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Articles Published or Accepted can be found at Publications page (up to Sept. 2009). Those articles in preparation/draft stage or Submitted, and any proposals, memos and presentations are on this page.

Irrotational motions of bubbles under the action of acceleration of added mass and viscous drag, J. Wang, T. Funada and D. D. Joseph. 2006.

We present elementary examples of how irrotational flows can be used to study the unsteady motions of bubbles in a viscous liquid suitable for the instruction of undergraduate students. A spherical gas bubble accelerates to steady motion in an irrotational flow of a viscous liquid induced by a balance of the acceleration of the added mass of the liquid with the viscous drag. The equation of rectilinear motion is linear and may be integrated giving rise to exponential decay with a decay constant 18 v t/a2 where v is the kinematic viscosity of the liquid and a is the bubble radius. The problem of decay to rest of a bubble moving initially when the forces maintaining motion are inactivated and the acceleration of a bubble initially at rest to terminal velocity are considered. We also discuss the motion of a massless cylindrical bubble under the combined action of Kutta-Joukowski lift, acceleration of added mass and viscous drag. The cylinder moves with a constant speed along a circular path if the liquid is inviscid, whereas it goes to rest in a spiral fashion when the viscous drag is added

Irrotational viscous pressure and the dissipation method, D. D. Joseph and J. C. Padrino. 2006.

Purely irrotational theories for the viscous effects on the oscillations of drops and bubbles, J. C. Padrino, T. Funada and D. D. Joseph. 2006.

In this paper, we apply two purely irrotational theories of the motion of a viscous fluid, namely, viscous potential flow (VPF) and the dissipation method to the problem of the decay of waves on the surface of a sphere. We treat the problem of the decay of small disturbances on a viscous drop surrounded by gas of negligible density and viscosity and a bubble immersed in a viscous liquid. The instantaneous velocity field in the viscous liquid is assumed to be irrotational. In VPF, viscosity enters the problem through the viscous normal stress at the free surface. In the dissipation method, viscosity appears in the dissipation integral included in the mechanical energy equation. Comparisons of the eigenvalues from VPF and the dissipation approximation with those from the exact solution of the linearized governing equations are presented. The results show that the viscous irrotational theories exhibit most of the features of the wave dynamics described by the exact solution. In particular, VPF and DM give rise to a viscous correction for the frequency that determines the crossover from progressive to standing waves. Good to reasonable quantitative agreement with the exact solution is also shown for certain ranges of modes and dimensionless viscosity: For large viscosity and short waves, VPF is a very good approximation to the exact solution. For small viscosity and long waves, the dissipation method furnishes the best approximation.

Irrotational Faraday Waves on a Viscous Fluid, T. Funada, J. Wang, D. D. Joseph and N. Tashiro. 2005.

An analysis of irrotational Faraday waves on an inviscid fluid was given by Benjamin and Ursell 1954. Here we extend the analysis of the same problem to purely irrotational waves on a viscous fluid. Following our earlier work on free surface problems, two irrotational theories are presented. In the first theory (VPF) the effects of viscosity enter only through the viscous normal stress term evaluated on the potential. In the second irrotational theory (VCVPF), a viscous contribution is added to the Bernoulli pressure; otherwise the second theory is the same as the first. The second theory VCVPF gives rise to the same damped Mathieu equation as the dissipation method. Pressure fields are not required and not used in the dissipation method. The dissipation method is a purely irrotational theory, though it depends on viscosity, in which only irrotational velocity fields are needed. The two purely irrotational theories VPF and VCVPF are not restricted to small viscosities; they are restricted to small vorticity and do not apply near no-slip wall where Vorticity is generated.

Our VCVPF and dissipation theories give the same damped Mathieu equation as the phenomenological approximation of Kumar and Tuckerman 1994. The damping term in VCVPF is twice the damping rate of VPF. The growth rates of unstable disturbances computed by VPF are uniformly larger than those computed by VCVPF (or equivalently by Kumar and Tuckerman). Comparisons with the exact solution and the Rayleigh-Taylor instability show that thresholds and growth rates for viscously damped waves are better described by VPF than VCVPF

The motion of a spherical gas bubble in viscous potential flow, D. D. Joseph and J. Wang. 2005.

A spherical gas bubble accelerates to steady motion in an irrotational flow of a viscous liquid induced by a balance of the acceleration of the added mass of the liquid with the Levich drag. The equation of rectilinear motion is linear and may be integrated giving rise to exponential decay with a decay constant where is the kinematic viscosity of the liquid and a is the bubble radius. The problem of decay to rest of a bubble moving initially when the forces maintaining motion are inactivated and the acceleration of a bubble initially at rest to terminal velocity are considered. The equation of motion follows from the assumption that the motion of the viscous liquid is irrotational. It is an elementary example of how potential flows can be used to study the unsteady motions of a viscous liquid suitable for the instruction of undergraduate students. Another example, considered here, is the purely radial irrotational motion of a viscous liquid associated with the motions of a spherical gas bubble. This gives rise to an exact potential flow solution of the Navier-Stokes equations in which the jump of the viscous component of the normal stress is evaluated on the potential flow. The equation of motion for the liquid is almost always called the Reyleigh-Plesset equation but the viscous terms were introduced by Poritsky (1951) and not by Plesset (1949). We show that when the normal stress equation is converted into energy equation in the conventional way used for inviscid fluid, the viscous normal stress term is converted into the viscous dissipation in the liquid evaluated on potential flow.

Irrotational analysis of the toroidal bubble in a viscous fluid, J. Wang and D. D. Joseph. 2005.

We consider the problem of the rise of a toroidal gas bubble previously considered by Pedley (1968). We add the irrotational viscous drag to the force wrench in the impulse equation. In this case, the impulse equation and the energy equation governing the rise of the bubble are the same. The solution of this equation is computed; after a transient state the system evolves to a steady state in which the diameter, toroidal radius and rise velocity are constant.

Solution of Mathieu's equation by Runge-Kutta integration, T. Funada, J. Wang, D. D. Joseph, N. Tashiro and Y. Sonoda. 2005.

In this note we shall show that Mathieu's equation for x(t)

may be efficiently and accurately integrated by the Runge-Kutta (RK) method under the initial conditions

for the even Mathieu functions cen(t, q) → cos (nt) as q → 0 and

for the odd Mathieu functions sen(t, q) → sin (nt) as q → 0.

The lift, drag and torque on an airfoil in foam modeled by the potential flow of a second-order fluid, J. Wang and D. D. Joseph. 2005.

We compute the irrotational streaming flow of a second-order fluid past a Joukowski airfoil. The pressure and extra stress are evaluated using the irrotational flow theory, and the lift, drag and torque on the airfoil are obtained by integration of the normal stress over the surface of the airfoil. Our calculation can give rise to a lift force opposite to what would be predicted from the classical theory of aerodynamics. The result is in qualitative agreement with the experiments of the flow of a foam past an airfoil by Dollet, Aubouy and Graner 2004.

Capillary Collapse and Rupture, T. S. Lundgren and D. D. Joseph. 2005.

The breakup of a liquid capillary filament is analyzed as a viscous potential flow near a stagnation point on the centerline of the filament towards which the surface collapses under the action of surface tension forces. We find that the neck is of parabolic shape and its radius collapses to zero in a finite time; the curvature at the throat tends to zero much faster than the radius, leading ultimately to a microthread of nearly uniform radius. During the collapse the tensile stress due to viscosity increases in value until at a certain finite radius, which is about 1.5 microns for water in air, the stress in the throat passes into tension, presumably inducing cavitation there.