Finite-time self-similar rupture in a generalized elastohydrodynamic lubrication model

2025-04-22 0 0 458.34KB 7 页 10玖币

侵权投诉

Finite-time self-similar rupture

in a generalized elastohydrodynamic lubrication model

William Changa, Hangjie Jib

aUniversity of Southern California, Los Angeles, 90089, CA, USA

bDepartment of Mathematics, North Carolina State University, Raleigh, 27607, NC, USA

Abstract

Thin ﬁlm rupture is a type of nonlinear instability that causes the solution to touch down to zero at ﬁnite

time. We investigate the ﬁnite-time rupture behavior of a generalized elastohydrodynamic lubrication model.

This model features the interplay between destabilizing disjoining pressure and stabilizing elastic bending

pressure and surface tension. The governing equation is a sixth-order nonlinear degenerate parabolic par-

tial differential equation parameterized by exponents in the mobility function and the disjoining pressure,

respectively. Asymptotic self-similar ﬁnite-time rupture solutions governed by a sixth-order leading-order

equation are analyzed. In the weak elasticity limit, transient self-similar dynamics governed by a fourth-order

similarity equation are also identiﬁed.

Keywords: high-order nonlinear PDEs, degenerate PDEs, singularities, thin ﬁlms

1. Introduction

This paper presents a study of the development of ﬁnite-time singularities in a one-dimensional sixth-

order partial differential equation for h(x,t)on a ﬁnite domain, 0 ≤x≤L,

∂h

∂t=∂

∂xhn∂

∂xB∂4h

∂x4−∂2h

∂x2+1

mhm,(1)

where the parameters B,m,n>0. This model is motivated by the work by Carlson and Mahadevan [1] on

adhesive elastohydrodynamic touchdown that occurs as an elastic sheet begins to adhere to a wall. The PDE

(1) ﬁts into the framework of classical lubrication theory which has been widely studied for the dynamics of

thin layers of slow viscous ﬂuids spreading over solid surfaces [2, 3]. Under the long-wave approximation,

the lubrication equation for the evolution of the thickness (or the height hof the free-surface) of the ﬂuid

layer can be derived from Navier-Stokes equations in the low Reynolds number limit,

∂h

∂t=∂

∂xM(h)∂p

∂x,(2a)

where the mobility function M(h) = hnwith n>0. Here, n=3 corresponds to the no-slip boundary

condition at the liquid-solid interface, and more general Navier slip condition can be incorporated via

M(h) = h3+λh2. Following the work of Young and Stone [4], we deﬁne the dynamic pressure pto in-

corporate the elastohydrodynamic effects,

p=B∂4h

∂x4−∂2h

∂x2+Π(h),B>0,(2b)

Email addresses: chan087@usc.edu (William Chang), hangjie_ji@ncsu.edu (Hangjie Ji)

Preprint submitted to Elsevier October 11, 2022

arXiv:2210.04405v1 [math.AP] 10 Oct 2022

where ∂4h/∂x4represents the elastic bending pressure due to long-wavelength sheet deformations, B>0

is a scaling parameter for the bending pressure, ∂2h/∂x2represents the surface tension between the elastic

sheet and liquid, and the disjoining pressure

Π(h) = A

hm,m>0,A=1

m>0 (2c)

characterizes the wetting property of the solid substrate, where A>0 is the Hamaker constant. For m=3,

Π(h) = A/h3corresponds to the van der Waals model [5] for the destabilizing intermolecular adhesion

pressure [1]. Other elastohydrodynamic lubrication models [4, 6] have also used the disjoining pressure

Π(h) = A(h−3−σ6h−9)with σ>0, where the two terms in ˜

Π(h)represent the repulsive and attractive

intermolecular forces corresponding to the standard Lennard-Jones potential. A similar form of the disjoining

pressure ˜

Π(h) = Ah−3(1−εh−1)with ε>0 is often used in thin ﬁlm models, setting a lower bound h=

O(ε)>0 for the ﬁlm thickness hand preventing thin ﬁlm rupture from happening [7–9].

Starting from positive and ﬁnite-mass initial data h0(x)>0 at time t=0, the dynamics of the model

(1) are governed by the interaction between the higher-order elastic bending pressure, the surface tension,

and the disjoining pressure. Following the work of Young and Stone [4], we consider the no-ﬂux boundary

conditions at x=0 and x=L,

hx=hxxx =hxxxxx =0,at x=0,L.(3)

The dynamics of (1) can also be described by a monotone decreasing energy functional

E=ZL

2∂2h

∂x22

2∂h

∂x2

+U(h)dx,with dE

dt =−ZL

hn∂p

∂x2

dx ≤0,(4)

where U(h)is the interaction potential that satisﬁes U0(h) = Π(h).

Thin ﬁlm rupture is a type of nonlinear instability that leads to ﬁnite-time singularities as the ﬁlm thick-

ness approaches zero at a point. That is, h→0 at an isolated point, x=xc, at a ﬁnite critical time t=tc. It

was shown in [10] that thin ﬁlm equations can yield self-similar rupture singularities driven by van der Waals

forces. Different types of ﬁnite-time rupture dynamics have been investigated in a family of generalized lu-

brication equations parametrized by exponents in conservative and non-conservative loss terms, respectively

[7, 11]. In this work, we focus on the impact of the sixth-order bending pressure and the fourth-order surface

tension terms on the rupture dynamics of the generalized elastohydrodynamic lubrication equation (1).

Finite-time singularities in thin ﬁlm equations can result from growth in spatial perturbations due to

strong instabilities. To perform a stability analysis of ﬂat ﬁlm solutions in (1), we perturb the spatially-

uniform base state h=¯

hby an inﬁnitesimal Fourier mode h(x,t) = ¯

h+δeikπx/L+λt+O(δ2), where kis

the wave number, λis the growth rate of disturbances, and the initial amplitude δ1. Substituting the

expansion into model (1) and linearizing about h=¯

hyields the dispersion relation

λ=−¯

hnkπ

L2"Bkπ

L4

+kπ

L2

−1

hm+1#.(5)

This relation indicates that the uniform ﬁlm ¯

h<hcis long-wave unstable with respect to perturbations asso-

ciated with any wave number k∈Z+, where the critical ﬁlm thickness hc=B(kπ/L)4+ (kπ/L)2−1/(m+1).

Moreover, the relation (5) also shows that the disjoining pressure Π(h) = 1/(mhm)is destabilizing, and both

the elastic bending pressure B∂4h/∂x4and the surface tension −∂2h/∂x2are stabilizing in the PDE (1).

The structure of the paper is as follows. In Section 2 we analyze the asymptotic self-similar rupture

solutions in (1), with a focus on the role of the bending pressure term. Numerical studies for the singularity

solutions are presented in Section 3, followed by concluding remarks in Section 4.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Finite-timeself-similarruptureinageneralizedelastohydrodynamiclubricationmodelWilliamChanga,HangjieJibaUniversityofSouthernCalifornia,LosAngeles,90089,CA,USAbDepartmentofMathematics,NorthCarolinaStateUniversity,Raleigh,27607,NC,USAAbstractThinlmruptureisatypeofnonlinearinstabilitythatcausesthesolut...

展开>> 收起<<

Finite-time self-similar rupture in a generalized elastohydrodynamic lubrication model.pdf

共7页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Finite-time self-similar rupture in a generalized elastohydrodynamic lubrication model

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: