Abstract
This article concerns the stability of a drop on a wall for which the contact angle, \(\theta _\mathrm{{{w}}}\), varies from place to place. Such a wall may allow unstable equilibria of the drop, i.e. ones for which small perturbations to equilibrium grow, making the equilibrium unrealisable in practice. This will be referred to as dynamic instability and is one of the two versions of instability considered. The other arises from consideration of potential energy, which is the sum of surface (liquid/gas, liquid/solid and solid/gas) components and the gravitational potential energy. Equilibria are extrema of the potential energy with respect to variations of drop geometry which preserve its volume. An equilibrium is said to be statically stable if it is a local minimum of the potential energy for volume-preserving perturbations of the drop. The relationship between static and dynamic stability is the main subject of this paper. The liquid flow is governed by the incompressible Navier–Stokes equations. To allow for the moving contact line, a Navier slip condition with slip length \(\lambda \) is used at the wall, as is a prescribed contact angle, \(\theta _\mathrm{{{w}}} =\theta _\mathrm{{{w}}} ({x,y})\), at the contact line, where x, y are Cartesian coordinates on the wall. The perturbation is assumed small, allowing linearisation of the governing equations and, in the usual manner of stability analysis, complex modes having the time dependency \(e^\mathrm{{{st}}}\) are introduced. This leads to an eigenvalue problem with eigenvalue s, the sign of whose real part determines dynamic stability/instability. A quite different eigenvalue problem, which describes static stability/instability is also derived. It is shown that, despite this difference, the conditions for dynamic and static instability are in fact the same. This conclusion is far from evident a priori but should be good news for interested numerical analysts because determination of static stability is much less numerically costly than a dynamic stability study, whereas it is the latter which gives a true determination of stability.
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References
Bonn D, Eggers J, Indekeu J, Meunier J, Rolley E (2009) Wetting and spreading. Rev Mod Phys 81:739–805
Vellingiri R, Savva N, Kalliadasis S (2011) Droplet spreading on chemically heterogeneous substrates. Phys Rev E 84:036305
Savva N, Groves D, Kalliadasis S (2019) Droplet dynamics on chemically heterogeneous substrates. J Fluid Mech 859:321–361
Bostwick JB, Steen PH (2015) Stability of constrained capillary surfaces. Annu Rev Fluid Mech 47:539–568
Finn R (1986) Equilibrium capillary surfaces. Springer, New York
Wu Y, Wang F, Ma S, Selzer M, Nestler B (2020) How do chemical patterns affect equilibrium droplet shapes? Soft Matter 16:6115
Brinkmann M, Kierfeld J, Lipowsky R (2004) A general stability criterion for droplets on structured substrates. J Phys A 37:11547–11573
Ewetola M, Ledesma-Aguilar R, Pradas M (2021) Control of droplet evaporation on smooth chemical patterns. Phys Rev Fluids 6:033904
Rabaud M, Moisy F (2020) The Kelvin–Helmholtz instability, a useful model for wind-wave generation? Comptes Rendus Mecanique 348(6–7):489–500
Moffatt HK (1964) Viscous and resistive eddies near a sharp corner. J Fluid Mech 18:1–18
Huh C, Scriven LE (1971) Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J Colloid Interf Sci 35:85–101
Sui Y, Ding Hang, Spelt PDM (2014) Numerical simulations of flows with moving contact lines. Ann Rev Fluid Mech 46:97–119
Hocking LM, Rivers AD (1982) The spreading of a drop by capillary action. J Fluid Mech 121:425–442
Cox RG (1986) The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J Fluid Mech 168:169–194
Afkhami S, Zaleski S, Bussmann M (2009) A mesh-dependent model for applying dynamic contact angles to VOF simulations. J Comput Phys 228:5370–5389
Sui Y, Spelt PDM (2013) Validation and modification of asymptotic analysis of slow and rapid droplet spreading by numerical simulation. J Fluid Mech 715:283–313
Maglio M, Legendre D (2014) Numerical simulation of sliding drops on an inclined solid surface. In: Computational and Experimental Fluid Mechanics with Applications to Physics, Engineering and the Environment, pp 47-69. Springer
Solomenko Z, Spelt PDM, Alix P (2017) A level-set method for large-scale simulations of three-dimensional flows with moving contact lines. J Comput Phys 348:151–170
Lawden DF (1962) An introduction to tensor calculus and relativity. Methuen & Co., Ltd, London
Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge
Kusumaatmaja H (2015) Surveying the free energy landscapes of continuum models: application to soft matter systems. J Chem Phys 142:124112
Acknowledgements
This work was carried out with support from the French ANR research agency, project number ANR-15-CE08-0031, also known as ICEWET.
