Concentration gradient generator

Two feed concentrations, the pump flow rates, the number of stages (2 to 5) and the serpentine geometry: the “Christmas tree” network is solved exactly, branch flow rates by the hydraulic network solver (series/parallel), concentrations by the mass balance at the nodes. Out come the concentration staircase of the outlets, the share of each feed in every stream and the pressure drop of each inlet. The perfect-mixing hypothesis is judged through τ = L / L_mix. Two inverse modes fit the pump split in the least-squares sense, by analytic inversion: design reads both feeds at the ends of the target profile, fit starts from feeds already entered.

The first formula is the mass balance at a recombination node, a conservation identity: the outgoing stream carries the flow-weighted average of the incoming concentrations. The second follows from the network resolution: with collectors of negligible resistance, the n identical serpentines of a stage see the same pressure difference and share the total flow equally. R is the hydraulic resistance of one rectangular serpentine. Pe compares the transverse diffusion time (w²/D) with the advection time (w/v); L_mix is the distance travelled while diffusion crosses the width w, to be compared with the length of one serpentine.

Christmas tree

a staged microfluidic network where each row of serpentines splits then recombines neighbouring streams; after S stages, the S + 1 outlets carry a staircase of concentrations.

Stage

a row of identical serpentines between two collectors; stage s has s + 1 channels, each carrying Q_total/(s + 1). The unrolled length of a serpentine is the length available for diffusion to mix the recombined streams.

Perfect mixing at a node

streams meeting in a collector leave at the flow-weighted average concentration. It is only defensible if each serpentine is longer than the mixing length.

Mixing length (L_mix)

channel length needed for transverse diffusion to homogenise the width w: L_mix = v·w²/D, as an order of magnitude.

Design, fit

the two inverse questions. The fit starts from vials already prepared: both feeds are entered, only the pump split is adjusted, and the residual deviation from the target profile is returned outlet by outlet. Design starts from an experiment yet to be set up: both feeds are read at the ends of the target profile, and the split is fitted by the same method. One and the same profile can be out of reach for one and exact for the other. Thus “80 · 60 · 40 · 20” misses by 20 with vials at 100 and 0, and lands exactly with vials at 80 and 20.

End theorem

the first outlet is fed by pure A and the last by pure B, φ = 1 and φ = 0 exactly, whatever the number of stages, if the split ρ = Q_A/(Q_A + Q_B) stays between 1/3 and 2/3. Outside that interval the tree saturates: below ρ < 1/3 the first collector draw takes more than pump A supplies and φ(outlet 1) = 3ρ; above 2/3, the mirror, φ(last) = 3ρ − 2. The feeds read at the ends are then no longer exact, and the result flags it. The bounds 1/3 and 2/3 are where the segments of the first collector change direction, the same fact that makes the share from A piecewise affine in ρ.

Total flow ceiling

the outlet profile does not determine the total flow rate: scaling both pumps by the same factor changes no concentration (the balance is homogeneous of degree zero). Homogenisation, however, bounds it. Since τ_s = n_s·L·h·D/(Q_total·w) and the most exposed row is the one sharing the flow into the fewest parts (n = 3, the first recombination row, whatever the stage count), requiring τ ≥ 1 everywhere amounts to Q_total ≤ 3·L·h·D/w. Below this ceiling, any flow rate suits the mixing; choosing one calls for another constraint, target residence time or available sample volume.

Share from A

fraction of an outlet’s flow originating from feed A, computed by the same mass balance with a tracer (1 on side A, 0 on side B). Each outlet concentration is its exact combination: c = φ·c_A + (1 − φ)·c_B.

• All serpentines share the same geometry, hence the same hydraulic resistance; a design with unequal channels falls outside the model.
• Horizontal collectors have negligible resistance compared with the serpentines: each collector is a pressure equipotential, and the channels of a stage share the flow equally.
• Mixing is perfect at every node: outlet concentrations are flow-weighted averages. This hypothesis is assessed through τ = L / L_mix.
• The fluid is Newtonian and incompressible, in established laminar flow; both feeds carry the same fluid, only the concentration differs.
• The mixing length L_mix = v·w²/D is an order of magnitude from a dimensional derivation: the exact prefactor depends on the homogeneity criterion chosen.

The resolution is exact under its hypotheses, and only there: all serpentines identical, collectors of negligible resistance, perfect mixing at every node, established laminar flow of an incompressible Newtonian fluid. A design with unequal serpentines, resistive collectors or non-rectangular channels falls outside this model. Perfect mixing is assessed as soon as the diffusion coefficient D is entered (it depends on the diffusing species). The ratio τ = L / L_mix says how many times the available length covers the homogenisation length, and the threshold is τ ≥ 1. Below it, the staircase is still computed as the perfect-mixing limit, not as the state of the device, and the result says so. Longer serpentines or lower flow rates raise τ. It is read on the limiting stage, the first recombination stage, whose channels carry the most flow. τ is the only variable mixing depends on: two devices with equal τ mix identically, whatever their lengths, velocities, widths and diffusion coefficients. No flow regime is inferred from the Reynolds number (no verified bound for a microchannel). The resistance formula requires h < w strictly: for a square section, the pressure drop is not computed. Concentrations are entered in any unit, the same for both feeds; mixing being linear, the outlets are written in that unit.

