Fig. 1 The rectangular domain and velocity boundary conditions for the problem described by Eqs. (1) – (3) More

Fig. 2 Comparison of u velocity profiles with experiment for the r  =   0 case. ( a ) Measurements scaled with σ o determined by fitting to the EFS (dashed line) as calculated by Forliti et al. [ 23 ] (squares), Liepmann and Laufer [ 24 ] (triangles), and Wygnanski and Fiedler [ 25 ] (circle... More

Fig. 3 Effect of nonlinearity. ( a ) Maximum gradient in η space of the u ( η ) ∼ f ′ ( η ) profile. ( b ) Vorticity thickness spreading rate σ δ ω ′ = ( 1 − r ) / | f ″ | max from which σ can be determined from measurement of δ ω ′ ... More

Fig. 4 Comparison of u velocity profiles for cases with r   >   0. The values of σ were determined from Eq. (A4) using vorticity thickness measurements reported in the experiments. ( a ) r  =   0.143, δ ω ′ = 0.134 ,   σ = 17.4 , and r  =   0.382, δ ω ′ = 0.... More

Fig. 1 Sketch of the flow configuration. Vibration of the lower plate generates movement of the upper plate with velocity U top which is one of the unknowns. More

Fig. 2 Variations of the upper plate velocity U top / A 2 as: functions of the wave phase speed c and wave number α ( a ), a function of α ( b ), and as a function of c ( c ; dashed lines show proportionality to c ). The data has been multiplied by 10 −3 in ... More

Fig. 3 Distributions of pressure at the lower plate (( a ): c = 5 ; ( b ): c = 250 ) and of the x -component of pressure force (( c ): c = 5 ; ( d ): c = 250 ). Dashed lines represent wave shape. More

Fig. 4 Distributions of the x -component of the shear force acting on the fluid at the lower plate σ X v , L / A 2 (( a ): c = 5 and ( b ): c = 250 ). Dashed lines represent wave shape. More

Fig. 5 Distributions of the u -velocity component at x / λ = 0 ,   0.25 ,   0.5 ,   0.75 for   A = 0.01 , α = 0.1 ,   2 ,   20 , and c = 5 More

Fig. 6 Variations of energy dissipation DIS / ( λ A 2 ) ( a ), the shear work done at the vibrating plate   VWORK / ( λ A 2 ) ( b ), and the pressure work done at the vibrating plate RHS 1 / ( λ A 2 ) ( c ) as functions of the vibration ... More

Fig. 7 Variations of the force F xvU / A 2 are required to prevent the movement of the upper plate as a function of α More

Fig. 8 Sketch of wave shapes subject to analysis More

Fig. 9 Variations of the upper plate velocity U top / B 2 for triangular waves made of distinct adjacent elements ( d = 0 ) as functions of the wave number α for different levels of tiltings T ( a ) and as functions of the tilt parameter T for alpha = 0.... More

Fig. 10 Variations of the upper plate velocity U top / B 2 as functions of the wave number α   for : the symmetric trianglar waves placed at distances d apart ( a ) and triangular tilted waves placed at a distance b  =   3λ/4 apart ( b ). Data for T = ± 0.125   ... More