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Appendix A: Mathematical details
Appendix A: Mathematical details
1.1 A.1 Derivation of (2.12)
The normal unit vector, \({\mathbf{n}}\), is originally only defined on the perturbed interface, \(d=\eta \), but can be extended using
The interface curvature follows from
evaluated at the interface. Since \(F=d-\eta \), \(\left| {\nabla d} \right| =1\) and \(\nabla d{\mathbf{.}}\nabla \eta =0\),
The second term on the right-hand side being of second order, it is neglected, hence \({\mathbf{n}}=\nabla F\) according to (A.1). Thus, (A.2) gives
correct to first order when evaluated at the interface.
For the equilibrium, we set \(\eta =0\) in (A.4) and \({\mathbf{u}}=0\) in (2.5) to derive
for the equilibrium pressure, \(p_{e} \), at the equilibrium interface, \(d=0\). Making the small displacement \(\eta {\mathbf{n}}\) to arrive at the perturbed interface and given that \(p_{e} -{\mathbf{G.x}}\) is constant,
correct to first order, where
On the other hand, applying (2.5) and (A.4) to the perturbed interface
Using (A.6),
It remains to express the term \(\nabla ^{2}\eta \) in terms of \(\eta \left( {q^{\alpha }} \right) \). To this end, let \(\Sigma \) be a region in the \(q^{\alpha }\) plane representing part of the equilibrium interface and V the volume in physical space for which \(q^{\alpha }\in \Sigma \) and \(0<d<\delta \), where \(\delta \) is an infinitesimal constant (see Figs. 6 and 7). The divergence theorem gives
The normal vectors of the parts of \(\partial V\) with constant d are directed parallel to \(\nabla d\). Since \(\nabla d{\mathbf{.}}\nabla \eta =0\), \(\partial \eta /\partial n=0\). Thus, these parts of \(\partial V\) do not contribute to (A.10). The remainder of \(\partial V\) gives
for the right-hand side of (A.10), where \(\partial S\) is the curve on the equilibrium interface corresponding to the boundary \(\partial \Sigma \) in the \(q^{\alpha }\) plane, \(\mathrm{{d}}s\) is elementary arc length along \(\partial S\) and \({\mathbf{N}}\) is the unit vector, tangential to the equilibrium interface and orthogonal to \(\partial S\), which is directed outwards from S (see Fig. 7). Thus, (A.10) implies
where S corresponds to \(\Sigma \).
From here until the end of this section, we restrict attention to the equilibrium interface. A point on the interface has position vector \({\mathbf{x}}\left( {q^{\alpha }} \right) \), of which derivatives, \(\partial {\mathbf{x}}/\partial q^{1}\) and \(\partial {\mathbf{x}}/\partial q^{2}\), are tangential to the interface and yield the metric tensor via
The components of this tensor form a symmetric, positive-definite matrix. Since \({\mathbf{N}}\) is tangential and \(\partial {\mathbf{x}}/\partial q^{1}\), \(\partial {\mathbf{x}}/\partial q^{2}\) span the space of such vectors,
It is convenient to define \(\nu _{\alpha } =g_{\alpha \beta } \mu ^{\beta }\), hence \(\mu ^{\alpha }=g^{\alpha \beta }\nu _{\beta } \), where \(g^{\alpha \beta }\) is the inverse of the matrix \(g_{\alpha \beta } \). Thus,
\({\mathbf{N.N}}=1\), (A.13), (A.15) and the definition of \(g^{\alpha \beta }\) imply
An infinitesimal displacement \(\mathrm{{d}}q^{\alpha }\) in the \(q^{\alpha }\) plane produces the displacement \(\mathrm{{d}}{\mathbf{x}}=\mathrm{{d}}q^{\alpha }\partial {\mathbf{x}}/\partial q^{\alpha }\) on \(S_{i} \), hence
Since (A.17) holds for any choice of \(\mathrm{{d}}q^{\alpha }\),
where \(\delta _{\alpha }^{\beta } \) is the Kronecker delta. Writing
Let \(\mathrm{{d}}q^{\alpha }\) represent an infinitesimal displacement along the curve \(\partial \Sigma \), in the sense, anticlockwise, indicated by the arrow in Fig. 6. The corresponding displacement, \(\mathrm{{d}}{\mathbf{x}}=\mathrm{{d}}q^{\alpha }\partial {\mathbf{x}}/\partial q^{\alpha }\), along \(\partial S\) is perpendicular to \({\mathbf{N}}\), hence, using (A.13), (A.15) and the definition of \(g^{\alpha \beta }\),
This result shows that \(\nu _{\alpha } \) provides a normal vector to \(\partial \Sigma \) in the \(q^{\alpha }\) plane, as indicated in Fig. 6, and implies
where \(\tau \ne 0\) is infinitesimal. Thus, the vector \(b_{\alpha } =\tau \nu _{\alpha } \) has components \(b_{1} =\mathrm{{d}}q^{2}\) and \(b_{2} =-\mathrm{{d}}q^{1}\). Given that the displacement along \(\partial \Sigma \) is anticlockwise, \(b_{\alpha } \), which is normal to \(\partial \Sigma \), is directed outwards from \(\Sigma \). Next, consider the infinitesimal displacement \(b_{\alpha } \) in the \(q^{\alpha }\) plane. This produces \(d{\mathbf{x}}=b_{\alpha } \partial {\mathbf{x}}/\partial q^{\alpha }\) in physical space. Equation (A.15) and \(b_{\alpha } =\tau \nu _{\alpha } \) give
hence
Because \(b_{\alpha } \) is directed outwards from \(\Sigma \), \(\mathrm{{d}}{\mathbf{x}}\) takes us from \(\partial S\) to a location on \(S_{i} \) just outside S. By definition, \({\mathbf{N}}\) is a normal vector directed outwards from S. Thus, \(\mathrm{{d}}{\mathbf{x.N}}>0\), hence, since \(b_{\alpha } b_{\alpha } >0\), (A.24) gives \(\tau >0\). Given that \(b_{\alpha } \) is directed outwards from \(\Sigma \), \(\nu _{\alpha } =\tau ^{-1}b_{\alpha } \) is an outwardly directed normal vector to \(\partial \Sigma \), as indicated in Fig. 6.
