Volume 28, Issue 9, September 2016

In the present work, the effect of an inflow sinusoidal excitation that is superimposed over the mean flow on the vortexshedding characteristics of a square cylinder is studied. The frequency of pulsation is varied around the natural vortexshedding frequency, and the amplitude of pulsation is varied moderately in comparison to the cylinder diameter, at a fixed Reynolds number (=100). A flow regime map is prepared and compared with the experimental results, which are available for a circular cylinder that is subjected to inline excitation. We correlate the spectra to the corresponding flow regime. Visualization of the vorticity contours reveals that the significant interaction of the baseregion vorticities with the main shear layer vorticities is important in the mechanism of formation of the several vortexshedding modes. The strength and sign of base region vorticity with respect to the shear layers has a fundamental role to play in the mechanism of formation. It is hypothesized that the similarity in vortexshedding modes across different excitation types, bluff body geometry, and for different parameters is due to the similarity in the underlying vorticity dynamics.
 LETTERS


Foam on troubled water: Capillary induced finitetime arrest of sloshing waves
View Description Hide DescriptionInterfacial forces exceed gravitational forces on a scale small relative to the capillary length—two millimeters in the case of an airwater interface—and therefore dominate the physics of submillimetric systems. They are of paramount importance for various biological taxa and engineering processes where the motion of a liquid meniscus induces a viscous frictional force that exhibits a sublinear dependence in the meniscus velocity, i.e., a power law with an exponent smaller than one. Interested in the fundamental implications of this dependence, we use a liquidfoam sloshing system as a prototype to exacerbate the effect of sublinear friction on the macroscopic mechanics of multiphase flows. In contrast to classical theory, we uncover the existence of a finitetime singularity in our system yielding the arrest of the fluid’s oscillations. We propose a minimal theoretical framework to capture this effect, thereby amending the paradigmatic damped harmonic oscillator model. Our results suggest that, although often not considered at the macroscale, sublinear capillary forces govern the friction at liquidsolid and liquidliquid interfaces.
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 ARTICLES

 Micro and Nanofluid Mechanics

Viscous slip coefficients for binary gas mixtures measured from mass flow rates through a single microtube
View Description Hide DescriptionThe viscous slip coefficient for heliumargon binary gas mixture is extracted from the experimental values of the mass flow rate through a microtube. The mass flow rate is measured by the constantvolume method. The viscous slip coefficient was obtained by identifying the measured mass flow rate through a microtube with the corresponding analytical expression, which is a function of the Knudsen number. The measurements were carried out in the slip flow regime where the firstorder slip boundary condition can be applied. The measured viscous slip coefficients of binary gas mixtures exhibit a concave function of the molar ratio of the mixture, showing a similar profile with numerical results. However, from the detailed comparison between the measured and numerical values with the complete and incomplete accommodation at a surface, it is inappropriate to estimate the viscous slip coefficient for the mixture numerically by employing separately measured tangential momentum accommodation coefficient for each component. The time variation of the molar ratio in the downstream chamber was measured by sampling the gas from the chamber using the quadrupole mass spectrometer. In our measurements, it is indicated that the volume flow rate of argon is larger than that of helium because of the difference in the tangential momentum accommodation coefficient.

Transient motion of and heat transfer in a rarefied gas between plane parallel walls with different surface properties
View Description Hide DescriptionTransient motion of and heat transfer in a rarefied gas between plane parallel walls with different surface properties are studied based on kinetic theory. It is assumed that one wall is a diffuse reflection boundary and the other wall is a Maxwelltype boundary, and the transient behavior of the gas caused by a sudden heating of one of the walls is studied. The linearized Boltzmann equation for a hardsphere molecular gas is numerically studied using the modified hybrid scheme of the characteristic coordinate and finite difference methods, to correctly describe the discontinuities in the velocity distribution function. The transient motion of the gas from an early time stage to the final timeindependent state is studied over a wide range of the mean free path and the accommodation coefficient of the boundary. Between the two transient flows caused by the heating of the respective walls, the values of the heat flow on the heated wall are different, whereas those on the unheated wall coincide identically. This property, which is a consequence of the symmetric relation of the linearized Boltzmann equation, is numerically confirmed over a wide range of the mean free path. The long time behavior of the heat flow on the walls is quite similar to that of the shear stress in the Couette flow problem, whereas a distinct wavy behavior is observed in an early time stage.