1. Elvesys (Elveflow) · Microfluidic reviews, Flow Resistance in Microfluidics: Principles, Calculations & Applications, consulté 2026-08-02.

   Location: Section « Introduction to microfluidics and flow resistance », équation « ΔP = QR_H », ΔP la différence de pression [Pa en unités SI], Q le débit volumique [m³/s], R_H la résistance hydraulique [Pa·s/m³]. · accessed 2026-08-02

2. Molecular Medicine Reports (Spandidos Publications) · Bao X., Li Z., Liu H. et al., Stimulation of chondrocytes and chondroinduced mesenchymal stem cells by osteoinduced mesenchymal stem cells under a fluid flow stimulus on an integrated microfluidic device, vol. 17, n° 2 (2018), p. 2277-2285, version PMC5783459.

   Location: Section « Materials and methods », sous-section « Numerical modeling of the shear stress », « R=[12ηL/(1–0.63(h/w)] × (1/h3w). In this formula, R is the hydraulic resistance of the rectangular microchambers, η the dynamic viscosity of the liquid, L the channel length, h and w (always h < w) the channel height and width, respectively. ». La parenthèse « always h < w » est la convention d'orientation du rapport d'aspect ET le domaine géométrique retenus par ce moteur. Le même paragraphe referme la chaîne sur la loi de Hagen-Poiseuille : « Δp=QR_H ». · accessed 2026-08-02 · DOI 10.3892/mmr.2017.8153 · Licence CC BY-NC-ND 4.0

3. Elvesys (Elveflow) · Microfluidic reviews, Flow Resistance in Microfluidics: Principles, Calculations & Applications, consulté 2026-08-02.

   Location: Section « The concept of flow resistance through examples », identité série « R_HT = R_H1 + R_H2 + R_H3 + R_H4 + R_H5 » (éléments mis bout à bout), et exemple de circuit combiné « R_HT = 1 / (1/R_H1 + 1/R_H2) + R_H3 » (deux éléments en parallèle, puis un troisième en série avec eux). · accessed 2026-08-02

4. Wikipedia (Wikimedia Foundation), Reynolds number, révision consultée : oldid 1362194896.

   Location: Section « Flow in a pipe », formule « Re = uD_H/ν = ρuD_H/µ = ρQD_H/(µA) = WD_H/(µA) », avec (unités SI de la source) : D_H « the hydraulic diameter of the pipe (the inside diameter if the pipe is circular) (m) » ; µ (mu) « the dynamic viscosity of the fluid (Pa·s = N·s/m² = kg/(m·s)) » ; ν (nu) « the kinematic viscosity (ν = µ/ρ) (m²/s) », explicitement DISTINCTE de µ. Jamais substituable sans diviser par ρ ; ρ (rho) « the density of the fluid (kg/m³) » ; u « the mean velocity of the fluid (m/s) ». Diamètre hydraulique défini, pour une section quelconque, par « D_H = 4A/P » (A l’aire de la section, P le périmètre mouillé du canal) : c’est cette définition, pas le diamètre d’un tube circulaire, qui rend la formule applicable à un canal microfluidique non circulaire. La source précise elle-même la limite de cette substitution : « the hydraulic diameter can be substituted for the diameter of a circular duct, with reasonable accuracy, if the aspect ratio AR of the duct cross-section remains in the range 1/4 < AR < 4 ». · accessed 2026-08-02 · Licence CC BY-SA 4.0

5. Bar-Meir, Genick · LibreTexts (Civil Engineering), « Fluid Mechanics », Kinematic Viscosity, consulté 2026-08-02.

   Location: Définition « ν = µ/ρ », dimensions [m²/s] · accessed 2026-08-02 · Licence GNU Free Documentation License 1.3

6. Sensors (MDPI), Shayor Ahmed Abrar, Kabir Md Emamul, Rifath Md Sartaj Ahamed, Rashid Adib Bin, Oh Kwang W., A Synergistic Overview between Microfluidics and Numerical Research for Vascular Flow and Pathological Investigations, vol. 24, n° 18 (2024), article 5872, version PMC11435959.

   Location: Section 1 « Introduction », « To model the flow of Newtonian fluid via a single microchannel, the Hagen–Poiseuille law may be used. The flow profile throughout the channel is assumed to follow a parabolic shape ». · accessed 2026-08-02 · DOI 10.3390/s24185872 · Licence CC BY 4.0

Provisional · Sources and assumptions are provided; the documentary review is incomplete.

Physical model. The result follows the significant figures of the least precise input.

revision 2 · reviewed on 20 September 2026

Revision log

1. revision 2 · 2026-09-20 · scope change

   Two inverse questions instead of one. The fit (feeds entered, split adjusted) is unchanged; the new design mode reads both feeds at the ends of the target profile. The homogenisation ceiling of the total flow rate, Q_total ≤ 3·L·h·D/w, is given when the diffusion coefficient is entered.

   No previous result changed: the direct mode and the fit return the same numbers, shared links carrying `dir=inverse` included.

2. revision 1 · 2026-08-25 · source update

   Tool created: Christmas-tree network solved by the hydraulic network solver, mass balance at the nodes, mixing criterion by Péclet number, six sources declared.

   No previous result changed: new tool.