The demands for increasingly smaller, more capable, and higher power density technologies have heightened the need for new methods to manage and characterize extreme heat fluxes. This work presents the use of an anisotropic version of the time-domain thermoreflectance (TDTR) technique to characterize the local heat transfer coefficient (HTC) of a water-cooled rectangular microchannel in a combined hot-spot heating and subcooled channel-flow configuration. Studies focused on room temperature, single-phase, degassed water flowing at an average velocity of ≈3.5 m/s in a ≈480 μm hydraulic diameter microchannel (e.g., Re ≈ 1850), where the TDTR pump heating laser induces a local heat flux of ≈900 W/cm2 in the center of the microchannel with a hot-spot area of ≈250 μm2. By using a differential TDTR measurement approach, we show that thermal effusivity distribution of the water coolant over the hot-spot is correlated to the single-phase convective heat transfer coefficient, where both the stagnant fluid (i.e., conduction and natural convection) and flowing fluid (i.e., forced convection) contributions are decoupled from each other. Our measurements of the local enhancement in the HTC over the hot-spot are in good agreement with established Nusselt number correlations. For example, our flow cooling results using a Ti metal wall support a maximum HTC enhancement via forced convection of ≈1060 ± 190 kW/m2 K, where the Nusselt number correlations predict ≈900 ± 150 kW/m2 K.

## Introduction

Temperature control is a critical regulatory process in wide variety of systems. Without it, sustainable operation is not possible in arguably everything from the functionality of biological organisms [1] to the reliability of electronic [2,3], photonic [4], and electro-chemical devices [5] to high-speed transportation [6] and materials manufacturing [7]. For today's technologies, there seems to be a ubiquitous trend toward increasingly smaller, more capable, and higher energy or power density devices. Subsequently, without concurrent advances in energy efficiencies, these smaller and/or more powerful devices require improved thermal management systems to maintain their temperatures within operational limits at higher heat flux conditions. Figure 1 outlines this growing challenge faced by the microelectronics industry for next-gen devices [3], where, for example, the heat fluxes within the next decade are expected to surpass 3000 MW/m2—which is nearly 50 times greater than the heat flux radiated by the Sun [8]. This work revisits hot-spot cooling in microchannels, focusing on the validation of our optical pump-probe method to characterize large, gradient-driven heat and mass transport.

For high heat flux thermal management, microscale cooling with liquids has become a promising alternative to traditional air cooling due to the liquids' larger heat capacity, thermal conductivity, and intrinsic ability to dissipate large amounts of thermal energy (heat)—or regulate fluctuations in surface temperature—via liquid–vapor (latent heat) phase transformations. In result, there has been significant interest by academia and industry on convective and phase-change heat transfer at the micro- and nanoscale, where hundreds of papers have been published on related liquid cooling processes including (but not limited to) single-phase flow [9], multiphase flow [10,11], flow boiling [12], pool boiling [13,14], spray cooling [15,16], heat pipes [17,18], thermosyphons [19], microdroplet evaporation [20], single-phase jet impingement cooling [21,22], and microjet impingement boiling [23,24].

The Holy Grail for all these liquid cooling techniques is an accurate, predictive understanding of the heat transfer coefficient ($h$ or $HTC$). In general, the cooling efficiency of any heat removal process is encapsulated by the $HTC$, which is a proportionality constant that couples the heat flux ($q$) to the temperature difference ($ΔT$) that drives the heat flow. The magnitude of the $HTC$ is dictated by several factors, including the velocity distribution of the flow field, the thermofluid properties of the coolant, and surface characteristics of the device (e.g., geometry, microstructure, temperature, and chemistry).

Table 1 summarizes the range in $h$ for a variety of different cooling methods. As shown, techniques based on phase-change heat transfer (e.g., boiling and evaporation) have, most commonly, improved $HTC$s relative to their single-phase (e.g., nonboiling) counterparts; however, these multiphase cooling methods also suffer from the reality that the added materials phase coincides with a much higher propensity to induce a critical or unstable “cooling regime.” In which case, the cooling performance of a multiphase system operating in a so-called unstable cooling regime typically coincides with an order of magnitude reduction in the $HTC$. A well-known example is the onset of the critical heat flux (CHF) during nucleate pool boiling, where at CHF (and at wall superheats beyond the CHF) the $HTC$ can decrease by several orders of magnitude [43]. Another well-known example is the onset of wall dryout during thin-film evaporation and nucleate flow boiling [4446].

The optimal cooling method is also dictated by several other factors such as system size, cost of operation, and desired control scheme (i.e., active or passive). For instance, spray cooling with water is currently the most effective process for dissipating large thermal loads (i.e., heat fluxes ∼10 MW/m2) from the surfaces of moderately sized systems (e.g., surface areas < 0.5 m2) [47,48], whereas jet impingement boiling is the optimal method for dissipating ultrahigh heat fluxes (e.g., heat fluxes in the range of 0.5–20 MW/m2) from sub-mm2 sized hot-spots [23,35].

To date, the largest $HTC$ s are observed with techniques based on jet impingement boiling. Interestingly, for subcooled jet impingement boiling, the $HTC$ at the edge of the stagnation zone is found to decrease with increasing wall temperature until the onset of nucleate boiling [24], supporting that the local maximum in the HTC is at the edge of the stagnation zone and coincides with the cooling region where no phase change and only sensible heat transfer takes place [49]. Within the stagnation zone, the thickness of the thermal boundary layer (BL) is at a minimum and the acceleration of flow field is at a maximum. Recently, Mitsutake and Monde [50] have shown that heat fluxes within 48% of the theoretical maximum can be obtained with jet impingement cooling. For reference, typically two-phase cooling methods achieve CHF values that are less than 10% of this theoretical limit (i.e., , where $qmaxktg$ is the maximum evaporative heat flux predicted by the kinetic theory of gases) [51]. Another interesting finding for spray or jet impingement boiling is that the addition of noncondensable gases (NCGs) to coolant can increase the overall $HTC$ [31,52]. This is a rather counter-intuitive result because the addition of NCGs should effectively decrease the heat capacity and thermal conductivity of the coolant and thereby reduce the sensible heat contributions to the HTC.