The stability of two layer dielectricelectrolyte microflow subjected to an external electric field
View Description Hide DescriptionThe twophase microflow of conductive (electrolyte) and nonconductive (dielectric) viscous liquids bounded by two solid walls in an external electric field is scrutinized. The lower solid wall, which is adjoined to the electrolyte, is a charged dielectric surface; the upper wall which bounds the dielectric is insulated. The problem has a steady onedimensional (1D) solution. The theoretical results for a pluglike velocity profile are successfully compared with available theoretical and experimental data from the literature. The linear stability of the steadystate flow is investigated numerically with spectral Galerkin’s method for solving linearized eigenvalue problem. This method was successfully applied for related problem of electroosmosis of ultrathin film. The numerical analysis provides insights on the coexistence of long and shortwave instabilities. The influence of control parameters such as the ratio of the viscosities of both liquids and the ratio of the channel heights on the stability of onedimensional flow was investigated for different values of external electric field. The influence of an external pressure gradient on the flow stability is also investigated. The experimental facts established by other authors, according to which the system destabilizes if the electroosmotic flow is oppositely directed to the external pressure gradient, is confirmed in this work. Otherwise stabilization takes place.

Temperature gradientinduced fluid pumping inside a singlewall carbon nanotube: A nonequilibrium molecular dynamics study
View Description Hide DescriptionIn this paper we investigate the fluid transport inside a singlewall carbon nanotube induced by a temperature gradient along the tube length, focusing on the effect of fluid–wall interaction strength. It is found that the fluid moves from the hot side of the nanotube towards the cold side. By increasing the fluid–wall interaction strength, the fluid volumetric flux assumes a maximum, increases, and then decreases. Fluid transport is pressuredriven in weak interactions; in contrast, in strong interactions, the fluid is broken into two parts in the radial direction. Fluid transport in the central regions of the tube is pressuredriven, while it is surfacedriven in the areas close to the wall.

Is the water flow more or less than that predicted by the NavierStokes equation in microorifices?
View Description Hide DescriptionMicrofluid mechanics is an important field in modern fluid mechanics. However, flows through microscale short tubes (microorifices) are not yet fully understood. Thus far, experiments on the flow through microorifices have been conducted by two methods: the pressuregiven method (PGM), in which the pressure is given and the rate of flow is measured, and the flowgiven method (FGM), in which the flow rate is given and the pressure is measured. According to conventional fluid mechanics, these two methods should give the same result; however, studies have found lower fluidity (lower flow rate) in PGM and higher fluidity (lower pressure drop) in FGM than that predicted by the NavierStokes equation, suggesting that the difference is caused by the method used. To clarify the cause of this difference, we examined the flow of ultrapure water (UPW) with elapsed time by PGM. UPW was passed through Ni or Ti microorifices with 20μm diameter at applied pressures of 501000 Pa. The difference in the shape and material of the orifices did not have a great effect on the flow property. The flow rate was frequently higher than that predicted at the start of the flow experiment; however, it subsequently fell and finally reached zero as time elapsed. This fact suggests that UPW inherently flows at velocities higher than those predicted by the NavierStokes equation; however, the flow is then resisted by something that develops over time. We removed an orifice in which flow had stopped from the experimental apparatus, observed it by phase contrast microscope and electron probe micro analyzer, and revealed that a visible membrane, a transparent latticelike structure, or nothing existed in the orifice. Dissolved air was reduced by deaerating the air from UPW (deaeration), bubbling UPW with Ar (Arbubbling), or preventing UPW from contact with air after UPW production (airprevention). Deaeration, Arbubbling, and airprevention reduced the probability of formation of the visible membrane. UPW treated by a combination of airprevention and Arbubbling showed no visible membrane. Furthermore, we passed UPW through an electrically grounded orifice (grounding) and found that grounding also reduces the probability of formation of the visible membrane. These findings suggest that the membrane formation was related to the presence of air dissolved in UPW and the action of charges generated in the flow. The reduction of the dissolved air by Arbubbling and airprevention provided a higher flow rate, although deaeration provided a slightly lower flow rate than seen in the case without deaeration. Grounding yielded a higher average flow rate. A combination of Arbubbling and grounding provided flow rates considerably larger than the predicted ones. We found a correlation between the probability of the membrane formation and the magnitude of the fall in flow rates. We concluded that the membranes, whether visible or invisible, came from the dissolved air by the action of charges generated at the orifice by the flow. Furthermore, the membrane developed naturally in PGM; in contrast, the membrane, even if it developed, was flushed away from the orifice in FGM because of the constant flow supplied. Therefore, the UPW flows in PGM with fluidity lower than the predicted value owing to the resistance of the membrane, whereas the UPW flows in FGM with fluidity higher than the predicted value owing to the inherent characteristics of UPW.

