Polymeric Rheological Metamaterials
2.3. Rheological cloak
2.3.1. Introduction
2.3.3.2. Simulation results without cloaks
The Navier-Stokes equations can become the simplified forms same as Eqns. 2.2.8-11 without the transient term ( ⁄ ) like
∇ ∙ = 0 (2.3.17)
∇ ∙ = ∇ (2.3.18)
= (∇ + ∇ ) (2.3.19) ,where is the second order stress tensor, is the velocity field, is the hydrostatic pressure, is the viscosity of a fluid, and ∇ is the nabla operator, under the assumptions of steady-state flow, an incompressible fluid, and neglecting gravity effect. In this section, two general cases without cloaks were simulated and analyzed; the bare case and the obstacle case.
Figures 2.3.4(a), (i-iii) show the simulation results for the bare case where no obstacle exists. The parallel pressure contours and the constantly distributed pressure field indicate that no perturbation occurs in the fluid. The parallel flow streamlines, uniform velocity fields, and uniform pressure gradient field imply the same characteristics about the bare case. In the obstacle case (Figs. 2.3.4(b), (i-iii)), the dense pressure contours were generated on the obstacle surface. Therefore, the pressure gradient field was perturbed and
51
a large pressure gradient was applied to the obstacle surface resulting in a drag force.
Accordingly, the flow streamlines were disturbed and the obstacle must be subjected to a resistant force by the fluid that strikes the obstacle surface.
Figure 2.3.4. Simulation results for (a) the bare case and (b) the obstacle case; (i) pressure fields, (ii) pressure gradient fields (py), and (iii) velocity fields.
52 2.3.3.3. Simulation results with cloaks
In this section, the simulation results of the cases with the rheological cloak are shown and discussed. At first, numerical simulation using the transformed viscosity tensor for the ideal case (Eqn. 2.3.15) was carried out. The transformed viscosity tensor for the ideal cloak ( ) has spatially varying components both and axis (Fig. 2.3.5(a)). The radial component varies from infinite viscosity at = to 2 mPa∙ s at = and the azimuthal component varies from 0 mPa∙s at = to 0.5 mPa∙s at = . Due to this spatial relationship, impedance at the boundary of the cloak ( = ) is perfectly matched with the background impedance although the space was distorted.
The simulation results with the ideal cloak are shown in Figs. 2.3.5(b-c). In the simulation, the transformed Navier-Stokes equations (Eqns. 2.2.29-30) were solved by using the PDE interface customized in COMSOL Multiphysics. The cloaking behavior is discussed later along with the simulation results of the reduced cloak (Fig. 2.3.6) because there are so many overlapping parts between the two results.
Unfortunately, the ideal cloak is almost impossible to be realized experimentally due to the extremely varying material parameters at both principal axes, and . Therefore, the reduced set of the transformed viscosity tensor ( ) was calculated as Eqn. 2.3.16. For impedance matching with the background area, the reduced transformed viscosity tensor ( ) should be scaled up 1.5 times. Each component of the reduced transformed viscosity tensor was plotted in Fig. 2.3.6(a). Unlike the transformed viscosity tensor for the ideal cloak ( ), it has the radial component varying from infinite at = to 1.5 mPa∙s at
= . More importantly, the azimuthal component does not vary and has a constant value of 0.375 mPa∙s. is the only varying component along the radial distance since becomes a constant. Since one of the components of both axes becomes a constant, the experimental conditions can be much more mitigated. Despite this material condition alleviation, the reduced cloak can induce the same dispersion relationship, except for outer boundary of the cloak.
The simulation results of the reduced case rheological cloak are shown in Figs.
2.3.6(b-c) for pressure, pressure gradient, and velocity fields. The obstacle encircled by the cloak was not affected by the applied pressure distribution (Fig. 2.3.6(b), (i)). In other
53
words, the deviatoric stress of a flowing fluid cannot penetrate into the central region but rather is directed around the obstacle. As a result, the pressure contour outside the cloak remains parallel. The pressure gradient outside the cloak has a constant value (Fig. 2.3.6(b), (ii)), as if there was no obstacle around as same with the bare case. It means that an external observer is unaware of something hidden in the fluid flow from a rheological point of view.
Also, this concealment removes the confusion of velocity fields and creates straight streamlines outside the cloak (Fig. 2.3.6(b), (iii)). Instead, the fluid momentum, which is excluded from both the center and the background, is compensated in the annular cloak region (a < < ), inducing a very high flow rate. Since the force of a flowing fluid does not transfer to the rheologically empty space (0 < < a), it is obvious that the drag to the surface of the placed object must be very small.
This cloaking phenomenon can be understood more intuitively in the simulation results of the cloak-only case that the obstacle is absent (Figs. 2.3.6(c), (i-iii)). In the cloaked region, pressure gradient along the flow direction (Py) is almost zero (Fig. 2.3.7(a)).
Furthermore, the magnitude of velocity field is close to zero unlike the fast flow velocity outside the cloak and it allows us to predict very tranquil flow inside the cloak (Fig. 2.3.6(c), (iii) and Fig. 2.3.7(b)). Most notably, it could be deduced that the cloaking behavior is independent of a shape of objects within the circular cloaked region. As a result, it was numerically proven that a drag-free fluidic space can be created if the cylindrical space is wrapped with the cloak with the scaled reduced transformed viscosity tensor.
In practice, how much drag is reduced by the cloak was evaluated by calculating and comparing the drag forces applied on the surface of the cylindrical obstacle (Fig. 2.3.7(c)).
The drag force ( ) acting on the obstacle surface was defined as
= (2.3.20)
, where is the y-axis stress acting on the obstacle surface and is the surface area of the obstacle. Both the viscous force and the pressure force are included in the stress term of . The drag coefficient ( ) was defined as
54
= 2
(2.3.21)
, where is the mass density of a fluid, is the flow velocity of the obstacle relative to a flowing fluid, and is the reference area of the obstacle.
The drag force on the obstacle was 120 μN, but it was reduced by 7 times to 18 μN after cloaked. Besides, the drag coefficient, which is a dimensionless quantity, was decreased by 10 times from 4569.7 to 440.8 by cloaking. These calculation results suggest the possibility of applications of the rheological cloak to the drag-free technology. The calculated drag values did not become zero because these are the mathematical iteration results. Naturally, if a smaller obstacle is placed in the cloak, the drag reduction rate will increase much higher (not shown in this dissertation).
55
Figure 2.3.5. Modeling of the ideal case rheological cloak. (a) Spatially varying viscosity tensor components (i) in radial and (ii) azimuthal axes. Simulation results for (b) the cloaked-obstacle case and (c) the cloak-only case of (i) pressure fields, (ii) pressure gradient fields (py), and (iii) velocity fields.
56
Figure 2.3.6. Modeling of the reduced case rheological cloak. (a) Spatially varying viscosity tensor components (i) in radial and (ii) azimuthal axes. Simulation results for (b) the cloaked-obstacle case and (c) the cloak-only case of (i) pressure fields, (ii) pressure gradient fields (py), and (iii) velocity fields.
57
Figure 2.3.7. Profiles of the simulation fields and drag values. Profiles of (a) the pressure gradient field (py) and (b) velocity field at the center line (the line at y=0.5 cm and z=25 μm). (c) Comparison of the drag forces and drag coefficients generated on the obstacle surface with or without the cloak.
58 2.3.4. Metamaterial design