The components of \(g^{\alpha \beta }\) are
where g is the determinant of the matrix \(g_{\alpha \beta } \), which is positive because \(g_{\alpha \beta } \) is positive definite. (A.16), (A.22), (A.25) and \(\tau >0\) imply
Since \(\mathrm{{d}}s=\left( {g_{\alpha \beta }\,\mathrm{{d}}q^{\alpha \, }\mathrm{{d}}q^{\beta }} \right) ^{1/2}\), (A.20), (A.22) and (A.26) give
Using Green’s theorem,
Consider a rectangular area element, \(\mathrm{{d}}q^{1}\,\mathrm{{d}}q^{2}\), in the \(q^{\alpha }\) plane. This corresponds to a small parallelogram on S with sides \({\mathbf{a}}_{1} \,\mathrm{{d}}q^{1}\) and \({\mathbf{a}}_{2}\, \mathrm{{d}}q^{2}\), where \({\mathbf{a}}_{\alpha } =\partial {\mathbf{x}}/\partial q^{\alpha }\). The area of this parallelogram is \(\mathrm{{d}}S=\left| {{\mathbf{a}}_{1} \times {\mathbf{a}}_{2} } \right| \mathrm{{d}}q^{1}\,\mathrm{{d}}q^{2}=\left( {\left| {{\mathbf{a}}_{1} } \right| ^{2}\left| {{\mathbf{a}}_{2} } \right| ^{2}-\left( {{\mathbf{a}}_{1} {\mathbf{.a}}_{2} } \right) ^{2}} \right) ^{1/2}\mathrm{{d}}q^{1}\,\mathrm{{d}}q^{2}\) according to the Lagrange identity. Employing (A.13), \(\left| {{\mathbf{a}}_{1} } \right| ^{2}\left| {{\mathbf{a}}_{2} } \right| ^{2}-\left( {{\mathbf{a}}_{1} {\mathbf{.a}}_{2} } \right) ^{2}=g_{11} g_{22} -g_{12}^{2} =g\), hence \(\mathrm{{d}}S=\sqrt{g} \mathrm{{d}}q^{1}\,\mathrm{{d}}q^{2}\), which is the relation between elementary areas in physical space and the \(q^{\alpha }\) plane. Using this result and (A.28), (A.12) yields
Finally, letting \(\Sigma \) shrink down to approach a point, \(\nabla ^{2}\eta =\Delta \eta \), hence (A.9) gives (2.12).
