The two different flow types investigated in this study. The cleaning fluid was first deposited onto the substrate before being removed by (a) an impinging water jet that fell vertically onto the substrate to rinse off the cleaning solution or (b) an aspiration jet connected to a vacuum above the substrate that siphoned the cleaning solution off of the surface. The diagrams are not drawn to scale as the thickness of the cleaning fluid layer and the size of the silica particles are exaggerated.
Schematic of the particle removal setup. Two rectangular glass coverslips are joined to make a 22 mm by 60 mm by 0.2 mm substrate that is placed with one end centered under the stagnation point. A fluorescence scanner was used to determine the difference in the concentrations of particles before and after the flow experiment to find an average removal over the area.
Rheology. (a) Apparent shear viscosity as function of shear rate under steady shear experiments. PAM8M was fit to a FENE-P model with input of and outputs of λ = 2.16 s, , and L 2 = 22 [ Wedgewood and Bird (1988) ]. (b) CaBER results showing apparent Trouton ratio as a function of Hencky strain. Gly and PAM10K samples showed approximate Newtonian behavior (Tr ≈ 3) while PAM5K, PAM8M, and Boger were highly elastic, showing significant strain hardening.
Siphoning average particle removal. Error bars are shown for a sample size of four trials each.
In situ particle removal video screenshots of water rinsing PAM8M. Screenshots were taken (a) 0.000 s, (b) 0.012 s, and (c) 0.044 s after impingement. The direction of flow is designated with an arrow.
Siphoning particle removal versus radial position for various flow rates. The particle removal efficiency depends both on the magnitude of the siphoning flow, increasing with increased flow rate, and on the radial position of the particles, decaying radially away from the stagnation point. A line, representing a lower limit of reliability, is presented to aid visualization.
Rinsing flow particle removal versus radial position. Complications in the flow field caused by the presence of the hydraulic jump and the elastic recoil could explain the fairly constant, but non-monotonic, behavior inside of the pseudo-steady state position of the hydraulic jump (∼2.5 cm). A line, representing a lower limit of reliability, is presented to aid visualization.
Schematic of the flow field simplification. (a) The actual geometry consists of a complicated siphoning experiment with a free surface and many particles on the surface, represented by the circle which is not drawn to scale. (b) The simplified geometry uses a singular sink flow in a semi-infinite domain near a no-slip wall. This simplification is justified only near the wall where the particles of interest reside and have a length scale orders of magnitude smaller than the separation of the sink from the wall.
Schematic of the flow field coordinate system. Adapted from Blake (1971) , the schematic shows a position P and its relative coordinates x from the origin, r from the sink, and w from the image sink. Note that the domain has been reduced to two dimensions by considering a ray in the axisymmetric flow having a value of x 2 = 0 and that the height of the sink above the substrate is the natural length scale used to nondimensionalize the coordinate system.
Schematic of the flow field image system for a sink near an infinite no-slip boundary. Adapted from Blake and Chwang (1974) , the schematic shows a singular sink near a no-slip wall. The image system consists of an equivalent image sink, a stresslet, and a sink-doublet. At zero Reynolds number (Stokes flow), this configuration produces an exact solution for a singular sink in a semi-infinite domain near a no-slip wall.
Weissenberg number using versus radial position for various flow rates. The onset of Wi being will vary radially with different sink flows, where increasing sink flows will become relevant further from the stagnation point.
Stress components versus radial position. The dominant component, , implies that the dumbbell is being stretched in the radial direction. The solvent contribution is plotted for comparison although the value of the shear stress is negligibly small relative to the polymer stresses (solution dimensionally correlates M = 514 ml/min, h = 1 μm).
Averaged dominant polymer stress (stars) and averaged particle removal (open circles) versus radial position. (a) The -component of the polymer contribution to the stress tensor has been integrated over the individual rectangular domains as described in Fig. 2 . The onset of substantial particle removal correlates with the presence of a large polymer stress driven by the local shear flow (solution dimensionally correlates M = 514 ml/min, h = 1 μm). (b) Particle removal versus averaged dominant polymer stress shows a power law dependency for removal above a certain level associated with noise from post-processing steps.
Physical properties of test fluids.
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