Theoretical model of confined thermoviscous flows for artificial cytoplasmic streaming

Recent experiments in cell biology have probed the impact of artificially induced intracellular flows in the spatiotemporal organization of cells and organisms. In these experiments, mild dynamical heating (a few kelvins) via focused infrared light from a laser leads to long-range, thermoviscous flows of the cytoplasm inside a cell. To extend future use of this method in cell biology, popularized as focused-light-induced cytoplasmic streaming (FLUCS), new quantitative models are needed to link the external light forcing to the produced flows and transport. Here, we present a fully analytical, theoretical model describing the fluid flow induced by the dynamical laser stimulus at all length scales (both near the scan path of the laser beam and in the far field) in two-dimensional confinement. We model the effect of the focused light as a small, local temperature change in the fluid, which causes a small change in both the density and the viscosity of the fluid locally. In turn, this results in a locally compressible fluid flow. We analytically solve for the instantaneous flow field induced by the translation of a heat spot of arbitrary time-dependent amplitude along a scan path of arbitrary length. We show that the leading-order instantaneous flow field results from the thermal expansion of the fluid and is independent of the thermal viscosity coefficient. This leading-order velocity field is proportional to the thermal expansion coefficient and the magnitude of the temperature perturbation, with far-field behavior typically dominated by a source or sink flow and proportional to the rate of change of the heat-spot amplitude. In contrast, and in agreement with experimental measurements, the net displacement of a material point due to a full scan of the heat spot is quadratic in the heat-spot amplitude, as it results from the interplay of thermal expansion and thermal viscosity changes. The corresponding average velocity of material points over a scan is a hydrodynamic source dipole in the far field, with direction dependent on the relative importance of thermal expansion and thermal viscosity changes. Our quantitative findings show excellent agreement with recent experimental results and will enable the design of new controlled experiments to establish the physiological role of physical transport processes inside cells.

Figure 1

(a) Sketch of setup; a laser focused on a fluid confined between two parallel plates creates a heat spot (red), which translates along a scan path and induces fluid flow. (b) Experimentally found trajectories of tracer beads in the fluid, due to repeated scanning of the heat spot, viewed from above, adapted with permission from Fig. 2(d) of Ref. [34] (The Optical Society). (c) Theoretical time-averaged trajectories of tracers in the fluid as derived in this study (colours indicate the average speed of tracers), with parameters corresponding to experiments in panel (b).

Figure 2

Setup of our theoretical model. A heat spot of characteristic radius $a$ translates at speed $U$ in the $x$ direction, along a scan path along $y=0$ from $x=-\ell$ to $x=+\ell$, causing inertialess fluid flow between the two no-slip boundaries at $z=0$ and $z=h$.

Figure 3

View from above of translating heat spot in Fig. 2, with schematic instantaneous flow streamlines.

Figure 4

Cartoon of the physical mechanism for the instantaneous flow field contribution ${\mathbf{u}}_{1,0}^{\text{(T)}}$ associated with translation of the heat spot (i.e., temperature perturbation) at order $\alpha$, based on the case of a constant-amplitude heat spot in Ref. [30]. The heat-spot shape (shown in red), which is specified by the function $\mathrm{\Theta }(x-t,y)$ in our mathematical model, translates in the positive $x$ direction. At the front of the heat spot, the heat spot is arriving, so the temperature is locally increasing, $\partial \mathrm{\Theta }/\partial t>0$, producing a hydrodynamic source. At the back, the heat spot is leaving, so instead $\partial \mathrm{\Theta }/\partial t<0$, resulting in a sink.

Figure 5

Polar coordinates with origin coinciding with the center of the heat spot at time $t$.

Figure 6

Streamlines for the flow field ${\mathbf{u}}_{1,0}^{\text{(S)}}(x-t,y)$ [as in Eq. (35) for the instantaneous flow at order $\alpha$] associated with the time-variation of the heat-spot amplitude. (a) Streamlines for $-5\le x-t,y\le 5$, close to the heat spot (near field), with magnitude of the velocity field $|{\mathbf{u}}_{1,0}^{\text{(S)}}|$ indicated by color. (b) Streamlines for $-50\le x-t,y\le 50$ to illustrate far-field behavior.

