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Review Article

# Recent Advances in Physics of Fluid Parametric Sloshing and Related Problems

[+] Author and Article Information
Raouf A. Ibrahim

Department of Mechanical Engineering,
Wayne State University,
Detroit, MI 48098
e-mail: Ibrahim@eng.wayne.edu

The Rayleigh–Taylor instability occurs at the interface between two plane-parallel layers of immiscible fluids, in which the more dense fluid is on top of the less dense one. The equilibrium is unstable to any perturbations or disturbances of the interface. This occurs if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards. The potential energy of the configuration is lower than the initial state and consequently the disturbance will grow and lead to a further release of potential energy, as the more dense material moves down under the gravitational field, and the less dense material is further displaced upward.

The dimension of the surface tension is $dyn/cm=0.001 N/m=0.001 J/m2$.

The Marangoni effect is the mass transfer along an interface between two fluids due to surface tension gradient. The Marangoni number may be regarded as proportional to (thermal-) surface tension forces divided by viscous forces and is given by the expression $Ma=((-dγ/dT)(LΔT/μα))$, where $L$ is a characteristic length (m), $μ$ is the dynamic viscosity (kg/(s · m)), $ΔT$ (in Kelvin K), $α=k¯/(ρcp)$ is the thermal diffusivity (m2/s), $k¯$ is the thermal conductivity (Watts/m K), and $cp$ is the specific heat at constant pressure (J/kg K).

The Péclet number (Pe) is defined as the ratio of the rate of advection (a transport mechanism of a substance or conserved property by a fluid due to the fluid's bulk motion) of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient. It is equivalent to the product of the Reynolds number and the Prandtl number, i.e., $Pe=LU/α=Re·Pr$. The Prandtl number Pr is the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity, i.e., $Pr=ν/α=cpμ/k¯$, where U is the velocity (m/s), D is the mass diffusion coefficient (m2/s), and $ν=μ/ρ$ is the kinematic viscosity (m2/s).

The capillary number (Ca) represents the relative effect of viscous forces versus surface tension acting across an interface between two immiscible liquids and is defined by the expression $Ca=μV/γ$, where V is a characteristic velocity.

Bond number measures the relative magnitudes of gravitational and capillary forces and is defined by the expression $Bo=ρgL2/γ$.

The Stokes number (Stk) is a dimensionless number corresponding to the behavior of particles suspended in a fluid flow. It is defined as the ratio of the characteristic time of a droplet to a characteristic time of the flow and may be given by the ratio $Stk=t˜U0/dc$ where $t˜$ is the relaxation time of the particle (the time constant in the exponential decay of the particle velocity due to drag), $U0$ is the fluid velocity of the flow, and $dc$ is the characteristic dimension of the obstacle. Particles with low Stokes number follow fluid streamlines (perfect advection) whereas for large Stokes number, the particle's inertia dominates so that the particle will continue along its initial trajectory.

Soliton is a wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in which the speed of the waves varies according to frequency. Dispersion and nonlinearity can interact to produce permanent and localized wave forms. Solitions are of permanent form and are localized within a region.

In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). In general, it is a linear partial differential equation, which may describe the wave function of the system, also called the quantum state or state vector.

A bifurcation that requires at least m control parameters to occur is called a codimension-m bifurcation.

The rhomboid is a parallelepiped, a solid figure with six faces in which each face is a parallelogram and pairs of opposite faces lie in parallel planes. Some crystals are formed in 3D rhomboids. This solid is also sometimes called a rhombic prism.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 14, 2014; final manuscript received December 24, 2014; published online May 25, 2015. Assoc. Editor: Shizhi Qian.

