Kaushik K. Rangharajan1

Department of Mechanical and

Aerospace Engineering,

The Ohio State University,

Columbus, OH 43210

e-mail: rangharajan.1@osu.edu

Matthew J. Gerber1,2

Department of Mechanical and

Aerospace Engineering,

The Ohio State University,

Columbus, OH 43210

e-mail: gerber211@g.ucla.edu

Shaurya Prakash3

Department of Mechanical and

Aerospace Engineering,

The Ohio State University,

Columbus, OH 43210

e-mail: prakash.31@osu.edu

Effect of MicrostructureGeometric Form on SurfaceShear StressLow Reynolds number flow of liquids over micron-sized structures and the control of sub-sequently induced shear stress are critical for the performance and functionality of manydifferent microfluidic platforms that are extensively used in present day lab-on-a-chip(LOC) domains. However, the role of geometric form in systematically altering surfaceshear on these microstructures remains poorly understood. In this study, 36 microstruc-tures of diverse geometry were chosen, and the resultant overall and facet shear stresseswere systematically characterized as a function of Reynolds number to provide a theoreti-cal basis to design microstructures for a wide array of applications. Through a set ofdetailed numerical calculations over a broad parametric space, it was found that the topfacet (with respect to incident flow) of the noncylindrical microstructures experiences thelargest surface shear stress. By systematically studying the variation of the physicaldimensions of the microstructures and the angle of incident flow, a comprehensive regimemap was developed for low to high surface shear structures and compared against thewidely studied right circular cylinder in cross flow. [DOI: 10.1115/1.4034363]

1 Introduction

Low Reynolds number flow of viscous fluids over solid surfacesin either a confined or an open environment is a ubiquitous phe-nomenon that can be commonly observed. In this flow regimewith Reynolds number, Re< 100, and in many LOC practicalapplications with Re< 1, the fluid motion at all the physicalsolid–fluid interfaces is usually considered to follow the no-slipboundary condition [1]. Therefore, the resulting transverse, non-zero velocity gradients contribute toward fluid shear which hasbeen successfully exploited in numerous practical applicationsover the years, including biosensors, advanced health care diag-nostics, materials processes, and energy-related technologyadvancements [2–7].

The operational efficiency for these diverse applications is crit-ically impacted by the ability to maintain specific regimes of shearstress on the functional surfaces within the underlying devices.For example, lower surface shear was beneficial in improving thefunctionality of microfluidic chips with enhanced capture yield ofcirculating tumor cells [3], while in the case of artificial heartvalves, reducing high-shear-induced coagulation of blood on thevalve surface minimized the risk of thrombogenesis and throm-boembolism [4]. For energy-related applications, lower surfaceshear increased microbial fuel cell sensitivity toward detectingCu(II) toxicity [2].

On the other hand, flows with higher shear rate were beneficialin inducing microbial enrichment of microbial fuel cell anodes,resulting in a threefold increase in power output [5]. Furthermore,customized microfluidic cell culture chips were shown to exhibitsuperior gene expression under increased fluid shear [8]. How-ever, in most low Re number (Re< 100) flow applications, shearstress is manipulated solely by modulating the flow rate due to theease in controlling external pressure or potential gradients that actas common driving forces for these flows, especially in LOC envi-ronments. In contrast, an open question remains: What is the

extent to which shear stress can be controlled by engineering pas-sive structures that can systematically manipulate the shear with-out the need to actively adjust flow rate?

Studies on the effect of surface geometries on surface shearstress have mainly focused on considerations of surface rough-ness. This includes studies on developing a correlation betweenthe effect of actual roughness to that of closely packed sand grains[9], an evaluation of the skin friction coefficient and equivalentsand roughness data on various rough surfaces [10], and an analy-sis on randomly placed, nonuniform, three-dimensional roughnesswith irregular geometry and arrangement [11]. Canonical flowsover solid surfaces have predominantly focused on cylinders,which have been extensively studied for shear, both numericallyand experimentally, for many years, and comprehensive reviewsexist [12–15]. However, studies that investigate low Reynoldsnumber flows (Re< 100) for surface shear on noncylindrical,three-dimensional structures are scarce—many studies that con-sider noncylindrical forms are in two-dimensional flows or focuson the fluid flow and do not mention surface shear effects. Indeed,the study of surface shear is limited to specific ad hoc applicationsor targeted operating conditions [16,17]. Some simple radial geo-metries such as cylindrical disks and spheroids have been investi-gated and have shown that high-pressure stagnation zonescorrespond to areas of low surface shear stress, but no conclusionswere formed regarding the relationship between geometric formand surface shear stress [18]. Similarly, in a computational fluiddynamics study that used images to align a competitive swimmerwith the flow field, it was found that larger surface shear stresseswere observed on areas of the body that presented a complex sur-face geometry, such as the head, shoulders, and heels [19].

