function [ome] = spin(r)

% Calculates empirically the spin of spheres colliding 
% with the flat, frictional base
% using the numerical simmulations described in Louge, M.Y. 
% "Computer simulations of rapid granular flows of spheres 
% interacting with a flat, frictional boundary", 
% Physics of Fluids 6(7), pp. 2253-2269 (1994).
% Unfortunately, because no general correlation for spin with r
% is yet available, the constants employed here are limited
% to the specific conditions and collision parameters
% (binary: e=0.972, mu=0.177, beta0=0.25; 
% sphere on wall: ew=0.80, muw=0.141, beta0w=0.35)
% that Louge & Keast used in: Louge, M.Y. and Keast S.C. 
% "On dense granular flows down flat frictional inclines",
% Physics of Fluids 15(5), pp. 1213-1233 (2001), see their eq. (49).
%
% Michel Louge 4/11/06
%
%	input	-   r           =	relative velocity at the contact point
%                                   divided by sqrt[3*granular_temperature_there]
%
%	output	-	ome         =   sphere angular velocity at the base 
%                                 *(D/2)/sqrt[3*T0]
%                                   where	D       = sphere diameter
%                                           T0      = granular temperature at base
%
%               For the measurement technique of these collisional parameters, see
%               S. Foerster, M. Y. Louge, H. Chang & K. Allia
%               "Measuremenfs of the collision properties of
%               small spheres,Ó Phys. Fluids 6, 1108 (1994).
%               For the measurements of the specific spheres used by Louge
%               & Keast, see A. Lorenz, C. Tuozzolo, and M. Y. Louge, 
%               "Measurements of impact properties of small, nearly 
%               spherical particles," Experimental Mechanics 37, 292-298 (1997).
%           

% 	other internal variables
%
%           -   a4          =   constant in Louge & Keast's fit of spin
%           -   a5          =   ditto
%           -   n3          =   empirical exponent in Louge & Keast's spin fit
%           -   omef        =   asymptotic dim(-) spin in Louge & Keast's fit
%

a4      =   0.692184944406366   ;
a5      =   1.29134880397588    ;
n3      =   0.924676214314275   ;
omef    =   0.693756539738975	;

ome     =  omef*(1-1/(1+r^n3*(a4+a5*r)))        ;
