Axisymmetric thermoviscous and thermal expansion flows for microfluidics

TL;DR

This work extends the theory of laser-induced thermoviscous and thermal-expansion-driven flows from 2D confined geometries to a fully 3D, unconfined fluid, incorporating bulk viscosity and solving for the temperature field under a translating heat spot. By combining a numerically obtained temperature input with a perturbative, axisymmetric flow analysis in small $\alpha$ (thermal expansion) and $\beta$ (thermal shear-viscosity change), the authors derive analytic expressions for the instantaneous flow during a single scan and the net tracer transport over a full scan, up to quadratic order. A key finding is that the net transport arises at orders $O(\alpha\beta)$ and $O(\alpha^2)$, consisting of competing thermoviscous and purely thermal-expansion mechanisms, while bulk viscosity influences only the pressure field and not the velocity. The far-field transport behaves as a 3D source-dipole with strength set by $\alpha$ and $\beta$, suggesting pathways to 3D micromanipulation of particles via scan-path design and feedback control. This model broadens the parameter space for microfluidic control, enabling three-dimensional net transport without physical channels and informing future experiments in FLUCS-like cytoplasmic streaming and micromanipulation.

Abstract

Recent microfluidic experiments have explored the precise positioning of micron-sized particles in liquid environments via laser-induced thermoviscous flow. From micro-robotics to biology at the subcellular scale, this versatile technique has found a broad range of applications. Through the interplay between thermal expansion and thermal viscosity changes, the repeated scanning of the laser along a scan path results in fluid flow and hence net transport of particles, without physical channels. Building on previous work focusing on two-dimensional microfluidic settings, we present an analytical, theoretical model for the thermoviscous and thermal expansion flows and net transport induced by a translating heat spot in three-dimensional, unconfined fluid. We first numerically solve for the temperature field due to a translating heat source in the experimentally relevant limit. Then, in our flow model, the small, localised temperature increase causes local changes in the mass density, shear viscosity and bulk viscosity of the fluid. We derive analytically the instantaneous flow generated during one scan and compute the net transport of passive tracers due to a full scan, up to quadratic order in the thermal expansion and thermal shear viscosity coefficients. We further show that the flow and transport are independent of bulk viscosity. In the far field, while the leading-order instantaneous flow is typically a three-dimensional source or sink, the leading-order average velocity of tracers is instead a source dipole, whose strength depends on the relative magnitudes of the thermal expansion and thermal shear viscosity coefficients. Our quantitative results reveal the potential for future three-dimensional net transport and manipulation of particles at the microscale.

Figures (11)

• Figure 1: Two-dimensional (A, B) and three-dimensional (C) thermoviscous and thermal expansion-driven net flows. (A) Trajectories of tracers in viscous fluid confined between parallel plates, induced by repeated scanning of six scan paths, in an experiment erben2024opto (top) and according to analytical modelling of net thermoviscous flows liao2023theoreticalerben2024opto (bottom). Scale bar: $15~µm$. (B) Control of 15 microparticles (white) with 8 scan paths (magenta) to form a humanoid figure in an experiment erben2024opto. The net thermoviscous flow predicted by theory liao2023theoreticalerben2024opto is shown in blue, with target positions in green. Scale bar: $15~µm$. Panels A and B adapted from Ref. erben2024opto and licensed under https://creativecommons.org/licenses/by/4.0/. (C) Theoretical trajectories of tracers in three-dimensional, unconfined fluid induced by scanning of a spherical heat spot (scan direction indicated by arrow), as derived in this work, due to thermoviscous effects (top) and due to purely thermal expansion-driven flows (bottom).
• Figure 2: Setup for our model of heat transport induced by a scanning laser. A spherical heat source of characteristic radius $b$ translates at speed $U$ in the $z$ direction, along a scan path from $z=-\ell$ to $z=\ell$ along the $z$ axis (cylindrical radial coordinate $r$), in unbounded, viscous fluid, causing a localised temperature perturbation.
• Figure 3: Temperature profile for numerical simulations of forced heat equation for scanning Péclet number $\text{Pe}_\text{scan}=0.63$, during one scan of the heat source ($-1.375\leq t \leq 1.375$). (A) Heat map showing spatial variation of temperature perturbation $\Delta T$ with cylindrical coordinates $r$ and $z$, at selected times. The centre of the heat source is indicated in white, with the scan path in black. (B) Amplitude of temperature perturbation $A(t)$ as a function of time. (C) Shape of temperature perturbation $\Delta T/A(t)$, i.e. temperature perturbation normalised by its peak, along the $z$ axis, at selected times. The location of the peak temperature on the $z$ axis is given by $z=z_\text{peak}(t)$. Colours change from pink to yellow as time $t$ increases. The dashed curve indicates the shape of the prescribed heat source [Eq. \ref{['eq:analytical_shape']}], for comparison.
• Figure 4: Shape of temperature perturbation $\Delta T/A(t)$ along the $z$ axis, at selected times, as obtained by numerical simulation of the forced heat equation for scanning Péclet number $\text{Pe}_\text{scan}=2$, during one scan of the heat source ($-1.375\leq t \leq 1.375$), for comparison with result for $\text{Pe}_\text{scan}=0.63$ in Fig. \ref{['fig:plots_numerics_unsteady_diffusion_translating_on_off_source_v6_16012025']}C. The dashed curve shows the shape of the prescribed heat source [Eq. \ref{['eq:analytical_shape']}].
• Figure 5: Streamlines of the flow $\mathbf{u}_{1,0}^\text{(S)}$ at order $\alpha$ associated with the switching-on of the spherical heat spot, with centre at $(r=0,z=t)$ in dimensionless coordinates (nondimensionalisation described in Sec. \ref{['sec:nondiml']}). Colour shows the magnitude $\vert \mathbf{u}_{1,0}^\text{(S)} \vert$.