Polymeric Rheological Metamaterials
Equations 2.4.6-7 Equations 2.4.6-7 can be described in terms of physical space variables only by substituting the radial variable in the virtual space ( ) to the physical space variable ( )
2.4.4.1. Multilayered concentrator
96 2.4.4. Metamaterial design
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Figure 2.4.5. A constructed geometry of the multilayered concentrator and its magnified view.
Table 2.4.1. Viscosity tensor components of the multilayered concentrator given to each layer.
Layer No. , (mPa∙s) , (mPa∙s)
Center region 1 1
1 0.222 14.38
2 0.222 9.56
3 0.222 6.96
4 0.222 5.36
5 0.222 4.32
6 0.222 3.6
7 0.222 3.06
8 0.222 2.66
9 0.222 2.36
10 0.222 2.1
Background 1 1
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Figure 2.4.6. Modeling of the multilayered rheological concentrator. (a) Spatially varying viscosity tensor components in (i) radial and (ii) azimuthal axes. Simulation results for (b) the multilayered concentrator and (c) bare cases of (i) pressure fields, (ii) pressure gradient fields (py), and (iii) velocity fields.
99 2.4.4.2. Unit cell modeling
The same strategy used in Section 2.3.4.2 was employed for designing effective viscosity unit cells and mapping the rheological metamaterial concentrator with the unit cells. In the unit cells for the metamaterial cloak, the length in axis was designed to be much longer than the length in axis for embodying anisotropic viscosity components. On the other hand, the unit cell required for the metamaterial concentrator should have a longer length on axis than axis. In accordance with the layer thickness of the multilayered concentrator, 300 by 300 by 50 μ m3 unit cells were modeled with theoretically shape- determined micropillars. The velocity field of the bare unit cell without any microstructure is shown in Fig. 2.4.7(a) and the effective viscosity of this unit cell is identical with the pristine water viscosity, 1 mPa∙ s. Figure 2.4.7(b) shows the velocity field of the same background unit cell used for the rheological metamaterial cloak.
The viscosity components given to each layer (Table 2.4.1) were multiplied by a factor of 5.4 to make the radial viscosity larger than 1 mPa∙s, from 0.222 mPa∙s to 1.2 mPa∙s (red lines in Fig. 2.4.8). Ten unit cells with micropillars of different shapes were constructed and their simulated velocity fields are shown in Fig. 2.4.7(c). The effective viscosity components embodied by each unit cell and mapping information are tabulated in Table 2.4.2. The length of the micropillars (l) was distributed from 285 μm to 265 μm, which functions to diversify the effective viscosity in axis from 72.06 mPa∙s to 11.66 mPa∙s.
On the other hand, thickness of all micropillars was set to 80 μm, and hence the azimuthal components varies only from 1.204 mPa∙s to 1.244 mPa∙s irrespective of the layer while remaining constant at almost 1.2 mPa∙s. The designed unit cells were mapped on the shell region ( < < ) and acted as an element of the rheological metamaterial concentrator.
The background region ( > ) was mapped with the 150 μm microcylinder to achieve impedance matching.
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Figure 2.4.7. Simulated velocity fields for (a) the bare unit cell without a microstructure, (b) the background unit cell, and (c) the 10 unit cells which are mapped on each layer of the rheological metamaterial concentrator. The left and right figures in (c) show the velocity fields when the pressure field is applied along each axis. Red lines indicate flow streamlines along the flow direction and means a length of the micropillar.
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Figure 2.4.8. Viscosity tensor components of the unit cells mapped on each layer of the rheological metamaterial concentrator; (a) in radial and (b) azimuthal axes.
Table 2.4.2. Effective viscosity unit cells mapped on each layer of the rheological metamaterial concentrator.
Layer No. l (μm) , (mPa∙s) , (mPa∙s)
Center region =150 2.758 2.758
1 285 1.2037 72.061
2 283 1.2059 53.617
3 281 1.2122 39.555
4 278 1.2145 27.785
5 276 1.2223 23.224
6 274 1.2240 20.568
7 272 1.2269 17.576
8 269 1.2421 14.653
9 267 1.2443 12.925
10 265 1.2444 11.655
Background =150 2.758 2.758
* means the radius of the microcylinder.
102 2.4.4.3. Designed metamaterial concentrator
The rheological metamaterial concentrator was designed by arranging the 10 anisotropic unit cells developed in Section 2.4.4.2 (Fig. 2.4.9). The layers 1 to 10 of the metamaterial concentrator consist of 24, 30, 37, 43, 49, 55, 62, 68, 74, and 81 correlating unit cells.