J. Fluids Eng 137(9), 090801 (Sep 01, 2015) (52 pages) Paper No: FE-14-1448; doi: 10.1115/1.4029544 History: Received August 14, 2014; Revised December 24, 2014; Online May 25, 2015

## Figures

Fig. 1

The physical interpretation of the resonance at half the excitation frequency as provided in Ref. [15]

Fig. 2

Waveforms of (a) cross waves, (b) sloshing modes, (c) oblique subharmonic waves, and (d) harmonic wave-trains [31]

Fig. 3

Amplitude–frequency response in a rectangular container (dimension of 17.78 cm × 22.86 cm) for mode (1,1) of fluid depth of 11.43 cm under two different excitation amplitudes: 0.9 mm (___ predicted branch, ▪ measured) and 1.55 mm (____ - ____ predicted, ◻ measured). (a) Without additive and (b) with additive [94].

Fig. 4

Dependence of fluid amplitude on excitation (a) amplitude and (b) frequency in a rectangular container (dimension of 17.78 cm × 22.86 cm): Solid curves are predicted points and solid and hollow symbols are measured points [94]. (a) Response amplitude–excitation amplitude for different excitation frequencies for mode (1,1) and fluid depth of 11.43 cm and (b) response amplitude-frequency for different values of fluid depth (h) and excitation amplitude (Z0) for mode (0,1).

Fig. 5

Stability zones on the plane of excitation/Galileo ratio A/G and wave number k showing a tricritical point and superharmonic waves in viscous sheet of water–glycerol mixture with Galileo number G = 2.7×103, capillary number Ca = 4.7×10-4, and excitation frequency ratio ϖ = 7.5. The shaded zones and zones bounded by dots are for subharmonic and harmonic, respectively, and for different rotational rates (a) 2ωrh2/ν = 0.0, (b) 2.3, (c) 2.9, and (d) 4.0 [108].

Fig. 6

Experimental measurement of liquid response amplitude of sloshing mode (1,1) as a function of excitation frequency for 1/2 subharmonic response for different values of excitation amplitude: Δ:Z0 = 0.65 mm, ○: Z0 = 1.31 mm, and □: Z0 = 2.18 mm [19]

Fig. 7

Experimental Faraday wave amplitudes dependence on excitation frequency. Solid symbols represent frequencies varied in small steps from low to high frequency at constant forcing amplitude. Hollow symbols represent frequency variations in the opposite direction. Three forcing amplitudes are used: 2.5 mm (▪ and □), 3.0 mm (▲ and Δ), 3.5 mm (♦ and ◇), and 3.5 mm. Solid/dashed lines with arrows represent excited Faraday waves with decreasing/increasing frequency [141].

Fig. 8

Comparison of prediction and experimental measurements of dependence of Faraday wave amplitude on the excitation frequency ratio at an excitation amplitude of 2.5 mm: ...... predicted using Henderson and Miles [67] theory; _____ numerical results. Symbols are experimental results [141]. (a) The fluid is treated water: ▪ and □ measured results with increasing and decreasing frequencies, respectively. (b) The fluid is treated water with Photo Flo in a ratio of 100:1. • and ○ represent measured results with increasing and decreasing frequencies, respectively.

Fig. 9

Time history records of two cases of steadily modulated Faraday waves [141]. (a) Excitation frequency of 3.30 Hz and excitation amplitude of 2.90 mm. (b) Excitation frequency of 3.32 Hz and excitation amplitude of 2.65 mm. Measurements were taken at the horizontal center of the tank [141].

Fig. 10

Stability boundary shows regions of various waveforms in Faraday wave experiments [181]

Fig. 11

(a) Profile of three different modes during period-tripled breaking. T is the temporal wave period before period tripling (twice the forcing period). Each crest feature appears at the end-walls of the tank 1:5 T after it appears at the centerline [181]. (b) Images showing the wave-amplitude modulations period tripling scenario [197].

Fig. 12

Experimental snapshots of three breaking modes with excitation amplitude F = 4:60 mm and f = 1.60 Hz. Time (unit: s) is shown in each frame. (a) Mode A with “leftward” plunging breaker, (b) double plunger to each side of the dimpled crest in mode B, and (c) Mode C with maximum elevation at t = 0:04 s [181].