The importance of the sensible heat contributions and NCGs to the HTC in two-phase cooling is not new. However, most studies correlate the boiling and evaporation performance to only the latent heat contributions, and mixed results are reported for NCGs [53,54]. In support, are the past spray cooling studies by Kim and coworkers [52,55] and the very recent pool boiling studies by Jaikumar and Kandlikar [56,57]. For the latter, the studies by Jaikumar and Kandlikar showed that the record HTC values of $h≈$ 800 kW/m2 K were observed with specific micropillar surfaces that presumably optimized the sensible cooling by minimizing nucleation and maximizing liquid convection at the base of the micropillars. We hypothesize that this sensible cooling effect at the base of the micropillars is directly correlated with the increased HTC observed within the stagnation zone for jet impingement boiling. In both cases, for example, the fluid flow field presumably induces a suppression of the thickness of the thermal BL, ultimately increasing the HTC for a prescribed heat flux.

These results (among others) warrant the need to better decipher the relative significance between the different cooling mechanisms that dictate phase-change heat and mass transfer phenomena, especially at the micro- and nanoscale and at time-scales fast enough to render transient changes in the hydrodynamic and thermal boundary layers [5861]. In microdomains, multiphase flow boiling and heat transfer is attributed to four key mechanisms: microlayer evaporation, interline evaporation, transient conduction, and microconvection [58]. For reference, the sensible heat contributions discussed previously are effectively regulated by the rate at which the coolant can be heated (i.e., the rates of microscale conduction and convection within the thermal BL). To accomplish this level of thermophysical characterization, new synchronized thermofluid diagnostics are needed that can combine high-fidelity temperature and flow-field measurements at spatial and temporal resolutions of <5 μm and <200 μs, respectively [60].

This work introduces our optical pump-probe approach to characterize the local (<5 μm) HTC in the thermal BL of flowing fluids using the optical diagnostic called time domain thermoreflectance (TDTR). TDTR is a well-established optical technique used by the thermal science community to characterize micro- and nanoscale heat transport (e.g., most frequently the thermal conductivity and interfacial thermal conductance). The TDTR technique uses two concentrically focused pump and probe laser beams to heat (with the pump) and then measure (with the probe) the temporal changes in heat transport in a sample [6264]. We employ the recently developed anisotropic version of TDTR, where nonconcentric beams are used to heat (pump) and measure (probe) the anisotropic thermal transport properties by spatially offsetting pump and probe beams in small increments [65,66]. For proof of principle, we studied single-phase water in rectangular microchannels [67]. This work also builds on past TDTR studies of droplet impingement and evaporation [58,68,69] and facilitates later thermophysical studies of multiphase heat and mass transport. The reminder of this manuscript describes the TDTR technique, fruitfulness of measuring the TDTR voltage ratio ($Vin/Vout$), and our corresponding methodology for HTC analysis to decipher the thermofluid transport inside and outside the thermal BL.

## Methodology

### Experimental Setup.

We investigated hot-spot cooling in individual microchannels with length, width, and height dimensions of $L≈$ 15 mm, $W≈$ 600 μm, and $H≈$ 400 μm, respectively. These channel dimensions correspond to a hydraulic diameter of $Dh≈$ 480 μm. Degassed distilled water was used as the working fluid/coolant.

Figure 2 provides a schematic overview of the experimental setup and TDTR measurement approach. Figure 2(a) shows our flow-loop methodology based on the use of a custom syringe pump design that incorporates fluid pumping via two identical syringes (36 mm, inner diameter) with bonded plunger ends. All reported experiments are for fluid flow in the indicated flow direction; however, the flow direction can be easily reversed and reversed flow has no noticeable effect for local measurements in the center of the microchannel (see Fig. S1, which is available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection). This is expected since the experiment is done at atmospheric pressure and flow direction is horizontal so there is no gravity effect on flow direction and the data. The current setup facilitates volumetric flow rates ranging from 0.2 mL/min to 55 mL/min, which corresponds to ranges in average flow velocity ($vavg$), mass flux ($G$), and Reynolds number ($ReD$) with our microchannel setup of 0.01 $≲vavg≲$ 3.8 m/s, 13.9 $≲G≲$ 3808 kg/m2/s, and 7 $≲ReD≲$ 2031, respectively. All reported experiments are based on a pumping rate of 50 mL/min, corresponding to $vavg≈$3.47 m/s, $G≈$3462 kg/m2/s, and $ReD≈$1850 using atmospheric pressure and room temperature water-inlet properties for the fluid. The camera setup facilitates flow visualization and alignment of the pump-probe lasers in the microchannel. For precise alignment, the microchannel sample stage is mounted on six-axis stage, providing three (3)-translational and three (3)-rotational axes (or degrees-of-freedom) for translation and alignment.

Figure 2(c) provides an expanded view of the construction and design of the microchannel sample stage. The TDTR optical bench setup shown in Fig. 2(b) is discussed in more detail in Sec. 2.3. As shown in Fig. 2(c), the microchannel consists of three primary pieces: an acrylic polymer substrate (1 in, diameter; 1/8 in, thick), a micropatterned polydimethyl-siloxane (PDMS) seal ($≈$400 μm, thick), and a metal-coated fused silica (FS) glass window (1 in, diameter; 1/16 in, thick). The microchannel is constructed by pressure sealing the acrylic substrate to the metal-coated FS window. The microchannel geometry (or cutout in the PDMS seal) is fabricated by laser ablation processing of the PDMS film. Laser ablation patterning is also used to make the fluid inlet and outlet ports ($≈$ 1 mm, diameter) in the acrylic substrate. After pressure sealing, the microchannel dimensions are verified using the camera imaging setup shown in Fig. 1(a). No leaking of the PDMS seal or flow loop is observed for the maximum allowable flow rates of 55 mL/min.

### Flow Field and Measurement Geometry.

The experimental setup is designed to measure the HTC in different regions of boundary layer (BL) flow. For flow over a flat wall, the thickness of the hydrodynamic and thermal BLs is commonly expressed as $δh=5x/(Rex)1/2$ and $δth=δh/(Pr)1/3$, respectively, [70]. For our experimental setup, the developing hydrodynamic BLs from each wall—i.e., top (acrylic) and bottom (metal on glass)—will converge before the middle of the microchannel. Thus, both the thermal and hydrodynamic BLs are thinner than that predicted using standard expressions, e.g., $δh=5x/(Rex)1/2$. Figure 2(d) shows this convergence scenario with correspondence to our pump-probe TDTR measurement setup. We stress, as depicted in Fig. 2(d), that our TDTR measurement region-of-interest (ROI) is located just after convergence of the hydrodynamic BLs.

