Low Reynolds number flow of liquids over micron-sized structures and the control of subsequently induced shear stress are critical for the performance and functionality of many different microfluidic platforms that are extensively used in present day lab-on-a-chip (LOC) domains. However, the role of geometric form in systematically altering surface shear on these microstructures remains poorly understood. In this study, 36 microstructures of diverse geometry were chosen, and the resultant overall and facet shear stresses were systematically characterized as a function of Reynolds number to provide a theoretical basis to design microstructures for a wide array of applications. Through a set of detailed numerical calculations over a broad parametric space, it was found that the top facet (with respect to incident flow) of the noncylindrical microstructures experiences the largest surface shear stress. By systematically studying the variation of the physical dimensions of the microstructures and the angle of incident flow, a comprehensive regime map was developed for low to high surface shear structures and compared against the widely studied right circular cylinder in cross flow.
Low Reynolds number flow of viscous fluids over solid surfaces in either a confined or an open environment is a ubiquitous phenomenon that can be commonly observed. In this flow regime with Reynolds number, Re < 100, and in many LOC practical applications with Re < 1, the fluid motion at all the physical solid–fluid interfaces is usually considered to follow the no-slip boundary condition . Therefore, the resulting transverse, nonzero velocity gradients contribute toward fluid shear which has been successfully exploited in numerous practical applications over the years, including biosensors, advanced health care diagnostics, materials processes, and energy-related technology advancements [2–7].
The operational efficiency for these diverse applications is critically impacted by the ability to maintain specific regimes of shear stress on the functional surfaces within the underlying devices. For example, lower surface shear was beneficial in improving the functionality of microfluidic chips with enhanced capture yield of circulating tumor cells , while in the case of artificial heart valves, reducing high-shear-induced coagulation of blood on the valve surface minimized the risk of thrombogenesis and thromboembolism . For energy-related applications, lower surface shear increased microbial fuel cell sensitivity toward detecting Cu(II) toxicity .
On the other hand, flows with higher shear rate were beneficial in inducing microbial enrichment of microbial fuel cell anodes, resulting in a threefold increase in power output . Furthermore, customized microfluidic cell culture chips were shown to exhibit superior gene expression under increased fluid shear . However, in most low Re number (Re < 100) flow applications, shear stress is manipulated solely by modulating the flow rate due to the ease in controlling external pressure or potential gradients that act as common driving forces for these flows, especially in LOC environments. In contrast, an open question remains: What is the extent to which shear stress can be controlled by engineering passive structures that can systematically manipulate the shear without the need to actively adjust flow rate?
Studies on the effect of surface geometries on surface shear stress have mainly focused on considerations of surface roughness. This includes studies on developing a correlation between the effect of actual roughness to that of closely packed sand grains , an evaluation of the skin friction coefficient and equivalent sand roughness data on various rough surfaces , and an analysis on randomly placed, nonuniform, three-dimensional roughness with irregular geometry and arrangement . Canonical flows over solid surfaces have predominantly focused on cylinders, which have been extensively studied for shear, both numerically and experimentally, for many years, and comprehensive reviews exist [12–15]. However, studies that investigate low Reynolds number flows (Re < 100) for surface shear on noncylindrical, three-dimensional structures are scarce—many studies that consider noncylindrical forms are in two-dimensional flows or focus on the fluid flow and do not mention surface shear effects. Indeed, the study of surface shear is limited to specific ad hoc applications or targeted operating conditions [16,17]. Some simple radial geometries such as cylindrical disks and spheroids have been investigated and have shown that high-pressure stagnation zones correspond to areas of low surface shear stress, but no conclusions were formed regarding the relationship between geometric form and surface shear stress . Similarly, in a computational fluid dynamics study that used images to align a competitive swimmer with the flow field, it was found that larger surface shear stresses were observed on areas of the body that presented a complex surface geometry, such as the head, shoulders, and heels .
