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 ( )
2.5. Rheological rotator
2.5.1. Introduction
2.5.3.1. Simulation conditions
In this section, simulation results of the four rotators using the viscosity tensors described in Eqns. 2.5.9-12 are explained. Most simulation conditions used in this section are same with the conditions demonstrated in Section 2.3.3.1. For the three rotators, the hexahedron block geometry and the concentric ring in Section 2.3 (Fig. 2.3.3) were used, and for the artificial tornado case, the inner radius was set to a small value of 10 μ m. Free tetrahedral meshes consisted of 111567 domain elements, 73272 boundary elements, and 1884 edge elements. The used element size parameters and control entities are same with the those described in Section 2.3.3.1.
The same customized PDE (partial difference equation) interface suggested in Section 2.3.3.1 solved the Navier-Stokes equations with the transformed viscosity tensors of the rheological rotators. The solver, boundary conditions, and material properties used in the analysis are the same for the rheological cloak.
119 2.5.3.2. Simulation results
Using the transformed viscosity tensors developed in Section 2.5.2, the feasibility of the rheological rotators was demonstrated with numerical simulation results. The simulation results of the four rheological rotators were analyzed with transformed coordinate grids, pressure fields, pressure gradient fields, and velocity fields with flow streamlines.
Figure 2.5.1 shows the numerical simulation results for the 180° rotator with ° (Eqn. 2.5.9). From Fig. 2.5.1(a), it can be found that the coordinate system of the rotating shell ( < < ) is largely warped while the inside and outside the 180° rotator maintains their original shape. A red point on the outer circle of the rotating shell is connected to a blue point on the inner circle on the opposite side through the shell with the twisted space.
Hence, the inner space surrounded by the 180° rotator has a physical quantity in opposite direction to that of the surrounding space.
Figure 2.5.1(b) shows the 180° inverted pressure field in the central region ( < ).
The pressure gradient in the central region is formed as a positive number from top to bottom, as opposed to the pressure gradient condition applied in the simulation. Therefore, the negative pressure gradient field (about -96 kN/m3) is found in the central region (Fig.
2.5.1(c)). It should be noted that the pressure gradient of the central region has the same magnitude as the surrounding pressure gradient (about 96 kN/m3), but only the sign becomes opposite.
Due to the negative pressure gradient, the flow direction of fluid is reversed as shown in Fig. 2.5.1(d). The length and direction of the black arrows indicate the flow rate and direction, respectively. When the flowing fluid enters the outer boundary of the 180° rotator, it accelerates greatly in a helical direction with spiral patterns and reaches the farthest side of the inner circle. The velocity magnitude of the central region and the background region was matched with each other at 30 mm/s.
The 90° rotator with ° (Eqn. 2.5.10) was demonstrated in Fig. 2.5.2. Similarly with the 180° rotator, the coordinate grid of the rotating shell is twisted to connect a red point on the outer circle of the rotating shell to another blue point on the inner circle on the perpendicular side. Due to this spatial distortion, the pressure gradient in the central region is aligned along negative x-direction that is perpendicular to the applied pressure fields (Fig.
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2.5.2(b)). Because the pressure fields are directed by 90°, the pressure gradient in y-axis (Py) has an almost zero value (Fig. 2.5.2(c)) and that in x-axis (Px) was calculated at about -97 kN/m3. According to this pressure gradient, the fluid in the central region flows in the direction of negative x axis (Fig. 2.5.2(d)).
The two rotators for 180° and 90° rotation require excessive space warpage, which is not feasible experimentally. For this reason, the 30° rotator ° (Eqn. 2.5.11) was modeled for experimental realization (Fig. 2.5.3). The inner and outer points of the rotating shell are connected to one another through much more gentle spiral patterns than in the previous two cases. Along the transformed coordinate space, the pressure, pressure gradient, and velocity fields were formed in an inclined direction of 30° (Figs. 2.5.3(a-c)).
A tornado is a rapidly rotating whirlwind that rotates very vigorously. By setting the center region very small ( =10 μm) and the rotation angle to be very large ( =1800°), the artificial tornado was modeled with an extremely distorted fluidic space as an extreme case (Fig. 2.5.4). Not only the flow rate in the 1800° rotator is very fast, but also the swing angle is also very large, since the fluid inside the rotator must be rotated by 1800° before reaching the inner central region. In spite of very severe distortion of the space, it can be seen that the surrounding space is not affected and maintains a uniform pressure and velocity fields.
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Figure 2.5.1. Simulation results of the 180° rheological rotator: (a) transformed coordinate system, (b) pressure fields, (c) pressure gradient fields, and (c) velocity fields.
Figure 2.5.2. Simulation results of the 90° rheological rotator: (a) transformed coordinate system, (b) pressure fields, (c) pressure gradient fields, and (c) velocity fields.
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Figure 2.5.3. Simulation results of the 30° rheological rotator: (a) transformed coordinate system, (b) pressure fields, (c) pressure gradient fields, and (c) velocity fields.
Figure 2.5.4. Simulation results of the 1800° rheological rotator as an artificial tornado:
(a) transformed coordinate system, (b) pressure fields, (c) pressure gradient fields, and (c) velocity fields.
123 2.5.4. Metamaterial design