Figure 7

Magnitude of the flow field at order $\alpha$ associated with the time-variation of the heat-spot amplitude, $|{\mathbf{u}}_{1,0}^{\text{(S)}}|$ [Eq. (49)], against the radial distance $r$ to the center of the heat spot, plotted on a log–log scale. The flow is a regularized source: it is zero at the center of the heat spot, while its far-field behavior is given by $|{\mathbf{u}}_{1,0}^{\text{(S)}}|\sim 1/r$.

Figure 8

Streamlines for the velocity field ${\mathbf{u}}_{1,0}^{\text{(T)}}(x-t,y)$ [as in Eq. (35) for the instantaneous flow at order $\alpha$] associated with translation of the heat spot in the positive $x$ direction (scan direction). (a) Streamlines for $-5\le x-t,y\le 5$, close to the heat spot (near field), with magnitude of the velocity field $|{\mathbf{u}}_{1,0}^{\text{(T)}}|$ indicated by color. (b) Streamlines for $-50\le x-t,y\le 50$ to illustrate far-field behavior.

Figure 9

Magnitude of the flow field at order $\alpha$ associated with translation of the heat spot, $|{\mathbf{u}}_{1,0}^{\text{(T)}}|$ [Eq. (51)], against the radial distance $r$ to the center of the heat spot, plotted on a log–log scale, for $\theta =0$, $\pi /8$, $\pi /4$, $3\pi /8$, and $\pi /2$. The far-field behavior is given by $|{\mathbf{u}}_{1,0}^{\text{(T)}}|\sim 1/r^2$.

Figure 10

Snapshots of the streamlines of the instantaneous fluid velocity field ${\mathbf{u}}_{1,0}$ at order $\alpha$, i.e., leading order, over the course of half a scan period, $-t_0\le t\le 0$, with $t_0=1.375$ and sinusoidal amplitude function given by Eq. (55). The heat spot translates in the positive $x$ direction. The center of the heat spot is indicated with a red dot, while the scan path is shown as a thick black line segment.

Figure 11

Same as Fig. 10 but with a trapezoidal heat-spot amplitude function instead of sinusoidal, illustrating the case where the switching-on and switching-off of the heat spot are rapid and the amplitude is constant for most of the scan period. Snapshots of the streamlines of the instantaneous fluid velocity field ${\mathbf{u}}_{1,0}$ at order $\alpha$, i.e., leading order, over the course of half a scan period, $-t_0\le t\le 0$, with $t_0=1.375$ and trapezoidal amplitude function given by Eq. (56) with switching-on time $t_{\text{s}}=0.4$. The heat spot translates in the positive $x$ direction. The center of the heat spot is indicated with a red dot, while the scan path is shown as a thick black line segment.

Figure 12

Streamlines for the flow ${\mathbf{u}}_{1,1}$ at order $\alpha \beta$, which includes the first effect of viscosity variation with temperature. The heat spot translates in the positive $x$ direction (scan direction). (a) Streamlines for $-5\le x-t,y\le 5$, close to the heat spot (near field). The magnitude of the velocity field divided by the square of the amplitude, $|{\mathbf{u}}_{1,1}{|/A\left(t\right)}^2=\left|{\mathbf{u}}_{1,1}^{\text{(T)}}\right|$, is indicated by color. (b) Streamlines for $-50\le x-t,y\le 50$ to illustrate far-field behavior.

Figure 13

Scaled magnitude of the order-$\alpha \beta$ velocity field, $|{\mathbf{u}}_{1,1}{|/A\left(t\right)}^2=\left|{\mathbf{u}}_{1,1}^{\text{(T)}}\right|$ [Eq. (83)], against the radial distance $r$ to the center of the heat spot, plotted on a log–log scale, for $\theta =0$, $\pi /8$, $\pi /4$, $3\pi /8$, and $\pi /2$. The far-field behavior is given by $|{\mathbf{u}}_{1,1}{|/A\left(t\right)}^2\sim 1/4r^2$.