1.2 A.2 Derivation of (2.14)
The condition that the liquid/gas interface meets the wall at angle \(\theta _\mathrm{{{w}}} \left( {x,y} \right) \) is
at the contact line, where \(n_{z} \) is the z-component of \({\mathbf{n}}\). As we saw earlier, \({\mathbf{n}}=\nabla d-\nabla \eta \) correct to first order in the perturbation. Thus,
Let \(\left( {x,y} \right) \) be a point on the equilibrium contact line and \(\bar{{\theta }}=\theta _\mathrm{{{w}}} \left( {x,y} \right) \) the associated contact angle (see Fig. 8). Setting \(\eta =0\) in (A.31), (A.30) gives
Since \(d\left( {x,y,z=0} \right) =0\), the point \(\left( {x+\mathrm{{d}}x,y+\mathrm{{d}}y} \right) \) lies on the perturbed contact line, \(d\left( {x,y,z=0} \right) =\eta \), provided
Applying (A.30) and (A.31) at \(\left( {x+\mathrm{{d}}x,y+\mathrm{{d}}y} \right) \),
Subtracting (A.32) and recalling that \(\bar{{\theta }}=\theta _\mathrm{{{w}}} \left( {x,y} \right) \),
Taking the displacement \(\left( {\mathrm{{d}}x,\mathrm{{d}}y} \right) \) perpendicular to the equilibrium contact line, it lies in the direction of the vector \(\left( {\frac{\partial d}{\partial x},\frac{\partial d}{\partial y}} \right) \) and (A.33) implies
With this displacement,
and
where
is a unit vector, normal to the equilibrium contact line and tangential to the wall, which is directed outwards from the wetted region (see Fig. 8). Thus, \(\varvec{\mathcal {N}}.\nabla \theta _\mathrm{{{w}}} \) is the normal derivative of \(\theta _\mathrm{{{w}}} \left( {x,y} \right) \) at the contact line. Since \(\left| {\nabla d} \right| =1\),
Let \({\mathbf{N}}\) be the unit vector shown in Fig. 8, which is tangential to the equilibrium interface and normal to the contact line. Keeping \(\left( {x,y} \right) \) constant, \(\mathrm{{d}}\eta =\mathrm{{d}}z\,\partial \eta /\partial z\) is the change in \(\eta \) for the increment \(\mathrm{{d}}z\). \(\nabla {{d}}. \nabla \eta =0\) implies that the component of \(\nabla \eta \) normal to the interface is zero. Thus, only the component of displacement, \(-\mathrm{{d}}z{\mathbf{N}}\sin \bar{{\theta }}\), parallel to the interface produces a change in \(\eta \), hence \(d\eta =-\mathrm{{d}}z\sin \bar{{\theta }}{\mathbf{N.}}\nabla \eta \). It follows that \(\partial \eta /\partial z=-\sin \bar{{\theta }}{\mathbf{N.}}\nabla \eta \) so (A.43) gives
as the contact-line condition, where
Taking the surface S, used in Sect. A.1 and shown in Fig. 7, to be the entire equilibrium interface, \(\partial S\) is the contact line and \({\mathbf{N}}\) is the vector defined above. Using (A.20), (A.44) yields (2.14). Note that, as stated following (2.14), its left-hand side is the derivative, \({\mathbf{N.}}\nabla \eta \), of \(\eta \), taken tangential to \(S_{i} \), normal to the contact line and outwards from \(S_{i} \).
1.3 A.3 A frequently used identity
The following identity will often be used in subsequent sections:
where \(f_{1} \) and \(f_{2} \) are any functions defined on the equilibrium liquid/gas interface \(S_{i} \) and C is the equilibrium contact line. This identity can be derived as follows. Using \(\mathrm{{d}}S=\sqrt{g} \mathrm{{d}}q^{1}\,\mathrm{{d}}q^{2}\) and (2.13),
where \(\Sigma _{i} \) represents the entire equilibrium interface in the \(q^{\alpha }\) plane. Writing
the first term can be treated using the two-dimensional divergence theorem in the \(q^{\alpha }\) plane:
where \(\mathrm{{d}}s_{q} \) is elementary arc length on \(\partial \Sigma _{i} \). Transforming to physical space, \(\partial \Sigma _{i} \) becomes the equilibrium contact line, C, and it can be shown (using (A.22), (A.26) and \(\mathrm{{d}}s=\left( {g_{\alpha \beta }\, \mathrm{{d}}q^{\alpha }\,\mathrm{{d}}q^{\beta }} \right) ^{1/2})\) that \(\sqrt{g} \mathrm{{d}}s_{q} /\sqrt{\nu _{\delta } \nu _{\delta } } =\mathrm{{d}}s\), where \(\mathrm{{d}}s\) is elementary arc length on C. Thus,
Again using \(dS=\sqrt{g} \mathrm{{d}}q^{1}\mathrm{{d}}q^{2}\),
Equations (A.47)–(A.51) give (A.46).
An application of (A.46) is the following. Let \(\tilde{{\eta }}\) be a function, possibly complex, on \(S_{i} \) such that (3.13) holds on C. It follows from (3.16) and (A.46) that
where * denotes complex conjugation.
1.4 A.4 Derivation of the energy Eq. (2.16)
Writing \({\varvec{\upsigma }}=-{p}'{\mathbf{I}}+2\,\text {Oh }{\mathbf{e}}\), where \({\mathbf{I}}\) is the identity tensor and \({\mathbf{e}}=\left( {\nabla {\mathbf{u}}+\left( {\nabla {\mathbf{u}}} \right) ^{\mathrm{{T}}}} \right) /2\) the strain-rate tensor, \({\varvec{\upsigma }}\) is the perturbation of the stress tensor. Equations (2.2) and (2.9) give
while (2.12) implies
on the interface. Employing (2.2) and the definitions of \({\varvec{\upsigma }}\) and \({\mathbf{e}}\), the identity \({\mathbf{u.}}\left( {\nabla .{\varvec{\upsigma }}} \right) =\nabla {\mathbf{.}}\left( {{\mathbf{u}}.{\varvec{\upsigma }}} \right) -{\varvec{\upsigma }} :\nabla {\mathbf{u}}\) yields \({\mathbf{u.}}\left( {\nabla .{\varvec{\upsigma }}} \right) =\nabla .\left( {{\mathbf{u}}.{\varvec{\upsigma }}} \right) -2\,\text {Oh }{\mathbf{e:e}}\). Using this result, scalar multiplying (A.53) by \({\mathbf{u}}\) and integrating over the drop, D, the divergence theorem leads to
Equation (2.3) and the definitions of \({\varvec{\upsigma }}\) and \({\mathbf{e}}\) give
as the wall contribution to the surface integral in (A.55). Note that \(S_\mathrm{{{w}}} \) is the equilibrium wetted area of the wall.