The heat loads in microelectronics are rarely spatially and temporally uniform, where partial, periodic, or hot-spot heating are most commonly realized [71]. If heating starts at a relatively large distance downstream from the channel inlet, then the thickness of the thermal BL is much less than the thickness of the hydrodynamic BL (i.e., $δth≪δh$). Figure 2(d) shows the developing thermal BL (with exaggerated thickness) induced via the absorption of pump laser light in the metal thin film, where $Tf$ and $TS$ correspond to the inlet fluid temperature and hot-spot surface temperature, respectively. Also depicted in Fig. 2(d), the heat from the hot-spot in the metal thin film is transported into both the coolant and the FS glass substrate. This study is interested in the heat removed by the flowing fluid/coolant (not the substrate), where the heat removed by the glass substrate serves as an unwanted heat sink in our experiments. Nevertheless, changing the coolant flow field will change the thicknesses of both $δh$ and $δth$, which subsequently will change the amount of heat removed by coolant (via both conduction and convection at the metal–coolant interface).

### TDTR Setup and Measurement Principle.

The TDTR measurement principle is based on measuring rate of heat removal from a metal thin film by its surroundings. For example, in this study, the cooling of a Ti thin film ($≈$64 nm in thickness) by flowing water (top) and the FS substrate (bottom) in Fig. 2(d). With regard to the pump-probe aspect of the TDTR method, consider a focused pulse train of laser light (i.e., the pump beam) that heats the surface of the metal. Now, each fs pulse of the focused pump beam induces a local temperature jump ($ΔT$) in the metal over an area, $A≈πw2$. Then, after each fs heating event, the metal dissipates heat to its surroundings. Thus, the metal thin film serves as both a heater and a thermometer, where the rate of heating is nearly instantaneous (e.g., fs heating) and the rate of cooling is dictated by the overall thermal conductance (or thermal effusivity—$eth$) of the surroundings. For example, the cooling rate becomes more rapid by increasing either the thermal conductivity ($Λ$) or the heat capacity ($Cp$) of the surroundings.

The thermometry aspect of TDTR is accomplished by the probe beam. For example, a short time delay after each pump heating event (e.g., $δt=τd$), the probe beam (also, a focused, pulse train of laser light) “probes” the change in temperature of the metal. The probe beam actually “probes” the change in reflectivity of the metal, which is coupled to the metal's local temperature by its thermoreflectance coefficient ($dR/dT$). Hence, the name of the TDTR technique: time-domain thermoreflectance.

Figure 2(b) shows our TDTR optical setup, using the two-tint methodology [63] to help filter (remove) pump laser light on the differential photodiode (PD) detector. In our TDTR setup, the laser source is a Coherent Chameleon femtosecond Ti:Sapphire laser (pulse frequency: 80.1 MHz, pulse width: 140 fs, central wavelength: 787 nm). The Chameleon laser output is split into two laser beams (pump and probe). The pump beam is frequency modulated at either $fmod=$ 9.81 MHz, 976 kHz, or 962 kHz using an electro-optic modulator (EOM). After the EOM, the pump beam is reflected (down and back) using an Au retroreflector on a mechanical delay stage. After the other optics, indicated in Fig. 2(b), the pump and probe beams are concentrically focused onto the metal thin film on the sample using a 20× Mitutoyo, infinity corrected, long working distance microscope objective.

The spatial variation in the pump path length by the delay stage is equivalent to a temporal time delay ($τd$) between each focused pump and probe pulse on the metal/sample. Our TDTR setup achieves pump-probe time delays of −120 ps $<τd<$ 3.3 ns. The focused beam waists of pump and probe beams on the metal/sample ($w$) are most frequently $≈$9.5 μm and $≈$8 μm, respectively. We adjust the incident pump and probe laser powers on the sample to maximize the measurement signal (for a minimum amount of probe power) while also ensuring that total dc temperature rise/heating of the pump-induced hot-spot is no more than 60 K (typically, $<$11 mW and $<$5 mW for the pump and probe, respectively).

We use a differential PD detector to measure the probe's thermoreflectance signal of the sample as a function of $τd$, where again this thermoreflectance signal is induced via the frequency-modulated heating by the pump beam. The time-domain voltage output of the detector is measured by a lock-in amplifier at a reference frequency equal to $fmod$, using triple-shielded RF coax cables and a resonant band-pass filter between the detector and the lock-in amplifier. The lock-in amplifier extracts the detector voltage signal into in-phase ($Vin$) and out-of-phase ($Vout$) voltage components in the frequency domain. These voltages as a function of pump-probe delay ($τd$) are then compared to the predictions of a TDTR thermal transport model to extract the thermal properties of the sample.

We use, as most commonly done by others, the in-phase to out-of-phase voltage ratio ($Vin/Vout$) to correlate the time-domain changes in the surface reflectivity to the thermal transport properties of the sample [6264]. In short, the TDTR voltage ratio ($Vin/Vout$) is the key measurement parameter for characterizing the thermal transport properties of a sample. This work shows how measurements of $Vin/Vout$ can be used to extract the local HTC of flowing and stagnant fluids.

The TDTR method does not require a calibration. Rather, the measurement accuracy is validated by reproducing thermal property data of known material systems using no free parameters in the TDTR thermal transport model. In this regard, the TDTR method is not limited by its measurement resolution; rather, TDTR is limited by its measurement precision (i.e., reproducibility of a measurement). In principle, the technique is capable of measuring a local, transient HTC within the range of 100 kW/m2/K $≲h≲$500 MW/m2/K over spatial measurement areas within 10–2500 μm2 and at a minimum temporal time-scale of $≈$100 μs. This predicted range of TDTR measurement space for the HTC is based on (i) a practical range in thermal conductivities that can be measured with the TDTR (e.g., 0.01 $≲Λ≲$ 3000 W/m/K), (ii) a practical range in the footprint/measurement area for the focused pump-probe lasers (e.g., 10$≲w2≲$ 2500 μm2), and (iii) the minimum time-constant setting ($τm$) of a MHz bandwidth lock-in amplifier (i.e., $τm=$ 100 μs). We note that this discussion did not consider the length scale that the HTC is probed within the thermal BL. This topic is addressed in Sec. 2.6. We also point out that the precision of our HTC measurements (discussed later in Sec. 3) was observed to be within $δh≈±$ 100 kW/m2/K.