Due to the scarcity of studies that investigate surface shear on distinguishable three-dimensional structural forms in low Reynolds number flows, no generalizations exist on how the surface structure actually manipulates or modulates shear stress under these conditions. Therefore, no generalizations exist on how the surface structure actually manipulates or modulates shear stress. However, in many applications, the geometry and layout of the physical structure in the flow path are critical for the device or system operation [20,21]. For example, many bioprocessing devices and bioreactors rely on the integrity of a biofilm adhered to their functional surfaces, and past works [22,23] have shown the dependence of biofilm morphology on surface shear stress. Surface structure geometry in several practical applications is typically chosen based on considerations such as ease of fabrication  or on what materials are commercially available  without much design focus on the flow–structure interaction, which actually governs the operation of these devices. Therefore, the resulting systems are inherently unoptimized for intended, shear-dependent functions because specific effects of geometry on surface shear stress were not explicitly considered.
Therefore, the purpose of this work is to provide a systematic analysis for low Reynolds number flow of water (a common working fluid in many LOC flows) over various microstructures on a flat, solid surface and subsequently characterize the induced fluid shear stress on the microstructure surface. Specifically, a regime map that directly correlates various microstructure shapes to regimes of surface shear stress has been developed. The regime map comprises a wide array of microstructures, such as cones, pyramids, rectilinear prisms, and other forms commonly found in engineering applications to provide a direct correlation between structure geometry, inlet flow conditions, and surface shear stress. Such regime maps can potentially provide starting point data to engineer higher operational efficiencies for applications due to the ability to now explicitly incorporate the geometric dependence of shear stress. In addition, the contribution of various microstructure facets (front, rear, side, and top of a structure) with respect to incoming flow toward overall fluid shear was also quantified as the structure was exposed to fully developed low Reynolds number flows. It is important to recognize that the focus of this work is on the study of external flows over microstructures where the wall constraints can be considered negligible.
In this paper, 36 geometric forms, each with the same height of 100 μm, were heuristically chosen based on common geometries found in a variety of engineering applications (as discussed in Sec. 1) and compared for relative shear stress, based on their detailed facet geometries with respect to incident or incoming flow (Fig. 1). The total wetted surface area for all the geometric shapes is plotted in Fig. S1, which is available under the “Supplemental Materials” tab on the ASME Digital Collection. Velocity gradients on individual facets are discussed for a cube, a common geometric form as a representative case. Using a cube of edge length of 100 μm as a base structure, the surface areas, edge lengths, and facet angles were systematically altered in incremental steps to study the effect of gradual changes in geometrical form over the base structure. In order to observe trends in surface shear stress based on the physical orientation of a structure with respect to the incident flow, a right pyramid and a cube were altered in systematic steps through one full rotation as observed with respect to the incident flow.
where: is the tensor contraction operator, and the superscript denotes the matrix transpose. Throughout this paper, “front facet” refers to the structural component facing the incident flow. Additionally, the overall and facet shear stress of all the microstructures discussed in this paper were nondimensionalized with respect to a reference case explicitly listed in Sec. 3. All the models were solved using the supercomputing cluster at the Ohio Supercomputer Center that employs 8328 HP Intel Xeon x5650 central processing units with 12 cores and 48 GB of memory per HP SL390 G7 node. The numerical model was validated with the past work , and good agreement was found on the spatial variation of velocity around the cylinders and the x-component of shear stress on the periphery of cylinder, as shown in Fig. S3, which is available under the “Supplemental Materials” tab on the ASME Digital Collection.
Results and Discussion
The 36 distinct geometric shapes considered in this study are divided into four categories based on traditional geometric definitions, namely: (1) rectilinear prisms (eight shapes with noncircular top), (2) radial prisms (11 shapes with top facet having a radial profile), (3) nonvertical prisms (ten shapes with top facet having varying heights), and (4) apex structures (seven shapes with no-top facet) as summarized in Fig. 1.
For a given Re, the overall shear stress of a microstructure, , was divided by the overall shear stress of a cylinder and reported as , thus nondimensionalizing the stress and providing a direct comparison to the cylinder, which has been one of the most common structures used in a variety of flow configurations as discussed in Sec. 1. Moreover, comparing the shear stress of the 36 cases chosen here with respect to the cylinder also facilitates ready comparison with the published data  and provides easy visualization to existing engineering applications that employ cylinders. Due to the nondimensionalization, =1 for a cylinder as shown in the regime map (Fig. 2). for a 100 μm diameter (and height) cylinder was calculated to be 7.5 × 10−5 N/m2 (Re = 0.001), 7.5 × 10−3 N/m2 (Re = 0.1), and 16.1 N/m2 (Re = 100). As the nondimensionalized shear stress for all the microstructures reported in this study was similar for Re = 0.001 and 0.1, results for Re = 0.1 will be discussed as a representative condition. The nondimensionalized overall shear stress, , for each structure (Fig. 2) as well as differences in front, top, side, and rear facets were examined (Figs. 3 and 4). The three-dimensional orientation of all the microstructures with respect to a global Cartesian coordinate frame and also the direction of incident flow are explicitly shown in Fig. 2.
