Well-Posedness of Hydrodynamics on the Moving Elastic Surface

Wang, Wei; Zhang, Pingwen; Zhang, Zhifei

doi:10.1007/s00205-012-0548-x

Cited by 17 publications

(9 citation statements)

References 24 publications

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“…Modern mathematical methods, such as the implicit interfaces methods [7, 10–12] and direct boundary integral and immersed boundary approaches [3, 9, 28, 32], calculate the elastic energy of a bilayer in terms of the principal curvatures of the neutral surfaces, assuming that lipid orientation is normal to the neutral surface. Mathematical studies minimizing the Helfrich mean curvature squared energy of surfaces have been presented [31, 38, 39]. These approaches describe the bilayer by one mathematical surface; they give an accurate determination of membrane energy when curvatures are small compared to inverse membrane thickness, as is the case for relatively large vesicles and liposomes.…”

Section: Introductionmentioning

confidence: 99%

Teardrop shapes minimize bending energy of fusion pores connecting planar bilayers

2013

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A numerical gradient flow procedure was devised to characterize minimal energy shapes of fusion pores connecting two parallel planar bilayer membranes. Pore energy, composed of splay, tilt, and stretching, was obtained by modeling each bilayer as two monolayers and treating each monolayer of a bilayer membrane as a freely deformable surface described with a mean lipid orientation field. Voids between the two monolayers were prevented by a steric penalty formulation. Pore shapes were assumed to possess both axial and reflectional symmetry. For fixed pore radius and bilayer separation, the gradient flow procedure was applied to initially toroidal pore shapes. Using initially elliptical pore shapes yielded the same final shape. The resulting minimal pore shapes and energies were analyzed as a function of pore dimension and lipid composition. Previous studies either assumed or confined pore shapes, thereby tacitly supplying an unspecified amount of energy to maintain shape. The shapes derived in the present study were outputs of calculations and an externally provided energy was not supplied. Our procedure therefore yielded energy minima significantly lower than those reported in prior studies. The membrane of minimal energy pores bowed outward near the pore lumen, yielding a pore length that exceeded the distance between the two fusing membranes.

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Section: Introductionmentioning

confidence: 99%

Teardrop shapes minimize bending energy of fusion pores connecting planar bilayers

2013

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“…Wang et al . [38] prove the local well-posedness of an elastic surface model consisting of a viscous incompressible membrane fluid, but neglect the interaction with the bulk fluid. Their work generalizes an earlier analysis by Hu et al .…”

Section: Introductionmentioning

confidence: 92%

“…Proof. We shall define an energy functional, E, such that 38) and such that there exist constants C 1 > 0 and C 2 > 0, where…”

Section: The Energy Estimatementioning

confidence: 99%

“…Their model for the elastic shell involves a bending stress which extremizes the Willmore energy functional, similar to here, and a membrane or surface energy that is a function of the local area ratio. Wang et al [40] prove the local well-posedness of an elastic surface model consisting of a viscous incompressible membrane fluid, but neglect the interaction with the bulk fluid. Their work generalizes an earlier analysis by Hu et al [27].…”

Section: Introductionmentioning

confidence: 99%

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Well-posedness of two-dimensional hydroelastic waves

Ambrose¹,

Siegel²

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A well-posedness theory for the initial-value problem for hydroelastic waves in two spatial dimensions is presented. This problem, which arises in numerous applications, describes the evolution of a thin elastic membrane in a two-dimensional (2D) potential flow. We use a model for the elastic sheet that accounts for bending stresses and membrane tension, but which neglects the mass of the membrane. The analysis is based on a vortex sheet formulation and, following earlier analyses and numerical computations in 2D interfacial flow with surface tension, we use an angle–arclength representation of the problem. We prove short-time well-posedness in Sobolev spaces. The proof is based on energy estimates, and the main challenge is to find a definition of the energy and estimates on high-order non-local terms so that an a priori bound can be obtained.

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“…There exist further results [11,17,18] concerning a Helfrich-type flow where the Lagrange parameters instead of volume and area are prescribed and which consequently should not be related directly to fluid vesicles. In [28] local-in-time existence and uniqueness for a homogeneous Newtonian surface fluid subject to Canham-Helfrich stresses is shown. While the bulk fluid is neglected the authors keep the inertial term in the equations for the surface fluid, yielding a kind of dissipative fourth order wave-type equation.…”

Section: Introductionmentioning

confidence: 97%