# Three-dimensional numerical study of flow physics of single-phase laminar flow through diamond (diverging–converging) microchannel

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## Abstract

Diamond microchannel is a non-uniform microchannel that has both diverging and converging flow passages of certain lengths. In this work, three-dimensional simulation and analysis of single-phase laminar fluid flow through diamond microchannel has been carried out for different geometrical and flow parameters. Water is taken as the working fluid. The pressure drop exhibits linear dependence on mass flow rate and varies inversely with inlet angle, width ratio, and hydraulic diameter, which is similar to a straight channel. A non-linear behavior is observed for microchannel with higher inlet angle and higher width ratio, which suggests flow transition because of the presence of recirculation and separation zones. The inlet angle and width ratio are identified as the critical parameters that characterize the flow. An appropriate length scale is defined to make the overall pressure drop of diamond microchannel same as an equivalent uniform microchannel. This characteristic length located at 1/7th of the total length of the microchannel from the inlet makes the pressure drop of microchannel independent of its geometric and flow parameters. The local flow behavior has been further analyzed with the help of pressure, velocity and shear stress profiles. These results are significant due to the relevance of diamond shape microchannel in the design of micromixers, microreactors, and micropumps.

## Keywords

Diamond microchannel Single-phase flow Equivalent hydraulic diameter Flow characterization## 1 Introduction

Microfluidic and nanofluidic devices have large applications in the field of engineering. During the early stage of microfluidics, most of the attention was given to uncover physics of fluid flow through microchannel with uniform cross-section [1, 2, 3, 4, 5, 6]. These studies have shown that new flow physics such as rarefaction, velocity and temperature jump at the solid–liquid interface, and compressibility effect at low Mach number are present at microscale flows with gas as the working fluid [2, 3]. In addition, other studies have shown that surface tension effects become significant at microscale in two-phase flows [7] and respiratory fluid mechanics [8]. Although flow through uniform microchannels has been largely understood, flow through non-uniform microchannels has not received sufficient attention. The fundamental flow phenomenon in non-uniform microchannel differs substantially from its uniform cross-section counterpart [9, 10]. Diverging–converging microchannels (diamond) are characteristic non-uniform channels and flow in diamond microchannels have numerous applications in medicine, engineering and biology, such as in heat transfer enhancement [11], flow rectification [12], DNA stretching [13], micromixer [14, 15, 16], particle separation [17], etc. These applications require a thorough, fundamental understanding of the flow physics in diamond microchannels especially from the viewpoint of design optimization.

Several attempts have been made to investigate flow in diverging–converging channels at the conventional and microscales, for which flow physics and different flow scenarios have been studied [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. It is observed that the flow through mild converging–diverging tube is laminar for Reynolds number, Re < 900 [18], while flow through converging–diverging tube with relatively large convergence/divergence angle shows flow instability and turbulence when Re > 400 [28]. In other converging–diverging channels, flow remains laminar for Reynolds number ranging from 100-1000 [19]. The above observations show that the flow in diverging–converging channels is substantially different from straight channels. Therefore, studying the effect of geometrical configuration on the nature of the flow is relevant. Compared to other varying cross-section microchannels, diverging–converging microchannels have received less attention [11, 29, 30, 31]. Further, it is interesting to see whether the flow properties in microchannel with diverging–converging configuration are dependent on aspect ratios, diverging–converging angles and Reynolds numbers. Most of the available studies have not presented the underlying physics and no clear understanding of the effect of various geometric and flow parameters on the flow in a diamond microchannel is available in the literature. These issues provided the motivation for undertaking the present work.

