Non-isothermal flows of liquid metals and melts: Qualitative features, asymptotic models, problems, exact and approximate solutions

• The study focuses on effects due to temperature dependence of the viscosity.
• Thermophysical properties of liquid metals and alloy melts are analyzed.
• A non-isothermal laminar flow model for liquid metals and melts is suggested.
• Asymptotic equations of motion for low Pr and various Re numbers are derived.
• Problem solution shows considerable differences compared to constant viscosity.

We analyze experimental thermophysical properties of liquid metals and alloy melts and show that the kinematic viscosity is essentially dependent on temperature, whereas the density, thermal conductivity, and specific heat capacity are only weakly dependent on temperature. Based on this fact, we formulate a mathematical model for non-isothermal laminar flows of liquid metals and melts with variable viscosity. We derive asymptotic equations of motion for low Prandtl numbers (liquid metals are characterized by Pr≃10−2⪡1Pr≃10−2⪡1) and different Reynolds numbers and obtain a number of exact and approximate analytical solutions expressible in elementary functions or representable in closed form. We look at a few specific fluid and thermodynamic problems and show that the dependence of viscosity on temperature significantly affects the drag coefficient in non-isothermal flows as compared to isothermal flows. We outline a few semi-empirical approximations of ν(T)ν(T) and show that the power-law formula ν=ν0(T0/T)kν=ν0(T0/T)k provides a very good accuracy for several liquid metals (including sodium and mercury). The asymptotic models, equations and formulas presented in the paper can be used to state and solve new non-isothermal hydrodynamic problems for liquid metal and melt flows.