Specifying the suitable boundary conditions is the basis for successful computation. In this section we first discuss some general aspects of boundary conditions, then we use them for the laser machining processes.

Three kinds of boundary conditions are typically encountered in heat conduction analysis, these are:

Given the boundary temperature T,

Given the boundary heat flux q, and

Given a boundary heat flux balance relation.

If the boundary temperature is given, there is no particular difficulty in modeling, one needs only specify the value of the boundary grids to the specified temperature.

If the boundary heat flux along direction x is given, the energy balance is:

where q is the boundary heat flux, k is the heat conductivity, T is temperature.

The more general case is a boundary energy balance relation. For example, sometimes the boundary inward heat flux q is specified in terms of a heat transfer coefficient h and a surrounding temperature T_f . Then the boundary condition is:

The above relation is the energy balance between boundary heat conduction and the boundary heat convection. If radiation and heat flux should also be considered, we have the following boundary conditions:

where is Stefan-Boltzmann constant, is the surface emissivity. This relation can be used to model simple 1D heat transfer in laser heat treatment if one specify q_in to be the absorbed laser energy.

Note that the above boundary conditions consider no effect of phase change. But in laser machining, phase change effect can not be neglected. To consider the phase change effect, the boundary mass flow q_mass (define leaving flux to be positive) must be known, then one can include the energy brought away by the mass flux. Let the energy brought away by unit mass to be e, the boundary condition is:

The energy brought away by unit mass is consisted of internal energy e_int and kinetic energy e_k. The mass flow is related to the surface recessing velocity v_n during machining. So we have:

It is clear that to deal with phase change problems, the properties at the solid-liquid-vapor interface must be studied in detail. The properties involved include interface temperature, particle velocity relative to the target, surface recessing velocity, latent heat, etc. This topic will be discussed in following sections, Section 3.5 and Section 3.6.

Figure 3.51: The configuration of laser machining process

In Level 2 Section 3.4 we have reviewed the basic fluid mechanics. The fluid flow boundary conditions are described here. The boundary conditions we are considering are solid-liquid interface, inlet and outlet , and the liquid-gas interface or the free surface.

Fig. 3.52 shows an SEM picture of a laser drilled hole on pure copper using 50 pulses at 6e8 W/cm² laser intensity, laser pulse duration time is 50ns, beam size is 4.5microns. The flow of molten layer is obvious. So the fluid flow boundary condition is also needed for laser machining processes.

First, for a solid, impermeable wall, there is no slip and no temperature jump between a viscous heat conducting fluid and the solid

V_fluid=V_wall and T_fluid=T_wall for solid wall

This boundary condition applies for laser machining when the dynamic motion of the molten layer is considered.

The second kind of boundary is the inlet or outlet boundary condition. The complete distribution of pressure, velocity, and temperature must be known at all times at inlet or outlet:

However the outlet distributions of the variables are usually unknown in practice, sometimes they should be the results of calculation, while the inlet flow is usually given. Then what shall we do with the outlet boundary condition when calculating the fluid field numerically? The answer is surprisingly simple: No boundary condition information is needed at the outflow boundary as long as the Peclet number is sufficiently large.

Peclet number is defined as:

where V is the flow velocity, L is the characteristic length of the flow field, is the diffusion coefficient. Peclet number is the ratio of the strengths of convection and diffusion. So if Peclet number is large enough, the influence of downstream to the calculation domain can be neglected.

The third and the most complex conditions occur at a liquid-gas interface. The liquid and the gas velocity at this interface must be equal, this is called the kinetic boundary condition:

There must be mechanical equilibrium across the interface as well. The viscous-shear stress must balance:

Also the pressure must balance the surface tension effect at the interface:

where is the surface tension coefficient, Rx and Ry are the liquid surface curvature radius along the x, y direction respectively.

And finally the energy balance at the liquid-gas interface should be considered, which has been discussed in the heat transfer part.

Figure 3.53 Boundary conditions in fluid flow analysis of laser machining process

In laser machining, a thin layer of liquid material may exist such as laser metal machining. In this case, thermocapillary convection or Marangoni Mechanism becomes important. When the surface tension variations arise due to temperature gradients along the interface, the shear stress acting on the interface induce fluid motions. This effect is especially important for laser welding processes when the dynamics of the welding pool has to be analyzed. This effect is also used to explain the surface morphology of the laser-induced bumps and fluid motion in laser drilling.

In this case the force balance boundary condition along the surface normal direction is:

where is the surface normal, is the surface tangential unit vector. This equation states: at the free surface, vaporization recoil pressure, surface tension and thermocapillary force are balanced.

We won't go into details for this part due to limited time and contents requirements. Interested reader can refer to relevant literatures.