Of particular interest is in defining the length scale (hydraulic diameter) required in the calculation of various non-dimensional numbers. Since the microchannel cross-section varies along the length, it is difficult to fix the appropriate value of the characteristic length scale [32, 33, 34]. Duryodhan et al. [35] provided a methodology to obtain the characteristic length scale for microchannel having a diverging or a converging cross-section. Based on a comprehensive study, they recommended that the value of hydraulic diameter be calculated at 33% and 28% of the total length of the microchannel from the narrow end for the diverging and converging configurations respectively. The above-mentioned length scale is not applicable to diamond microchannel with both diverging and converging sections; however, it interesting to see if the procedure for obtaining the length scale suggested by them is applicable to this more complex geometric configuration.

The specific objectives of this work are to study the local and global behavior of single-phase laminar liquid flow passing through diamond microchannel and obtain the critical parameters to characterize the flow. Towards this, a three-dimensional numerical study on laminar single-phase flow through diamond microchannel has been carried out. The effects of hydraulic diameter, total length, inlet diamond angle, width ratio on flow behavior have been studied in detail. The results are discussed with the help of pressure and velocity profiles, and wall shear stress. An appropriate length scale is also identified based on the methodology proposed in the literature.

## 2 Physical description of the problem

### 2.1 Geometric configuration of different non-uniform microchannels

*δ*) (Eq. 1), which is the ratio of length of the diverging section to the total length of the microchannel

*δ*= 0.5 (symmetric diamond microchannels). That is, in this study we are not considering asymmetric diamond channel to avoid the asymmetric transfer characteristic of a flow passage, or fluidic diodicity [12].

Geometric configuration of different types of non-uniform cross-section microchannels based on length ratio (*δ*)

### 2.2 Diamond microchannel

*W*

_{i}and

*W*

_{o}are the inlet and outlet width of the microchannel (taken to be equal here) to avoid complexity as described before,

*α*is the inlet diamond angle,

*β*is the outlet diamond angle,

*W*

_{b}is the larger width (interface width) of the microchannel,

*L*

_{d}and

*L*

_{c}are lengths of the diverging and converging section of the microchannel,

*L*is the total length of the microchannel (which is the sum of the lengths of diverging and converging sections),

*δ*is the length ratio (Eq. 1), and

Geometrical and flow parameters considered for numerical analysis

Parameter | Range |
---|---|

Inlet diamond angle ( | 4°–16° |

Outlet diamond angle ( | 4°–16° |

Total length of the microchannel ( | 10–30 mm |

Length ratio ( | 0.5 |

Width ratio ( | 3.3–15 |

Depth of the microchannel ( | 50–200 μm |

Inlet and outlet width ( | 200 μm |

Volume flow rate (\(\dot{Q}\)) | 0.5–5 ml/min |

## 3 Mathematical formulation

*x, y, z*are coordinates along the length (mm), depth (µm) and width (µm),

*u, v, w*are the corresponding flow velocities (m/s),

*ρ*and

*µ*are density (kg/m

^{3}) and dynamic viscosity (N-s/m

^{2}) of the fluid.

Boundary conditions are taken as constant mass flow rate at the inlet, constant pressure at the outlet, and no slip at the inner walls. Pressure based solver is used with semi implicit method for pressure linked equations (SIMPLE) scheme for pressure–velocity coupling. Second order discretization scheme is employed for pressure equation whereas momentum equation is discretized using quadratic upstream interpolation for convective kinetics (QUICK). Absolute convergence criterion is set as 10^{−5} for the residuals of all the equations.

Grid independent study for numerical simulations

Number of cells | Δ | % deviation |
---|---|---|

50,000 (10 × 10 × 500) | 243.83 | 4.13 |

2,40,000 (20 × 20 × 600) | 253.90 | 1.22 |

5,00,000 (25 × 25 × 800) | 257.86 | 0.38 |

9,00,000 (30 × 30 × 1000) | 258.84 |

### 3.1 Validation

*α*= 8°,

*H*= 200 µm,

*L*= 20 mm and

*α*= 4°,

*H*= 240 µm,

*L*= 20 mm. It is observed from Fig. 3 that the numerical results are in good agreement for microchannels with

*α*= 4° (9% average and 12% maximum deviation) and

*α*= 8° (12% average and 18% maximum deviation) with respect to the experimental data. The deviation can be attributed to some difference in the inlet and exit conditions; for example, there are inlet and exit reservoirs that have not been modelled in the simulations.

