Fluidic Shaping: Control & Optimization

• Fluidic Shaping is a set of techniques that uses multiphase flow, capillarity, and hydrodynamics to dynamically form and optimize fluid domains in both engineered and biological systems.
• The phase field approach leverages modified Navier–Stokes equations and adaptive finite element methods to achieve topology optimization and accurate diffuse interface resolution.
• Programmable techniques such as dielectrophoretic, thermocapillary, and magnetic interface shaping enable precise fabrication in optics, soft robotics, and space-based applications.

Fluidic shaping is a set of techniques and physical principles that enable the dynamic formation, control, and optimization of fluid domains, interfaces, or boundaries—either in living systems, engineered fluids, or the design of devices—by leveraging the intrinsic physics of multiphase flow, capillarity, hydrodynamics, and their coupling with mechanical, electrical, and chemical driving forces. The field spans applications from topology optimization of fluid domains, programmable shaping of interfaces for optics and microfluidics, biologically-driven morphogenesis, to precision design of soft structures and robotic actuators.

1. Phase Field Approaches to Fluidic Shape and Topology Optimization

A central paradigm in computational fluidic shaping is the phase field method for shape and topology optimization of fluid domains governed by the incompressible stationary Navier–Stokes equations (Garcke et al., 2014). The phase field variable, denoted φ, defines the fluid region inside a fixed holdall domain Ω: φ ≈ +1 in free fluid, φ ≈ –1 in a penalized “porous” region, and values in between within diffuse interfaces.

The shape optimization problem is thus relaxed from a sharp boundary free-boundary problem to an optimal control problem over φ and fluid velocity u, described by the coupled system:

• Modified Navier–Stokes equations with penalization:

$$\alpha(\phi) \mathbf{u} - \mu \Delta \mathbf{u} + (\mathbf{u}\cdot \nabla)\mathbf{u} + \nabla p = \mathbf{f} \quad \text{in } \Omega,$$

$$\nabla \cdot \mathbf{u} = 0 \quad \text{in } \Omega$$

with Dirichlet boundary conditions.

• The penalty term $\alpha(\phi)$ enforces Darcy-type suppression of velocity in the non-fluid region, with $\alpha(1) = 0$ and $\alpha(-1) \gg 1$.
• The objective functional, incorporating regularization, is

$$J_\varepsilon(\phi, \mathbf{u}) = \int_\Omega \left[ \frac{1}{2}\alpha(\phi)|\mathbf{u}|^2 + f(x, \mathbf{u}, \nabla\mathbf{u}) \right]\,dx + \frac{\gamma}{2}\int_\Omega |\nabla \phi|^2\,dx + \frac{\gamma}{\varepsilon}\int_\Omega \psi(\phi)\, dx,$$

where the gradient and potential terms comprise a Ginzburg–Landau regularization that penalizes interfacial perimeter and ensures well-posedness and regularity in the optimal structure.

The dynamics are driven by a mass-conserving $H^{-1}$ gradient flow—yielding a generalized Cahn–Hilliard system coupled to the Navier–Stokes equations and their adjoint. Discretization uses Taylor–Hood elements for velocity/pressure and Lagrange elements for φ, with adaptive mesh refinement guided by residual-based error estimators localizing mesh density to sharp diffuse interfaces.

This framework naturally allows topological changes in fluid domains, demonstrated numerically in cases from treelike structure formation (via phase field evolution from an initially homogeneous porous state), to optimization of classic shapes for drag minimization (e.g., the “rugby ball” paradigm), to complex channel morphologies for musical embouchure design. Performance is quantified via dissipative power, computed drag integrals, and geometric metrics such as “circularity” for the resulting shapes. The use of adaptive residual estimators ensures interface resolution and computational efficiency.

2. Mechanisms and Principles in Biological Fluidic Shaping

Fluid flow plays a direct role in morphological self-organization in living organisms, particularly in network-shaped systems such as slime mould Physarum polycephalum and fungi (Alim, 2018). Here, contractile actin cortex induces peristaltic, low Reynolds number ($Re \approx 0.002$), oscillatory flows throughout a tubular network. The local Poiseuille velocity profile and shear stress shape the architecture, dynamically driving dilation and constriction of tubes, which in turn modulate spatial and temporal flow characteristics.