Figure 3 shows the predicted $Vin/Vout$ ratio as a function of pump-probe delay ($τd$) with comparisons to measured data for both air and nonflowing (stagnant) water in contact with a Ti metal-coated FS glass window. The model predictions (lines) are based on literature thermal property data for the fluid (air or water), Ti thin film, and FS glass substrate. The schematic in Fig. 3 corresponds to the materials and measurement configuration, where the pump-probe beams pass through the FS glass substrate and then heat the “backside” of the Ti thin film. The data in Fig. 3 show that the magnitude of $Vin/Vout$ is larger for the more thermally conductive fluid—i.e., water (as opposed to air) in the microchannel. Also, for these “backside” TDTR measurements, oscillations in $Vin/Vout$ are observed (see Fig. 3)—presumably due to Brillouin backscattering in the glass substrate [64]. We point out the oscillation peak at 100 ps because this study uses $Vin/Vout$ measured at a single delay time (i.e., $τd≈$100 ps) to predict the HTC of flowing fluids. Thus, our measured fluid thermal conductivities and corresponding HTC predictions will be slightly overestimated (e.g., 5–20%, with and without fluid flow) based on $Vin/Vout$ measured at solely $τd≈$100 ps. Conversely, underestimates are found using $Vin/Vout$ measured at solely $τd≈$80 ps because an oscillation valley exists at that delay time.

As illustrated in Fig. 3 (schematic), the heat load from the hot-spot (laser) is transferred into both the fluid and the FS glass substrate. If the fluid is air, then nearly all the heat goes into the substrate (e.g., $ethair≪ethFS$). Whereas, if the fluid is water, then heat load is nearly split equally between FS substrate and the water coolant (e.g., $ethwater≈ethFS$). We note that the HTC measurement sensitivity can be improved by replacing the FS substrate with a different optically transparent, thermally resilient substrate having an ultralow thermal conductivity (or eliminating the substrate altogether). Due to the lack of a practical alternative to FS glass, all studies are conducted with microchannels on metal-coated FS glass.

### HTC Predictions Via TDTR.

The dependence of $Vin/Vout$ on changes in the thermal effusivity of the fluid/sample is an essential attribute of the TDTR method for characterizing the HTC of stagnant, flowing, or evaporating fluids. The following is the derivation of the HTC in terms of the fluid's thermal effusivity. We start with the standard expression for the HTC
$h=qΔT,$
(1)
where for the TDTR method, the heat flux into the fluid is
$q=P̃laserπw2$
(2)
and the temperature difference between the metal surface and the fluid outside the thermal BL (due to AC pump heating at $ω=2πfmod$) is
(3)
Equation (3) is based on the solution by Carslaw and Jaeger for periodic surface heating in a semi-infinite solid [72,73]. Equation (3) is still valid for stagnant fluids, where here $P̃laser$ represents the average laser power of the modulated pump beam at $ω$ that is absorbed by the metal thin film and is transported as heat into the fluid over the heating area ($πw2$). Inserting Eqs. (2) and (3) into Eq. (1), this heating power per unit area cancels out and we have the following HTC equation for the TDTR method:
(4)

We included the right-hand term in Eq. (4) to emphasize that the HTC is proportional to the thermal effusivity of the fluid; in particular, $eth$ within the TDTR measurement region (i.e., $eth$ within the thermal BL of the pump induced hot-spot). In Eq. (4), $C$ is a constant, $fmod$ is the modulation frequency of pump beam (e.g., 962 kHz), and $tc$ is a critical (or fundamental) time-scale in a TDTR experiment for the metal thin film to exchange thermal energy with its surroundings.

Alternatively, Eq. (4) could be derived by setting $C=1$ and relating $tc$ to the thermal diffusivity of the fluid/surroundings, $tc=[ℓth/2]2/Dth$, where
$ℓth=2Dth/ω=Dth/πfmod$
(5)

is a fundamental length scale in TDTR (called the thermal penetration depth) [73] that corresponds to the average depth of thermal energy exchange between the fluid/surroundings and an interface that is periodically heated at $ω$.

In this study, we use Eq. (4) and the measured TDTR ratio data to extract the HTC. In short, we measure $Vin/Vout$ at different delay times and different flow-field conditions. Then, we use the thermal effusivity of the fluid as a fitting parameter to relate the TDTR model predictions to our measurements of $Vin/Vout$.

Figure 4 shows the predicted dependence of $Vin/Vout$ on both Fig. 4(a) the thermal effusivity ($eth$) and Fig. 4(b) the thermal diffusivity ($Dth$) of the surroundings. In particular, $Vin/Vout$ for a variety of different top-layer materials (e.g., solid, liquid, or gas) for the measurement schematic is shown in Fig. 3. These data are provided to emphasize that both (i) the magnitude of $Vin/Vout$ at a given pump-probe delay ($τd$) and (ii) the cooling rate of the Ti metal (e.g., $Δ(Vin/Vout)/Δτd$) are mainly dictated by the thermal effusivity of the surroundings—e.g., $ethfluid=[ΛfluidCpfluid]1/2$. For these predictions, the thermal properties of the FS substrate ($ΛFS=$ 1.32 W/m K, $CpFS=$ 1.64 J/cm3 K), Ti thin film ($ΛTi=$ 20 W/m K, $CpTi=$ 2.38 J/cm3 K), and volumetric heat capacity of the sample/fluid are held constant, while $Λfluid$ is varied to represent the range in $eth$ (or $Dth$) of different sample/fluid systems. We used a constant heat capacity of either $Cpfluid=$ 4.15 J/cm3 K (lower solid line in Fig. 4(a) and upper solid line in Fig. 4(b)) or $Cpfluid=$ 1.2 J/cm3 K (upper solid line in Fig. 4(a) and lower solid line in Fig. 4(b)) because they represent upper and lower limits of $Cp$ for various solids and liquids at room temperature.

