gρlD2/σ, where D is the pipe diameter, ρl, ρg, and σ are densities and surface tension. Their plots give rise to power laws in Eö; the compositions of these separate power laws emerge as bipower laws for two separate flow regions for large- and small-buoyancy Reynolds numbers.