By using a small-time series expansion technique, the thermal effect of surface tension (Gibbs-Thomson effect) on the early-stage phase growth of a spherical nucleus immersed in an infinite subcooled liquid is studied in this paper. The result shows that surface tension greatly reduces the incipient growth rate of the solid nucleus. Critical value of surface tension is found beyond which the decreasing of the phase growth rate with time becomes non-monotonic. Analytical expression for the phase growth rate in terms of relevant physical parameters is also derived under the condition of small degree of undercooling.
Normalised quadratic control, a modification of the conventional quadratic control method has recently been proposed to exponentially stabilise a homogeneous bilinear system wliose open-loop eigcnvalucs all fall on the imaginary axis. The normalised quadratic control method is now extended to a broader class of bilinear systems whose open-loop eigenvalues may fall to the left or right of the imaginary axis. It is shown that for bilinear systems that are already open-loop stable. the normalised quadratic control can add to the system an extra exponcntial decay rate. For open-loop unstable bilinear systems. the normalised quadratic control can stabilise the system if the extra exponcntial decay rate provided by the control is larger than the open-loop system's growth rate.
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