50×50 number of the isotropic unit cells composed of 150 μm cylinders were mapped in the background for impedance matching. The overall size of the designed concentrator is the same as for the continuous media and multilayered cases. The pillar-arrayed model of the designed metamaterial concentrator was drawn by using CATIA 5.18. After inserting the designed pillar-arrayed concentrator into the hexahedron block, the Boolean operation removed the patterned space, creating a fluid domain model with solid microstructures.
Numerical simulation for the designed metamaterial concentrator was carried out to confirm the concentrating effect. Almost the same simulation conditions as those described in Session 2.3.4.3 were used. The other conditions are described as follows. Completed free tetrahedral meshes consisted of 1304955 domain elements, 553384 boundary elements, 77321 edge elements. The maximum and minimum element size were 52 μm and 10 μm, orderly. The maximum element growth rate was 1.13, the curvature factor was 0.5, and the resolution of narrow regions was 0.8.
Figure 2.4.10 shows the simulation results of pressure fields, velocity fields, and flow streamlines for the designed metamaterial concentrator. All simulation results are consistent with the previous simulation results of the continuous media and multilayer cases as shown in Fig. 2.4.11. The concentrated pressure field in the central region is found in Fig.
2.4.10(a), (i), while the uniformly distributed pressure field in the bare case is shown in Fig. 2.4.10(b), (i). The pressure contours are also created more densely in the central compressed space. It could be seen that the velocity field of the central region inside the metamaterial concentrator greatly increased compared to the outside region or the region without the concentrator (Figs. 2.4.10(a-b), (ii)). A number of streamlines were condensed in the central region, which represents concentrated fluid momentum (Figs. 2.4.10(a-b), (iii)). As in the cloaking metamaterial, the velocity field and flow streamlines outside the metamaterial concentrator were not perturbed at all, regardless of the spatial variation. The unit cell strategy used to create the rheological metamaterial concentrator can be evaluated as successful from the simulation results of fluid flow fields that are concentrated in the
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central region and have no effect on the surrounding space.
The magnified view of the velocity fields shows the increased flow rate more clearly (Fig. 2.4.12). To quantitatively analyze the concentrating effect, the average velocity of the unit cell (⟨ ⟩ ) was calculated for the region with or without the concentrator, where
⟨∙⟩ means the volume averaging operator for the unit cell. The dashed rectangles in Fig.
2.4.12 indicate the unit cells subject to the calculation of the velocity fields. The mean velocity value (⟨ ⟩ ) increased about 2 times by encircling the central region by the metamaterial concentrator, from 8 mm/s to 17 mm/s.
Unfortunately, the degree of amplification was lower than the 3-fold increase of the ideal concentrator performance, calculated from the continuous media simulation results.
It is speculated that the approximation error in the effective viscosity unit cell design might cause this inconsistency, on a similar rationale that can not to create the space of a perfect zero flow rate in the metamaterial cloak development. Obviously, the error in the assumptions can be reduced if the metamaterial concentrator is designed and built with smaller unit cells. However, in this dissertation, the metamaterial concentrator was designed under the present unit cell conditions due to fabrication and experimental limitations.
From the velocity analyses, theoretical hydraulic kinetic energy of the unit cell region ( , ) could be calculated based on Eqn. 2.4.13 as
, = ⟨ ⟩
2 (2.4.13) , where is the mass density of a fluid, and is the volume of a unit cell. After the calculation, the value in the bare region was evaluated to be 1.44e-7 μJ. And the value in the central concentrated region was estimated to be 6.50e-7 μJ, which is a 4.5 times higher value than that of the bare region. Naturally, the relatively low increase rate of originates from the same reason as in the velocity field calculation.
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Figure 2.4.9. Design of the rheological metamaterial concentrator of a pillar-arrayed case with configuration and a magnified view.
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Figure 2.4.10. Simulation results for (a) the designed rheological metamaterial concentrator and (b) bare cases of (i) pressure fields, (ii) velocity fields, and (iii) flow streamlines.
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Figure 2.4.11. Comparison of the simulation results for (a) the continuous media, (b) multilayered, and (c) pillar-arrayed rheological concentrator cases; (i) pressure fields, (ii) velocity fields and (iii) flow streamlines.
Figure 2.4.12. Magnified images of the simulated velocity fields in the central region; (a) with and (b) without the rheological concentrator. Dashed squares indicate the unit cell.
107 2.4.5. Experimental realization