11TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY - PIV15 Sata Barbara, Califoria, Sept. 14-16, 2015 ABSTRACT HELE-SHAW RHEOMETRY BY MEANS OF PARTICLE IMAGE VELOCIMETRY Sita Drost & Jerry Westerweel Laboratory for Aero & Hydrodyamics, Delft Uiversity of Techology, The Netherlads j.westerweel@tudelft.l I this paper, we apply a ovel approach to determie the flow behavior idex of a power-law fluid by meas of a microfluidic device ad particle image velocimetry. The cocept of this method is based o a mathematical aalysis of Hele-Shaw flow of a power-law fluid. INTRODUCTION I this paper we briefly review the mai mathematical cocepts of Hele-Shaw flow of a o- Newtoia fluid i a Hele-Shaw flow cell, ad we reproduce the results of a earlier paper [3] as a validatio of the method. Sice o aalytical solutio exists for Hele-Shaw flow of a Herschel- Flows of liquids i microfluidic devices combie a high deformatio rate with a low Reyolds umber, which makes that such devices offer ew possibilities for rheometry. I additio, because of their small dimesios, microfluidic devices require low sample volumes ad remai optically trasparat provided. A recet overview of developmets i microfluidic rheometry is give by Pipe & McKiley [1]. A Hele-Shaw flow cell is a microfluidic cofiguratio where the flow occurs betwee two parallel plates, where the gap betwee the two plates is much smaller tha the lateral dimesios. Uder such coditios, the flow for a Newtoia fluid, averaged over the gap alog the directio ormal to the plates, behaves as a irrotatioal potetial flow, despite the fact that the Reyolds umber (based o the distace betwee the two plates) is typically very low (i.e., creepig flow). This property has bee used i the past to demostrate the streamlie patters of the twodimesioal flow aroud objects of a iviscid fluid. I the case of a o-newtoia fluid, Hele- Shaw flow becomes more complex. Arosso ad Jafalk [2] provide a mathematical aalysis for Hele-Shaw flow of a ielastic power-law fluid, where the shear stress τ of the fluid as a fuctio of the shear rate! γ! ca be described by:! τ (! γ ) = K! γ, (1) where K is the cosistecy, ad the power idex. I a recet paper [3] we implemeted this approach by drivig a o-newtoia fluid through a glass microfluidic chip with a 100:1 cotractio. The flow patter ca be measured by meas of particle image velocimetry, ad a fit of the mathematical model to the measured flow patter yields the fluid power idex. A further developmet is to exted this method to a fluid that has a yield stress τ0, such as a Herschel-Bulkley fluid, where the shear stress ca be described by:! τ (! γ ) = τ 0 + K! γ. (2) Examples of Herschel-Buckley fluids are drillig muds (used i oil exploratio) ad Heiz ketchup. A complicatio is that may of these fluids are ot optically trasparet, so that it is ot possible to perform PIV measuremets i bulk flows. This is where a microfluidic Hele-Shaw flow cell provides a suitable solutio, as the separatio betwee the plates is of the order of 100 µm, so that most fluids remai sufficietly optically trasparat to make PIV possible.
Herschel-Buckley fluid, we compare the flow patters agaist umerical solutios of the equatios. To apply the method to a Herschel-Bulkley fluid, we used a drillig mud that cotais lapoite (2% wt) ad barite (25% wt). This mud has a yield stress ad is shear thiig, i.e. < 1. It was ot possible to use the cotractio flow cell used i Ref. [3] as the exit became blocked by the barite particles. (Also, it appeared to be very difficult remove ay trapped air bubbles.) Istead we used a Hele-Shaw flow cell with a cylider i its ceter. THEORY The covetioal aalysis of Hele-Shaw flow of a Newtoia fluid results i a irrotatioal iviscid flow, i.e. potetial flow, where the velocity u averaged over the gap width is give by:! u = h2, with:!, ad:!, (3) 3η p * 2 ψ = 0 2 p = 0 where η is the fluid viscosity, 2h the gap width, p the pressure, ad ψ the stream fuctio. For a power-law fluid (1), the equatio that correspod to those i (3), become [2,3]: ( ) = 0 ( ψ 1 1 ψ ) = 0! p 1 1 p, ad:!, (4) ad the gap-averaged velocity is the give by: 1! u = 2 p. (5) 2 +1 K h 1+1 This expressio is valid for flows of viscoelastic fluids with a Weisseberg umber Wi 1. For = 1 these expressios reduce to velocity ad the Laplace equatios i (3) for a Newtoia Hele- Shaw flow. The power-law idex ca be determied directly from the gap-averaged velocity field i (5). This equatio ca be rewritte i terms of the velocity compoets u = (u,v) ad the gradiets of these velocity compoets ito [3]:! ( 1) ( u 2 + v 2 ) 2 3 v v v 2 2uv (6) x y u u2 y + ( u2 + v 2 ) 2 1 v x u y = 0. The velocity compoets ad the spatial derivatives ca be measured usig PIV. The gap-averaged velocity is measured oly whe the depth-of-correlatio is sufficietly large i compariso to the gap width [4]. However, it should be oted that the magitude of the velocity is ot importat here, as the value of is related to the shape of the streamlies [3]. This approach was validated usig Figure 1. Streamlies resultig from PIV measuremets, superimposed o a image of the Hele-Shaw cell (flow is from right to left). Top: Glycerol, 20 µl/mi, middle: Xatha gum solutio, 20 µl/mi, bottom: PEG-PEO Boger fluid, 5 µl/mi. The part of the flow where the streamlies are show is also the part that was used for the fittig procedure. The axes represet the PIV iterrogatio widow umber. From: [3].