The interfacial contribution to (A.55) follows from (2.11) and (A.54) as
Applying (A.46),
Using (2.14),
Symmetry of \(g^{\alpha \beta }\) implies
Finally,
as the interfacial contribution to (A.55), where \(E_\mathrm{{{s}}} \) is given by (2.18). Combining (A.55), (A.56) and (A.62) yields (2.16).
1.5 A.5 Derivation of (3.2) and (3.3)
Using Lagrange multipliers for the constraints (3.1), we look for extrema of
Symmetry of \(g^{\alpha \beta }\) implies
for the variation of \(Q\left[ \omega \right] \) due to the infinitesimal variation, \(\delta \omega \), of \(\omega \). Using (A.46),
The condition for an extremum is \(\delta Q=0\) for any \(\delta \omega \). Thus, we obtain (3.2) and (3.3). The constraints (3.1) also need to be imposed. Of these, the first contributes to the eigenvalue problem, whereas the second provides a normalisation condition.
1.6 A.6 Derivation of (3.4) and (3.6)
The first of Eq. (3.1), together with (3.2) and (3.3), gives
on \(S_{i} \) and
on C. Multiplying (A.67) by \(\omega _{m} \), integrating over \(S_{i} \) and using (A.66) with n replaced by m,
Using (A.46),
Permuting n and m and subtracting, symmetry of \(g^{\alpha \beta }\) yields
If \(\zeta _{n} \ne \zeta _{m} \), (A.71) gives orthogonality of \(\omega _{n} \) and \(\omega _{m} \). In the case of a degenerate eigenvalue, the associated eigenfunctions can be orthogonalised, so (3.4) holds for all \(n\ne m\). It also applies when \(n=m\), thanks to the normalisation resulting from the second equation of (3.1).
where
1.7 A.7 Derivation of (3.15)
Let \({{\tilde{\varvec{\upsigma }}}}=-\tilde{{p}}{\mathbf{I}}+2\,\text {Oh }{{\tilde{\mathbf{e}}}}\), where \({{\tilde{\mathbf{e}}}}=\left( {\nabla {{\tilde{\mathbf{u}}}}+\left( {\nabla {{\tilde{\mathbf{u}}}}} \right) ^{\mathrm{{T}}}} \right) /2\). (3.8) and (3.9) imply
while (3.12) gives
on the interface. Taking the complex conjugate of (A.75), scalar multiplying by \({{\tilde{\mathbf{u}}}}\) and integrating over the drop D, \({{\tilde{\mathbf{u}}.}}\left( {\nabla {{.\tilde{\varvec{\upsigma }}}}^{*}} \right) =\nabla {\mathbf{.}}\left( {{{\tilde{\mathbf{u}}.\tilde{\varvec{\upsigma }}}}^{*}} \right) -2\,\text {Oh }{{\tilde{\mathbf{e}}:\tilde{{\mathbf{e}}}}}^{*}\) and the divergence theorem lead to
Equation (3.10) and the definitions of \({{\tilde{\varvec{\upsigma }}}}\) and \({{\tilde{\mathbf{e}}}}\) give
as the wall contribution to the surface integral in (A.77).
Using (3.11), (A.52) and (A.76),
for the interfacial contribution to (A.77). Equation (3.15) follows from (A.77)–(A.79).
1.8 A.8 Nonexistence of \(s\ne 0\) modes such that \(s\rightarrow 0\) as \(\text {Oh}\rightarrow {\text {Oh}}_{c} >0\)
In order to derive a contradiction, suppose an \(s\ne 0\) mode which approaches \(s=0\) as \(\text {Oh}\rightarrow \text {Oh}_{c} >0\). The limit would be an \(s=0\) mode. Unless \(\zeta =0\) is an eigenvalue there are no such modes and we already have a contradiction. The case in which \(\zeta =0\) is an eigenvalue is more complicated and is treated below.