Figure 14

Streamlines for the two terms that make up the velocity field ${\mathbf{u}}_{1,1}^{\text{(T)}}=-\mathbf{\nabla }{p}_{1,1}^{\text{(T)}}+\mathrm{\Theta }{\mathbf{u}}_{1,0}^{\text{(T)}}$ at order $\alpha \beta$, associated with the translation of the heat spot in the positive $x$ direction (scan direction). The magnitude of each velocity field is indicated by color. (a) Streamlines of the potential flow $-\mathbf{\nabla }{p}_{1,1}^{\text{(T)}}(x-t,y)$ for $-5\le x-t,y\le 5$. The potential flow decays algebraically in the far field, whereas the flow $\mathrm{\Theta }{\mathbf{u}}_{1,0}^{\text{(T)}}$ decays exponentially, so the potential flow dominates the far field of the flow ${\mathbf{u}}_{1,1}^{\text{(T)}}$ at order $\alpha \beta$. (b) Streamlines of the flow at leading order associated with translation of the heat spot, modulated by the heat-spot shape, $\mathrm{\Theta }(x-t,y){\mathbf{u}}_{1,0}^{\text{(T)}}(x-t,y)$, for $-5\le x-t,y\le 5$. The envelope $\mathrm{\Theta }$ suppresses the magnitude away from the heat spot exponentially. The contribution of the flow $\mathrm{\Theta }{\mathbf{u}}_{1,0}^{\text{(T)}}$ to the velocity field ${\mathbf{u}}_{1,1}^{\text{(T)}}$ at order $\alpha \beta$ is thus mostly confined to the near field. In the near field, the flow $\mathrm{\Theta }{\mathbf{u}}_{1,0}^{\text{(T)}}$ has larger magnitude than the potential flow $-\mathbf{\nabla }{p}_{1,1}^{\text{(T)}}$, producing the leftward flow seen in the near field of the flow ${\mathbf{u}}_{1,1}^{\text{(T)}}$.

Figure 15

Schematic explanation of the physical mechanism for the flow ${\mathbf{u}}_{1,1}^{\text{(T)}}$ at order $\alpha \beta$, the leading-order effect of viscosity variation, on the $x$ axis. We begin with the flow ${\mathbf{u}}_{1,0}^{\text{(T)}}$ associated with the translation of the heat spot at leading order (i.e., order $\alpha$), explained in the cartoon in Fig. 4. Then, we consider the first effect of thermal viscosity changes, at order $\alpha \beta$. Viscosity decreases locally due to heating. Intuitively, this amplifies the leading-order flow in the heat spot (compared with if the viscosity were instead its reference value), producing the compressible contribution $\mathrm{\Theta }{\mathbf{u}}_{1,0}^{\text{(T)}}$ to the order-$\alpha \beta$ flow. However, the order-$\alpha \beta$ flow is incompressible, which is enforced by the pressure $p_{1,1}^{\text{(T)}}$ associated with heat-spot translation at order $\alpha \beta$. This adds a potential flow $-\mathbf{\nabla }{p}_{1,1}^{\text{(T)}}$; mathematically, this arises from the parallel-plates geometry. The flow ${\mathbf{u}}_{1,1}^{\text{(T)}}$ at order $\alpha \beta$ is the combination of these two effects, amplification and incompressibility.

Figure 16

Divergence of potential flow fields $-\mathbf{\nabla }{p}_{1,0}^{\text{(T)}}$ at order $\alpha$ (left) and $-\mathbf{\nabla }{p}_{1,1}^{\text{(T)}}$ at order $\alpha \beta$ (right) associated with translation of the heat spot in the positive $x$ direction (scan direction). Red indicates a source (positive divergence); blue indicates a sink (negative divergence); green indicates that the magnitude of the divergence is small. (a) Filled contour plot for $\mathbf{\nabla }·(-\mathbf{\nabla }{p}_{1,0}^{\text{(T)}})$. (b) Filled contour plot for $\mathbf{\nabla }·(-\mathbf{\nabla }{p}_{1,1}^{\text{(T)}})$.