Figure 2. Schematic of the microfluidic Hele-Shaw flow cell fitted with a cylider. All dimesios are i mm. three differet classes of model fluids: a Newtoia fluid ( = 1), a ielastic shear-thiig powerlaw fluid ( = 0.86), ad a Boger fluid (i.e., a viscoelastic fluid with costat viscosity, = 1). The fluids were pushed through a 100:1 cotractio Hele-Shaw flow cell [3]. I all three cases, satisfactory results were obtaied (see Figure 1), with values of deviatig at most 4% from values measured usig covetioal rheometry. For a Herschel-Bulkley fluid (2) with a yield stress, there exists o aalytical solutio comparable to (6). Istead we solved the Navier-Stokes equatios umerically for differet values of, ad the compare the streamlies to the flow patter observed i the flow cell. EXPERIMENTAL For the Hele-Shaw rheometry of a drillig mud we modified the flow geometry from a cotractio flow to a Hele-Shaw flow aroud a cylidrical object, as show i Figure 2. The cell is 10 mm wide, ad 40 mm log, with a gap width of 100 µm. I the ceter of the flow cell is a cylider with a diameter of 4 mm. This has the advatage that we could avoid cloggig, as occurred i the cotractio flow, ad also it was easier to remove trapped air bubbles. The gap width of the flow cell is aroud 100 µm, which is small eough for sufficiet light to pass through the flow cell. I the case of the specific drillig mud we used, the barite particles, which have a size i the rage of several µm up to several tes of µm, act as flow tracers. Figure 3 shows a sample image take i the cotractio flow cell. Figure 3. Image of drillig mud iside a microfluidic Hele- Shaw cotractio flow cell. The fluid layer is 100 µm ad is sufficietly trasparet to trasmit light. Red arrow idicate flow as a result of compressig the trapped air bubbles. It was however at this stage ot yet possible to acquire multi-frame exposure sequeces for the purpose of PIV aalysis of the images. The flow would rather suddely start whe a pressure was applied (as a result of the yield stress), ad it was at this stage ot possible to defie a proper exposure time delay ad/or frame rate. We therefore limit the presetatio of the results to streak photographs. Sice oly the directio of the flow is ecessary to recostruct the streamlies, we would pla to use a auto-correlatio PIV aalysis of the visualizatio images to determie the directio of the streaks from the shape of the image auto-correlatio peak.
Figure 4. Compariso of the observed streamlies with umerical solutios of the Hele-Shaw flow streamlies for a Herschel-Bulkley fluid with power idex = 1 (left) ad with = 0.5 (right). I order to determie the power-idex of the fluid we fitted the observed streamlies to precomputed umerical solutios of the flow i the cell. I Figure 4 we compare the observed flow patters agaist the solutio for a Newtoia fluid ( = 1) agaist that of a shear-thiig fluids ( < 1). The red lie is a highlighted streak patter. From this it is evidet that the flow patter deviates from that of a Newtoia fluid, so that the fluid must be clearly o-newtoia. A further compariso with several pre-computed streamlie patters, idicated that the solutio with = 0.5 gave the best fit to the observed streamlie patter. Whe the fluid passes through the cell aroud the cylidrical object, the shear stress ear the frot ad rear stagatio poits is below the yield stress. This leads to two early triagular regios, oe ear each of the stagatio poits. The size of these regios depeds o the relative magitude of the yield stress with respect to the applied pressure. This provides a meas to determie the value of the yield stress whe the pressure is measured. At the time of the measuremet, we were ot able to obtai absolute pressure measuremets. CONCLUSION We describe the use of a Hele-Shaw flow cell to perform o-lie rheometry of o-newtoia fluids. I previous work [3] it was demostrated that this approach gives reliable results for the power-idex of a power-law fluid, provided that the flow is i a o-elastic regime (Wi 1). A thi microfluidic Hele-Shaw flow cell has the advatage that it ca be used also for fluids that are otrasparet i bulk, such as drillig muds. These fluids typically have a yield stress, which is a extesio to the method described previously [3]. We cosidered the flow i a Hele-Shaw flow cell with cylider i its ceter. For a Herschel-Buckley fluid there exists o uique aalytical solutio for the streamlie patter i a Hele-Shaw flow cell. Istead, the observed streamlies were compared agaist umerical solutios of the Navier-Stokes equatios of the flow. This idicated the clear o-newtoia characteristics of the fluid. Furthermore, the relative magitude of the yield stress could be obtaied by evaluatig the size of the solidified regios ear the frot ad rear stagatio poits of the cylider. REFERENCES
1. Pipe, C.J., ad G.H. McKiley, Microfluidic rheometry, Mech. Res. Commu. 36 (2009) 110 120 2. Arosso, G., ad U. Jafalk, O Hele-Shaw flow of power-law fluids, Eur. J. Appl. Math. 3 (1992) 343 366 3. Drost, S. ad J. Westerweel, Hele-Shaw rheology J. Rheol. 57 (2013) 1787-1801. 4. Kloosterma, A., C. Poelma, ad J. Westerweel, Flow rate estimatio i large depth-of-field micro-piv Exp. Fluids 50 (2011) 1587 1599