Nonlinear effects on electrophoresis of a charged dielectric nanoparticle in a charged hydrogel medium
View Description Hide DescriptionThe impact of the solid polarization of a charged dielectric particle in gel electrophoresis is studied without imposing a weakfield or a thin Debye length assumption. The electric polarization of a dielectric particle due to an external electric field creates a nonuniform surface charge density, which in turn creates a nonuniform Debye layer at the solidgel interface. The solid polarization of the particle, the polarization of the double layer, and the electroosmosis of mobile ions within the hydrogel medium create a nonlinear effect on the electrophoresis. We have incorporated those nonlinear effects by considering the electrokinetics governed by the StokesBrinkmanNernstPlanckPoisson equations. We have computed the governing nonlinear coupled set of equations numerically by adopting a finite volume based iterative algorithm. Our numerical method is tested for accuracy by comparing with several existing results on freesolution electrophoresis as well as results based on the DebyeHückel approximation. Our computed result shows that the electrophoretic velocity decreases with the rise of the particle dielectric permittivity constant and attains a saturation limit at large values of permittivity. A significant impact of the solid polarization is found in gel electrophoresis compared to the freesolution electrophoresis.

Lightinduced phenomena in onecomponent gas: The transport phenomena
View Description Hide DescriptionThe article presents the theory of transport processes in a onecomponent gas located in the capillary under the action of resonant laser radiation and the temperature and pressure gradients. The expressions for the kinetic coefficients determining heat and mass transport in the gas are obtained on the basis of the modified Boltzmann equations for the excited and unexcited particles. The Onsager reciprocal relations for cross kinetic coefficients are proven for all Knudsen numbers and for any law interaction of gas particles with each other and boundary surface. Lightinduced phenomena associated with the possible nonequilibrium stationary states of system are analyzed.
 Interfacial Flows

Selfsimilarity and scaling transitions during rupture of thin free films of Newtonian fluids
View Description Hide DescriptionRupture of thin liquid films is crucial in many industrial applications and nature such as foam stability in oilgas separation units, coating flows, polymer processing, and tear films in the eye. In some of these situations, a liquid film may have two free surfaces (referred to here as a free film or a sheet) as opposed to a film deposited on a solid substrate that has one free surface. The rupture of such a free film or a sheet of a Newtonian fluid is analyzed under the competing influences of inertia, viscous stress, van der Waals pressure, and capillary pressure by solving a system of spatially onedimensional evolution equations for film thickness and lateral velocity. The dynamics close to the spacetime singularity where the film ruptures is asymptotically selfsimilar and, therefore, the problem is also analyzed by reducing the transient partial differential evolution equations to a corresponding set of ordinary differential equations in similarity space. For sheets with negligible inertia, it is shown that the dominant balance of forces involves solely viscous and van der Waals forces, with capillary force remaining negligible throughout the thinning process in a viscous regime. On the other hand, for a sheet of an inviscid fluid for which the effect of viscosity is negligible, it is shown that the dominant balance of forces is between inertial, capillary, and van der Waals forces as the film evolves towards rupture in an inertial regime. Real fluids, however, have finite viscosity. Hence, for real fluids, it is further shown that the viscous and the inertial regimes are only transitory and can only describe the initial thinning dynamics of highly viscous and slightly viscous sheets, respectively. Moreover, regardless of the fluid’s viscosity, it is shown that for sheets that initially thin in either of these two regimes, their dynamics transition to a late stage or final inertialviscous regime in which inertial, viscous, and van der Waals forces balance each other while capillary force remains negligible, in accordance with the results of Vaynblat, Lister, and Witelski.

