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analysis analytic function aX 3Y cold plate cold surface immersed conformal mapping procedure conformal transformations constant of integration constant temperature cooled plate immersed coordinates determine the shape Dimensional Solidification Dimensionless heat flow dimensionless physical plane dimensionless temperature elliptic integral equation 12 equations 20 finite width flowing liquid flowing warm liquid freezing temperature frozen interface frozen layer formed frozen region frozen-layer configur frozen-layer thickness Heat Conduction heat flow lines heat flux Heat Transfer immersed in flowing insulated intermediate mapping plane isotherms Laplace equation length scale parameter Lewis Research Center LIQUID By Robert Liquid Flowing liquid interface MAPPING FOR STEADY NASA parameter k/ha)[(tj plate of finite potential function potential plane real axis rectangle relation Robert Siegel Lewis shape of two-dimensional shown in figure Siegel Lewis Research solidified layers formed STEADY TWO-DIMENSIONAL SOLIDIFICATION steady-state frozen layer SURFACE IN FLOWING Temperature derivative plane temperature distribution thin layers two-dimensional solidified layers values Vl b2 ythin