equation(5) |am→|=|bm→|=|cm→|=32S Thus the function equation(6) (|bm→|−32S)2+(|am→|−32S)2+(|cm→|−32S)2will equal zero if the distances are correctly found by the algorithm. Therefore, a function of m→ is defined, equation(7) y=f(m→)≡(|bm→|−32S)2+(|am→|−32S)2+(|cm→|−32S)2so by minimization of the f(m→), the centre of the cube can be identified. By tracking the three tracer positions at the corners a, b and c respectively, the motion of the centre of the cube m can

be found. This represents the solid translational motion. From Fig. 2B, the velocity of “a” relative to “m” (Smith & Smith, 2000, pp. 254–269) is equation(8) r˙a=ua×rawhere uaua is angular velocity, and ua = (ωx, ωy, ωz). The actual velocity of “a” will therefore be equation(9) Selleckchem NVP-BKM120 R˙a=R˙m+ua×raThus equation(10) Va=Vm+ua×(a→−m→) In a similar way, equation(11) Vb=Vm+ub×(b→−m→) equation(12) Vc=Vm+uc×(c→−m→)where the velocity is calculated by three successive locations as follows. equation(13) Vx(ti)=12(x(ti+1)−x(ti)ti+1−ti+x(ti)−x(ti−1)ti−ti−1) In a similar way, the velocity in y and z directions can be obtained. For

a rigid body, the angular velocity of any point in the rigid body round the mass-centre should be same, and described by ω. If a function of ω is defined as equation(14) y=f(ω)≡|Va−Vm−ω×(a→−m→)|2+|Vb−Vm−ω×(b→−m→)|2+|Vc−Vm−ω×(c→−m→)|2the ω can be calculated by the minimization of (14). Then the RAD001 ic50 observed internal spin rate of the cube can be calculated as in Eq. (15). why equation(15) N=|ω|2π To find how the cube spin varies with their position, the can was divided by several 2 mm × 2 mm × 119 mm cuboids, the solid spin was calculated by using the average for the cube which the centre of the cube was captured by the cuboid, as described in (Yang et al., 2008b). Thus, the average cube spin rate N¯ was given by equation(16) N¯j=1l∑i=1lN(j,i)where N(j, i) denoted the instantaneous spin rate for the ith position of the cube in the jth cuboid. The statistic internal spin rate of the cube, (i) average of internal spin rate (μ)

and (ii) the standard deviation of internal spin rate (σ), were obtained by the following equations: equation(17) μ=1k∑j=1kN¯j equation(18) σ=∑j=1k(N¯j−μ)2k The experiments similar to those in Yang et al. (2008a), tracking 3 tracers in three liquids, were performed. The cans throughout this study were supplied by Stratford Foods Ltd, Stratford UK and measured 119 mm high with a diameter of 100 mm. The experiments were designed for the observation of the effect of solids fraction and liquid viscosity on solids rotational and translational motions. The liquids used were water, dilute golden syrup and golden syrup with viscosities of 0.001, 2 and 27 Pa s, respectively. For each liquid, the experiments were carried out at four solids fractions, which were 10, 20, 40 and 50% (v/v).