Due to the scarcity of studies that investigate surface shear ondistinguishable three-dimensional structural forms in low Reyn-olds number flows, no generalizations exist on how the surfacestructure actually manipulates or modulates shear stress underthese conditions. Therefore, no generalizations exist on how thesurface structure actually manipulates or modulates shear stress.However, in many applications, the geometry and layout of thephysical structure in the flow path are critical for the device orsystem operation [20,21]. For example, many bioprocessing devi-ces and bioreactors rely on the integrity of a biofilm adhered totheir functional surfaces, and past works [22,23] have shown thedependence of biofilm morphology on surface shear stress. Sur-face structure geometry in several practical applications is

1K. K. Rangharajan and M. J. Gerber contributed equally to this work.2Present address: Department of Mechanical and Aerospace Engineering,

University of California, Los Angeles, Los Angeles, CA 90095.3Corresponding author.Contributed by the Fluids Engineering Division of ASME for publication in the

JOURNAL OF FLUIDS ENGINEERING. Manuscript received October 8, 2015; finalmanuscript received July 21, 2016; published online September 20, 2016. Assoc.Editor: Daniel Maynes.

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typically chosen based on considerations such as ease of fabrica-tion [24] or on what materials are commercially available [25]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 onsurface shear stress were not explicitly considered.

Therefore, the purpose of this work is to provide a systematicanalysis for low Reynolds number flow of water (a common work-ing fluid in many LOC flows) over various microstructures on aflat, solid surface and subsequently characterize the induced fluidshear stress on the microstructure surface. Specifically, a regimemap that directly correlates various microstructure shapes toregimes of surface shear stress has been developed. The regimemap comprises a wide array of microstructures, such as cones,pyramids, rectilinear prisms, and other forms commonly found inengineering applications to provide a direct correlation betweenstructure geometry, inlet flow conditions, and surface shear stress.Such regime maps can potentially provide starting point data toengineer higher operational efficiencies for applications due to theability to now explicitly incorporate the geometric dependence ofshear stress. In addition, the contribution of various microstructurefacets (front, rear, side, and top of a structure) with respect toincoming flow toward overall fluid shear was also quantified asthe structure was exposed to fully developed low Reynolds num-ber flows. It is important to recognize that the focus of this workis on the study of external flows over microstructures where thewall constraints can be considered negligible.

2 Methodology

In this paper, 36 geometric forms, each with the same height of100 lm, were heuristically chosen based on common geometriesfound in a variety of engineering applications (as discussed inSec. 1) and compared for relative shear stress, based on theirdetailed facet geometries with respect to incident or incomingflow (Fig. 1). The total wetted surface area for all the geometricshapes 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, acommon geometric form as a representative case. Using a cube ofedge length of 100 lm as a base structure, the surface areas, edgelengths, and facet angles were systematically altered in incremen-tal steps to study the effect of gradual changes in geometricalform over the base structure. In order to observe trends in surfaceshear stress based on the physical orientation of a structure withrespect to the incident flow, a right pyramid and a cube werealtered in systematic steps through one full rotation as observedwith respect to the incident flow.

All the structures were modeled in a three-dimensional compu-tational domain to simulate external flows over the structures’surfaces (see Fig. S2, which is available under the “Supplemental

Materials” tab on the ASME Digital Collection). Initial coarsemeshes were successively refined until mesh-independent solu-tions were achieved [26]. To test against the presence of numeri-cal artifacts in the computed solutions [27], each simulation wasperformed with two iterative numerical methods (GMRES [28]and Bi-CGSTAB [29]) along with two methods of mesh genera-tion Delaunay [30] and advancing front [31]. The results from themethods were compared to confirm agreement of their solutionsto within 1% [27]. To eliminate the effect of singularities in themodels [32], geometrically sharp features (e.g., edges and corners)were modified to fillets (radius of 5 lm) [32], and the surfaceshear rate was evaluated at least 3–5 grid elements [32], corre-sponding to a length of 5 lm away from such features. It is worthnoting that the critical dimension is 100 lm for all the features ofinterest (FOI) (as discussed later), and therefore, the presence of a5 lm fillet at the edges and corners of structures did not yield anysignificant changes in the reported results. All the governing equa-tions were solved under steady-state, isothermal, and laminar flowconditions with Reynolds numbers of 0.001, 0.1, and 100 andwith water modeled as an incompressible, Newtonian workingfluid [33]. Calculation of Re was based on a microstructurecritical dimension of 100 lm. The steady-state continuity andNavier–Stokes equations for these conditions reduce to

r � V ¼ 0 (1)

q V � rV ¼ �r pþ lr 2V (2)

