| 1. |
- Nika, Grigor, et al.
(författare)
-
Asymptotics for Dilute Emulsions with Surface Tension
- 2015
-
Ingår i: Journal of Elliptic and Parabolic Equations. - : Springer Science and Business Media LLC. - 2296-9020 .- 2296-9039. ; 1:1, s. 215-230
-
Tidskriftsartikel (refereegranskat)abstract
- We consider an emulsion formed by two newtonian fluids, one being dispersed in the other under the form of droplets, in the presence of surface tension. We investigate the dilute case where the droplet size aɛ is much smaller than the distance e between the droplets’ centers. We prove that the limit behavior when $ɛ → 0$ is described by the unperturbed Stokes flow and estimate the order of convergence rate of the velocity to be $a_n/2^ɛ$. We improve the convergence result and determine the first corrector in the velocity expansion. Taylor’s and Einstein’s viscosity formulas are recovered.
|
|
| 2. |
- Nika, Grigor, et al.
(författare)
-
Design of multi-layer materials using inverse homogenization and a level set method
- 2019
-
Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 346, s. 388-409
-
Tidskriftsartikel (refereegranskat)abstract
- This work is concerned with the micro-architecture of multi-layer material that globally exhibits desired mechanical properties, for instance a negative apparent Poisson’s ratio. We use inverse homogenization, the level set method, and the shape derivative in the sense of Hadamard to identify material regions and track boundary changes within the context of the smoothed interface. The level set method and the shape derivative obtained in the smoothed interface context allows to capture, within the unit cell, the optimal micro-geometry. We test the algorithm by computing several multi-layer auxetic micro-structures. The multi-layer approach has the added benefit that contact during movement of adjacent “branches” of the micro-structure can be avoided in order to increase its capacity to withstand larger stresses.
|
|
| 3. |
- Nika, Grigor, et al.
(författare)
-
Dilute emulsions with surface tension
- 2016
-
Ingår i: Quarterly of Applied Mathematics. - : American Mathematical Society (AMS). - 0033-569X .- 1552-4485. ; 74:1, s. 89-111
-
Tidskriftsartikel (refereegranskat)abstract
- We consider an emulsion formed by two newtonian fluids in which one fluid is dispersed under the form of droplets of arbitrary shape in the presence of surface tension. We consider both cases of droplets with fixed centers of mass and of convected droplets. In the non-dilute case, for spherical droplets of radius aϵ of the same order as the period length $ϵ$, the two models were studied by Lipton-Avellaneda (1990) and Lipton-Vernescu (1994). Here we are interested in the time-dependent, dilute case when the characteristic size of the droplets aϵ, of arbitrary shape, is much smaller than ϵ. We study the limit behavior when $ϵ→0$ in each of these two models. We establish a Brinkman type law for the critical size $a_ϵ=O(ϵ^3)$ in the first case, whereas in the second case there is no “strange” term, and in the limit the flow is unperturbed by the droplets.
|
|
| 4. |
- Nika, Grigor, et al.
(författare)
-
Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension
- 2016
-
Ingår i: Discrete and Continuous Dynamical Systems. Series S. - : American Institute of Mathematical Sciences. - 1937-1632 .- 1937-1179. ; 9:5, s. 1553-1564
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we are interested in a problem of dilute emulsions of two immiscible viscous fluids, in which one is distributed in the other in the form of droplets of arbitrary shape, with non-uniform surface tension due to surfactants. The problem includes an essential kinematic condition on the droplets. In the periodic homogenization framework, it can be shown using Mosco-convergence that, as the size of the droplets converges to zero faster than the distance between the droplets, the emulsion behaves in the limit like the continuous phase. Here we determine the rate of convergence of the velocity field for the emulsion to that of the velocity for the one fluid problem and in addition, we determine the corrector in terms of the bulk and surface polarization tensors.
|
|