## 4 Variation in pressure drop

### 4.1 Theory

*L*is the total length of channel,

*ρ*is the density of liquid,

*V*is the average velocity,

*f*is the friction factor,

*D*

_{h}is the hydraulic diameter, and

*x*is the flow direction. Since the cross-section changes along the flow direction, the hydraulic diameter at each location can be expressed as:

*W(x)*is the variable width of the microchannel along the flow direction and

*H*is constant depth of the microchannel. The average velocity can be expressed as:

*A(x)*is the area of cross-section at

*x*location. Reynolds number Re(

*x*) along the flow direction can be written as:

*γ*is the aspect ratio at the specified location

Using the above equations, the theoretical pressure drop is calculated for different geometrical and flow parameters and compared with numerical pressure drop as discussed in the next section. The pressure drop is calculated over the flow rate range of 0.5–5 ml/min (8.33 × 10^{−6} to 8.33 × 10^{−5} kg/s).

### 4.2 Effect of inlet angle

### 4.3 Effect of width ratio

*L*= 20 mm, and

*H*= 100 µm. The width ratio has been changed while keeping all other parameters constant.

It is observed from Fig. 5 that for a given flow rate, pressured drop increases with increase in width ratio. In addition, it can be seen that at larger width ratio, i.e. *ɛ *= 15, the pressure drop is a non-linear function with flow rate and it shows a large deviation from the theoretical pressure drop, whereas other cases are in good agreement with the theoretical pressure drop. The reason for this non-linearity is due to the appearance of flow reversal and separation zones, as explained in detail in Sect. 6.

### 4.4 Effect of depth and length

## 5 Characteristic length scale for diamond microchannel

*α*= 8°,

*L*= 20 mm and

*H*= 100 µm for a flow rate of 3 ml/min. The theoretical pressure drop is found to be 34% lesser than the actual pressure drop. Table 4 shows the hydraulic diameters calculated theoretically based on different methods and made a comparison of pressure drop between uniform and diamond microchannels for the above- mentioned case. The pressure drop for uniform microchannel is predicted by using Eq. 11 by employing the theoretically calculated hydraulic diameter by the above-mentioned methods. From Table 4, it is observed that the hydraulic diameters obtained by different theoretical methods are not the appropriate length scales for this problem and hence we need to look for better location to predict the theoretical pressure drop.

Theoretically calculated hydraulic diameters using different methods and pressure drop comparison between diamond and uniform microchannel

Sl. no. | Methodology | Characteristic location | Pressure drop comparison |
---|---|---|---|

1 |
| x = 0 | Δ |

2 | \(D_{h} = \frac{1}{L}\mathop \smallint \limits_{0}^{L} D_{h} \left( x \right)dx\) | x = L/3.6 | Δ |

3 | \(D_{h} = \frac{{2W_{avg} H}}{{\left( {W_{avg} + H} \right)}}\) | x = L/4 | Δ |

4 | \(D_{h} = \frac{{2W_{max} H}}{{\left( {W_{max} + H} \right)}}\) | x = L/2 | Δ |

### 5.1 Equivalent hydraulic diameter

The algorithm followed for estimating the equivalent hydraulic diameter is as follows:

### *Step 1*

Calculate the value of Poiseuille number (*f∙*Re) at different locations in the microchannel and compare the calculated value of *f∙*Re with theoretical *f∙*Re obtained by using Eq. 11.

### *Step 2*

Locate the width at which the numerically obtained Poiseuille number coincides with the theoretical value, which is called characteristic width.

### *Step 3*

Identify the location at which the characteristic width is located from the inlet of the channel and obtain the characteristic length.

