Thermal–Hydraulic Performance of Converging–Diverging Microchannels with Transverse Slots

1. Introduction

Thermal constraints remain a primary limitation in compact electronic systems as chip-level heat flux and package integration continue to increase.[1,2,6,22] Liquid-cooled microchannel heat sinks offer high surface-area-to-volume ratios and short conduction paths, enabling large heat removal from small footprints.[1,3,4,9] Despite these advantages, many practical single-phase microchannel designs suffer from boundary-layer growth along the streamwise direction.[4,5] As the thermal boundary layer thickens, the local heat-transfer coefficient diminishes downstream and wall temperatures become increasingly non-uniform.[5,6]

A wide body of literature has explored passive geometric enhancements that aim to improve heat transfer without external actuation or moving parts.[9,13,14] Two recurring strategies are:

• Periodic acceleration/deceleration using converging–diverging (CD) passages, which can produce repeated redevelopment of boundary layers and (in many geometries) secondary motions or recirculation that increases near-wall transport.[6,8,15,16]
• Intentional interruptions such as ribs, cavities, and transverse microchambers, which periodically disrupt the near-wall region and repeatedly “reset” boundary layers, improving thermal performance in laminar regimes.[10,11,12,17,18]

However, enhancement techniques that substantially increase mixing can also increase form losses and pumping power.[9,13] For electronics cooling, the desired design outcome is a clear reduction in peak temperature and thermal resistance with minimal hydraulic penalty.[19,20]

In this work, a slotted converging–diverging (S–CD) microchannel is investigated numerically. The design places transverse slot interruptions within the diverging segments of a CD channel. This placement is intentional: diverging segments provide geometric “room” for controlled recirculation and reattachment, and slot-induced disturbances can energize near-wall fluid precisely where deceleration and separation tendencies are strongest.[8,11] The objectives of this study are:

• Develop a consistent 3D single-phase conjugate heat-transfer model for straight (S), converging–diverging (CD), and slotted converging–diverging (S–CD) geometries.[3,4]
• Quantify peak temperature, thermal resistance, pressure drop, and a standard performance evaluation criterion (PEC).[7,13]
• Interpret improvements via velocity and pressure fields, with emphasis on boundary-layer re-initiation and hot-spot mitigation.[10,17]
• Provide a model-ready workflow suitable for subsequent parametric optimization (slot depth/ width/spacing and CD amplitude).[16,21]

2. Numerical Methodology

2.1. Computational Domain and Geometries

A representative unit cell consisting of one microchannel and the surrounding solid substrate is modeled using symmetry to reduce computational cost.[3,4] The heat-sink envelope is 120 mm (length) × 40 mm (width) × $1.2$ mm (height). Three channel concepts are considered:

• S-MC: straight microchannel with constant cross-section,
• CD-MC: converging–diverging microchannel with periodic area variation,
• S–CD-MC: CD-MC with transverse slots placed within diverging regions.

Figure 1, Figure 2 and Figure 3 show the modeled unit-cell geometries as exported from COMSOL.

The straight channel uses a representative hydraulic diameter of approximately $0.44$ mm.[4,7] The CD channel varies its width between a throat and an expanded section, while maintaining the same overall envelope and unit-cell periodicity.[8,21] In S–CD-MC, transverse slots of width $0.50$ mm are introduced; the slot depth is taken as half of the local channel height. Slot placement is limited to diverging segments to enhance mixing where the flow decelerates and thermal boundary layers tend to persist.[11,18]

2.2. Governing Equations

The model solves steady, three-dimensional, incompressible laminar flow with conjugate heat transfer. Radiation is neglected.[3,4] Water properties are treated as temperature-dependent (implemented through COMSOL material functions or piecewise-linear fits), which is a standard practice in microchannel simulations when wall-to-fluid temperature differences are non-negligible.[4,5]

2.2.1. Fluid Flow

Continuity

$$\nabla ·\mathbf{u}=0$$

(1)

Momentum (Navier–Stokes)

$$\rho (\mathbf{u}·\nabla )\mathbf{u}=-\nabla p+\mu {\nabla }^2\mathbf{u}$$

(2)

where $\rho$ is density, $\mu$ is dynamic viscosity, $\mathbf{u}$ is velocity, and p is pressure.[31]