Expanding Taylor bubble under constant heat flux
View Description Hide DescriptionModelization of nonisothermal bubbles expanding in a capillary, as a contribution to the understanding of the physical phenomena taking place in Pulsating Heat Pipes (PHPs), is the scope of this paper. The liquid film problem is simplified and solved, while the thermal problem takes into account a constant heat flux density applied at the capillary tube wall, exchanging with the liquid film surrounding the bubble and also with the capillary tube outside medium. The liquid slug dynamics is solved using the LucasWashburn equation. Mass and energy balance on the vapor phase allow governing equations of bubble expansion to be written. The liquid and vapor phases are coupled only through the saturation temperature associated with the vapor pressure, assumed to be uniform throughout the bubble. Results show an overheating of the vapor phase, although the particular thermal boundary condition used here always ensures an evaporative mass flux at the liquidvapor interface. Global heat exchange is also investigated, showing a strong decreasing of the PHP performance to convey heat by phase change means for large meniscus velocities.
 Viscous and NonNewtonian Flows

Surface fractionation effects on slip of polydisperse polymer melts
View Description Hide DescriptionThe slip behavior of several highdensity polyethylenes with broad range of molecular weight (MW) including bimodals is studied as a function of molecular weight (MW) and its distribution. A formulation similar to the double reptation theory is used to predict the slip velocity of the studied polymers as a function of MWD coupled with a model of surface molecular weight fractionation. While surface fractionation has a minor effect on slip of narrow to moderate MWD polymers (particularly unimodal), its role is significant for broad bimodal MWD polymers. The entropy driven migration of short chains toward the die wall has a profound effect and should be considered in order to calculate the effective MWD on the boundary layer and thus the correct magnitude of wall slip.

Electroosmotic and pressuredriven flow of viscoelastic fluids in microchannels: Analytical and semianalytical solutions
View Description Hide DescriptionIn this work, we present a series of solutions for combined electroosmotic and pressuredriven flows of viscoelastic fluids in microchannels. The solutions are semianalytical, a feature made possible by the use of the Debye–Hückel approximation for the electrokinetic fields, thus restricted to cases with small electric doublelayers, in which the distance between the microfluidic device walls is at least one order of magnitude larger than the electric doublelayer thickness. To describe the complex fluid rheology, several viscoelastic differential constitutive models were used, namely, the simplified PhanThien–Tanner model with linear, quadratic or exponential kernel for the stress coefficient function, the JohnsonSegalman model, and the Giesekus model. The results obtained illustrate the effects of the Weissenberg number, the JohnsonSegalman slip parameter, the Giesekus mobility parameter, and the relative strengths of the electroosmotic and pressure gradientdriven forcings on the dynamics of these viscoelastic flows.
 Particulate, Multiphase, and Granular Flows

Rise of an argon bubble in liquid steel in the presence of a transverse magnetic field
View Description Hide DescriptionThe rise of gaseous bubbles in viscous liquids is a fundamental problem in fluid physics, and it is also a common phenomenon in many industrial applications such as materials processing, food processing, and fusion reactor cooling. In this work, the motion of a single argon gas bubble rising in quiescent liquid steel under an external magnetic field is studied numerically using a VolumeofFluid method. To mitigate spurious velocities normally generated during numerical simulation of multiphase flows with large density differences, an improved algorithm for surface tension modeling, originally proposed by Wang and Tong [“Deformation and oscillations of a single gas bubble rising in a narrow vertical tube,” Int. J. Therm. Sci. 47, 221–228 (2008)] is implemented, validated and used in the present computations. The governing equations are integrated by a secondorder space and time accurate numerical scheme, and implemented on multiple Graphics Processing Units with high parallel efficiency. The motion and terminal velocities of the rising bubble under different magnetic fields are compared and a reduction in rise velocity is seen in cases with the magnetic field applied. The shape deformation and the path of the bubble are discussed. An elongation of the bubble along the field direction is seen, and the physics behind these phenomena is discussed. The wake structures behind the bubble are visualized and effects of the magnetic field on the wake structures are presented. A modified drag coefficient is obtained to include the additional resistance force caused by adding a transverse magnetic field.

