The formal development of a theory of viscoelastic surface fluids with bending resistance - their kinematics, dynamics, and rheology are discussed. It is relevant to the mechanics of fluid drops and jets coated by a thin layer of immiscible fluid with rather general rheology. This approach unifies the hydrodynamics of two-dimensional fluids with the mechanics of an elastic shell in the spirit of a Cosserat continuum. There are three distinct facets to the formulation of surface continuum mechanics. Outlined are the important ideas and results associated with each: the kinematics of evolving surface geometries, the conservation laws governing the mechanics of surface continua, and the rheological equations of state governing the surface stress and moment tensors

Waxman, A. M.

English

NASA Technical Reports Server

Mecharlics of couple-stress fluid coatings 
Allen M. Waman 
Biomedical Engineering & Instrumentation, National Institutes of Health 
Bldg. 13, Rm. 3W-13, Rethesda, MD 20205 
Abstract 
We outline here, the formal development of a theory of viscoelastic surface fluids with I 
bending resiatance - their kinematics, dynamics, and rheology. It is relevant to the mech- 
anics of fluid drops and jets coated by a thin layer of imiacible fluid with rathr general I rheology. This approach unifies the hydrodynamics of two-dimensional fluids with the mechan- ics of an elastic shell in the spirit of a Cosserat continuum. ? 
Introduction 
--
Recently. ~ a x n a n l * ~  has developed a formal theory of viscoelastic surface fluids in which I bending resistance was incorporated in a purely phenomenological way. Motivation for this I 
two-dimensional continuum theory stems from a variety of applications: interfacial stability f emulsion rheology, red blood cell deformability, and coated drop and jet mechanics.The mech- 
anics of Newtonian surface fluids, accounting for the evolving surface geometry, was first ! 
considered by ~criven3 (as detailed by Arisfh). Extension to viscoelastic surface rheologies2 
and the inclusion of bending resistance1 in the formalism then followed. 
Bending rigidity arises from the finite thickness structure of the fluid coating, e.g. 
surface tension at the multiple interfaces of a compound drop or jet, a layer of normally 
oriented rod-like molecules such as those which form the lipid bilayer membrane of bio- 
logical cells, and electrically charged or polarized monolayers at a fluid interface. What- 
ever the molecular origins of the bending rigidity may be, the associated bending moments 
(or couple-stresses) may be included in the mechanics of the surface phase in a purely 
phenomenological way. However, it would clearly be of interest to see if averaging tech- 
niques could indeed reduce the mechanics of finite thickness fluid coatings to that of 
couple-stress surface fluids. Such averagir.g methods unc.eriie the development and success of 
elastic shell theory.5 The direct approach which we have adopted is motivated by the notion 
of a Cosserat surface which has been exploited by t!le shell theorists for some time now.6 
We view our model continuum as a two-dimensi dl viscoelastic fluid, isotropic in the 
surface, and asfiociate with each material point on this surface a 'director' (viz. an arrow) 
oriented along the local normal with its center of mass located at the surface. Changes in 
surface shape imply a reorientation of these directors which manifests itself dynamically in 
two ways: reorientation corresponds to curvature changes which generate bending moments, in 
addition the rate of reorientation corresponds to an internal angular momentum of the sur- 
face phase over and above any surface vorticity. We shall see that the director dynamics 
enters into the surface equations of motion through an asymmetric surface stress tensor and 
a transverse shearing stress. 
There are three distinct facets to the formulation of surface continuum mechbnics and we 
shall try to outline here the important ideas and results associated with each: the kinemat- 
ics of evolving surface geometries, the conservation laws governing the mechanics of surface 
continua, and the rheological equations of state g verning the surface stress and monent 
tensors. Further details may be found elsewhere. ' 1 '  
Evolving surface geometries 
As the surface phase 1s enerally located at the interface between two bulk fluids, mot- 
ions in the bulk lead to a 8istortion of the inrerface and hence, an evolution of the sur- 
face phase geometry. In order to discuss the mechanics of surface continua we must be able 
to track the surface as it moves through space. In addition, since various key geometrical 
quantities (e.g. metric and curvature tensors) enter into the dynamical equations, it is 
useful to derive evolution equations for these quantities. But first we must establish a co- 
ordinate system on the surface. Following ~criven3, tle construct a set of 'fixed surface co- 
ordinates' (r (d-1,2) which label geometric oints on the surface. As the surface evolves, 
these fired coordinates move through space a!ong the local normal to the surface; they are 
unaffected by any flow of the surface material tangential to the surface. Associated with 