In Fig. 2, a shear stress regime map for the overall shear stress, , as a function of geometry is presented for Re = 0.1 (with Re = 0.001 being similar to Re = 0.1) and 100. Rectilinear prisms have varying in the range of 0.73–0.88 for Re = 0.1, suggesting that the overall shear on a cubic prism is only 73% of the shear stress seen by a cylinder under similar flow conditions and critical dimensions. In rectilinear prisms, when the fluid contacts the front facet, the velocity gradient is larger but limited only to the edges and small everywhere else, in contrast to curvilinear prisms, where the velocity drop is extended out over nearly the entire surface area of the structure, resulting in higher surface shear stress. This is the first of many results that suggest a fluid flow over sharply angled features (such as rectilinear prisms) will, in general, result in lower surface shear stress when compared to curvilinear surfaces. At Re = 100, rectilinear prisms have varying over a broader range of 0.62–0.97. Nonvertical prisms exhibited an even wider spread in the magnitude of (0.38–0.80 compared to a cylinder) for Re = 0.1, but was confined between 0.37 and 0.72 at Re = 100. Radial prisms exhibited larger regimes of in magnitude (0.82–1.1 at Re = 0.1 and 0.74–1.2 at Re = 100); however, apex forms exhibited lower regimes of with respect to the rectilinear and radial prisms with the exception of the cone at Re = 0.1, where was estimated to be 0.92 or nearly the same as the cylinder, despite not having a top facet. Apex structures due to minimal cross-sectional area (normal to the flow field) would allow for increased momentum of the fluid flow over the entire structure reducing the velocity gradients over their surface. If true, then structures 19, 23, 25, and 26 (all forms with cross-sectional area less than 1 μm2) should also experience less surface shear stress in the Re = 100 case, which is demonstrated by the results.
The regime map study thus clearly indicates that although the overall shear stress experienced by a microstructure is similar at Re = 0.001 and 0.1, further extrapolation of shear stress at higher Re (100) is not obvious and therefore was explicitly evaluated (Fig. 2). In short, the regime map provides data trends for low Reynolds number flow that can be used for selection of geometric categories for a variety of applications as discussed in Sec. 1.
Velocity Gradients and Impact on Shear Stress.
Nondimensionalized shear stress experienced by individual facets (front, rear, side, and top facets) for all the 36 microstructures is plotted at Re = 0.1, 100 in Figs. 3, and 4, respectively. Shear stress experienced at each facet was nondimensionalized with of a cylinder at the respective Re, as discussed in Sec. 3.1.
In Fig. 3, critical facet stress of 0.5 (after nondimensionalization, i.e., 50% of the overall shear on a cylinder) was benchmarked for front and rear facets, as the transition between high and low shear stress structures. Similarly, as the top facet was found to experience higher shear stress in comparison to all other facets, a critical facet stress of 1.5 was chosen to differentiate between high shear and low shear structures. The of side facets, though greater than 0.5, varied over a narrow range for all the microstructures (average 0.79 ± 0.13). Similarly, critical shear stresses of 1 for the front facet, 0.3 for the rear facet, 1 for the side facet, and 1.5 for the top facet were chosen to differentiate high shear and low shear structures at Re = 100, as shown in Fig. 4. This classification between “high shear” and “low shear” structures in Figs. 3 and 4 was implemented to aid in the selection of microstructures for shear-dependent applications. Nondimensionalized facet shear stress as a function of Re is plotted for all the microstructures in Fig. S4, which is available under the “Supplemental Materials” tab on the ASME Digital Collection.