Figure 17

Streamlines for the velocity field ${\mathbf{u}}_{2,0}^{\text{(S)}}(x-t,y)$ [as in Eq. (98) for the instantaneous flow at order ${\alpha }^2$] associated with the time-variation of the heat-spot amplitude, for $-5\le x-t,y\le 5$, with magnitude of the velocity field $|{\mathbf{u}}_{2,0}^{\text{(S)}}|$ indicated by color. The flow decays exponentially in the far field.

Figure 18

Streamlines for the flow field ${\mathbf{u}}_{2,0}^{\text{(T)}}(x-t,y)$ [as in Eq. (92) for the instantaneous flow at order ${\alpha }^2$] associated with translation of the heat spot in the positive $x$ direction (scan direction). (a) Streamlines for $-5\le x-t,y\le 5$, close to the heat spot (near field), with magnitude of the velocity field $|{\mathbf{u}}_{2,0}^{\text{(T)}}|$ indicated by color. (b) Streamlines for $-50\le x-t,y\le 50$ to illustrate the far-field behavior.

Figure 19

Magnitude of the flow field at order ${\alpha }^2$ associated with translation of the heat spot, $|{\mathbf{u}}_{2,0}^{\text{(T)}}|$, against the radial distance $r$ to the center of the heat spot, plotted on a log–log scale, for $\theta =0$, $\pi /8$, $\pi /4$, $3\pi /8$, and $\pi /2$. The far-field behavior is given by $|{\mathbf{u}}_{2,0}^{\text{(T)}}|\sim 1/4r^2$.

Figure 20

Sketch of a material point initially at position ${\mathbf{X}}_0$ when the heat spot is at the start of the scan path $x=-\ell$ at time $t=-t_0$.

Figure 21

Trajectories of material points during one scan at leading order, i.e., order $\alpha$. The material points start at position $(0,Y_0)$, on the $y$ axis, at time $t=-t_0$ when the heat spot begins the scan from $x=-\ell$, for selected values $Y_0=0$, 0.5, 1, 2, 3, and 5. (a) Plot of the $x$ component $\mathrm{\Delta }{X}_{1,0}$ of the displacement of the material point at order $\alpha$ as a function of time $t$, during one scan, i.e., $-t_0\le t\le t_0$. (b) Same as panel (a) but for the $y$ component $\mathrm{\Delta }{Y}_{1,0}$ of the displacement of the material point at order $\alpha$. (c) Plot of the same trajectories in the $(\mathrm{\Delta }{X}_{1,0},\mathrm{\Delta }{Y}_{1,0})$ plane. The length of the scan path is chosen to be $2\ell \equiv 2t_0=2.75$, to match microfluidic experiments [34].

Figure 22

Cartoon summary of zero net displacement at order $\alpha$ of a material point, depicted as a black circle. A heat spot translating along a scan path from $x=-\ell$ to $x=\ell$ induces a velocity field experienced by the material point initially at position vector ${\mathbf{X}}_0$. (Here, the material point starts at the midpoint of the scan path, ${\mathbf{X}}_0=\mathbf{0}$.) This results in a small displacement of the material point at time $t$. The rightward displacement of the material point due to the switching-on and switching-off of the heat spot at the ends of the scan path is canceled out by the leftward displacement due to the translation of the heat spot, correct to order $\alpha$.

Figure 23

Streamlines for the average velocity of a material point at order $\alpha \beta$ over a scan period, $\mathrm{\Delta }{\mathbf{X}}_{1,1}({\mathbf{X}}_0;t_0)/2t_0$. The heat spot translates in the positive $x$ direction (scan direction). The scan path is indicated with a thick black line segment. (a) Streamlines for $-5\le X_0,Y_0\le 5$, close to the scan path (near field), with magnitude of the average velocity of material points $|\mathrm{\Delta }{\mathbf{X}}_{1,1}({\mathbf{X}}_0;t_0)/2t_0|$ indicated by color. (b) Streamlines for $-50\le X_0,Y_0\le 50$, which illustrate the far-field behavior.