The interplay between flow and network geometry is governed by classic relations:

$$Q = \frac{\pi R^4 \Delta P}{8 \eta L}, \quad \kappa_{\text{eff}} = \kappa + \frac{U^2 R^2}{48\kappa},$$

where $Q$ is volumetric flow, $\kappa_{\text{eff}}$ is the effective diffusivity via Taylor dispersion, and network adaptation follows Murray’s law for balanced shear forces.

Fluidic shaping thus emerges from feedback loops integrating peristaltic advection, network geometry, and mechanical adaptation, operating across scales from intracellular oscillations up to organismal-level migration and nutrient transport. Quantitative experimental measures reveal that peristaltic advective transport far outpaces pure diffusion on relevant timescales, establishing the functional and evolutionary optimization enabled by fluidic morphodynamics.

3. Programmable and Engineered Fluidic Interface Shaping

Programmable fluidic shaping is realized in a range of physical mechanisms, including:

• Dielectrophoretic Shaping: Surface-patterned electrodes generate Maxwell stress distributions on liquid films, enabling spatial modulation of the interface through electrical control (Gabay et al., 2021). The governing balance,

$$\gamma h''(x) - \rho g h(x) + f_{\mathrm{DEP}}(x) = \Delta P,$$

captures the competition of surface tension, hydrostatic pressure, and localized DEP force (often modeled as a Gaussian) to produce adaptive, reconfigurable topographies. These deformations are validated at micron scale using digital holographic microscopy.

• Thermocapillary Shaping: Projected light patterns (via DMD) generate spatiotemporal temperature gradients on a liquid film, creating Marangoni (thermocapillary) flows due to local surface tension gradients $\sigma = \sigma_0 - \beta(\Theta - \Theta_0)$. Inverse solutions to the thin-film evolution equation yield closed-form mappings from desired topographies to required substrate temperature fields (Eshel et al., 2021). This enables rapid, high-fidelity fabrication of diffractive optical elements with sub-nanometric surface quality.
• Magnetically Driven Interface Shaping: Droplets of nonmagnetic liquid on a ferrofluid–air interface are shaped in real time by programmable activation of surrounding electromagnets, with data-efficient Bayesian optimization determining the magnetic field configuration that minimizes the RMSE between the observed and target droplet shape, as measured by angular radii (Harischandra et al., 18 May 2025). This method realizes shapes ranging from primitive geometric forms to complex, letter-like outlines.

4. Fluidic Shaping for Optical Fabrication and Space-Based Applications

Fluidic shaping has become a disruptive technology for the fabrication of optical components, particularly lenses and freeform surfaces, eliminating the need for traditional grinding/polishing (Frumkin et al., 2020, Elgarisi et al., 2021, Luria et al., 7 Oct 2025). The fundamental process consists of injecting a curable liquid polymer into a rigid or 3D-printed border or frame, and immersing it in an immiscible liquid whose density can be matched to the polymer (neutral buoyancy). Surface tension, dominance in microgravity, and geometric boundary conditions drive the interface towards minimum energy configurations, generally characterized by constant mean curvature.

The shape is predicted analytically via free energy minimization. In rotational symmetry, the interface profile follows Bessel or spherical cap solutions, with the Bond number $Bo = |\Delta \rho| g R_0^2/\gamma$ controlling the transition from sphere to asphere:

$$F = 2\sigma \int_{0}^{R_0} \left[ \sqrt{1 + [h'(r)]^2} + \sqrt{1 + [h_b'(r)]^2} \right] r \, dr + \text{gravity} + \text{volume constraints}.$$

Notably, fluidic shaping is scale-invariant in microgravity or under neutral buoyancy. Experiments performed on the ISS (Luria et al., 7 Oct 2025) and parabolic flight (Luria et al., 2022) have demonstrated centimeter- to 172-mm-class lenses with sub-nanometric surface roughness ($R_q \approx 0.8$–$1.1$ nm), matching theoretical models and confirming preservation of optical quality across size scales.

For ophthalmic applications, the technique has been adapted for fully prescription-correctable eyewear in low-resource settings. By controlling polymer volume and elliptical border eccentricity, both the spherical and cylindrical corrections can be achieved analytically and fabricated rapidly (<10 min per lens pair), relying solely on low-cost infrastructure and low-power UV LEDs (Elgarisi et al., 14 Jun 2024).

