## 2 Model
### Governing Equations  
Heat conduction is supposed to be the principal mechanism of heat transfer in the target and is described by the heat equation in enthalpy form [14; 15]. \[\frac{\partial h}{\partial t}+u_{s}\frac{\partial h}{\partial z}=\frac{ \partial}{\partial z}\bigg{(}\lambda\frac{\partial T}{\partial z}\bigg{)} \tag{1}\]  
where \(h\) is the enthalpy per unit volume, \(t\) the time, \(\lambda\) the thermal conductivity, and \(T\) the temperature. The second term in the left hand side of Equation (1) is responsible for advection in the frame-moving relative medium. In the case of net evaporation, the interface recesses, the medium of the target moves in the positive direction of axis Z, and \(u_{s}>0\). In the case of net condensation, the interface rises, the medium of the target moves in the negative direction of axis Z, and \(u_{s}<0\). The volumetric specific heats in solid \(c_{s}\) and liquid \(c_{l}\) phases are approximately taken as constants. Then, the thermal equation of state [14],  
\[T=\left\{ \begin{array}{ccc}h/c_{s}&,&h\leq c_{s}T_{m}\\ T_{m}&,&c_{s}T_{m}<h<c_{s}T_{m}+q_{m}\\ T_{m}+(h-c_{s}T_{m}-q_{m})/c_{l}&,&h\geq c_{s}T_{m}+q_{m}\end{array} \right. \tag{2}\]  
relates the enthalpy and the temperature, where \(T_{m}\) is the melting point and \(q_{m}\) the latent heat of melting per unit volume. Gas-dynamic equations of continuity (3) and (4), momentum (5) and energy (6) [15],  
\[\frac{\partial n_{v}}{\partial t}+\frac{\partial(n_{v}u)}{\partial z}=0 \tag{3}\]  
\[\frac{\partial n_{g}}{\partial t}+\frac{\partial(n_{g}u)}{\partial z}=0 \tag{4}\]  
\[\frac{\partial(pu)}{\partial t}+\frac{\partial(pu^{2}+p)}{\partial z}=0 \tag{5}\]  
\[\frac{\partial(E+\rho u^{2}/2)}{\partial t}+\frac{\partial[u(E+\rho u^{2}/2+p )]}{\partial z}=0 \tag{6}\]  
are applied at \(z>0\), where \(n_{v}\) and \(n_{g}\) are the number densities of vapor and ambient gas, respectively, \(u\) the gas velocity, \(\rho\) the gas density, \(p\) the pressure and \(E\) the internal energy per unit volume. In the gas mixture [15],  
\[\rho=m_{v}n_{v}+m_{g}n_{g}, \tag{7}\]  
\[p=(n_{v}+n_{g})kT, \tag{8}\]  
\[E=\bigg{(}\frac{n_{v}}{\gamma_{v}-1}+\frac{n_{g}}{\gamma_{g}-1}\bigg{)}kT \tag{9}\]  
where \(m_{v}\) and \(m_{g}\) are the molecular masses and \(\gamma_{v}\) and \(\gamma_{g}\) the adiabatic exponents of vapor and ambient gas, respectively. Mass balance at the interface, \(z=0\), is [15]  
\[n_{t}u_{s}=n_{v}u, \tag{10}\]  
where \(n_{t}\) is the number density in the target. If laser energy is absorbed at the target surface with the heat effect per unit time per unit area equal to absorbed laser energy flux \(K_{a}\), the energy balance at \(z=0\) is [15]\[K_{a}+u_{s}h(T_{s})-\lambda\frac{\partial T}{\partial z}\Big{|}_{z=0-0}=un_{p}U+u(E+ pu^{2}/2+p) \tag{11}\]  
where \(T_{s}\) is the temperature of the target surface and \(U\) the binding energy of target atoms at \(T=0\). Note that gas temperature at the interface \(T(0)\) is necessary to calculate the right hand side of Equation (11). Temperature \(T(0)\) is generally different from \(T_{s}\) because of a temperature jump across the Knudsen layer formed near the interface at strong evaporation or condensation. Binding energy \(U\) is the principal part of the latent heat of evaporation \(q_{v}\). In addition to Equations (10) and (11), two boundary conditions at \(z=0\) are necessary at evaporation and only one boundary condition is applied at condensation [26]. Gas-dynamic boundary conditions of evaporation/condensation relate the parameters of target surface with the parameters of vapor formed at the outer edge of the Knudsen layer. The target parameters are its surface temperature \(T_{s}\) and saturated vapor pressure \(p_{s}\) at \(T_{s}\).