Fig. 13

Time sequence of snapshots taken video camera (25 frames per second) for free surface under vertical excitation shaking [185]

Fig. 14

Surface profiles of “standing waves” at maximum elevation for a sequence of different initial amplitudes for a cosine surface velocity distribution and initial uniform water depth of h = 1 and L = 2π (Refs. [185,186-185,186])

Fig. 15

Surface wave collapses and resulting jets in a cylindrical tank with an inner diameter of 12.7 cm filled to a depth of 6.5 cm under vertical excitation: (a) Snapshot showing the collapse of a surface wave depression and the subsequent upward jet caused by self-focusing of the kinetic energy associated with a near singularity under excitation frequency of 7.84 Hz and excitation amplitude of 2.39 m/s2, which was increased to 3.23 m/s2. Once the standing wave is sufficiently tall, it produces a deep depression which collapses to a singularity and jet. (b) Snapshot of fluid of viscosity of 0.26 cm2/s. The driving frequency and acceleration amplitude are 8.00 Hz and 4.64 m/s2, respectively [195].

Fig. 16

Instability boundary for Fluorocarbon (FC-72) fluid. The solid curve corresponds to the theoretical curve with damping ratio ζ=0.0022. Bifurcation points are bifurcation to mode (2,1), detuning parameter, β1 = -0.962; a excitation frequency ratio, Ω/2ω01 = 0.94; wave-breaking bifurcation, β2 = -1.015,Ω/2ω01 = 0.974; unsteady wave motion, β3 = -0.753, Ω/2ω01 = 0.987; bifurcation to mode (3,1), β4 = 0.875, Ω/2ω01 = 1.02. - - -: experimental wave breaking threshold and · · ·: onset of unstable wave motion [197].

Fig. 17

Snapshots of free surface ((a) and (c)–(e)) obtained in container R = 5 cm and image (f) obtained in container R = 2.5 cm, showing the last stable wave amplitude and the following (half a period later) wave depression (cavity) below, for glycerin–water (a), water (c), and FC-72 ((d) and (e)). (b) The wave crest for water, with parasitic capillary waves, taken at 0.186 wave period before the maximum wave amplitude is reached, shown in (c); (a) Ω/2π = 8.85 Hz, excitation amplitude ratio Z0/R = 0.0368; (c) Ω/2π = 8.86 Hz, Z0/R = 0.0154; (d) Ω/2π = 8.77 Hz, Z0/R = 0.0179; (e) Ω/2π = 8.35 Hz, Z0/R = 0.0143, (f) R = 2.5 cm, FC-72, Ω/2π = 11.65 Hz, Z0/R = 0.0175 [197].

Fig. 18

Behavior of ethanol drop deposited on silicon oil: (a) sketch of the vertical section of the floating drop in the absence of oscillations and (b) It is circular at rest, and (c)–(e) three successive images showing the temporal evolution of the drop under parametric excitation frequency of 130 Hz [198]

Fig. 19

Demonstration of effective and geometric symmetries: (a) the extension by reflection of a Neumann boundary condition solution on (O,x) leads to periodic boundary condition on (-π,π). (b) The relation between the physical domain Π and the extended domain Π˜ [211].

Fig. 20

Liquid surface images in a square tank produced by (a) pure (3,2) mode and (b) superposition of the (3,2) and (2,3) modes with equal amplitudes. The images were averaged over one period of the forcing period [225].

Fig. 21

Stability boundaries of liquid free surface in a square tank (6.17 × 6.17 cm): (a) regions of three modes and their degenerate modes. The resonances are asymmetric: subcritical on the left and supercritical on the right, (b) expanded view near the (3,2)–(2,3) resonance. Three primary regions are shown: the flat surface, mixed states in region B, and pure states in region D. The intermediate regions A and C are characterized by co-existence of different types of fixed points (flat or mixed in A, mixed or pure in C) which are realized for different initial conditions [225].