Let \(\omega _{n} \) be one of the \(\zeta =0\) eigenfunctions introduced in Sect. 3.1. Thus, \(\omega _{n} \), which is real and corresponds to an \(s=0\) mode, satisfies
with constant \(\phi _{n} \), and
on the contact line. Define \({\mathbf{v}}_{n} \left( {{\mathbf{x}}} \right) \) and \(\chi _{n} \left( {{\mathbf{x}}} \right) \) via
and
on \(S_{i} \), where \({\mathbf{t}}\) is any tangent vector to \(S_{i} \). Taking two independent choices for \({\mathbf{t}}\), (A.88) gives two boundary conditions. Equations (A.83)–(A.88) can be interpreted as follows. Equations (A.84) and (A.85) mean that \({\mathbf{v}}_{n} \left( {{\mathbf{x}}} \right) \) and \(\chi _{n} \left( {{\mathbf{x}}} \right) \) are the velocity and pressure of a steady, incompressible Stokes flow within D. This flow is subject to the Navier conditions (A.86) on the wall and Eqs. (A.87), (A.88) on \(S_{i} \). Equation (A.87) specifies the normal component of velocity as \(\omega _{n} \), while (A.88) means that the tangential components of the surface force are zero. Equation (A.80) is required for a solution. This follows from integration of (A.85) over D, use of the divergence theorem, \(v_{nz} =0\) on the wall and (A.87) on \(S_{i}\). Equation (A.83) makes the solution for \(\chi _{n} \left( {{\mathbf{x}}} \right) \), which would otherwise be only determined up to an additive constant, unique.
Consider a mode with \(s\ne 0\) and let \({\mathbf{v}}=s^{-1}{{\tilde{\mathbf{u}}}}\), \(\chi =s^{-1}\left( {\tilde{{p}}-\tilde{{p}}_{0} } \right) \), where the constant \(\tilde{{p}}_{0} \) is such that
with
on the wall,
on the equilibrium interface,
at the equilibrium contact line and
The normal and tangential components of (A.94) give
and
where \({\mathbf{t}}\) is any tangent vector to \(S_{i} \).
Using (A.91), (A.90) can be rewritten as
where \({{\hat{\varvec{\upsigma }}}}=-\chi {\mathbf{I}}+2\,\text {Oh }{ \hat{{\mathbf{e}}}}\) and \({\hat{{\mathbf{e}}}}=\left( {\nabla {\mathbf{v}}+\left( {\nabla {\mathbf{v}}} \right) ^{\mathrm{{T}}}} \right) /2\). Equation (A.94) gives
on \(S_{i} \). Scalar multiplying (A.99) by \({\mathbf{v}}_{n} \) and integrating over D,
Using \({\mathbf{v}}_{n} {\mathbf{.}}\left( {\nabla {{.\hat{\varvec{\upsigma }}}}} \right) =\nabla {\mathbf{.}}\left( {{\mathbf{v}}_{n} {{.\hat{\varvec{\upsigma }}}}} \right) -2\,\text {Oh }{{\hat{\mathbf{e}}}}_{n} {:\hat{{\mathbf{e}}}}\), where \({{\hat{\mathbf{e}}}}_{n} =\left( {\nabla {\mathbf{v}}_{n} +\left( {\nabla {\mathbf{v}}_{n} } \right) ^{\mathrm{{T}}}} \right) /2\), and the divergence theorem,
The contribution of \(S_\mathrm{{{w}}} \) to the surface integral can be evaluated using \({{\hat{\varvec{\upsigma }}}}=-\chi {\mathbf{I}}+2\text { Oh }{\hat{{\mathbf{e}}}}\), \({\hat{{\mathbf{e}}}}=\left( {\nabla {\mathbf{v}}+\left( {\nabla {\mathbf{v}}} \right) ^{T}} \right) /2\) and \(v_{z} =v_{nz} =0\). Thus,
Equations (A.80), (A.87) and (A.100) imply
Using (A.46), (A.82), (A.95) and symmetry of \(g^{\alpha \beta }\),
hence
which is zero according to (A.81) and (A.96). Thus, (A.102) and (A.103) give
for any \(n\in N\), where N denotes the set of n for which \(\zeta _{n} =0\).
Suppose the given \(s\ne 0\) mode approaches \(s=0\) as \(\text {Oh}\rightarrow \text {Oh}_{c} \) and is normalised using
In the limit \(\text {Oh}\rightarrow \text {Oh}_{c} \), \(s\rightarrow 0\), \(\tilde{{\eta }}\rightarrow \tilde{{\eta }}_{c} \), \({\mathbf{v}}\rightarrow {\mathbf{v}}_{c} \), \(\chi \rightarrow \chi _{c} \) and \(\tilde{{p}}_{0} \rightarrow \tilde{{p}}_{0c} \), where, according to (A.95)–(A.97), \(\tilde{{\eta }}_{c} \) and \(\tilde{{p}}_{0c} \) satisfy the \(s=0\) problem, (3.13), (3.14) and (3.21). Equations (A.89)–(A.93) and (A.98) give
and
on \(S_{i} \). Given \(\tilde{{\eta }}_{c} \), (A.109)–(A.114) determine \({\mathbf{v}}_{c} \) and \(\chi _{c} \). Using (A.86), (A.107) has the limiting form
for \(n\in N\), where \({\hat{{\mathbf{e}}}}_{c} =\left( {\nabla {\mathbf{v}}_{c} +\left( {\nabla {\mathbf{v}}_{c} } \right) ^{\mathrm{{T}}}} \right) /2\).