In Fig. 4, predictions are provided for two different pump-probe delay times ($τd=$ 100 ps (solid-lines) and $τd=$ 3 ns (dashed-lines)). We point out that for low thermal effusivity samples/fluids—e.g., $ethair<$ 0.01 kW s1/2/m2 K in Fig. 4(a)—the TDTR ratio converges to that of the Ti-coated FS substrate in vacuum. Moreover, for low thermal effusivity samples, the cooling rate is relatively small, where cooling rate of the metal is directly correlated with the decay rate in the TDTR ratio (i.e., $d(ΔT)/dt∝Δ(Vin/Vout)/Δτd$). However, for ultrahigh thermal effusivity samples (e.g., diamond), this decay rate or difference between $Vin/Vout$ at $τd=$ 100 ps (open-diamond) and $Vin/Vout$ at $τd=$ 3 ns (filled-diamond) is the maximum predicted. We also note that the magnitude of this difference is systematic with increases in $eth$, whereas (as illustrated in Fig. 4(b)) the cooling rate of the Ti metal thin film is not systematic with increases in the thermal diffusivity of the sample. In summary, the TDTR model predicts that both $Vin/Vout$ and $Δ(Vin/Vout)/Δτd$ are directly proportional to the thermal effusivity of the fluid; therefore, so should the HTC (as indicated by Eq. (4)), especially for heat transport in single-phase fluids.

### Anisotropic TDTR Measurements.

In Secs. 2.12.4, we described our setup and measurement principle for the traditional TDTR method. The traditional TDTR method (based on two concentrically focused pump and probe beams) is most commonly used to measure the through- (or cross-) plane thermal conductivity ($Λ⊥$) of the sample (i.e., $Λ$ in the perpendicular ($⊥$) direction from the metal thin film). The TDTR method and modified versions can also be used to measure the in-plane thermal conductivity ($Λ||$), which is of interest for studies of materials with thermal transport anisotropy [74,75]. Recent work by Feser et al. [65,66] has proposed the approach of using spatially offset (or nonconcentrically focused) pump and probe beams to measure both $Λ⊥$ and $Λ||$. In their method (which we call “Anisotropic TDTR”), the pump beam heats the metal thin film and then the probe beam senses the rate of surface temperature change (decay) at different lateral locations.

Figure 5 illustrates the anisotropic TDTR method with additional illustrations related to the thermal and hydrodynamic BLs of fluid flow field. As shown in Fig. 5, by spatially offsetting the pump and probe beams, the anisotropic TDTR method can probe heat transport inside and outside the “pump-induced” thermal BL. In our experiments, the probe beam is actually at a fixed location in the microchannel and we displace the pump beam upstream and downstream of the probe. However, for simpler illustration and descriptions, later we show the opposite to help emphasize our probing of heat transport upstream and downstream the “pump-induced” thermal BL (see Figs. 5(b) and 5(c), respectively). For reference, these displacements are small and are typically at most twice the pump's focused beam waist (i.e., $|Δx|≤2w$, where $w≈$9.5 μm). In our setup, pump beam displacements relative to the probe can be produced along both the $x$- and $y$-axis directions. Displacements of the pump beam are accomplished by rotating the polarized beam splitter (PBS) shown in Fig. 2(b) with a custom two-axis (stepper-motor controlled) galvo stage. The galvo stage has a displacement resolution along the $x$-axis (i.e., flow-field axis) of $≈$ 0.0935 μm/μ-step. For reference, 25 μ-steps of the $x$ -axis stepper motor corresponds to a 1/4 /4 offset of the pump relative to the probe.

### Fluid Flow Field in TDTR Measurement Region.

For both the traditional and anisotropic TDTR methods, the heat transport measurements are described to take place within a region of thickness $ℓth$ from the metal thin film (see Eq. (5)). This thickness (or depth) in a TDTR measurement is also illustrated in Fig. 5(a) with respect to the flow-field velocity. For reference, $ℓth≈$225 nm for room temperature TDTR studies with water on Ti-coated glass and $fmod≈$ 962 kHz, where increasing the modulation frequency to $fmod≈$ 9.81 MHz corresponds to $ℓth≈$ 70 nm. In either case, this is a very thin region and the maximum flow field velocity we can obtain within this short distance ($ℓth$) from the metal surface is $vℓth≈$0.016 m/s (based on a Hagen–Poiseuille flow field). However, the flow field outside $ℓth$ still influences the heat transfer within $ℓth$. Nevertheless, this estimate for $vℓth$ is based on our microchannel testing conditions/geometry (see Sec. 2.1) and a fully developed flow profile with no-slip at the metal/fluid interface. For reference, the maximum flow velocity in the center of the channel is $vmax≈$6.94 m/s (i.e., the flow 200 μm from the metal/glass wall, using and a volumetric flow rate of 50 mL/min from the syringe pump).

For comparative purposes, we compare this flow-field velocity in the TDTR measurement region (i.e., $vℓth$) to the velocity that thermal energy propagates by heat conduction in the fluid (e.g., the group velocity— $vg$). Considering only the real part of the group velocity, thermal energy within $ℓth$ propagates at $vg=4πfmodDth$ [76], which for our experiments with near room temperature water and $fmod=$ 962 kHz corresponds to $vg≈$2.7 m/s. This group velocity for thermal energy transport is a factor of 100 greater than $vℓth$; yet, $vg$ is still 61% and 21% less than $vmax$ and $vavg$, respectively. The latter is pointed out because if we consider $ℓth$ as the TDTR measurement region, then a flowing fluid outside $ℓth$ (i.e., the “fluid surroundings” outside the $ℓth$ boundary is comprised of higher velocity and lower temperature water) would still be influencing the heat and mass transport within $ℓth$. However, for a stagnant fluid, vg is much greater than vmax$(vg≫vmax)$, and thus, only conductive heat transport is dominant—which justifies our previous claim that Eq. (3) is still valid for a stagnant fluid.

## Results and Discussion

### Differential Measurements of the HTC Using Anisotropic TDTR.

In this section, we describe our initial TDTR studies of flowing water in microchannels, where a local hot-spot (induced by the pump beam) is cooled by either stagnant or flowing water. Our initial studies used Al-coated glass substrates to mirror past studies with impinging droplets [58]. However, our experimental results with Al thin films and flowing fluids required higher pump laser powers (e.g., >30 mW) and were inconsistent due to bubble nucleation and pitting/corrosion of the Al metal thin film in the measurement region. These results correlate well with past droplet impingement studies, where a new droplet impingement site was required for each TDTR experiment of an impinging and evaporating droplet (see Sec. 3.6 in Ref. [58]). Therefore, we replaced the Al with a Ti thin film, which avoided such problematic pitting effects during our single-phase cooling studies with moderate pump laser powers (e.g., <30 mW).