*Po*=

*f∙*Re is given by

*P*and \(\dot{m}\) are the pressure drop and mass flow rate,

*f*and Re are Fanning friction factor and Reynolds number,

*D*

_{h}

*, A*are, respectively, hydraulic diameter, area of cross-section, which are calculated at characteristic location. The theoretical

*f∙*Re values are calculated using Eq. 11. From Fig. 9,

*W*

_{c}is the characteristic width which is located at

*L/n*distance from the inlet of the microchannel width

*W*

_{i}and

*L*’

_{c}(=

*L/n*) refers to the characteristic length at which the equivalent hydraulic diameter is calculated.

### 5.2 Variation in Poiseuille number

*f∙*Re) is calculated using Eq. 17 and it is compared with the theoretical

*f∙*Re obtained using Eq. 11. Figure 12 shows the Poiseuille number variation with Reynolds number for different cases for which hydraulic diameter is taken at the modified characteristic location. It is observed that

*f∙*Re is almost invariant with Reynolds number and in good agreement with the Shah and London [37] correlation. In addition,

*f∙*Re value lies within an absolute average error of 7% compared to theoretical

*f∙*Re values (maximum deviation is 28%). The equivalent hydraulic diameter at the characteristic location provides frictional characteristics equivalence between diamond microchannel and uniform microchannel.

## 6 Flow characterization and critical parameters

In previous studies for microchannels with non-uniform cross-section, the averaged channel size is employed for flow characterization [40, 41]. For the geometric configuration considered in this work, hydraulic diameter *D*_{h}=* 4A/P′* appears to be a proper parameter, where *A* and *P′* are the cross-sectional area and the perimeter at the characteristic location (at L/7th location from inlet) obtained in Sect. 5 and the corresponding value of Re can be calculated using this value of hydraulic diameter.

*α*or large width ratio

*ɛ*leads to anomalous behavior in pressure drop variation because of the strong flow velocity variation in the channel which enhances flow instability. For better understanding of this behavior, we are considering flow resistance

*R*

_{f}which is defined as the ratio of overall pressure drop across the microchannel to the given flow rate [23, 42]:

*R*

_{f}

*as*a function of Re for different inlet angles and width ratios. It is also known from the theory that for laminar flows, the flow resistance is independent of Reynolds number. From the figure, there exists a critical Re, Re

_{c}, which depends on the geometry of the microchannel. This is consistent with the previous findings in the literature for microchannels with non-uniform cross-sections [18, 19, 20, 21].

_{c}is about 185 for inlet angle of 12° and 135 for 16° respectively. At this value,

*R*

_{f}deviates from its mean value by 5.5% for both angles. From Fig. 13b, it is about 139 for highest width ratio i.e. for

*ɛ*=15 which is obtained at the flow rate of 2.5 ml/min and

*R*

_{f}shows 10% deviation from its mean value whereas for all other cases flow remains in laminar region as flow resistance shown to be independent of Reynolds number. This flow transition explains the anomalous behavior of pressure drop in Figs. 3 and 4 for higher inlet angle (

*α*) and higher width ratio (

*ɛ*). It also shows that the effect of width ratio is more significant than inlet angle on flow resistance, as confirmed further by velocity streamline plots shown in Fig. 14.

From Fig. 14, the presence of recirculation zones and flow separation is evident for Re > Re_{c}. Because of this, flow deviates from laminar behavior and hence uniform microchannel theory for laminar flows cannot be applied beyond this limit. This is consistent with Fig. 13 where *f∙*Re values for higher angle shows large deviation from its corresponding theoretical value. Furthermore, previous studies [23] mentioned that Re_{c}> 100 for the converging–diverging cross-section microchannel employed in their study.

Although the presence of recirculation zones causes the change in the flow behavior, they are restricted to the entry region of the microchannel. These vortices, however may arise at the junction of the inflection region (region at which channel curvature changes from diverging to converging section) and becomes important for further increase in inlet angle and width ratio [43, 44].