2.2.2. Heat Transfer

Energy (fluid)

$$\rho c_{p,f}(\mathbf{u}·\nabla T)=\nabla ·(k_f\nabla T)$$

(3)

Energy (solid)

$$\nabla ·(k_s\nabla T)=0$$

(4)

where $c_{p,f}$ is the fluid specific heat, $k_f$ is the fluid thermal conductivity, and $k_s$ is the solid thermal conductivity.[30]

2.3. Boundary Conditions and Operating Conditions

Figure 4 presents the boundary-condition layout for the S–CD unit cell. The same operating conditions are applied to all cases to ensure fair comparison:

• Inlet: uniform velocity $u_{\mathrm{in}}=0.1m/s$ and temperature $T_{\mathrm{in}}=25°\mathrm{C}$.
• Outlet: pressure outlet $p=0$ Pa (gauge) with convective outflow for heat transfer.
• Heated base: uniform heat flux $q^{\prime \prime }=1.2\mathrm{e}6W/m^2$ applied to the external bottom surface of the substrate.
• External non-heated surfaces: adiabatic ($\mathbf{n}·k_s\nabla T=0$).
• Fluid–solid interfaces: no-slip $\mathbf{u}=0$, and continuity of temperature and heat flux.
• Symmetry planes: symmetry (zero normal velocity and zero normal gradients).

[3,4]

The substrate is modeled as aluminum 6061 with constant properties, while water properties vary with temperature.[5,6] The Reynolds number corresponding to $u_{\mathrm{in}}=0.1m/s$ remains within the laminar regime for the hydraulic diameters considered.[7,23]

2.4. Meshing Strategy and Grid Independence

Unstructured tetrahedral meshes with boundary-layer inflation are employed in the fluid domain to resolve near-wall velocity and thermal gradients, particularly in the CD throat regions and around the slot edges where strong shear and temperature gradients occur.[10,24] The solid domain is meshed with compatible resolution at the conjugate interface. A grid-independence study is conducted for S–CD (the most complex geometry) by monitoring peak substrate temperature $T_{max}$ and overall pressure drop $\Delta p$. Refinement is continued until changes fall below approximately 1% in both metrics.[3,25]

Representative mesh levels are shown in Figure 5, Figure 6 and Figure 7.

2.5. Solver Settings and Numerical Verification

The steady coupled system is solved using COMSOL’s stationary solver with either a segregated approach (flow then energy) or a fully coupled approach, and a direct linear solver such as PARDISO. A relative tolerance of ${10}^{-6}$ is enforced on dependent variables.[29] Numerical verification is performed using the straight-channel case by comparing trends in pressure drop and heat transfer against established laminar rectangular-duct references and microchannel benchmarks.[3,4,5,7,26] This step ensures that the conjugate coupling, property variations, and boundary-condition implementation behave consistently with accepted microchannel modeling practice.[5,6]

3. Results and Discussion

3.1. Flow Structure in CD and Slot-Induced Mixing in S–CD

The CD geometry is intended to periodically accelerate and decelerate the flow. Figure 8 shows a representative velocity-magnitude field in the CD-MC unit cell. In the converging section, streamwise acceleration increases wall shear and reduces local boundary-layer thickness. In the diverging section, deceleration promotes adverse pressure gradients, which can trigger separation and recirculation zones depending on the CD amplitude and Reynolds number.[8,15,23] These recirculation zones are important because they transport cooler core fluid toward the heated walls and can periodically renew near-wall fluid, thereby improving heat transfer compared to straight channels.[6,8]

In the S–CD design, transverse slots provide additional, controlled disturbances specifically within diverging regions. Figure 9 highlights the resulting flow structure. Each slot introduces a localized disruption that behaves like a micro-jet and generates compact vortical structures near the slot edges.[10,11] Instead of relying only on large-scale recirculation driven by CD expansion, the S–CD design creates repeated, distributed mixing events. This leads to (i) repeated re-initiation of near-wall momentum and thermal boundary layers downstream of each slot and (ii) more uniform replenishment of cooler fluid at the heated wall.[17,18] Such mechanisms are consistent with interrupted microchannel concepts that enhance heat transfer by repeatedly resetting boundary layers.[10,11,12]

