OBRABOTKAMETALLOV Vol. 24 No. 3 2022 TECHNOLOGY In this work, the numerical simulation used the mathematical model previously developed by the authors [15], which considers the interaction of solid and liquid metal. There are two phases for this: Ωl – liquid и Ωs – solid, the combination of which represents the entire study area – Ω. The solid phase, in turn, consists of a wire Ωwire and substrate Ωsub. The motion of a metallic melt can be described as the motion of a viscous incompressible fl uid. In the general case, the system of equations will consist of differential equations describing the evolution of density ρ, velocities u and temperature T in the form of balance laws (mass, momentum, and energy balance equations, respectively): 1 0 0 , u , , , , , wire l v s s p rad d , dt d p dt d d , , dt dt dT c s s dt u R f f f g R u R q R (1) where u is velocity, ρ is density, fν is viscous forces, fs is surface tension force, fv is vapor pressure force, g is acceleration of gravity, cp is specifi c heat, q is heat fl ux, k is heat transfer coeffi cient, sυ is evaporation heat loss, srad is heat loss due to radiation. Density ρ and pressure P are related by means of the equation of state: 7 2 0 0 0 1 7 ( ) , c P (2) where c0 and ρ0 are sound propagation velocity and density at zero applied tension, respectively. For incompressible fl uids, the viscous forces will take the following form: 2 , v f u (3) where η is dynamic viscosity. Following the continuum approach of Brackbill and Cote [16], based on Continuous Surface Force (CSF), the surface tension effects are treated as volume forces in equation (1), distributed over an interfacial volume of fi nite width. The surface tension force is the sum of the normal and tangential components: – ( ) , s f kn I nn (4) where α is surface tension coeffi cient, k = Δn is surface curvature, n is surface normal, I is unit tensor, ( ) d T dT . The dependence of the surface tension coeffi cient on temperature is selected as linear: 0 0 0 ( ) ( ), T T T (5) where α0 is surface tension coeffi cient at temperature T0. This dependence describes the Marangoni effect. In addition to the standard capillary effects, the high temperatures inherent in additive manufacturing processes cause the metal to evaporate, which leads to vapor pressure forces and heat loss due to evaporation. Usually a phenomenological model is used to simulate these processes [17, 18]: 1 1 f ( )n ( ) exp , p T p T , p T C C T T (6) where Tυ is boiling point, constants Cp = 0.54pa and T C h R include atmospheric pressure pa, molar latent heat of fusion hυ and molar gas constant R.
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