Separation of chiral particles in a rotating electric field
View Description Hide DescriptionWhen a particle having a permanent dipole is placed in a rotating electric field in a fluid, it will migrate along the direction of the angular vector of the field. The average velocity of the particle is calculated based on a Brownian motion framework that accounts for the translationrotation coupling. An explicit expression is given for the average velocity of the particle as a function of the hydrodynamic mobility tensors and the frequency of the field. The expression indicates that the chiral particle and its mirrorimage particle move in opposite direction (which accounts for the right/left molecular separation in racemic solutions), and that the migration velocity has a peak at a certain frequency determined by the geometry of the particle. Calculations are made for a twisted ribbon particle and the relation between the migration velocity and the structure of the twisted ribbon particle is discussed.
 Laminar Flows

Lifting a large object from an anisotropic porous bed
View Description Hide DescriptionAn analytical study of two dimensional problem of lifting an object from the top of a fully saturated rigid porous bed is discussed. It is assumed that the porous bed is anisotropic in nature. The flow within the gap region between the object and the porous bed is assumed to be governed by Stokes equation while the flow within the porous bed is governed by Brinkman equation. The breakout phenomenon for different kinds of soil is reported. The effect of mechanical properties like anisotropic permeability, grain diameter size, and porosity on streamlines, velocity, and force is analyzed. Relevant comparison with C. C. Mei, R. W. Yeung, and K. F. Liu [“Lifting a large object from a porous bed,” J. Fluid. Mech. 152, 203–215 (1985)] and Y. Chang, L. H. Huang and F. P. Y. Yang [“Twodimensional liftup problem for a rigid porous bed,” Phys. Fluids, 27, 053101 (2015)] is done.

Critical role of blockage ratio for flame acceleration in channels with tightly spaced obstacles
View Description Hide DescriptionA conceptually laminar mechanism of extremely fast flame acceleration in obstructed channels, identified by Bychkov et al. [“Physical mechanism of ultrafast flame acceleration,” Phys. Rev. Lett. 101, 164501 (2008)], is further studied by means of analytical endeavors and computational simulations of compressible hydrodynamic and combustion equations. Specifically, it is shown how the obstacles length, distance between the obstacles, channel width, and thermal boundary conditions at the walls modify flame propagation through a combshaped array of parallel thin obstacles. Adiabatic and isothermal (cold and preheated) side walls are considered, obtaining minor difference between these cases, which opposes the unobstructed channel case, where adiabatic and isothermal walls provide qualitatively different regimes of flame propagation. Variations of the obstructed channel width also provide a minor influence on flame propagation, justifying a scaleinvariant nature of this acceleration mechanism. In contrast, the spacing between obstacles has a significant role, although it is weaker than that of the blockage ratio (defined as the fraction of the channel blocked by obstacles), which is the key parameter of the problem. Evolution of the burning velocity and the dependence of the flame acceleration rate on the blockage ratio are quantified. The critical blockage ratio, providing the limitations for the acceleration mechanism in channels with combshaped obstacles array, is found analytically and numerically, with good agreement between both approaches. Additionally, this combshaped obstaclesdriven acceleration is compared to finger flame acceleration and to that produced by wall friction.

Interaction of monopoles, dipoles, and turbulence with a shear flow
View Description Hide DescriptionDirect numerical simulations have been conducted to examine the evolution of eddies in the presence of largescale shear flows. The numerical experiments consist of initialvalueproblems in which monopolar and dipolar vortices as well as driven turbulence are superposed on a plane Couette or Poiseuille flow in a periodic twodimensional channel. The evolution of the flow has been examined for different shear rates of the background flow and different widths of the channel. Results found for retrograde and prograde monopolar vortices are consistent with those found in the literature. Boundary layer vorticity, however, can significantly modify the straining and erosion of monopolar vortices normally seen for unbounded domains. Dipolar vortices are shown to be much more robust coherent structures in a largescale shear flow than monopolar eddies. An analytical model for their trajectories, which are determined by selfadvection and advection and rotation by the shear flow, is presented. Turbulent kinetic energy is effectively suppressed by the shearing action of the background flow provided that the shear is linear (Couette flow) and of sufficient strength. Nonlinear shear as present in the Poiseuille flow seems to even increase the turbulence strength especially for high shear rates.