Next, the role of velocity gradients in contributing to the overall shear was evaluated by considering a cube of side 100 μm as a representative structure. The velocity gradients were nondimensionalized with respect to the overall shear rate of a cylinder at Re = 0.1 (7.89 s−1). The nondimensionalized velocity gradients that contribute to shear in the X, Y, and Z directions across the top facet were calculated and are shown in Fig. 5 for a cube of side 100 μm at Re = 0.1 as a representative case. As discussed in Sec. 2, data were plotted beginning at 5 μm from each edge in order to eliminate the effects of singularities. Therefore, each cross section of the cube reported in Fig. 5 has an area of 90 μm × 90 μm instead of 100 μm × 100 μm. The effect of this strategy is discussed and shown to be valid in the supporting information section (Fig. S5, which is available under the “Supplemental Materials” tab on the ASME Digital Collection).
The velocity vector is given by , where , , and are the unit vectors along the coordinate axes (shown in Fig. 5). Since the top facet is located in the X–Y plane, and due to the symmetric structure of the cube, the nondimensionalized velocity gradient is expected to dominate along Z (velocity gradient∼O(10 deg)) and was found to be negligible across the X,Y direction (velocity gradient ∼ O(10−15)) as shown in Fig. 5. The gradient of the velocity along Z is attributed to the no-slip boundary condition at the top facet, which leads to the development of a transverse boundary layer. This transverse boundary layer and the resulting velocity gradient are the primary causes of the surface shear stress calculated on the facet. Since the incident flow is in the X direction, (magnitude of ) is greater than , and therefore, the magnitude of is smaller by an order of magnitude compared to , as also shown in Fig. 5. In addition, contribution of toward can be ignored when averaged, given the symmetry involved in the structure. Similarly, the average of across the top facet was estimated to be zero (not shown) and therefore does not contribute toward the shear stress experienced by the top facet. From Fig. 5, exhibits the maximum magnitude compared to all other velocity gradients that contribute toward shear stress. The negligible magnitude of all other velocity gradients, ∼ O(10−15) as observed in Fig. 5 suggests that the variation of velocity in a direction normal to a particular facet predominantly influences the magnitude of shear stress at that facet. Therefore, the surface shear stress is largest along the edges of the facet (∼1.5 times larger than the average over the facet) and peaks near the corners of the facet (∼2.5 times larger than the average over the facet). Thus, one would hypothesize that structures having multiple edges and corners should experience higher overall shear stress, but as can be seen from Figs. 3 and 4, this is not the case. This discrepancy arises as the actual surface area of edges and corners is negligible compared to the overall facet area and therefore contributes minimally (<2%) to the overall average shear stress.
Figure 6 shows the nondimensionalized velocity gradients that primarily influence plotted across the front ( being the normal unit vector), side ( being the normal unit vector), and rear ( being the normal unit vector) facets for a cube with edge length 100 μm at Re = 0.1. Orientation of the cube with respect to the three-dimensional axis and incoming flow is as shown in Fig. 5. Since > , the gradient of (experienced only at the front and rear facets) is expected to be less than the gradient of (experienced only at the side facet). Therefore, as shown in Fig. 6, the shear rate experienced by the side facet is about 2.5 times greater than the front and rear facet shear stresses for a cube. This trend is in agreement with the results reported for Re = 0.1 for a cube and suggests that, in any structure with facets of similar orientation and shape to those of a cube, the side facets will strongly contribute to the overall shear stress for the structure (and referring to Figs. 2 and 3, this appears to be the case: structures with overall form similar to that of the cube experience comparable magnitudes of surface shear stress). However, for noncubic geometries, it is clear from the regime map (Fig. 2) that the overall geometric form affects the shear stress on individual facets differently (Figs. 3 and 4) and therefore justifies the need for such a regime map in designing next-generation LOC devices with embedded microstructures [3,36–39].
Parametric Manipulation of Physical Dimensions.
As shown in Fig. 7, four physical dimensions of a cube were incremented to progressively alter its shape, to achieve the final structures shown by the third image in each sequence at the top of each figure panel. The shear stress on each individual facet in this discussion is now nondimensionalized with respect to or the shear stress on the front facet of a cube, i.e., the facet of the geometry facing the incoming flow experienced by an unaltered cube at Re = 100, and reported as in Figs. 7(a)–7(d), to facilitate easy comparison to the unaltered case. For Re = 100, of the unaltered cube was calculated to be 12.5 N/m2.