Fig. 22

Surface mode contours: (a) (m,n) = (4,3) and (b) (m,n) = (7,2) [246]

Fig. 23

Stability boundaries of modes (4,3) and (7,2) in terms of the excitation amplitude Z0 and frequency f0 of a circular container of radius 6.35 cm filled with 1 cm layer of water. Shaded regions belong to slow periodic and chaotic oscillations involving competition between these modes [220].

Fig. 24

(A) Time history records showing the transition from periodic to chaotic oscillations and (B) corresponding power spectra under excitation frequency of 16.05 Hz and three different excitation amplitudes [220]

Fig. 25

Time series of relaxation oscillations showing slow drifts along branches of both equilibrium states with fast jumps between them [260]

Fig. 26

Threshold measurements for oil at 55°F in a square container of width of 11.4 cm and depth of 0.7 cm. The solid line is the finite depth numerical calculation. The dashed line represents the threshold curve in the absence of mode quantization (only the tongue minima are shown). The insets show wave patterns at onset for different frequencies [304].

Fig. 27

Surface pattern of essentially parallel lines was observed for fluid mixture of 88% glycerol and 12% water and kinematic viscosity of ν≈1.00 cm2/s under parametric excitation of single frequency of 80 Hz [301]

Fig. 28

Dependence of velocity of wave drift (measured by node displacement) on the parametric excitation acceleration for two different profiles of side walls. 1—Vertical profiles of the side wall and the structure of the shear flow and 2—vertical wall with corner step near the bottom [312]

Fig. 29

Experimental images of Faraday ripples in laboratory experiment: (a) a target with four dislocations (two positive and two negative); (b) one dislocation is attracted to the target core, spiral is formed; (c) all dislocations are attracted to the center and annihilated, perfect target reappeared (images (a)–(c) are separated by 2.0 s); and (d) asymptotic state of another experiment where a three-armed spiral was formed and rotated for a long time (one period of a standing wave corresponds to two white and two dark stripes on the photos due to time averaging) [312]

Fig. 30

Surface wave patterns in a stadium shaped container at various external frequencies [313]

Fig. 31

Regions of surface wave patterns in terms of fluid viscosity and excitation frequency, experimental results are shown by × for stripes pattern, □ for square pattern, Δ for hexagonal patter. Alternating × and □ indicate mixed stripe–square patterns [318].

Fig. 32

Bifurcation diagrams showing the dependence of the spatial amplitudes on the normalized acceleration at different excitation frequencies for (a) mode 10 and (b) mode 11, the excitation frequency is Ω = 12(2π) rad/s [326,364]

Fig. 33

Snapshots of the square Faraday patterns over a unit cell at excitation frequency of 12 Hz and acceleration of 30 m/s2: (a) maximum surface elevation and (b) minimum surface elevation [326]

Fig. 34

Snapshots of hexagonal Faraday patterns: (a)–(c) at excitation frequency of 12 Hz and acceleration of 39.3 m/s2 for three different temporal phases: (a) down hexagons; (b) pattern near the minimal surface elevation; and (c) up hexagons [326]

Fig. 35

Fluid free surface hexagons pattern produced by two-frequency parametric excitation 4ω and 5ω components for (a) ω/2π = 14.6 Hz, φ = 75 deg, and χ = 45 deg and (b) ω/2π = 28.0 Hz, φ = 68.4 deg, and χ = 72 deg [301]

Fig. 36

Standing wave pattern takes the form of a quasi-crystalline wave with 12-fold rotational symmetry of a layer of silicone oil under two-frequency parametric excitation. The brightest regions are locally horizontal, whereas darker colors indicate inclined regions [329].

Fig. 37

Samples of fluid surface wave patterns in a cylindrical container of diameter of 44 cm and fluid depth of 2 cm under parametric excitation showing (a) square symmetry under excitation frequency f = 45 Hz, (b) hexagonal symmetry at f = 30 Hz, (c) eightfold quasi-periodic at f = 29 Hz, and (d) tenfold quasi-periodic f = 27 Hz [204]

Fig. 38

Different surface patterns: (a) rhombic lattice with an angle θ between the primary wave-vectors, (b) hexagonal superlattice with an angle of 21.8 deg between the most closely spaced wave-vectors. (c) and (d) 12-fold and 14-fold quasi-lattices, up to 11th-order and 7th-order, respectively [203,316].