Figure 6 shows our anisotropic TDTR scans of the Ti-coated FS substrate with both stagnant air (filled circles) and stagnant water (open circles) in the microchannel. These data serve as a baseline for local HTC measurements using our differential TDTR measurement methodology, where these anisotropic TDTR scans with both stagnant air and stagnant water are needed for later HTC analysis with flowing water. We note that translating the overall pump-probe measurement ROI to a location outside the microchannel (i.e., onto the PDMS seal using the six-axis sample stage) showed increases in $Vin/Vout$ indicative of a polymer in contact with the Ti metal. For the experiments in Fig. 6, the pump-probe delay and modulation frequency were fixed at $τd=$ 100 ps and $fmod=$ 962 kHz, respectively. Measurements at longer delay times (e.g., $τd>$ 500 ps) had more measurement noise and experiments with decreased modulation frequencies (e.g., $fmod<$ 900 kHz) did not correlate well with the TDTR model predictions.

Figure 7 shows anisotropic TDTR scans at $τd=$ 100 ps with both stagnant water (circle symbols) and flowing water (star symbols) in the microchannel. Figure 7(b) shows $Vin/Vout$ measured at different pump-probe offsets ($Δx$) relative to the pump heating waist ($w$). We refer to the experiments without fluid flow as “stagnant water”; however, there still may be considerable microconvection in the vicinity of the micron-sized hot-spot induced by the pump beam, where fluid flow in the channel will magnify this microconvection in the probe measurement ROI. Figure 7(c) shows our corresponding measurements/predictions of the fluid thermal effusivity and HTC at different pump-probe offsets. As shown, water flow in the microchannel increases the effective thermal effusivity of the fluid (relative to that of the stagnant fluid).

The $Vin/Vout$ ratio data shown in Fig. 7(b) were measured after acquiring the air data (filled-circles) in Fig. 6. For example, after the air experiments, the microchannel was filled with water. Then, for a given pump-probe offset ($Δx$), starting with concentrically focused beams ($Δx=$ 0 μm), the in-phase ($Vin$) and out-of-phase ($Vout$) TDTR signals for stagnant water ($ReD$ = 0) and then flowing water ($ReD$ = 1850) were repeatedly measured, including several of these dual-scan measurements at offsets ranging within −20 μm $<Δx<$ 20 μm (or 2 $w≲Δx≲$2 $w$). Then, the TDTR ratio (i.e., $Vin/Vout$ in Fig. 7(b)) was computed for our subsequent predictions of the HTC (i.e., the data in Fig. 7(c)).

Currently, we do not have a validated bidirectional TDTR model for the anisotropic method, where bidirectional refers to heat transport (from the metal) into both the fluid and glass substrates. However, we do have a bidirectional TDTR model for through-plane thermal transport based on concentrically focused pump and probe beams (see Fig. 3). Therefore, for HTC analysis, we have employed a differential measurement/analysis scheme. This differential scheme consists of using this traditional bidirectional TDTR model to fit an effective through-plane thermal effusivity ($etheff$) to $Vin/Vout$ measured at different pump-probe offsets. The data in Fig. 7(c) are the results of this fitting process for $etheff$ (left axis) and the corresponding HTC (right axis—via Eq. (4)).

We note that before we could quantify $etheff$ (or the HTC) of stagnant or flowing water, we need to know the effective thermal effusivity of the FS substrate as a function of pump-probe offset (i.e., $etheff(Δx)|FS$). We obtain $etheff(Δx)|FS$ via TDTR model fits of the measured ratio data for air/Ti/FS in Fig. 6, where the properties of the air, Ti thin film, and heat capacity of FS substrate are held constant, such that $Λ⊥FS$ is the only TDTR model fitting parameter. This approach produces values for $Λ⊥FS$ at each pump-probe offset (or equivalently $etheff(Δx)|FS$ because heat capacity was held constant in this analysis). Alternatively, we have also obtained $etheff(Δx)|FS$ by fitting $CpFS$ while keeping the other model parameters fixed at literature values. Both approaches yield the same $etheff(Δx)|FS$ results. This same fitting procedure is used to predict $etheff$ of stagnant water and flowing water as a function of $Δx$ (i.e., the data in Fig. 7(c)). However, in this case, the anisotropic thermal effusivity data for the FS substrate ($etheff(Δx)|FS$) are now a known input to the TDTR model at each respective $Δx$ offset—hence, this is the differential aspect of our anisotropic TDTR measurement methodology.

Figure 8 shows the measured TDTR ratio and corresponding HTC results (via our differential measurement methodology) for concentric pump-probe alignment ($Δx/w≅$0) as a function of the fluid flow rate in microchannel. The data are provided for two different pump-probe delay times (e.g., $τd=$ 100 ps and $τd=$ 500 ps). As expected, our measurements at both delay times yield the same trends in HTC results. These data are provided to emphasize that our measured HTC enhancement due to forced convection over the pump-induced hot-spot is systematic with the magnitude of the water flow rate in the microchannel.

### Anisotropic TDTR With Different Metal Thin Films.

To demonstrate the applicability and meaningfulness of our HTC measurements using the anisotropic TDTR methodology, we conducted several studies with different metal thin-film materials deposited on FS glass substrates. As discussed in Ref. [66], the anisotropic TDTR method is more sensitive to the in-plane thermal transport (e.g., $Λ||$) using highly focused pump-probe beams and low thermal conductivity metal thin films. In this regard, metal alloys with large thermoreflectance coefficients are ideal. The NbV alloy used by Feser and Cahill is one such thin-film alternative to Al [66]. Moreover, of particular importance to our water flow studies, NbV alloys have corrosion resistance properties that are superior to Ti. In addition to Ti and NbV, we also used a complex metal alloy consisting of Hf, Gd, and HyMu80 alloy (which we call Hf80 due to its highest Hf content). This Hf80 metal alloy not only has a low thermal conductivity (e.g., $Λ≅$5.6 W/m K), but it is incredibly robust, facilitating later TDTR studies of flow boiling and jet impingement with extreme hot-spot heat fluxes. The supplemental file, which is available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection, provides additional information and TDTR results for water and air in contact with these NbV and Hf80 alloy thin films on FS substrates.

Figure 9 shows our anisotropic TDTR results for Hf80-coated FS substrates with both stagnant and flowing water in the microchannel. In comparison to the Ti thin-film data (Fig. 7), these TDTR ratio data with the Hf80 thin film (Fig. 9) have considerably more measurement error, especially at pump-probe offsets ($Δx$) greater than one pump beam waist ($w$). For this reason, we only show our analysis results for $eth$ and $HTC$ in Fig. 9(b) for the boxed region in Fig. 9(a). This magnified view also helps show that high- $Re$ flow in the microchannel has an effect on the TDTR ratio, especially with concentrically focused pump-probe beams (i.e., $(Δx/w)≈$0). We stress that similar to our results with a Ti metal thin film (Fig. 7), we observe a maximum increase in the TDTR ratio (or HTC) when “probing” within the developing thermal BL (i.e., probing within 0 $<(Δx/w)<$ 1/4, which is $≈$1–2 μm downstream the center of the pump induced hot-spot).