In Fig. 7(a), the results of incrementing 0 deg ≤ ≤ 26.6 deg are shown ( is the angle of tilt for each side of the cube), which transforms a cube into a right pyramid as it increases. It was found that the shear stress at the top facet, , increased with a quadratic dependence on ( = 0.98, where is the coefficient of determination) for the systematic translation of a cube to a pyramid. This scaling is expected because as the flat faces of the cube were systematically altered to approach triangular cross sections, the surface area of the top facet continued to decrease.
Moreover, the front facet exhibits a strong linear correlation ( =−0.91, where is the linear correlation coefficient) between decreasing and increasing . Though the front and rear facets have the same area, the shear stress experienced by the rear facet is an order of magnitude lower than that of the front facet. Additionally, was found to be invariant with a change in surface area, unlike , which implies that the shear stress experienced by each facet is strongly influenced by the overall geometric form and Re. At = 0, all the facets of a cube have equal surface area and from Eq. (4), the overall stress is dominated by , which has the maximum magnitude, which is in agreement with the discussion in Sec. 3.2. However, at = 26.6 deg, the top surface vanishes (or collapses to a point as the pyramid tip) and therefore does not contribute to overall stress. Unlike the side and rear facets whose shear stress is invariant to , decreases linearly and therefore from Eq. (4), decreases linearly as observed in Fig. 7(a), suggesting that the front facet is instrumental in dictating with an increase in .
In Fig. 7(b), results are shown for incrementing 0 ≤ ≤ 50 μm, i.e., increasing cube edge curvature () to round-out the cube and eventually reach a cylinder. The results for the shear stress on the front curves suggest that the “sharper” a geometric feature is (in this case, smaller values of ), the more surface shear stress it will exhibit, as expected in Ref.  and from previous discussions, above. Also, with increase in , the surface area of the top facet decreases by 21.5% from a square ( = 0 μm) to a circle ( = 50 μm), compared to a 100% decrease in the case discussed in Fig. 7(a). In addition, the top facet experiences the highest transverse velocity gradients when > 25 μm, as indicated by a higher value of in comparison to other facets in Fig. 7(b). Therefore, coupled with a significant contributed area and shear stress toward the estimation of ( contributes 36.0–41.5% to overall shear with increase in ), the top facet dictates the variation of with .
In Fig. 7(c), the results of incrementing the angle of tilt, , of the front face of a cube are shown. Therefore, the area of the front and side facets varies with , and the area of the rear facet remains fixed as shown in Fig. 7(c). Again, shows a quadratic increase (R2 = 0.97) for the evaluated values of . Despite an overall increase in from 0 deg to 45 deg, thereby increasing the front facet area by 41%, only increases by 0.86%. In comparison, the percent contribution of to is 16.9 ± 0.25% for all the values of , suggesting is independent of . Together, these results suggest that the top and front facets dictate for all the values of .
In Fig. 7(d), results are shown for incrementing 0 ≤ ≤ 50 μm, i.e., increasing the curvature () of the cube's top facet. As is increased, the height (and therefore area) of the front facet was systematically reduced to maintain the total height of the structure at 100 μm. The results for the front curve and the top facets confirmed previous results that sharper geometric features result in larger values of along expected trends . exhibits only a 3.4% overall decrease, while shows a strong linear correlation (r=−0.99) with , decreasing 41.5% overall, and remaining within 21.5% of . It is important to note that the area of the newly formed front curve increases as increases.
A common trend observed in all the cases was that the magnitude of shear stress on the rear facet, , remains small (∼3%) and is mostly unaltered by the changes to structure morphology, in agreement with the regime map (Re = 100, Fig. 2(b)). Similarly, minimal changes in the value of suggest that in most low Reynolds number based flows, the surfaces exposed to the incoming fluid (typically the front and top facets) dictates the performance characteristics. Also, a decrease in the area of the front facet when transformed from a cube (Figs. 7(a), 7(b), and 7(d)) results in a decrease in . However, decreasing the surface area of the top facet (Figs. 7(a)–7(d)) leads to a quadratic increase in the magnitude of most likely due to the increase in the amount of area exposed to a large velocity gradient near the edges of the facet. Overall, the results suggest that to produce a structure with larger overall surface shear stress, smaller top facets, rounder top facets, and flatter front facets normal to the incident flow are preferred, but of most importance is the form of the top facet. Thus, the trends previously observed and discussed in relation to Fig. 2 are further illuminated. In conclusion, the overall results in Fig. 7 suggest that the shear stress experienced by various facets is strongly influenced by the overall geometric form of microstructures.
