Microfluidic lab chip - 2D channel map

What is modelled

Two streams enter a microchannel through a Y-junction, one carrying a dye (or a protein, or an ion) and one clear. Pumps push the liquid at a set flow rate; the flow in the channel is pressure-driven, steady and laminar, so the streams run side by side and can only exchange dye by molecular diffusion across the interface. The model computes the pressure needed, the velocity profile across the width and the concentration field along the channel, then shows tracer particles carried through it.

Equations

• Hydraulic resistance of a w x h rectangular duct (exact series): R = 12 μ L / ( w h3 [ 1 − ∑odd n 192 h / (n5 π5 w) tanh(n π w / 2h) ] ) and ΔP = R Q.
• Depth-averaged velocity across the width from the exact Stokes solution, u(y) ∝ ∑odd n n−4 [ 1 − cosh(nπy/h) / cosh(nπw/2h) ], scaled so the mean is U = Q / (w h). It is nearly flat (plug-like) in a wide shallow channel and falls to zero in a thin layer at the side walls.
• Dye transport: u(y) ∂c/∂x = D ∂2c/∂y2 with no flux through the walls, marched along the channel with Crank-Nicolson steps. The scheme conserves the dye flux ∑ u c dy exactly.
• Inlet: stream A (c = 1) occupies the part of the width that carries the fraction QA/Q of the flow, so unequal flow rates move the interface off-centre.
• Numbers shown: Reynolds number Re = ρ U Dh / μ, Peclet number Pe = U w / D, diffusion length √(2 D t) and the mixing index 1 − σ(x)/σ(0) with the flux-weighted standard deviation of c.

Simplifications

• Concentration is averaged over the depth and axial diffusion is neglected (valid for Pe well above 10). Taylor dispersion and the depth structure of the interface are ignored, so very deep channels or very fast flows are less accurate.
• The two inlet branches are assumed to merge into a fully developed profile immediately; real junctions have an entrance length and some stretching of the interface.
• The chip is drawn about 25 times wider than a real 100-200 um channel so it can be seen; the physics uses the sizes you set. Constant fluid properties, Newtonian liquid, rigid walls, no electrokinetic effects.
• Diffusivities are typical room-temperature values (small dyes about 4 x 10^-10 m²/s, proteins about 7 x 10^-11); the viscosity slider scales the pressure but not D, which in reality also changes with viscosity.

Model checks

• The dye flux is constant along the whole channel to rounding error.
• With a uniform velocity the marched field matches the exact error-function solution of the two-stream diffusion problem.
• The series resistance matches an independent finite-difference solution of the duct cross-section; the common 1 − 0.63 h/w formula is quoted for comparison.
• Thousands of random-walk particles in the real flow reproduce the field's mixing at the outlet within statistical error.