### 6.1 Discussion

The disturbances in the flow pattern usually create flow instabilities [45]. Flow separation has a high tendency to destabilize the flow. The present geometry displays a flow separation zone at the entry region of the microchannels for the higher inlet angle and the higher width ratio (Fig. 14). However, the accelerating flow through converging microchannel causes the separated streamlines to reattach to the wall along the flow direction resulting in restricting the recirculation zones to the entry region of the microchannel. The flow reversal region is not strong enough to create the flow disturbance, however, the flow starts deviating from the laminar behaviour as evident from Fig. 13. This indicates a flow transition but this does not imply that the flow is turbulent. In other words, a new complex but still laminar flow evolves; however, for further increase in the values of *α*, *ɛ* and Re, the flow pattern tends to become unstable and may even transit to turbulent flow regime.

## 7 Flow physics

To understand the internal flow physics, it is necessary to understand the local distribution of pressure, velocity and shear stress for flow through diamond microchannels.

### 7.1 Local pressure distribution

*P*is local centerline pressure along the flow direction and

*P*

_{in}and

*P*

_{out}are centerline pressures at inlet and outlet respectively. The length of the microchannel is 20 mm and the flow rate considered for this simulation is 5 ml/min.

From Fig. 15, for straight microchannel, the pressure decreases along the flow direction with constant negative slope that indicates the fully developed flow. For microchannels with inlet angles of 4°–12°, the static pressure decreases along the flow direction, however, the dynamic pressure decreases and increases in diverging and converging sections. This is because the flow decelerates and accelerates in diverging and converging sections (Fig. 17). Consequently, the steep pressure drop is observed in the region 0 < *x/L* < 0.2 and this steepness increases as inlet angle increases. For the microchannel with *α* = 16°, the adverse pressure gradient occurs because of an increase in static pressure in the region 0 < *x/L* < 0.2. The increase in the static pressure is large enough to cause the excessive momentum loss inside the boundary layer, which leads to the occurrence of flow separation near the wall. In addition, the presence of steep pressure gradients has significant implications for the bubble dynamics in diamond microchannel. A bubble in a diamond microchannel moves easily especially near the entrance because of the presence of steep pressure gradient.

### 7.2 Velocity distribution

Figures 17a, b illustrate the effect of inlet angle on \(u.\frac{du}{dx}\) (convective deceleration and acceleration). The centerline velocity was considered for this purpose and the flow rate is 5 ml/min. From Fig. 17a, it is observed that the microchannel with inlet angle of 4° has larger deceleration rate (by 4%) and larger acceleration rate (by 32%) when compared to the microchannel with inlet angle of 16° although there is a sharp increase in acceleration rate for larger angle after *x/L* = 0.95. Figure 17b shows the enlarged view of the nature of \(u.\frac{du}{dx}\) in the inflection zone (0.4 < *x/L* < 0.6). It is noted that acceleration and deceleration for larger angle remains almost zero, which indicates that both the local and material acceleration are zero and nearly constant velocity is maintained in this region.

*L*= 20 mm and flow rate of 5 ml/min. Velocity profile taken at mid-plane of the microchannel is non-dimensionalized using the maximum velocity, which is at the midpoint of each plane. The velocity profile is parabolic at

*x/L*= 0 as flow enters in fully developed condition (Fig. 18a). Figure 18b shows an underdeveloped parabolic profile at

*x/L*= 0.2. The velocity profile remains flat throughout the channel, which indicates that the flow is in developing condition (Figs. 18c, d, e). This is because of the variation of Reynolds number (Re

_{x}) in the microchannel and its relation with the hydro-dynamically developing/entrance length (

*x*

_{h}). It is known that the entrance length (

*x*

_{h}) of the channel flow is directly proportional to Re

_{x}[46]

In the converging section, because of flow acceleration, the Reynolds number increases rapidly towards the outlet of the microchannel. Hence, it is difficult to obtain developed flow and form a parabolic velocity profile along the width near the exit of the microchannel. Therefore, the velocity profile remains flat towards the exit of the microchannel. In addition, there is a difference in velocity profiles at *x/L* = 0.2 and *x/L* = 0.8 even though they are at the same location from the narrow ends of both diverging and converging section. This shows the difference in rate at which flow decelerates and accelerates in diverging and converging sections.