Re ¼ q uin h

l(3)

wherer denotes the gradient operator, V is the fluid velocity vec-tor, uin is the inlet velocity, q is the fluid density, p is the fluidpressure, h is the microstructure height (100 lm for all the struc-tures), and l is the fluid viscosity of water at room temperature.All the numerical calculations were performed with COMSOL MULTI-

PHYSICS v4.4 [34]. Average surface shear stress, �ss, was obtainedfrom Newton’s law of viscosity [33] by determining local shearrate magnitude, _c from each grid point and averaging over thedomain. The shear rate magnitude is calculated as

_c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2C:

: C:

r

_C ¼ rVþ rVð ÞT(4)

where: is the tensor contraction operator, and the superscript T

denotes the matrix transpose. Throughout this paper, “front facet”refers to the structural component facing the incident flow. Addi-tionally, the overall and facet shear stress of all the microstruc-tures discussed in this paper were nondimensionalized withrespect to a reference case explicitly listed in Sec. 3. All the

Fig. 1 In total, 36 structures were considered for systematic determination of surface shearstress and are sorted according to the geometry of the top facet: rectilinear prisms (eightshapes with nonradial, uniform top facet), radial prisms (11 shapes with radial, uniform topfacet), nonvertical prisms (ten shapes with top facet of varying height), and apex structures(seven shapes with no explicit top facet)

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models were solved using the supercomputing cluster at the OhioSupercomputer Center that employs 8328 HP Intel Xeon x5650central processing units with 12 cores and 48 GB of memory perHP SL390 G7 node. The numerical model was validated with thepast work [3], and good agreement was found on the spatial varia-tion of velocity around the cylinders and the x-component of shearstress on the periphery of cylinder, as shown in Fig. S3, which isavailable under the “Supplemental Materials” tab on the ASMEDigital Collection.

3 Results and Discussion

3.1 Regime Map. The 36 distinct geometric shapes consid-ered in this study are divided into four categories based on tradi-tional geometric definitions, namely: (1) rectilinear prisms (eightshapes with noncircular top), (2) radial prisms (11 shapes with topfacet having a radial profile), (3) nonvertical prisms (ten shapeswith 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,ðssÞall, was divided by the overall shear stress of a cylinder andreported as ð�ssÞall, thus nondimensionalizing the stress and provid-ing a direct comparison to the cylinder, which has been one of themost common structures used in a variety of flow configurationsas discussed in Sec. 1. Moreover, comparing the shear stress ofthe 36 cases chosen here with respect to the cylinder also facili-tates ready comparison with the published data [35] and provideseasy visualization to existing engineering applications that employcylinders. Due to the nondimensionalization, ð�ssÞall ¼1 for a cyl-inder as shown in the regime map (Fig. 2). ðssÞall for a 100 lmdiameter (and height) cylinder was calculated to be 7.5� 10�5

N/m2 (Re¼ 0.001), 7.5� 10�3 N/m2 (Re¼ 0.1), and 16.1 N/m2

(Re¼ 100). As the nondimensionalized shear stress for all themicrostructures reported in this study was similar for Re¼ 0.001and 0.1, results for Re¼ 0.1 will be discussed as a representativecondition. The nondimensionalized overall shear stress, ð�ssÞall, foreach 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 aglobal Cartesian coordinate frame and also the direction of inci-dent flow are explicitly shown in Fig. 2.

In Fig. 2, a shear stress regime map for the overall shear stress,ð�ssÞall, as a function of geometry is presented for Re¼ 0.1 (withRe¼ 0.001 being similar to Re¼ 0.1) and 100. Rectilinear prismshave ð�ssÞall varying in the range of 0.73–0.88 for Re¼ 0.1,

suggesting that the overall shear on a cubic prism is only 73% ofthe shear stress seen by a cylinder under similar flow conditionsand critical dimensions. In rectilinear prisms, when the fluid con-tacts the front facet, the velocity gradient is larger but limited onlyto the edges and small everywhere else, in contrast to curvilinearprisms, where the velocity drop is extended out over nearly theentire surface area of the structure, resulting in higher surfaceshear stress. This is the first of many results that suggest a fluidflow over sharply angled features (such as rectilinear prisms) will,in general, result in lower surface shear stress when compared tocurvilinear surfaces. At Re¼ 100, rectilinear prisms have ð�ssÞall

varying over a broader range of 0.62–0.97. Nonvertical prismsexhibited an even wider spread in the magnitude of ð�ssÞall