Since \(\tilde{{\eta }}_{c} \) and \(\tilde{{p}}_{0c} \) satisfy the \(s=0\) problem, (3.13), (3.14) and (3.21), \(\tilde{{\eta }}_{c} \) can be expressed as a linear combination of the \(\zeta =0\) eigenfunctions of (3.1), (3.2) and (3.3) , i.e.
where the coefficients \(c_{n} \) may be complex and, using (3.4) and (A.108),
Comparing (A.83)–(A.88) with (A.109)–(A.114), we see that
Using the first of Eqs. (A.118) in (A.115),
where
is a real, square, symmetric matrix defined for \(n,\;m\in N\). Let \(\bar{{c}}_{n} \) be real and \({\bar{{\mathbf{v}}}}=\sum \nolimits _{n\in N} {\bar{{c}}_{n} {\mathbf{v}}_{n} } \), then
where \({\bar{{\mathbf{e}}}}=\left( {\nabla {\bar{{\mathbf{v}}}}+\left( {\nabla {\bar{{\mathbf{v}}}}} \right) ^{\mathrm{{T}}}} \right) /2\). Equation (A.121) is obviously positive or zero. If it were zero, then \({\bar{{\mathbf{e}}}}=0\), hence \({\bar{{\mathbf{v}}}}\) is a combination of a translation and a rotation. On the other hand, (A.86) implies
thus, \({\bar{{\mathbf{v}}}}=0\), hence \(\bar{{c}}_{n} =0\). We conclude that \(A_{nm} \) is positive definite. It follows from (A.119) that \(c_{n} =0\), which is incompatible with (A.117). This contradiction means \(s\ne 0\) modes cannot approach \(s=0\) as \(\text {Oh}\rightarrow \text {Oh}_{c} >0\).
1.9 A.9 Some properties of inviscid modes
Given (3.30), \(\nabla {\mathbf{.}}\left( {\psi \nabla \psi ^{*}} \right) -\left| {\nabla \psi } \right| ^{2}=\psi \nabla ^{2}\psi ^{*}=0\). Integrating over D, the divergence theorem, (3.31) and (3.32) give
On the other hand, (3.14) and (3.33) implies
Equations (A.123)–(A.125) yield
The integral on the left-hand side cannot be zero, otherwise \(\psi \) is constant and \(\tilde{{\eta }}=0\) from (3.32). (3.23) implies \(\psi =0\), hence \(\tilde{{p}}_{0} =0\) from (3.33). Thus, all unknowns would be zero, which is not allowed for an eigenvalue problem. Symmetry of \(g^{\alpha \beta }\) makes the right-hand side of (A.126) real. We conclude that \(\sigma \) is real. For each eigenvalue \(\sigma _{k} \), \(\psi _{k} \), \(\tilde{{\eta }}_{k} \) and \(\tilde{{p}}_{0k} \) are chosen real from here on. Because the integral on the left-hand side of (A.126) is positive, the \(\psi _{k} \) can be normalised such that (3.34) holds when \(k=l\).
Equation (3.30) implies \(\nabla {\mathbf{.}}\left( {\psi _{k} \nabla \psi _{l} } \right) -\nabla \psi _{k} {\mathbf{.}}\nabla \psi _{l} =\psi _{k} \nabla ^{2}\psi _{l} =0\). Integrating over D, the divergence theorem, (3.31) and (3.32) give
Using (3.13), (A.46) and symmetry of \(g^{\alpha \beta }\),
Equations (A.127)–(A.129) yield
Thus,
where the matrix \(\tilde{{E}}_{kl} \) is given by (3.35) and is symmetric. Using symmetry of \(\tilde{{E}}_{kl} \), permutation of k, l and subtraction gives
When \(\sigma _{k} \ne \sigma _{l} \), (A.132) implies
If \(\sigma _{k} =\sigma _{l} \) is a degenerate eigenvalue, its eigenfunctions can be orthogonalised such that (A.133) applies for \(k\ne l\). Thus, (A.133) holds for all \(k\ne l\). Given the normalisation referred to above, we obtain (3.34). (3.36) follows from (3.34) and (A.131).
1.10 A.10 Nonexistence of \(s\ne 0\) modes such that \(s\rightarrow 0\) as \({\text {Oh}}\rightarrow 0\)
Modes which continue to be affected by viscosity as \(\text {Oh}\rightarrow 0\) are discussed in the main text. A mode of this type has \(s=O\left( {\text {Oh}} \right) \) and hence approaches \(s=0\) as \(\text {Oh}\rightarrow 0\), but it is decaying and hence unimportant from a stability point of view. Here, we consider modes of the other type, i.e. those which approach an inviscid limit.