*L*= 20 mm, and flow rate of 5 ml/min. The velocity profile along the depth of the microchannel is observed to be parabolic at all locations along the streamwise direction for all the microchannels. This implies that the flow remains in the fully developed condition along the depth of the microchannel irrespective of the location in microchannel.

### 7.3 Wall shear stress variation

*L*= 20 mm. The flow rate is 5 ml/min for this analysis.

From Fig. 20, it is observed that both bottom wall shear stress and sidewall shear stress decreases with increase in inlet angle even though there is a sharp increase in the wall shear stress for larger angle. This is because velocity along the flow direction decreases with increase in inlet angle (Fig. 16). In addition, the variation of shear stress along the flow direction justifies the continuous variation of velocity gradient at the wall along the flow direction. For the case of 4°, the bottom wall shear stress decreases in the diverging section and increases in the converging section, whereas on the sidewall, the shear stress drops suddenly at the entry and then maintains almost constant throughout the diverging region and increases in the converging region. The variation of bottom wall shear stress and sidewall shear stress in case of 16° is similar to that of smaller inlet angle with sharp increase in the shear stress towards the outlet of the microchannel. This is because of flow acceleration towards the outlet of the microchannel, which leads to a sharp increase in the velocity gradient towards the exit of the microchannel.

## 8 Conclusions

*δ*) is proposed to generalise the different cases of non-uniform microchannels. For the geometric and flow parameters considered in this study, the following observations have been made:

- 1.
The pressure drop varies inversely with inlet angle, width ratio and hydraulic diameter has significant influence. It is also observed that the numerically obtained pressure drop is in good agreement with the theoretically calculated pressure drop, which implies that the existing theory for uniform microchannels can also be applied to present geometrical configuration by using an appropriate characteristic length scale.

- 2.
The equivalent hydraulic diameter is calculated at 1/7th of the length of the microchannel from its inlet is proposed as the length scale for diamond microchannel. This can be used to design and fabricate corresponding equivalent uniform microchannel that has same pressure drop as diamond microchannel. The Hagen-Poiseuille equation can be applied to diamond microchannel using this length scale, its location is shown to be independent of geometric, and flow parameters considered in this study.

- 3.
Inlet angle (

*α*) and width ratio (*ɛ*) are parameters that characterize the flow through diamond microchannel. It is observed that effect of width ratio has more significance than inlet angle on flow resistance. - 4.
For 0° <

*α*< 12° and 3.3 <*ɛ*< 11.5, the pressured drop can be predicted for the diamond microchannel using the straight microchannel theory in laminar flow regime for Re in the range of 0–400 whereas for 12° <*α*< 16° and 11.5 <*ɛ*< 15, it can be applied for Re < Re_{c}. - 5.
Flow physics in diamond microchannel is explained with the help of local pressure, velocity, and the wall shear stress distribution. The presence of steep pressure gradients at the entry region of microchannel provides an easy movement of bubble through microchannel that may have significant implications in the study of bubble dynamics in multiphase flows. The present configuration may also have significant implications in the study of particle dynamics. The appearance of recirculation zones may contribute in entrapment of particles [47, 48, 49, 50].

- 6.
Flow remains in developing condition along the width throughout the microchannel even though flow enters the channel in fully developed condition. It is confirmed with the help of velocity profiles along the width at different locations along the flow direction. Flow however remains in fully developed condition along the depth throughout the microchannel.

- 7.
Velocity varies along the flow direction because of the variable cross-section; however, the flow decelerates and accelerates in both diverging and converging flow passages at different rate even though both diverging and converging passages of same size. This continuous variation of velocity gradient along the flow direction is explained further with the help of wall shear stress distribution.

## Notes

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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