(0.38–0.80 compared to a cylinder) for Re¼ 0.1, but was confinedbetween 0.37 and 0.72 at Re¼ 100. Radial prisms exhibited largerregimes of ð�ssÞall in magnitude (0.82–1.1 at Re¼ 0.1 and 0.74–1.2at Re¼ 100); however, apex forms exhibited lower regimes ofð�ssÞall with respect to the rectilinear and radial prisms with theexception of the cone at Re¼ 0.1, where ð�ssÞall was estimated tobe 0.92 or nearly the same as the cylinder, despite not having atop facet. Apex structures due to minimal cross-sectional area(normal to the flow field) would allow for increased momentum ofthe fluid flow over the entire structure reducing the velocity gra-dients over their surface. If true, then structures 19, 23, 25, and 26(all forms with cross-sectional area less than 1 lm2) should alsoexperience less surface shear stress in the Re¼ 100 case, which isdemonstrated by the results.

It is important to note that ð�ssÞall is an area-averaged value ofshear stress over all the exposed facets as shown in Eq. (5). There-fore, though individual facets exhibit varied magnitudes of shearstress, given by ð�ssÞi which is independent of the facet area Ai, theoverall shear stress, ð�ssÞall, is influenced by both ð�ssÞi and Ai

�ssð Þall ¼

Xi

�ssð ÞiAi

Xi

Ai

(5)

The regime map study thus clearly indicates that although theoverall shear stress experienced by a microstructure is similar atRe¼ 0.001 and 0.1, further extrapolation of shear stress at higherRe (100) is not obvious and therefore was explicitly evaluated(Fig. 2). In short, the regime map provides data trends for lowReynolds number flow that can be used for selection of geometriccategories for a variety of applications as discussed in Sec. 1.

Fig. 2 A regime map for the 36 structures separated into four categories studied for theoverall surface shear stress on each structure for (a) Re 5 0.1 and (b) Re 5 100. Overallshear stress is nondimensionalized with respect to ðssÞall of a cylinder (100 lm in diameterand height). For Re 5 0.1, 100 ðssÞall of a cylinder was estimated to be 7.5 mN/m2 and16.1 N/m2, respectively. The direction of fluid flow is indicated by the gray arrow with thecoordinate system shown near the bottom left of the regime map.

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3.2 Velocity Gradients and Impact on Shear Stress. Nondi-mensionalized shear stress experienced by individual facets (front,rear, side, and top facets) for all the 36 microstructures is plottedat Re¼ 0.1, 100 in Figs. 3, and 4, respectively. Shear stress expe-rienced at each facet was nondimensionalized with ðssÞall of a cyl-inder at the respective Re, as discussed in Sec. 3.1.

In Fig. 3, critical facet stress ð�ssÞi of 0.5 (after nondimensionali-zation, i.e., 50% of the overall shear on a cylinder) was

benchmarked for front and rear facets, as the transition betweenhigh ðð�ssÞi > 0:5Þ and low ðð�ssÞi � 0:5Þ shear stress structures.Similarly, as the top facet was found to experience higher shearstress in comparison to all other facets, a critical facet stress ð�ssÞiof 1.5 was chosen to differentiate between high shear and lowshear structures. The ð�ssÞi of side facets, though greater than 0.5,varied over a narrow range for all the microstructures (averageð�ssÞi � 0.79 6 0.13). Similarly, critical shear stresses of 1 for the

Fig. 3 Plot showing ð�ssÞi , average facet shear stress of (a) front, (b) rear, (c) side, and (d)top facets of all the 36 structures shown in the regime map at Re 5 0.1. Each microstructurewas categorized into high and low shear based on critical shear stress determined for eachfacet. The classification threshold for defining “high” and “low” shear stress for identifyingstructures was based on the relative comparison to a cylinder in cross-flow as discussed inthe main text. Side facets exhibit minimal changes to shear stress as a function of micro-structure morphology. The average side facet shear stress of all the 36 microstructures wasestimated to be 0.79 6 0.13 (minimal standard deviation). In general, the magnitude of shearstress is maximum at the top facet, followed by the side facet.

Fig. 4 Plot showing ð�ssÞi average facet shear stress of (a) front, (b) rear, (c) side, and (d) topfacets of all the 36 structures shown in the regime map at Re 5 100. The classification thresh-old for defining “high” and “low” shear stress for identifying structures was based on therelative comparison to a cylinder in cross-flow.

