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.

Decay of free-gravity waves

It is generally believed that the major effects of viscosity are associated with vorticity. This belief is not always well founded; major effects of viscosity can be obtained from purely irrotational analysis of flows of viscous fluids. Here we illustrate this point by comparing irrotational solutions with Lamb's (1932) exact solution of the problem of the decay of free-gravity waves. Excellent agreements, even in fluids 107 more viscous than water, are achieved for the decay rates n(k) for all wavenumbers k, excluding a small interval around a critical value kc, where progressive waves change to monotonic decay.

Introduction

Lamb (1932, §348, §349) performed an analysis of the effect of viscosity on free-gravity waves. He computed the decay rate by a DM, using the irrotational flow only. He also constructed an ES for this problem, which satisfies both the normal and shear stress conditions at the interface.

Joseph and Wang (2004) studied Lamb's problem by using the theory of VPF and obtained a dispersion relation that gives rise to both the decay rate and the wave velocity. They also used VCVPF to obtain another dispersion relation. Because VCVPF is an irrotational theory, the shear stress cannot be made to vanish. However, the shear stress in the energy balance can be eliminated in the mean by the selection of an irrotational pressure that depends on viscosity.

The usual criterion for cavitation is that cavities will form in a liquid when and where the pressure falls below a critical value. In the ideal case, the cavitation threshold is the vapor pressure. The pressure in an incompressible viscous liquid is not a thermodynamic or material property; it is the average stress (actually the negative of the average stress, which is positive in tension). The viscous part of the stress is proportional to the rate of strain, which has a zero average with positive and negative values on the leading diagonal in the principal coordinates. It follows that in motion the liquid will develop stresses that are both larger and smaller than the average value. The theory of stress-induced cavitation seeks to relate the fracture or cavitation of a liquid to its state of stress rather than to its average stress. This kind of theory requires that the state of stress be monitored in the evolving field of motion to determine when and where the liquid will fracture. The theory can be thought of as an application area for Navier–Stokes fluid dynamics that can be studied by VPF when the flows are irrotational or nearly irrotational. The link between the theory of stress-induced cavitation and VPF is the fact that viscous stresses can be computed on irrotational motions.

High-Reynolds-number flows may be approximated by an outer irrotational flow and small layers on the boundary and narrow wakes where vorticity is important. The irrotational flow gives rise to an extra viscous dissipation over and above the dissipation in the boundary layer. At high Reynolds numbers the viscous dissipation in the irrotational flow outside is a very small fraction of the total that vanishes asymptotically as the Reynolds number tends to infinity.

Prandtl's boundary-layer theory is asymptotic and does not account for the viscous effects of the outer irrotational flow. Viscous effects on the normal stresses at the boundary of a solid cannot be obtained from Prandtl's theory. It is very well known and easily demonstrated that, as a consequence of the continuity equation, the viscous normal stress must vanish on a rigid solid. The only way that viscous effects can act on a boundary is through the pressure, but the pressure in Prandtl's theory is not viscous. It is determined by Bernoulli's equation in the irrotational flow and is imposed unchanged on the wall through the thin boundary layer. Therefore the important pressure drag cannot be calculated from Prandtl's theory. In addition, the mismatch between the irrotational shear stress and the shear stress at the outer edge of the boundary layer given by Prandtl's theory is not resolved.

The theory of potential flow is a topic in both the study of fluid mechanics and in mathematics. The mathematical theory treats properties of vector fields generated by gradients of a potential. The curl of a gradient vanishes. The local rotation of a vector field is proportional to its curl so that potential flows do not rotate as they deform. Potential flows are irrotational.

The mathematical theory of potentials goes back to the 18th century (see Kellogg, 1929). This elegant theory has given rise to jewels of mathematical analysis, such as the theory of a complex variable. It is a well-formed or “mature” theory, meaning that the best research results have already been obtained. We are not going to add to the mathematical theory; our contributions are to the fluid mechanics theory, focusing on effects of viscosity and viscoelasticity. Two centuries of research have focused exclusively on the motions of inviscid fluids. Among the 131,000,000 hits that come up under “potential flows” on Google search are mathematical studies of potential functions and studies of inviscid fluids. These studies can be extended to viscous fluids at small cost and great profit.

The fluid mechanics theory of potential flow goes back to Euler in 1761 (see Truesdell, 1954, §36). The concept of viscosity was not known in Euler's time. The fluids he studied were driven by pressures, not by viscous stresses.