For reference, the incident pump laser powers on the Ti (Figs. 7 and 8) and Hf80 (Fig. 9) metals were both $≈$ 10.5 mW. This corresponds to average hot-spot heat fluxes into the fluid of $q¯Ti≈$837 W/cm2 and $q¯Hf80≈$934 W/cm2, where $qCHF≈$1000 W/cm2 is a common CHF value for subcooled boiling on uniformly heated surfaces with water [77]. And thus, as expected, we can easily induce vapor bubble nucleation with more focused or increased laser power beams. On this note, we observe significantly improved TDTR signal-to-noise ratios by increasing the pump-probe laser powers (which would seem beneficial for our Hf80 studies in Fig. 9). However, at laser powers $≳$20 mW, we chaotically observed either (i) vapor bubble nucleation and growth at the pump-induced hot-spot or (ii) $Vin/Vout$ ratio data (in the absence of bubble nucleation) that required TDTR model fits with exaggerated thermofluid properties. This work is focused on validation of our technique with single-phase fluids, so laser powers $<$20 mW were used. Again, the Hf80 results are provided because this metal thin-film material is stable at high heat fluxes, which is favorable for future studies of hot-spot boiling in cross-flow or jet-impingement boiling. The results and discussion on how vapor bubble nucleation and growth influence our anisotropic TDTR measurements (or the HTC measured) are adjourned for a latter publication.

### Predicted Enhancement in the Local HTC Due to Forced Convection.

In Secs. 3.1 and 3.2, we showed that the Anisotropic TDTR method can be used to measure (or predict) the local HTC; in particular, the local HTC around a micron-sized hot-spot with and without forced convection (see Figs. 7(c) and 9(b)). However, our predicted HTC values are an order of magnitude greater than the maximum HTC values observed by others (see, for comparison, the single-phase HTC data in Table 1). In hindsight, this is expected because the TDTR method characterizes the HTC over very small length scales (e.g., $2w≈$19 μm and $ℓth≈$225 nm), where it is well known that the HTC is inversely proportional to the thermal BL thickness, which is also dependent on the size of the heat source [78]. Correspondingly, this length-scale correlation with the HTC is reflected by the Nusselt number, $Nu=hLc/ΛftBL$, where $ΛftBL$ is the fluid's thermal conductivity within the thermal BL (tBL) and $Lc$ is a characteristic length dictated by the cooling/heating configuration (e.g., heater width, length, and pipe diameter, etc.). Below, we show that our local HTC measurements can be predicted by combining well-established and experimental-specific Nusselt number correlations.

For our experiments, we predict the local HTC to follow:
$h=h0+h↑=ΛftBL2w[Nu0+Nu↑]$
(6)
where the characteristic length is diameter of the hot-spot ($Lc=2w$) and we separate the HTC (Nusselt number) into two components. The first component, $h0$ ($Nu0$), represents the local HTC for stagnant water in a TDTR experiment—i.e., that associated with both heat conduction and natural microconvection. Whereas, the second component, $h↑$ ($Nu↑$), represents the local enhancement in the HTC due to increased microconvection caused by the flowing fluid over the hot-spot. Explicitly, we use the following expressions for each component:
(7)
(8)

where the stagnant-fluid component (Eq. (7)) is purely based on the TDTR experimental conditions (see Eq. (4)) and the forced-convection component (Eq. (8)) is based on the product of the normal distribution ($N(ϵ¯,σ2)$) and the pioneering Nusselt number correlation by Incropera et al. [79] for single-phase convective heat transfer in a rectangular channel with a flush-mounted square heater (hence, the subscript [79] with 0.13). For the normal distribution in Eq. (8), we use $σ2=$ 1 (i.e., a variance of $w$) and slightly downstream expectation (i.e., $ϵ¯=(δx/w)=$ 0.25) to account for our anisotropic HTC observations with flowing fluids. Separating the Nusselt number into two components (i.e., one “constant” stagnant-fluid component and another “functional” forced-convection component) is quite common [80]. However, usually the stagnant-fluid component is an additional fitting parameter while, for TDTR, it is directly measured (and/or it has an explicit expression). We also note that, in principle, additional terms could be added to Eq. (6) to account for boiling, evaporation, or chemical reactions.

Figure 10 compares our measured enhancement in the HTC (i.e., $h↑TDTR=hflow−hstag$) to the HTC enhancement predicted (i.e., $h↑Eq.(6)$) due to high-$Re$ water flow over the hot-spot in the microchannel. The lines are the predictions and the symbols are our measured data for the two different metal thin films studied (Ti: filled circles, Hf80: open circles). Fair agreement is found between the Ti thin-film HTC data and the Nusselt number predictions using $C=$0.18 [78]. A maximum enhancement in the HTC is observed at a location slightly downstream the center of the pump hot-spot (e.g., a downstream distance of $Δx≈$ 5 ± 3 μm (or $Δx/w≈$ 0.52 ± 0.32), which also represents the presumed region of rapid thermal BL growth). The Hf80 thin-film data do not exhibit a systematic HTC enhancement peak and that, combined with the increased measurement noise for Hf80, have led to poor correlations with the Nusselt number predictions. The Hf80 data also show negative HTC enhancements for downstream probing at $Δx/w≳$1, where negative values of $h↑$ correspond to the fluid heating the metal. Nevertheless, aside from this fluid heating effect with Hf80, our anisotropic TDTR studies with both metal thin films demonstrate that there is an overall HTC enhancement due to forced convection (especially for upstream probing, where the flowing fluid can only cool the metal in the “probe measurement ROI”).

