Table 6 Common boundary conditions for the heat and mass transfer in macroscale models.
From: Multiscale computational understanding and growth of 2D materials: a review
Navier-Stokes | |
 No slip on walls | u = 0 |
 Normal inflow velocitya | u = −U0·n |
 Laminar inflowb | \({\mathrm{L}}_{{\mathrm{ent}}}\nabla _{\mathrm{t}} \cdot \left[ { - {\mathrm{p}}{\mathbf{I}} + {\upmu}\left( {\nabla _{\mathrm{t}}{\mathbf{u}} + \left( {\nabla _{\mathrm{t}}{\mathbf{u}}} \right)^{\mathrm{T}}} \right)} \right] = - {\mathrm{p}}_{{\mathrm{ent}}}{\mathbf{n}}\) |
 Mass Flowc | \(\displaystyle-\,{\int}_{\partial {\mathrm{\Omega }}} {\frac{{\uprho }}{{{\uprho }}_{{\mathrm{st}}}}} - \left( {{\mathbf{u}}\, \cdot \,{\mathbf{n}}} \right){\mathrm{d}}_{{\mathrm{bc}}}{\mathrm{dS}} = {\mathrm{Q}}_{{\mathrm{sccm}}}\) |
 Zero outlet pressure | \(- {\mathrm{p}}{\mathbf{I}} + {\upmu}( {\nabla {\mathbf{u}} + ( {\nabla {\mathbf{u}}} )^{\mathrm{T}}} ) - \frac{2}{3}{\upmu}\,( {\nabla \cdot {\mathbf{u}}} ){\mathbf{I}} = 0\) |
Heat transfer | |
 Inlet | −n · q = 0 |
 Thermal insolation | −n · q = 0 |
 Inflow heat fluxd | \(- {\mathbf{n}} \cdot {\mathbf{q}} = - {\mathrm{q}}_0\frac{{{\mathrm{A}}\left( {{\mathbf{u}} \cdot {\mathbf{n}}} \right)}}{{{\int}_{\mathrm{S}} {\left| {{\mathbf{u}} \cdot {\mathbf{n}}} \right|} {\mathrm{ds}}}} + {\uprho}\left( {{\mathrm{h}}_{{\mathrm{in}}} - {\mathrm{h}}_{{\mathrm{ext}}}} \right){\mathbf{u}} \cdot {\mathbf{n}}\) |
 Wall temperature | T = T(r) |
Flow-assisted diffusion | |
 No flux at furnace walls | −n · Ni = 0 |
 Specified concentration | c = c0(r) |
 Outflow | \(- {\mathbf{n}} \cdot {\mathrm{D}}_{\mathrm{i}}\nabla {\mathrm{c}}_{\mathrm{i}}\) |
 Inflow | c = cin |