these fixed coordinates are local tangent vectors 9. which then define a metric tensor for 
the surface Q S ~ . B .  A base vector triad at each point on the surface consists of these 
tangent vectors I(, and the local unit normal E to the surface. Thus, for example, the vcl- 
ocity y of the surface phase through space may be decomposed according to y m f l & , + v ( @ j ~ .  
https://ntrs.nasa.gov/search.jsp?R=19820015595 2020-03-21T16:55:21+00:00Z
Here, y* are the contravariant components of surface velocity a~rd v@" (a scalar) is the com- 
ponent of velocity in the direction of the local normal. 
Let the surface be embedded in an inertial space described by general coordinates X I  
(i=1,2,3) with corresponding base vectors #, The surface location is expressed by a rela- 
tion between the fixed surface coordinates and the space coordinates: # S X ' ( U * , ~ ) .  It can be 
shown that the evolution of the surface geometry through space is governed by the equation 
where the local rormal to the surface has been deccmpocred as prnif;. That is, the fixed sur- 
face coordinates move through space in the direction of the local normal and do so at a rate 
given by the normal component of velocity of the surface phase. One may also obtain evolu- 
tion equations for the metric (hC) and curvature (hh) tensors assoctated with the fixed 
surface coordinate system. 
Equation (2) expresses the fact that a normal velocity distribution over a curved surface 
leads to a stretching of the fixed surface coordinates (viz, the radial expansion of a 
spherical surface). Equation ( 3 )  describes geometric shape changes via an evolving curvature 
The first term, being the second covariant derivative of the normal velocity over the sur- 
face, leads to new geometric forms (a first derivative wo~ld only e:..prpss a tilting of the 
surface). Tht second term in (3) incorporates the effects of a chan in8 surface metric in ! the shape changes (e.g. an expanding sphere has a changing radius o curvature though it re- 
mains spherical). Simple evolution equations may also be derived for the tangent and normal 
vectors as well as the Christoffel symbols of the fixed surface coordinate systen~. As may be 
seen from equations (1) - (3), the evolution of the surface geonetry may be decoupled from 
the tangential flow of the surface phase in so far as it depends only on the nora,al compon- 
ent of velocity. However, the normal velocity is implicitly coupled to the tangential flow 
through the equations of motion governing the surface phase. 
Surface equations of motion 
The surface equations of conservation of mass, momentum, and angular momentum may all be 
derived in the fixed surface coordinate system through the use of the Reynolds transport 
theorem generalized to surface flows. If we may neglect mass exchange between tbe surface 
phase and the neighboring bulk fluids on the timescales of interest, then the conservation 
of surface mass leads to the following continuity equation for the surface mass density )C: 
The first two terms in (4) resemble those found in the continuity equation for bulk fluids. 
The third term is associated with the stretching 3f the fixed surface coordinates (cf. eq.2) 
'd=&g being the local mean cyrvature 05 the surface. If we bring this term to the right-hand 
side of (4) it appears as a mass sink (H4 0 for a sphere) in that it represents a fixed 
amount of mass being spread over an ever increasing surface area (for H 4 0  and v % O )  
The conserl.ation of linear surface momentum leads us to the follo~ling equations of motion 
for the surface continuum. They may be thought of as dynamical boundary conditions which 
couple the adjacent bulk phases. 
The right-hand sides of ( 5 )  closely resemble the equilibrium shell equations. 5 They repre- 
sent the net tan ential and normal forces actin on an element of surface. The surface 
stress tensor 4 and tmansrerse shear stress8 will be discussed further below; they mani- 
fest themselves in local stress veLtors (forces per unit length) acting on a curve bounding 
a small element of surface, T.0 leading t~ an in-plane stress, I* corresponding to a stress 
normal to the surface alone the bounding curve. These are usually termed internal #tresses. 
External stresses act on a small elenrent of surface and arise from body Zorces ( e . g .  gravi- 
tational and electrostatic forces) represented in ( 5 )  by ( j *Gafi+to*g,  and from the neigh- 
boring bulk fluids exerting normal and tangential stresses on i!5, t urface phase embodied in f=i"b*fm#. Expressions for C and fwJ are given elsewhere. On the left-hand sidee 
of (5) we have the tangential and normal components of surface acceleration given by 
In addition to the intrinsic derivatives of velocity in equaticL-.- i 6 ) ,  t:aere are terms re- 
sembling centripetal and Coriolis accelerations. They arise fr,:i, ti-0 tame varying base vec- 
tors associated with the evolving surface coordinate system, i . ;  evolving sl~rface is a 
non-inertial reference frame. In equation (5b), we see how tal : stresses cons ire with 
the cur7rature to generate a normal fcrce (a generalization of place  condition^. Siml- 
larly, the transverse shear in (5a) generates a tangentisl fc - .a the curvature. 
Considerations of angular momentum conservation Lead to exp.essions for the transverse 