Figure 24

Magnitude of the average Lagrangian velocity $|\mathrm{\Delta }{\mathbf{X}}_{1,1}({\mathbf{X}}_0;t_0)/2t_0|$ at order $\alpha \beta$ of material points against the distance $|{\mathbf{X}}_0|$, plotted on a log–log scale, for $\varphi =0$, $\pi /8$, $\pi /4$, $3\pi /8$, and $\pi /2$. Here, the initial position of a material point is written as ${\mathbf{X}}_0=\left|{\mathbf{X}}_0\right|(cos\varphi ,sin\varphi )$, i.e., in polar coordinates with origin at the midpoint of the scan path. In the far field, the magnitude $|\mathrm{\Delta }{\mathbf{X}}_{1,1}({\mathbf{X}}_0;t_0)/2t_0|$ follows an inverse square law $\sim 1/|{\mathbf{X}}_0{|}^2$.

Figure 25

Streamlines for the average velocity of a material point at order ${\alpha }^2$ over a scan period, $\mathrm{\Delta }{\mathbf{X}}_{2,0}({\mathbf{X}}_0;t_0)/2t_0$. The heat spot translates in the positive $x$ direction (scan direction). The scan path is indicated with a thick black line segment. (a) Streamlines for $-5\le X_0,Y_0\le 5$, close to the scan path, with magnitude of the average velocity of material points $|\mathrm{\Delta }{\mathbf{X}}_{2,0}({\mathbf{X}}_0;t_0)/2t_0|$ indicated by color. (b) Streamlines for $-50\le X_0,Y_0\le 50$ to indicate far-field behavior.

Figure 26

Magnitude of the average velocity $|\mathrm{\Delta }{\mathbf{X}}_{2,0}({\mathbf{X}}_0;t_0)/2t_0|$ at order ${\alpha }^2$ of material points against the distance $|{\mathbf{X}}_0|$, plotted on a log–log scale, for $\varphi =0$, $\pi /8$, $\pi /4$, $3\pi /8$, and $\pi /2$. As at order $\alpha \beta$, the initial position of a material point is written as ${\mathbf{X}}_0=\left|{\mathbf{X}}_0\right|(cos\varphi ,sin\varphi )$, i.e., in polar coordinates with origin at the midpoint of the scan path. In the far field, the magnitude $|\mathrm{\Delta }{\mathbf{X}}_{2,0}({\mathbf{X}}_0;t_0)/2t_0|$ follows an inverse square law $\sim 1/|{\mathbf{X}}_0{|}^2$.

Figure 27

Comparison between experimental and theoretical results for trajectories of tracer beads over many scans of the heat spot. (a) Experimentally found trajectories of tracer beads in the midplane (halfway between the parallel plates), from data gathered over a time period of $100\mathrm{s}$, adapted with permission from Fig. 2(d) of Ref. [34] (The Optical Society). (b) Theoretical leading-order trajectories of material points in the midplane ($z=h/2$) over many scans, with fully dimensional units; the average speed of material points is indicated with color. During each scan, the heat spot translates in the positive $x$ direction in our theory (scan direction). In both panels, the scan path is indicated with a thick black line segment.

Figure 28

Comparison between experimental data and theory for average speed of tracer beads. (a) Experimental data [34]. Log–log scatter plot of speed of tracer beads in the midplane averaged over $0.5\mathrm{s}$ vs distance of tracer bead from the origin (the midpoint of the scan path). Data from Fig. 2(f) in Ref. [34] replotted. (b) Theoretical leading-order prediction for panel (a), now dimensional. To generate the data points, the tracers are placed at initial positions lying on a square grid with spacing of $0.3\mu \mathrm{m}$. In panels (a) and (b), the black vertical line marks the distance from the origin corresponding to the end of the scan path, i.e., $\ell =5.5\mu \mathrm{m}$. (c) Theoretical prediction. Log–log plot of leading-order speed of a material point averaged over one scan vs distance of the material point from the origin, along radial lines at fixed angles $\varphi =0$, $\pi /8$, $\pi /4$, $3\pi /8$, and $\pi /2$ to the $x$ axis. Parameters are the same as for panel (b).