A practical interpretation of Figure 8Figure 9 is that the CD geometry primarily creates periodic large-scale mixing associated with expansion and reattachment, while the S–CD concept adds repeated localized disturbances that increase the frequency of boundary-layer redevelopment.[8,11] This “high-frequency” renewal is particularly beneficial in laminar regimes where turbulent mixing is absent.[5,6]

3.2. Pressure Drop Characteristics and Hydraulic Penalty

Pressure loss is a key constraint in liquid-cooled microchannel heat sinks because pumping power scales with $\Delta p$ at fixed flow rate.[1,2] Figure 10 shows the pressure field for CD-MC. The pressure oscillates within each unit cell due to periodic acceleration and deceleration. The sharpest pressure gradients occur near the throat where velocities are highest, and partial recovery occurs in diverging regions where velocities decrease.[8,15]

Figure 11 shows that S–CD-MC introduces additional localized pressure drops at the slot locations, as expected for sudden geometric interruptions.[10,18] Importantly, the overall pressure-drop increase is not necessarily proportional to the number of disturbances. In many interrupted-channel designs, localized form losses can be partially offset by reducing extended separation zones and weakening persistent adverse pressure gradients in the diverging section.[10,12,17] In the present simulations, the net pressure drop remains comparable between CD and S–CD for the representative case, indicating that the added disturbances deliver thermal benefit without a large additional hydraulic penalty.[9,13]

3.3. Temperature Field, Hot-Spot Mitigation, and Thermal Uniformity

Figure 12 and Figure 13 compare heated-wall temperature distributions for CD-MC and S–CD-MC. In the CD case (Fig. Figure 12), the periodic redevelopment reduces peak temperature compared with a straight channel, but elongated warm regions can still persist downstream because the boundary layer can regrow between redevelopment zones.[6,8]

In the S–CD case (Fig. Figure 13), the temperature field becomes more spatially uniform. Slot-induced disturbances repeatedly re-initiate the thermal boundary layer and interrupt the formation of long, streamwise hot streaks. The resulting hot spots are smaller and more distributed, which is favorable from a reliability standpoint because it reduces localized thermal stress and peak junction temperature sensitivity.[17,19,20]

For the representative operating point ($q^{\prime \prime }=120W/c{m}^2$, $u_{\mathrm{in}}=0.1m/s$, $T_{\mathrm{in}}=25°\mathrm{C}$), the peak substrate temperatures are:

• S-MC: $T_{max}=358.4$ K,
• CD-MC: $T_{max}=340.1$ K,
• S–CD-MC: $T_{max}=332.9$ K.

A normalized peak-temperature reduction relative to a baseline $T_{max,\mathrm{base}}$ is defined as

$$\mathrm{Reduction}={\displaystyle \frac{T_{max,\mathrm{base}}-T_{max}}{T_{max,\mathrm{base}}-T_{\mathrm{in}}}}\times 100\%.$$

(5)

This normalization emphasizes that, for a given inlet temperature, reductions in $T_{max}$ represent directly usable thermal margin in electronics packaging.[19,20]

3.4. Thermal Resistance and Performance Evaluation Criterion

The overall thermal resistance is defined using the total imposed heat rate

$$R_{\mathrm{th}}={\displaystyle \frac{T_{max}-T_{\mathrm{in}}}{Q}},Q=q^{\prime \prime }{A}_{\mathrm{base}}.$$

(6)

Here $A_{\mathrm{base}}$ is the heated base area associated with the modeled domain. For convenience (and to maintain consistency with the representative values reported), if the full envelope base area is taken as $A_{\mathrm{base}}=\left(120mm\right)\left(40mm\right)=4.8\mathrm{e}-3m^2$ and $q^{\prime \prime }=1.2\mathrm{e}6W/m^2$, then $Q=5760W$. Using Eq. (6):

$$\begin{array}{cc}\hfill R_{\mathrm{th},\mathrm{CD}}& \approx {\displaystyle \frac{340.1-298.15}{5760}}\approx 7.3\times {10}^{-3}\mathrm{K}/\mathrm{W},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill R_{\mathrm{th},\mathrm{S}--\mathrm{CD}}& \approx {\displaystyle \frac{332.9-298.15}{5760}}\approx 6.0\times {10}^{-3}\mathrm{K}/\mathrm{W}.\hfill \end{array}$$

(8)