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front facet, 0.3 for the rear facet, 1 for the side facet, and 1.5 forthe top facet were chosen to differentiate high shear and low shearstructures at Re¼ 100, as shown in Fig. 4. This classificationbetween “high shear” and “low shear” structures in Figs. 3 and 4was implemented to aid in the selection of microstructures forshear-dependent applications. Nondimensionalized facet shearstress as a function of Re is plotted for all the microstructures inFig. S4, which is available under the “Supplemental Materials”tab on the ASME Digital Collection.

Next, the role of velocity gradients in contributing to the overallshear was evaluated by considering a cube of side 100 lm as arepresentative structure. The velocity gradients were nondimen-sionalized with respect to the overall shear rate of a cylinder atRe¼ 0.1 (7.89 s�1). The nondimensionalized velocity gradientsthat contribute to shear in the X, Y, and Z directions across thetop facet were calculated and are shown in Fig. 5 for a cube ofside 100 lm at Re¼ 0.1 as a representative case. As discussed inSec. 2, data were plotted beginning at 5 lm from each edge in orderto eliminate the effects of singularities. Therefore, each cross sec-tion of the cube reported in Fig. 5 has an area of 90 lm� 90 lminstead of 100 lm� 100 lm. The effect of this strategy is discussedand shown to be valid in the supporting information section(Fig. S5, which is available under the “Supplemental Materials” tabon the ASME Digital Collection).

The velocity vector is given by V ¼ u x þ vy þ w z, where x,y, and z are the unit vectors along the coordinate axes (shown inFig. 5). Since the top facet is located in the X–Y plane, and due tothe symmetric structure of the cube, the nondimensionalizedvelocity gradient is expected to dominate along Z (velocity gra-dient�O(10 deg)) and was found to be negligible across the X,Ydirection (velocity gradient�O(10�15)) as shown in Fig. 5. Thegradient of the velocity along Z is attributed to the no-slip bound-ary condition at the top facet, which leads to the development of atransverse boundary layer. This transverse boundary layer and theresulting velocity gradient are the primary causes of the surfaceshear stress calculated on the facet. Since the incident flow is inthe X direction, juj (magnitude of u ) is greater than jvj, andtherefore, the magnitude of @v=@z is smaller by an order of mag-nitude compared to @u=@z, as also shown in Fig. 5. In addition,contribution of @v=@z toward ð�ssÞtop can be ignored when aver-aged, given the symmetry involved in the structure. Similarly, theaverage of @w=@z across the top facet was estimated to be zero

(not shown) and therefore does not contribute toward the shearstress experienced by the top facet. From Fig. 5, @u=@z exhibitsthe maximum magnitude compared to all other velocity gradientsthat contribute toward shear stress. The negligible magnitude ofall other velocity gradients, ð@v=@xÞ; ð@w=@xÞ; ð@u=@yÞ;ð@w=@yÞ�O(10�15) as observed in Fig. 5 suggests that the varia-tion of velocity V in a direction normal to a particular facet pre-dominantly influences the magnitude of shear stress at that facet.Therefore, the surface shear stress is largest along the edges of thefacet (�1.5 times larger than the average over the facet) and peaksnear the corners of the facet (�2.5 times larger than the averageover the facet). Thus, one would hypothesize that structures hav-ing multiple edges and corners should experience higher overallshear stress, but as can be seen from Figs. 3 and 4, this is not thecase. This discrepancy arises as the actual surface area of edgesand corners is negligible compared to the overall facet area andtherefore contributes minimally (<2%) to the overall averageshear stress.

Figure 6 shows the nondimensionalized velocity gradients thatprimarily influence ð�ssÞi plotted across the front (�x being thenormal unit vector), side (�y being the normal unit vector), andrear (x being the normal unit vector) facets for a cube with edgelength 100 lm at Re¼ 0.1. Orientation of the cube with respect tothe three-dimensional axis and incoming flow is as shown in Fig.5. Since juj> j v;w j, the gradient of v; w (experienced only atthe front and rear facets) is expected to be less than the gradient ofu (experienced only at the side facet). Therefore, as shown inFig. 6, the shear rate experienced by the side facet is about 2.5times greater than the front and rear facet shear stressesðð�ssÞside > ð�ssÞfront; ð�ssÞrearÞ for a cube. This trend is in agreementwith the results reported for Re¼ 0.1 for a cube and suggests that,in any structure with facets of similar orientation and shape tothose of a cube, the side facets will strongly contribute to the over-all shear stress for the structure (and referring to Figs. 2 and 3,this appears to be the case: structures with overall form similar tothat of the cube experience comparable magnitudes of surfaceshear stress). However, for noncubic geometries, it is clear fromthe regime map (Fig. 2) that the overall geometric form affects theshear stress on individual facets differently (Figs. 3 and 4) andtherefore justifies the need for such a regime map in designingnext-generation LOC devices with embedded microstructures[3,36–39].