Fig. 39

Schematic diagrams of resonant triads at the critical point. The wave-vectors satisfy k3 = k1+k2: (a) k3 = |k3|< k1 = |k1| = |k2| and (b) k1 < k3 < 2k1 [218].

Fig. 40

Experimental patterns: Patterns (a) and (b) were both obtained using Dow–Corning silicone oil with viscosity of 47 cSt and layer depth of 0.35 cm, while pattern (c) was observed for a 23 cSt oil layer of depth of 0.155 cm. All three patterns are obtained with excitation containing two frequencies in the ratio of 2:3. Pattern (a) was obtained under driving frequencies of 50 and 75 Hz, pattern (b) with frequencies of 70 and 105 Hz, and pattern (c) with 40 and 60 Hz [344].

Fig. 41

Two temporal phases of an eightfold QP observed for three-frequency driving ((a) and (b)). This state was observed in a 50/75/100 Hz experiment with driving amplitude ratio a1:a2:a3 = 0.16:0.36:0.48 and a phase difference of 180 deg between the 100-Hz component and the two other components [331].

Fig. 42

Phase diagrams for (a) excitation frequency ratio m:n = 4:5, with ω/2π = 20 Hz and φ5 = 16 deg; (b) excitation frequency ratio m:n = 6:7, with ω/2π = 16.5 Hz and φ7 = 40 deg [349]

Fig. 43

Phase diagrams for varying a4 and a5 for (a) two-frequency excitation m:n = 4:5 and (b) three-frequency excitation m:n:p = 4:5:2, where ω/2π = 20 Hz and φ5 = 16 deg. (*) Hexagons, (□) QPs, ( + ) squares and 2MS (two-mode superlattice patterns), and (•) disordered. Unmarked regions belong to a flat surface state [349].

Fig. 44

Time records of parametric excitation f(ωt) and their corresponding stability boundaries on acceleration amplitude a versus wave number k for (a) n = -2, (b) n = -0.3, (c) n = 0.5, and (d) n = 1. The plots are for ω = 20π rad/s, h = 0.3 cm, excitation acceleration amplitude a is units of g, fluid density ρ = 0.96 g/cm3, kinematic viscosity ν = 46 cm2/s, and surface tension γ = 20 dyn/cm. The resonance tongues labeled by H and SH refer to regions with harmonic or subharmonic instabilities, respectively. Their envelope shown by dashed lines change with n [295].

Fig. 45

Example of square pattern showing the height of interface over the horizontal coordinates, when the height is maximal. Resolution in x, y, and z is 80 × 80 × 160. Note that the vertical scale is stretched with respect to the horizontal scale: (a) at a given time t and (b) at another instant, time 0.24 × 2T after time t [363].

Fig. 46

Dependence of (a) mean and (b) mean square response of the first antisymmetric sloshing mode on excitation spectral density level [410]

Fig. 47

Dependence of the mean square of the first symmetric sloshing mode on excitation spectral density parameter [410]

Fig. 48

Comparison of measured and predicted of liquid free surface response probability density function [410]. (a) First antisymmetric mode and (b) first symmetric mode.

Fig. 49

Time evolution of the ensemble average 〈|X|〉 for different values of the spectrum width s: ______: sinusoidal acceleration, _ _ _ _: s= 0.1; - - - -: s = 0.4; ......: s = 0.8 [428]

Fig. 50

Bifurcation diagram based on the ensemble average values of the equilibrium 〈|X|〉 for different values of the spectrum width: s: ______: sinusoidal acceleration, _ _ _ _: s= 0.1; - - - -: s = 0.4; ......: s = 0.8 [428]

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