Large-scale concepts for space telescopes rely on fluidic shaping to assemble continuous liquid primary mirrors beyond the limits imposed by launch shroud dimensions. The fluidic pathway enables diffraction-limited, post-prime-focus optical architectures; a ~1-m demonstrator (“FLUTE~1”) has been proposed to validate key principles en route to tens-of-meter scale telescopes (Biancalani et al., 2 Oct 2025). The capillary length under microgravity,

$l_c = \sqrt{\gamma/|\Delta\rho|g},$

becomes arbitrarily large, allowing unconstrained surface smoothness over vast apertures.

5. Fluidic Shaping in Soft Robotics and Actuated Metasurfaces

Fluidic shaping is foundational in soft robotics and the creation of morphing metasurfaces (Zou et al., 2023, Kahak et al., 11 Mar 2025). Soft fluidic actuators are shaped by internal fluid pressure; their geometric deformation can be directly sensed and controlled by monitoring the equilibrium pressure or flow, enabling feedback for sorting, gripper control, and tactile shape or stiffness assessment, all without embedded electronic sensors.

In fluidic Kirigami metasurfaces, laser-cut plastic sheets with engineered cut patterns are wrapped around pressure-proof chambers. Upon global pressurization, each Kirigami unit cell deforms predictably according to nonlinear mechanics determined by the local cut geometry. Analytical and piecewise linear models (e.g., nonlinear Euler–Bernoulli or Castigliano-based for curved beams) allow precise map from design parameters (cell size, ligament width/gap, number of rings) to out-of-plane displacement and force characteristics. As a result, a single device can serve as either an acoustic holography lens—with sub-wavelength control of out-of-plane thickness map for ultrasonic wave steering—or a precision haptic interface, programmable for specified force–displacement profiles (Kahak et al., 11 Mar 2025).

6. Biological and Bioinspired Fluidic Shaping: Differential Swelling and Network-Controlled Morphogenesis

Beyond direct engineering, fluidic shaping encompasses mechanisms by which controlled transport and storage of fluid within an embedded network produce differential swelling, buckling, and irreversible morphogenesis, as in flower blooming (Luo et al., 2022). Network models couple a resistor–capacitor hydraulic network (defining the time-dependent spatial distribution of stored fluid content) to a mechanical spring network (with bond rest lengths modulated by local hydration). The resulting shape, including out-of-plane buckling and establishment of Gaussian curvature, is determined by the interplay between fluid redistribution and mechanical constraints.

Simulation and regression analyses reveal that in sparse, hierarchical flow networks, regions of maximal stored fluid content reliably correlate with local positive Gaussian curvature, providing a quantitative blueprint for designing shape-morphing membranes and deployable structures.

7. Methodological, Computational, and Application Considerations

The efficacy and practicality of fluidic shaping approaches are determined by their mathematical underpinning, numerical realization, and experimental validation:

• Computational Strategies: Adaptive finite element discretization, error estimation via residual-based indicators, and differentiable simulation frameworks (with cut-cell handling, signed distance function parameterization, and co-design of geometry and control policies) underpin robust optimization and learning in complex geometries (Garcke et al., 2014, Li et al., 22 May 2024).
• Physical and Scaling Regimes: Surface tension dominates interface formation in microgravity or neutral-buoyancy regimes; the Bond number and capillary length control the transition from capillarity- to gravity-governed surface profiles.
• Physical Robustness and Limitations: Thermo-chemical effects, particularly in polymer curing under microgravity, necessitate advanced thermal management and tailored protocols; precise liquid handling and boundary pinning are critical for defect suppression.
• Application Domains: Fluidic shaping enables rapid prototyping of optics (lenses, saddles, doublets, freeform phase masks), adaptive soft actuators, deployable morphing structures, lab-on-a-chip reconfigurability, as well as novel strategies in bioinspired robotics and morphogenesis.
• Future Prospects: Scalable space-based manufacturing of ultra-large, high-quality optical systems, real-time adaptive laboratory and clinical devices, and new classes of programmable, self-sensing morphing surfaces are direct beneficiaries of advances in fluidic shaping.

Fluidic shaping thus represents a unifying principle and set of strategies for the precision control and optimization of fluid–structure systems, operating from molecular to architectural scales, with applications spanning scientific, medical, bioengineering, and aerospace domains.