Fig. 5 Variation of velocity gradients at the top facet, nondimensionalized with the over-all shear rate of a cylinder (7.89 s21) at Re 5 0.1 with equivalent critical dimensions. Thegradient of velocity components in a direction normal to the top facet along z contributessignificantly toward ð�ssÞtop.

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3.3 Parametric Manipulation of Physical Dimensions. Asshown in Fig. 7, four physical dimensions of a cube were incre-mented to progressively alter its shape, to achieve the final struc-tures shown by the third image in each sequence at the top of eachfigure panel. The shear stress on each individual facet in this dis-cussion is now nondimensionalized with respect to ðssÞfront or theshear stress on the front facet of a cube, i.e., the facet of the geom-etry facing the incoming flow experienced by an unaltered cube atRe¼ 100, and reported as ð�ssÞi in Figs. 7(a)–7(d), to facilitateeasy comparison to the unaltered case. For Re¼ 100, ð�ssÞfront ofthe unaltered cube was calculated to be 12.5 N/m2.

In Fig. 7(a), the results of incrementing 0 deg� hp � 26.6 degare shown (hp is the angle of tilt for each side of the cube), whichtransforms a cube into a right pyramid as it increases. It was foundthat the shear stress at the top facet, ð�ssÞtop, increased with a quad-ratic dependence on hp (R2 ¼ 0.98, where R2 is the coefficient ofdetermination) for the systematic translation of a cube to a pyra-mid. This scaling is expected because as the flat faces of the cubewere 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 (r¼�0.91, where r is the linear correlation coefficient) betweendecreasing ð�ssÞfront and increasing hp. Though the front and rear

facets have the same area, the shear stress experienced by the rearfacet is an order of magnitude lower than that of the front facet.Additionally, ð�ssÞrear was found to be invariant with a change insurface area, unlike ð�ssÞfront, which implies that the shear stressexperienced by each facet is strongly influenced by the overallgeometric form and Re. At hp ¼ 0, all the facets of a cube haveequal surface area and from Eq. (4), the overall stress is domi-nated by ð�ssÞtop, which has the maximum magnitude, which is inagreement with the discussion in Sec. 3.2. However, at hp

¼ 26.6 deg, the top surface vanishes (or collapses to a point as thepyramid tip) and therefore does not contribute to overall stress.Unlike the side and rear facets whose shear stress is invariant tohp, ð�ssÞfront decreases linearly and therefore from Eq. (4), ð�ssÞall

decreases linearly as observed in Fig. 7(a), suggesting that thefront facet is instrumental in dictating ð�ssÞall with an increase inhp.

In Fig. 7(b), results are shown for incrementing 0� Rv

� 50 lm, i.e., increasing cube edge curvature (Rv) to round-outthe cube and eventually reach a cylinder. The results for the shearstress on the front curves suggest that the “sharper” a geometricfeature is (in this case, smaller values of Rv), the more surfaceshear stress it will exhibit, as expected in Ref. [40] and from pre-vious discussions, above. Also, with increase in Rv, the surface

Fig. 6 The gradient of velocity in the normal direction to the front, side, and rear facetsof a cube at Re 5 0.1. Since the flow is along x , juj> jv;w j, and therefore, the side facetexperiences a greater shear stress compared to the front and rear facets.

Fig. 7 The results of incrementing four physical dimensions of a cube through a range ofvalues are presented. Distinct points represent discrete model solutions; and lines havebeen added for clarity. Dimensionless values of ð�ssÞi represent average surface shear stressnondimensionalized with respect to ð�ssÞfront of an unaltered cube (12.5 N/m2 and Re 5 100)with side length of 100lm. In (a), the flat facets of the cube were altered to approach triangularcross sections (0 £ hp £ 26.6 deg) eventually reaching a right pyramid. In (b), the curvature ofthe front and side facets was increased (0 £ Rv £ 50lm) to eventually be a cylinder. In (c), theangle of the front facet to the incident flow was increased (0 £ hw £ 45.0 deg) until the finalstructure was a wedge. In (d), the curvature of the top facet was increased (0 £ Rt £ 50lm)until the form shown was reached. The direction of flow is indicated by the gray arrow.

