PSI - Issue 69

Mohammadjavad Abdollahzadeh et al. / Procedia Structural Integrity 69 (2025) 2–19

8

f " =− )( ' ! , ! ! " (A " τ "" )+ ! ! # &A # τ "# (+ ! ! $ (A $ τ "$ )- f # =− )( ' ! , ! ! " &A " τ "# (+ ! ! # &A # τ ## (+ ! ! $ &A $ τ #$ (- f $ =− )( ' ! , ! ! " (A " τ "$ )+ ! ! # &A # τ #$ (+ ! ! $ (A $ τ $$ )-

(5)

(6)

(7)

in the preceding equations τ ij (where i and j are representative of x, y, and z respectively) stands for the shear stress, a factor engendered due to friction between fluid elements attributable to fluid viscosity. The associated equations are explained as follows, where μ serves as an indicator of viscosity [57]. τ "" =−2µ, ! ! % " − ' - 6 ! ! % " + ! ! + # + ! ! , $ 7- (8) τ ## =−2µ, ! ! + # − ' - 6 ! ! % " + ! ! + # + ! ! , $ 7- (9) τ $$ =−2µ, ! ! , $ − ' - 6 ! ! % " + ! ! + # + ! ! , $ 7- (10) τ "$ =−µ, ! ! % $ + ! ! , " - (11) τ "# =−µ, ! ! + " + ! ! % # - (12) τ #$ =−µ, ! ! + $ + ! ! , # - (13) Two additional key parameters feature prominently in the momentum equation. The first of these concerns the buoyancy force, which is expressed by utilizing the Boussinesq approximation, represented by Equation (14). In this equation β is the thermal expansion coefficient and T ref is the reference temperature [40]. f . 88⃗ = ρgβ(T − T /01 ) (14) The term 'mushy zone' labels a region within a material wherein solid and liquid phases coexist concurrently. Even though the scenario is not strictly equivalent, liquid metal flowing through the mushy zone is assumed to resemble flow through a porous medium for the purpose of analysis. Within the context of this study, the morphology of the mushy zone is denoted by the constant 'C', the proportion of solid phase is represented by 'FS', and the small constant 'B' is employed to obviate any potential division by zero [40]. Z=C 2 "# ('42 " ) $ 6. (15) The subsequent equations demonstrate the use of a linear weight-averaged approach for estimating the properties of materials within the mushy zone. This includes parameters such as density, thermal conductivity (K), and specific heat capacity (C p ) [ 56 ] . In the following equations, 'fs' represents the fraction of the solid phase while 'f l ' stands for the fraction of the liquid phase. A schematic representation detailing the cellular configuration and the distinct phases is provided in Fig.4. ρ= f 7 ρ 7 +f 8 ρ 8 (16) k= f 7 k 7 +f 8 k 8 (17)