Effect of Altering Angle of Rotation.
The effect of changing the angle of orientation with respect to the incident flow for a pyramid, , and cube, , was investigated (Fig. 8). The axis of rotation for both the pyramid and cube is shown in Fig. 8. To facilitate comparison between the rotated structures and their nonrotated forms, the reported values of shear stress were nondimensionalized with at = 0 deg and = 0 deg.
In Fig. 8(a), the pyramid's feature of interest (FOI = front facet at = 0 deg) exhibits a global minimum at = 180 deg. The maximum values of occur at = 60 deg and 300 deg. Therefore, the magnitude of shear stress at the front is maximum when angled at ± 60 deg from the incident flow. From Fig. 2, we see that, in general, structures 5 and 8 (which have angled, square front facets) experience larger surface shear stress than the cube, whose front facet is normal to the flow. This result indicates that the surface shear stress on a structural form can be manipulated and achieved purely by changing its orientation with respect to the incoming flow. The shear stress exhibits a smooth, sinusoidal distribution of both and as is varied, i.e., the overall and facet shear stress is symmetric as the orientation of the structure with respect to incoming flow is varied. Figure 8(b) shows the effect of changing for a cube. The maximum values of occur at = 50 deg and 310 deg (i.e., ±50 deg with respect to the incident flow, which agrees closely with the results for the pyramid). For the top facet, the maximum values of occur at = 45 deg, 135 deg, 225 deg, and 315 deg, corresponding to orientations that exhibit a maximum gradient in velocity for the same top facet surface area.
Summary and Conclusions
A systematic numerical analysis for the 36 diverse geometries was analyzed to estimate overall and individual facet shear stress at Reynolds numbers (Re) of 0.001, 0.1, and 100, which spans five orders of magnitude for relatively low Re flows typically seen in many viscous flow and LOC applications. The structures were divided into four categories based on the geometry of the top facet, namely: rectilinear prisms, radial prisms, nonvertical prisms, and apex structures.
Overall shear stress of a microstructure was nondimensionalized with the overall shear stress of a cylinder at a given Re and reported in a regime map. The results indicate that the nondimensionalized facet and overall shear stress for all the 36 structures were found to be the same at Re = 0.001 and 0.1. However, as Re was increased to 100, facet and overall shear stress was found to vary in comparison to Re = 0.1. Since the low Reynolds number flow is incident in one-direction, the magnitude of the transverse velocity components is lower compared to the axial component (which is along the direction of flow). Therefore, facets that include the gradient of axial velocity toward the estimation of shear rate experience a higher magnitude of shear stress compared to facets that take into account the transverse component of velocity. In addition, by changing the angle of rotation, critical angles were found for a cube and pyramid for which the surface shear stress was maximum in magnitude. The results of this work are expected to provide a broad basis for choosing microstructures that are essential for shear-dependent applications.
The computational facilities at the Ohio Supercomputing Center (OSC) are acknowledged for the support. The U.S. Department of Energy (DOE) is acknowledged for the partial funding support for the personnel through ARPA-E under Grant No. DE-AR0000282. The discussions with Tong Lin are also acknowledged.
area of facet of interest
fluid pressure (components vary spatially)
linear correlation coefficient
radius of curvature of top facet
radius of curvature of fillet
coefficient of determination
- Re =
X component of
Y component of
Eulerian fluid velocity (components vary spatially)
Z component of
- X =
global Cartesian coordinate in the direction of inlet flow (see Fig. 2)
unit vector along X
- Y =
global Cartesian coordinate perpendicular to the flow
unit vector along Y
- Z =
global Cartesian coordinate perpendicular to the flow
unit vector along Z
average shear rate
angle of inclination of each facet of cube
angle of incidence of front facet normal to the incident flow
fluid viscosity (water in this study)
fluid density (water in this study)
average shear stress of facet i (N/m2)
shear stress of facet i (N/m2)
nondimensionalized average shear stress of facet i
nondimensionalized overall shear stress of microstructure
nondimensionalized average shear stress of front facet
nondimensionalized average shear stress of rear facet
nondimensionalized average shear stress of side facet
nondimensionalized average shear stress of top facet
angle of rotation of cube
angle of rotation of pyramid