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14, pp. 135-144, January 1968. [7] D. S. Abrams and J. M. Prausnitz. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems. , vol. 21, pp. 116-128, January 1975. [8] A. Fredenslund, R. L. Jones, J. M. Prausnitz. Group contribution estimation of activi‐ ty coefficients in nonideal liquid mixtures. , 1975, 21 (6), 1086-1099. [9] C. C. Chen, and Y. Song. Solubility modeling with a non-random two-liquid segment activity coefficient model.
5772/61901 Figure 1. Threshold radiuses of the water droplets in different impact velocities 29 30 Recent Advances in Thermo and Fluid Dynamics Defining RM as characteristic radius of the drop at zero impact velocity as follows: RM = 31/3 b1/3 k -2/3 (7) where k −1 = σ / ρg is the capillary length and b is the radius of the fiber, and also characteristic velocity of the drop as: UM = 4 gRM (8) Variation of the dimensionless threshold radius versus dimensionless impact velocity is plotted in Figure 2, which shows a good agreement with the experimental and theoretical data presented by Lorenceau et al.
28) and the eigenvectors form a complete orthonormal basis in a Hilbert space, when the eigen‐ value spectrum is nondegenerate. Thus, the spectral representation of the operator would be O ( t ) = O ( t ) d = å lOi ( t ) Y Oi ( t ) Y 0 i ( t ) , d = å Y Oi ( t ) Y 0 i ( t ) i³0 i³0 (29) 47 48 Recent Advances in Thermo and Fluid Dynamics where δ is the unity operator and ΨOi (t ) | is the dual of | ΨOi (t ) , so ΨOi (t ) | ΨOi (t ) = 1. Since the observable is Hermitian, the transformation of the eigenvector | ΨOi (0) at t = 0 to the eigenvector | ΨOi (t ) at time t will be represented by an unitary operator U(t ).