sh ar stress $* and the antismetric part -6 the ~urface stress tensor T-3. (We decompose 4 into a sum of symmetric and antisymnetric parts, e * T m + T * # t )  Me find 
where flH is the surface moment (or cocple-stress) tensor. In equations (7) and (a), is 
the contravariar~t alternating tensor of the surface, 6 and H~ are components of any ext.er,- 
nalLy imposed torque on the surface phase (e g. magnetic couples), ?nd & represents the 
tangential components of internal angular momentum associated with tbe tumbling motion of 
the directors. A ccmplicated expreesion may be derived for g,, but what is important is 
that it is determined entirely by the velocity fitld of the surface phase and the surface 
geomeLry (along with a presumed moment of inertia). Thus, it introduces no new unknowns into 
the equations of motlon. In arriving at (8) we have assumed that each director spins about 
its local normal at a rate equal tc one-half the local surface vorticjty. That is, they are 
viscously coupled to their surface phase environments and hencz, there is no normal compon- 
ent of internal angular momentum. We may use (7) and (8) to eliminate f and TNCI  from the 
equations of motion (5). These equations simplify enormously for slow flows where we may ne- 
glect all terms associated with the inertia of the surface phase. It remalns for us to give 
expressions for the synnnettic surface stress and moment tensors T(*U and @. 
Surface rheology 
We concern ourselves here with surface fluids which are isotr9pic in the surface, and 
summarize the rheologicyl laws discussed in detail by Waxman. l,2 Allowing the surface phase 
to support an in-plane hydrostatic stress' in the absence of any motion, we write 
Here,Jr is an isotropic surface pressure (or minus the net surface tension). It is related 
to the drnsity, temperature, and chemistry of the surface phase and neighboring bulk phases 
via a thermodynamic equation of state. For incompressible surface contirua,r becomes a dy- 
namic variable to be solved for along with the suzface velocity field. The ;ynmetric tensor 
y* embodies the viscoue ar.d elastic components of stress. (An explicit dependence of on 
# a'.ready represents an area elasticity.) 
The Newtonian surface fluid is the simplest e:-zmple of a viscous surface phaee. IL is de- 
scribed by a linear relation between stress and rate-of-strain; 
with L ~ n d  4 being coefficients of surface dilational and ehear viscosity. resy~ctively. The 
surface rate-of-strain is given by 
The first term represents the rate-of-deformntion due to gradients in the tangential flow, 
the second term rrpresents the geometric straining associated with the evolving suri'ace 
metric (cf . eq. 2) . 
A simple viscoelastic surface fluid which exhibits both stress relaxation in a finite 
time and delayed elasticity is ths 'corotational Jeffreys surface fluid' described L, 
Here, c4*' is of the r'orm (lob), and A ) ~ > o  are stress relaxation and strain retardation 
time constants. The time-derivative operator in (12) represents a rate-of-change as seen 
from a frame which is translating and cxotating (but not codeforming) with an element of 
surface msterial. It is the surface analogue of the Jaumann time-derivative. Its d.,rivation 
and cropertiss are discussed in detail in the work of ~axman.2 Equation (12) is a quasi- 
linear rhe lo ical law of the rate-type; nonlinear modifications of (12) have also been 
discussed. 9 
A simple law for the bending moment tensor is motivated by Hookean elasticity. It de- 
scribes a surface capable of storing potential energy in tending and relates the moment 
tensor to a measure of bending strain in a linear fashion; 
M* = cyrr  Klr * (13) 
Again, c**C is of the form (lob) with L acd 6 representing independent (positive moduli 
of bending rigidity. An appropriate chcice of bending strain for surface fluids is 3 
That is, Kw measures the deviation of urvature from a comparison curvature ar6 which rep- 
resents an lnitial reference curvature ia evolved forward in time in a corotational way. 
7iscoelastic moment relations of the rate-type may be constructed from (13) through the use 
of the surface corotational time-derivative operator. 
It is hoped that the dynamical fornuletion outlined here for couple-stress surface fluids 
will provide a useful approximation to the dynamics of thin fluid coatings in evolving ge- 
ometries. Application to the mechanics of cell membranes is anticipated in the near future. 
References 
1. Waxman, A.M. "Dynamics of a Couple-Stress Fluid Membrane" (preprint), 1981. 
2. Waxman. A.M. "A Corotational Time Derivative for Surface Tensors. Constitutive 
- -  .
Relations and a New Measure of Bending Strain" J. Non-Newtonian Fluid ~ech(in press), 1981 
3. Scriven, L.F..  "Dynamics of a Fluid Interface" Chem. Eng. Sci., vo1.12, p.98, 1960. 
4. Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice- 
Hall. 1962. 
5. Green, A.E. & Zerna, W. Theoretical elastic it^, 2nd edn, Oxford Univ. Press, 1968 
6. Green, A.E., Naghdi, P.M. & Wainwri t, W.L. A General Theory of a Cosserat 
Surface" Arch. Rat. Mech. Anal., vo1.20, p.%7, 1965. 


Mechanics of couple-stress fluid coatings

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