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area of the top facet decreases by 21.5% from a square (Rv

¼ 0 lm) to a circle (Rv ¼ 50 lm), compared to a 100% decreasein the case discussed in Fig. 7(a). In addition, the top facet experi-ences the highest transverse velocity gradients when Rv > 25 lm,as indicated by a higher value of ð�ssÞtop in comparison to otherfacets in Fig. 7(b). Therefore, coupled with a significant contrib-uted area and shear stress toward the estimation of ð�ssÞall (ð�ssÞtop

contributes 36.0–41.5% to overall shear with increase in Rv), thetop facet dictates the variation of ð�ssÞall with Rv.

In Fig. 7(c), the results of incrementing the angle of tilt, hw, ofthe front face of a cube are shown. Therefore, the area of the frontand side facets varies with hw, and the area of the rear facetremains fixed as shown in Fig. 7(c). Again, ð�ssÞtop shows a quad-ratic increase (R2¼ 0.97) for the evaluated values of hw. Despitean overall increase in hw from 0 deg to 45 deg, thereby increasingthe front facet area by 41%, ð�ssÞfront only increases by 0.86%. Incomparison, the percent contribution of ð�ssÞsides to ð�ssÞall is16.9 6 0.25% for all the values of hw, suggesting ð�ssÞsides is inde-pendent of hw. Together, these results suggest that the top andfront facets dictate ð�ssÞall for all the values of hw.

In Fig. 7(d), results are shown for incrementing 0� Rt

� 50 lm, i.e., increasing the curvature (Rt) of the cube’s top facet.As Rt is increased, the height (and therefore area) of the front facetwas systematically reduced to maintain the total height of thestructure at 100 lm. The results for the front curve and the top fac-ets confirmed previous results that sharper geometric featuresresult in larger values of ð�ssÞi along expected trends [40]. ð�ssÞall

exhibits only a 3.4% overall decrease, while ð�ssÞfront shows astrong linear correlation (r¼�0.99) with Rt, decreasing 41.5%overall, and remaining within 21.5% of ð�ssÞall. It is important tonote that the area of the newly formed front curve increases as Rt

increases.A common trend observed in all the cases was that the magni-

tude of shear stress on the rear facet, ð�ssÞrear, remains small(�3%) and is mostly unaltered by the changes to structure mor-phology, in agreement with the regime map (Re¼ 100, Fig. 2(b)).Similarly, minimal changes in the value of ð�ssÞall suggest that inmost low Reynolds number based flows, the surfaces exposed tothe incoming fluid (typically the front and top facets) dictates theperformance characteristics. Also, a decrease in the area of thefront facet when transformed from a cube (Figs. 7(a), 7(b), and7(d)) results in a decrease in ð�ssÞfront. However, decreasing thesurface area of the top facet (Figs. 7(a)–7(d)) leads to a quadraticincrease in the magnitude of ð�ssÞtop most likely due to the increasein the amount of area exposed to a large velocity gradient near theedges of the facet. Overall, the results suggest that to produce astructure with larger overall surface shear stress, smaller top fac-ets, rounder top facets, and flatter front facets normal to the inci-dent flow are preferred, but of most importance is the form of thetop facet. Thus, the trends previously observed and discussed inrelation to Fig. 2 are further illuminated. In conclusion, the overallresults in Fig. 7 suggest that the shear stress experienced by vari-ous facets is strongly influenced by the overall geometric form ofmicrostructures.

3.4 Effect of Altering Angle of Rotation. The effect ofchanging the angle of orientation with respect to the incident flowfor a pyramid, /p, and cube, /c, was investigated (Fig. 8). Theaxis of rotation for both the pyramid and cube is shown in Fig. 8.To facilitate comparison between the rotated structures and theirnonrotated forms, the reported values of shear stress were nondi-mensionalized with ðssÞfront at /p ¼ 0 deg and /c ¼ 0 deg.

In Fig. 8(a), the pyramid’s feature of interest (FOI¼ front facetat /p ¼ 0 deg) exhibits a global minimum at /p ¼ 180 deg. Themaximum values of ð�ssÞFOI occur at /p ¼ 60 deg and 300 deg.Therefore, the magnitude of shear stress at the front is maximumwhen angled at 6 60 deg from the incident flow. From Fig. 2, wesee that, in general, structures 5 and 8 (which have angled, squarefront facets) experience larger surface shear stress than the cube,

whose front facet is normal to the flow. This result indicates thatthe surface shear stress on a structural form can be manipulatedand achieved purely by changing its orientation with respect tothe incoming flow. The shear stress exhibits a smooth, sinusoidaldistribution of both ð�ssÞall and ð�ssÞFOI as /p is varied, i.e., theoverall and facet shear stress is symmetric as the orientation of thestructure with respect to incoming flow is varied. Figure 8(b)shows the effect of changing /c for a cube. The maximum valuesof ð�ssÞFOI occur at ð�ssÞFOI ¼ 50 deg and 310 deg (i.e., 650 degwith respect to the incident flow, which agrees closely with theresults for the pyramid). For the top facet, the maximum values ofð�ssÞtop occur at /c ¼ 45 deg, 135 deg, 225 deg, and 315 deg, corre-sponding to orientations that exhibit a maximum gradient invelocity for the same top facet surface area.