Numerical investigation of flowinduced rotary oscillation of circular cylinder with rigid splitter plate
View Description Hide DescriptionNumerical results of fluid flow over a rotationally oscillating circular cylinder with splitter plate are presented here. Different from the previous examinations with freely rotatable assembly, the fluid and structure interactions are treated as a coupled dynamic system by fully considering the structural inertia, stiffness, and damping. The hydrodynamic characteristics are examined in terms of reduced velocity Ur at a relatively low Reynolds number Re = 100 for different plate lengths of L/D = 0.5, 1.0, and 1.5, where Ur = U/(Df n), Re = UD/υ and f n = (κ/J)^{0.5}/2π with U the free stream velocity, D the diameter of the circular cylinder, υ the fluid kinematic viscosity, f n the natural frequency, J the inertial moment, κ the torsional stiffness, and L the plate length. Contrast to the freely rotating cylinder/plate body, that is, in the limit of κ → 0 or Ur →∞, remarkable rotary oscillation is observed at relatively low reduced velocities. For the typical case with L/D = 1.0, the maximum amplitude may reach five times that at the highest reduced velocity of Ur = 15.0 considered in this work. At the critical reduced velocity Ur = 4.2, notable hydrodynamic jumps are identified for the rotation amplitude, response frequency, mean drag coefficient, lift amplitude, and vortex shedding frequency. Moreover, the phase angle between the fluid moment and rotary oscillation abruptly changes from 0 to π at Ur = 6.5. Due to the combined effect of fluid moment, rotation response, and phase difference, the natural frequency of the rotating body varies in flow, leading to a wide regime of lockin/synchronization (Ur ≥4.2, for L/D = 1.0). The phenomenon of rotation bifurcation, i.e., the equilibrium position of the rotary oscillation deflects to a position which is not parallel to the free stream, is found to only occur at higher reduced velocities. The longer splitter plate has the lower critical reduced velocity. The occurrence of bifurcation is attributed to the antisymmetry breaking of the wake flow evolution. The resultant asymmetric mean pressure distribution on the splitter plate gives rise to the net lift force and the deviated moment on the assembly, leading to the offset mean position of splitter plate. The global vortex shedding is identified to be the classic 2S mode for both cases with and without the bifurcation, although the second vortex formation and the shedding pattern in the near wake for the bifurcate case are different from the nonbifurcate case with lower reduced velocities.

Nearbody vorticity dynamics of a square cylinder subjected to an inline pulsatile free stream flow
View Description Hide DescriptionIn the present work, the effect of an inflow sinusoidal excitation that is superimposed over the mean flow on the vortexshedding characteristics of a square cylinder is studied. The frequency of pulsation is varied around the natural vortexshedding frequency, and the amplitude of pulsation is varied moderately in comparison to the cylinder diameter, at a fixed Reynolds number (=100). A flow regime map is prepared and compared with the experimental results, which are available for a circular cylinder that is subjected to inline excitation. We correlate the spectra to the corresponding flow regime. Visualization of the vorticity contours reveals that the significant interaction of the baseregion vorticities with the main shear layer vorticities is important in the mechanism of formation of the several vortexshedding modes. The strength and sign of base region vorticity with respect to the shear layers has a fundamental role to play in the mechanism of formation. It is hypothesized that the similarity in vortexshedding modes across different excitation types, bluff body geometry, and for different parameters is due to the similarity in the underlying vorticity dynamics.
 Instability and Transition

The correlation between wake transition and propulsive efficiency of a flapping foil: A numerical study
View Description Hide DescriptionWe study numerically the propulsive wakes produced by a flapping foil. Both pure pitching and pure heaving motions are considered, respectively, at a fixed Reynolds number of Re = 1700. As the major innovation of this paper, we find an interesting coincidence that the efficiency maximum agrees well with the 2D3D transition boundary, by plotting the contours of propulsive efficiency in the frequencyamplitude parametric space and comparing to the transition boundaries. Although there is a lack of direct 3D simulations, it is reasonable to conjecture that the propulsive efficiency increases with Strouhal number until the wake transits from a 2D state to a 3D state. By comparing between the pure pitching motion and the pure heaving motion, we find that the 2D3D transition occurs earlier for the pure heaving foil than that of the pure pitching foil. Consequently, the efficiency for the pure heaving foil peaks more closely to the wake deflection boundary than that of the pure pitching foil. Furthermore, since we have drawn the maps on the same parametric space with the same Reynolds number, it is possible to make a direct comparison in the propulsive efficiency between a pure pitching foil and a pure heaving foil. We note that the maximum efficiency for a pure pitching foil is 15.6%, and that of a pure heaving foil is 17%, indicating that the pure heaving foil has a slightly better propulsive performance than that of the pure pitching foil for the currently studied Reynolds number.