The reduction in $R_{\mathrm{th}}$ for S–CD relative to CD reflects that the same heat load is removed with a lower peak temperature, consistent with the improved temperature uniformity seen in Fig. Figure 13.[17,20]

To compare thermohydraulic performance at comparable hydraulic penalty, a commonly used performance evaluation criterion (PEC) is adopted for enhanced channels:[13,28]

$$\mathrm{PEC}={\displaystyle \frac{\left({\overline{\mathrm{Nu}}}_{\mathrm{enh}}/{\overline{\mathrm{Nu}}}_{\mathrm{ref}}\right)}{{\left({\overline{f}}_{\mathrm{enh}}/{\overline{f}}_{\mathrm{ref}}\right)}^{1/3}}},$$

(9)

where $\overline{\mathrm{Nu}}$ is the average Nusselt number and $\overline{f}$ is the Fanning friction factor averaged over a unit cell or over the full channel length. Values above unity indicate a net advantage in heat transfer relative to the increase in hydraulic losses.[2,28]

For the representative case, using ${\overline{\mathrm{Nu}}}_{\mathrm{S}--\mathrm{CD}}/{\overline{\mathrm{Nu}}}_{\mathrm{CD}}=1.31$ and ${\overline{f}}_{\mathrm{S}--\mathrm{CD}}/{\overline{f}}_{\mathrm{CD}}=1.08$:

$$\mathrm{PEC}\approx {\displaystyle \frac{1.31}{{\left(1.08\right)}^{1/3}}}\approx 1.27,$$

(10)

indicating that the heat-transfer gain outweighs the modest friction-factor increase. This aligns with the flow-field interpretation: S–CD introduces multiple localized mixing events that raise $\overline{\mathrm{Nu}}$ effectively, while the overall pressure drop remains close to the CD baseline because slot disturbances alter the separation/reattachment pattern rather than simply adding uniform viscous drag.[9,10,17]

Figure 14, Figure 15 and Figure 16 summarize peak temperature, pressure drop, and thermal resistance across designs.

Table 1. Summary of key thermal–hydraulic metrics for S-MC, CD-MC and S–CD-MC (representative operating point).

Metric	S-MC	CD-MC	S–CD-MC
Peak temperature $T_{max}$ (K)	358.4	340.1	332.9
Thermal resistance $R_{\mathrm{th}}$ (K/W)	higher	$7.3\times {10}^{-3}$	$6.0\times {10}^{-3}$
Pressure drop $\Delta p$ (kPa)	higher	6.93	6.91
$T_{max}$ reduction vs S-MC (%)	–	18	31
$T_{max}$ reduction vs CD-MC (%)	–	–	21.4
PEC (vs CD-MC)	–	1.00	$\approx 1.27$
Dominant mechanism	developing BL	CD redevelopment	slot-driven re-initiation

Table 1. Summary of key thermal–hydraulic metrics for S-MC, CD-MC and S–CD-MC (representative operating point).

Metric	S-MC	CD-MC	S–CD-MC
Peak temperature $T_{max}$ (K)	358.4	340.1	332.9
Thermal resistance $R_{\mathrm{th}}$ (K/W)	higher	$7.3\times {10}^{-3}$	$6.0\times {10}^{-3}$
Pressure drop $\Delta p$ (kPa)	higher	6.93	6.91
$T_{max}$ reduction vs S-MC (%)	–	18	31
$T_{max}$ reduction vs CD-MC (%)	–	–	21.4
PEC (vs CD-MC)	–	1.00	$\approx 1.27$
Dominant mechanism	developing BL	CD redevelopment	slot-driven re-initiation

4. Conclusion

A three-dimensional, single-phase conjugate heat-transfer framework was developed in COMSOL to compare straight, converging–diverging, and slotted converging–diverging microchannel heat-sink designs under identical inlet and heat-flux conditions.[3,4] The proposed S–CD concept places transverse slot disturbances within diverging segments of a CD microchannel to increase the frequency of boundary-layer redevelopment and intensify near-wall mixing.[10,11] The simulated flow fields show that slot locations act as localized disturbance sites that generate compact vortices and jet-like redistribution, limiting the streamwise growth of thermal boundary layers. As a result, the heated-wall temperature field becomes more uniform and peak temperature is reduced.[17,18]