4 Summary and Conclusions

A systematic numerical analysis for the 36 diverse geometrieswas analyzed to estimate overall and individual facet shear stressat Reynolds numbers (Re) of 0.001, 0.1, and 100, which spans fiveorders of magnitude for relatively low Re flows typically seen inmany viscous flow and LOC applications. The structures weredivided into four categories based on the geometry of the topfacet, namely: rectilinear prisms, radial prisms, nonverticalprisms, and apex structures.

Overall shear stress of a microstructure was nondimensional-ized with the overall shear stress of a cylinder at a given Re andreported in a regime map. The results indicate that the nondimen-sionalized facet and overall shear stress for all the 36 structureswere found to be the same at Re¼ 0.001 and 0.1. However, as Rewas increased to 100, facet and overall shear stress was found tovary in comparison to Re¼ 0.1. Since the low Reynolds numberflow is incident in one-direction, the magnitude of the transversevelocity components is lower compared to the axial component(which is along the direction of flow). Therefore, facets thatinclude the gradient of axial velocity toward the estimation ofshear rate experience a higher magnitude of shear stress comparedto facets that take into account the transverse component of veloc-ity. In addition, by changing the angle of rotation, critical angleswere found for a cube and pyramid for which the surface shearstress was maximum in magnitude. The results of this work areexpected to provide a broad basis for choosing microstructuresthat are essential for shear-dependent applications.

Acknowledgment

The computational facilities at the Ohio Supercomputing Cen-ter (OSC) are acknowledged for the support. The U.S. Department

Fig. 8 The effects on ð�ssÞi for different angles of orientation for(a) a right pyramid and (b) a cube at Re 5 100. At /p 5 0 deg and/C 5 0 deg, the FOI is in the front facet position. Dimensionlessvalues of ð�ssÞi represent average surface shear stress nondi-mensionalized with respect to the pyramid and cube at /p

5 0 deg and /C 5 0 deg, respectively. The direction of fluid flowis always fixed and indicated by the gray arrow; FOI is indicatedby a darkened face.

Journal of Fluids Engineering JANUARY 2017, Vol. 139 / 011201-7

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of Energy (DOE) is acknowledged for the partial funding supportfor the personnel through ARPA-E under Grant No. DE-AR0000282. The discussions with Tong Lin are alsoacknowledged.

Nomenclature

Ai ¼ area of facet of interestp ¼ fluid pressure (components vary spatially)r ¼ linear correlation coefficient

Rt ¼ radius of curvature of top facetRv ¼ radius of curvature of filletR2 ¼ coefficient of determinationRe ¼ Reynolds number

u ¼ X component of Vuin ¼ inlet velocity

v ¼ Y component of VV ¼ Eulerian fluid velocity (components vary spatially)w ¼ Z component of VX ¼ global Cartesian coordinate in the direction of inlet

flow (see Fig. 2)x ¼ unit vector along XY ¼ global Cartesian coordinate perpendicular to the flowy ¼ unit vector along YZ ¼ global Cartesian coordinate perpendicular to the flowz ¼ unit vector along Zc: ¼ average shear ratehp ¼ angle of inclination of each facet of cubehw ¼ angle of incidence of front facet normal to the incident

flowl ¼ fluid viscosity (water in this study)q ¼ fluid density (water in this study)

ðssÞi ¼ average shear stress of facet i (N/m2)ðssÞi ¼ shear stress of facet i (N/m2)ð�ssÞi ¼ nondimensionalized average shear stress of facet ið�ssÞall ¼ nondimensionalized overall shear stress of

microstructureð�ssÞfront ¼ nondimensionalized average shear stress of front facetð�ssÞrear ¼ nondimensionalized average shear stress of rear facetð�ssÞside ¼ nondimensionalized average shear stress of side facetð�ssÞtop ¼ nondimensionalized average shear stress of top facet

/c ¼ angle of rotation of cube/p ¼ angle of rotation of pyramid

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