If in-plane thermal transport in the metal is not significant, then both metal thin films are expected to yield the same local HTC enhancement results because the flow-field conditions were identical ($ReD=$ 1850, $Tfinlet=$ 25 °C). This is pointed out because the flowing water is expected to both cool and heat the metal thin-film wall of the microchannel (e.g., upstream cooling and downstream heating of the metal wall relative to the central pump-induced hot-spot). This is depicted in Fig. 10(a) by the skewed pump and probe heating distributions (dotted lines). Thus, the overall thermal energy exchange between the fluid and the metal heater/thermometer is dictated by both the thermal effusivity of the metal and thermal effusivity of the fluid. For reference, $ethTi/ethwater≈$4.3 and $ethHf80/ethwater≈$ 2.0, indicating that the Hf80 metal will conduct less in-plane heat from the hot-spot (relative to Ti); and thus, Hf80 metal will see more in-plane heat from the flowing fluid (relative to Ti). Our HTC (or $Nu$) predictions (Eqs. (6)(8)) do not account for thermal effusivity of the metal. In addition, we assumed a Gaussian profile for the metal wall temperature. Therefore, improved experiments and predictions would benefit from both (i) continuum-level modeling of the metal wall temperature at different flow rates and (ii) additional anisotropic TDTR experiments at longer pump-probe delay times (e.g., both $τd=$ 100 ps and $τd=$ 3 ns, as shown in Fig. 4). The former would improve our estimates of the local model parameters (e.g., $Rex$, $Prx$, $μx$, $Λx$, etc.) while the latter would help decipher the relative heating or cooling contributions at different pump-probe offsets.

## Conclusion and Future Directions

The anisotropic TDTR method is shown to be a useful technique for characterizing anisotropic heat transport at submicron length scales. This work supports that the technique can be extended—via a differential measurement methodology—to characterize both the conductive and convective heat transfer contributions to fluid-flow cooling of a laser heated microchannel wall with subcooled water and moderate Reynolds number flow-field conditions. We show that our local HTC measurements can be predicted (with relatively good agreement using a Ti metal thin-film heater/thermometer) using a two-component Nusselt number correlation, where the first component represents the HTC due to both heat conduction and natural microconvection of the stagnant fluid and the second component accounts for the HTC enhancement due to forced convection. However, our results with other thin-film heaters/thermometers having lower thermal conductivities were not predicted well by this two-component correlation, presumably due to wall heating effects by the coolant downstream the pump-induced hot-spot. In this regard, future studies would benefit from (i) in situ experiments that can independently characterize the temperature distribution of the channel wall, (ii) additional anisotropic TDTR experiments at multiple pump-probe delay times (e.g., both $τd=$ 100 ps and $τd=$ 3 ns), (iii) studies directly with microchannel-structured Si heat sinks or microchannels fabricated on ultralow thermal conductivity substrates to maximize the net heat transferred into the fluid, and (iv) other wall heating configurations (i.e., eliminating the pump beam as the hot-spot heating source) such as uniform wall heating or the use of a third laser beam (for hot-spot heating). However, the latter would limit our proposed capability of using the stagnant TDTR measurements to directly predict the HTC distribution of the stagnant fluid.

## Acknowledgment

This material is based on research partially sponsored by the U.S. Office of Naval Research under Grant No. N00014-15-1-2481. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of U.S. Office of Naval Research or the U.S. Government. The authors thank Joseph P. Feser (University of Delaware) and John G. Jones (Air Force Research Lab) for their gracious depositions of the NbV and Hf80 thin-film coatings, respectively.

## Nomenclature

• =

area of pump-laser hot-spot =  $πw2$

•
• =

constant

•
• =

volumetric heat capacity

•
• =

hydraulic diameter = 480 μm

•
• $Dth$ =

thermal diffusivity

•
• $eth$ =

thermal effusivity

•
• =

modulation frequency, MHz

•
• =

mass flux

•
• =

heat transfer coefficient (HTC)

•
• =

microchannel height = 400 μm

•
• =

local HTC for a stagnant fluid

•
• =

local HTC enhancement ($↑$) due to forced convection

•
• $L$ =

microchannel length = 15 mm

•
• $Lc$ =

characteristic length scale for heat transfer = 2$w$ (this study)

•
• $ℓth$ =

thermal penetration depth

•
• $Nu$ =

Nusselt number =

•
• $Nu0$ =

local $Nu$ for a stagnant fluid

•
• $Nu↑$ =

local $Nu$ enhancement ($↑$) due to forced convection

•
• $P̃laser$ =

laser power converted into heat

•
• $Pr$ =

Prandtl number =  $Cpν/Λ$

•
• $q$ =

heat flux

•
• $qCHF$ =

critical heat flux (CHF)

•
• $R$ =

reflectance of the metal

•
• $ReD$ =

Reynolds number based on hydraulic diameter =

•
• $tc$ =

characteristic time-scale for heat transfer

•
• $Tf$ =

fluid temperature

•
• $TS$ =

surface/wall temperature

•
• $Tf∞$ =

fluid temperature at the microchannel inlet = 25 °C

•
• $vg$ =

group velocity of TDTR thermal waves

•
• $vavg$ =

average fluid velocity

•
• $Vin$ =

in-phase TDTR voltage

•
• $vmax$ =

maximum fluid velocity in the center of the microchannel

•
• $Vout$ =

out-of-phase TDTR voltage

•
• $vℓth$ =

fluid velocity at a perpendicular depth $ℓth$ from the hot-spot

•
• $Vin/Vout$ =

TDTR voltage ratio

•
• $w$ =

beam waist, $1/e2$ radius of the focused pump laser

•
• $W$ =

microchannel width = 600 μm

•
• $x$ =

flow direction, distance from the microchannel inlet

•
• $z$ =

perpendicular distance from the metal wall into the channel

### Greek Symbols

Greek Symbols

• $δh$ =

thickness of the hydrodynamic boundary layer

•
• $δth$ =

thickness of the thermal boundary layer

•
• $ΔT$ =

temperature difference between the fluid and metal wall/surface

•
• $ΔTAC$ =

amplitude of the temperature oscillations in metal due to $AC$ pump heating

•
• $Δx$ =

pump-probe offset

•
• $Δx/w$ =

offset ratio

•
• $Λ$ =

thermal conductivity

•
• =

dynamic viscosity of the fluid in the thermal BL

•
• $μ∞$ =

dynamic viscosity of the fluid at the microchannel inlet

•
• $ν$ =

kinematic viscosity

•
• $τd$ =

pump-probe time-delay

•
• $ω$ =

angular heating frequency

### Acronyms

Acronyms

• $BL$ =

boundary layer

•
• $fs$ =

femtosecond

•
• $FS$ =

fused silica

•
• $HTC$ =

heat transfer coefficient

•
• $ktg$ =

kinetic theory of gases

•
• $ROI$ =

region of interest

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