Evaporation-induced moisture damage of asphalt mixtures: Microscale model and laboratory validation

Multiphase macroscale models for macrosegregation and columnar to equiaxed transition during alloy solidification

“…Next, in equations (3-7) and (3)(4)(5)(6)(7)(8), the convective fluxes on the left-hand side (i.e., d g v and e l g v ) and the interfacial mass generation sources due to phase-change on the right-hand side (i.e., sd  and ed  ) need to be rewritten in terms of the mixture velocity m l l g  v v and phase fractions, respectively. To substitute sd  and ed  , one first need to write the continuity equations for the solid and extra-dendritic liquids, which read (2, 10, 48, 49)…”

“…and env w is the average growth velocity of the envelope. Substituting ed Γ from the first equality in equation (3)(4)(5)(6)(7)(8)(9) and noting that…”

“…Similarly, sd Γ and ed Γ on the left-hand side of equation (3-6) can be substituted using equations (3)(4)(5)(6)(7)(8) and (3)(4)(5)(6)(7)(8)(9), respectively, and the last term can be substituted from equation (3-10) to get…”

“…Next, equation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) needs to be re-written in the conservative form, where the inner product on the left-hand side is substituted by a divergence, and in terms of the mixture velocity m v only.…”

“…Next, equations for the solute balance in the inter-dendritic and extra-dendritic liquids (i.e., equations (3)(4)(5)(6) and (3-7)) are used to derive the equation for the solute balance in the liquid for the truncated-Scheil-type and Scheil-type model.…”