The analysis given here has many similarities with that of Sect. A.8. One significant difference is that \({\mathbf{v}}_{n} \) and \(\eta _{n} \) are replaced by \({\mathbf{v}}_{k} =\nabla \psi _{k} \) and \(\tilde{{\eta }}_{k} \), where \(\psi _{k} \), \(\tilde{{\eta }}_{k} \) are \(\sigma =0\) eigenfunctions of the inviscid problem (3.13), (3.14), (3.23) and (3.30)–(3.33). As in Sect. A.8, given an \(s\ne 0\) mode, let \({\mathbf{v}}=s^{-1}{{\tilde{\mathbf{u}}}}\) and \(\chi =s^{-1}\left( {\tilde{{p}}-\tilde{{p}}_{0} } \right) \), where the constant \(\tilde{{p}}_{0} \) is determined by (A.89). Equaions (A.90)–(A.107), with \({\mathbf{v}}_{n} \), \(\eta _{n} \) and \({\hat{{\mathbf{e}}}}_{n} \) replaced by \({\mathbf{v}}_{k} =\nabla \psi _{k} \), \(\tilde{{\eta }}_{k} \) and \({\hat{{\mathbf{e}}}}_{k} =\left( {\nabla {\mathbf{v}}_{k} +\left( {\nabla {\mathbf{v}}_{k} } \right) ^{\mathrm{{T}}}} \right) /2\), follow as before.
Letting \(\psi =s^{-1}\chi \), (A.89)–(A.96) give
with
on the wall,
on the equilibrium interface,
at the equilibrium contact line and
Equation (A.107), with \({\mathbf{v}}_{n} \), \({\hat{{\mathbf{e}}}}_{n} \) replaced by \({\mathbf{v}}_{k} \), \({\hat{{\mathbf{e}}}}_{k} \), gives
where \({\hat{{\mathbf{e}}}}=\left( {\nabla {\mathbf{v}}+\left( {\nabla {\mathbf{v}}} \right) ^{\mathrm{{T}}}} \right) /2\). Recalling that \({\mathbf{v}}_{k} =\nabla \psi _{k} \), where \(\psi _{k} \) is a \(\sigma =0\) inviscid eigenfunction, (A.142) applies for all k for which \(\sigma _{k} =0\). We denote the set of those k by K.
Suppose an \(s\ne 0\) mode approaches an inviscid limit with \(s=0\) as \(\text {Oh}\rightarrow 0\). In order that the viscous term in (A.135) be negligible in the limit, \(s^{-1}\text {Oh}\rightarrow 0\). Normalising the mode using (A.108), \(\tilde{{\eta }}\rightarrow \tilde{{\eta }}_{c} \), \({\mathbf{v}}\rightarrow {\mathbf{v}}_{c} \), \(\tilde{{p}}_{0} \rightarrow \tilde{{p}}_{0c} \) and \(\psi \rightarrow \psi _{c} \). Equation (A.134) implies
Equation (A.135) and \(s^{-1}\text {Oh}\rightarrow 0\) give
Thus,
according to (A.136). As discussed following Eq. (3.30), the inviscid problem only allows one wall boundary condition, rather than the three expressed by (A.137) for the viscous problem. The first two equations of (A.137) drop out in the inviscid limit, leaving \(v_{z} =0\), hence
on the wall. Eqations (A.138)–(A.141) yield
on the equilibrium interface,
at the equilibrium contact line and
Equations (A.148)–(A.150) show that \(\tilde{{\eta }}_{c} \) and \(\tilde{{p}}_{0c} \) satisfy the \(s=0\) problem, (3.13), (3.14) and (3.21), while (A.143) and (A.145)–(A.147) correspond to (3.23) and (3.30)–(3.32) and determine \(\psi _{c} \) given \(\tilde{{\eta }}_{c} \). Since \(s^{-1}\text {Oh}\rightarrow 0\), (A.142) implies
for all \(k\in K\).
That \(\tilde{{\eta }}_{c} \) and \(\tilde{{p}}_{0c} \) satisfy the \(s=0\) problem, (3.13), (3.14) and (3.21), indicates that \(\tilde{{\eta }}_{c} \) is a \(\sigma =0\) eigenfunction, hence
Equations (A.143), (A.145)–(A.147) and the corresponding equations for \(\psi _{k} \) and \(\tilde{{\eta }}_{k} \) imply
so
according to (A.144). Scalar multiplying by \(\nabla \psi _{l} \), where \(l\in K\), integrating over D and using (3.34), \({\mathbf{v}}_{l} =\nabla \psi _{l} \) and (A.151),
This result means that \(\tilde{{\eta }}_{c} =0\), which is incompatible with the normalisation (A.108). Thus, we have a contradiction and conclude that \(s\ne 0\) modes which approach an inviscid limit with \(s=0\) as \(\text {Oh}\rightarrow 0\) do not exist.
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Scott, J.F. Dynamic and static stability of a drop attached to an inhomogeneous plane wall. J Eng Math 135, 4 (2022). https://doi.org/10.1007/s10665-022-10220-z
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DOI: https://doi.org/10.1007/s10665-022-10220-z