Resolving hydrodynamic singularities through molecular interactions Len Pismen

Resolving hydrodynamic singularities through molecular interactions Len Pismen Technion, Haifa, Israel Supported by Israel Science Foundation Minerva Center for Nonlinear Physics of Complex Systems 3/11/2021 Paris, July 2003 1

Cusp on a free interface 3/11/2021 Paris, July 2003 2

Hydrodynamic problem boundary conditions kinematic condition �� tangential stress normal stress complex flow potential Stokes solution: boundary conditions 3/11/2021 Paris, July 2003 3

Conformal transformation conformal transformation � � Ca at � 3/11/2021 Paris, July 2003 4

Hydrodynamic solution must vanish at q=-p/2 defines parameter � as a function of Ca=u 0 � Ca � 3/11/2021 Paris, July 2003 5

Hydrodynamic solution – flow pattern Ca = 0. 1 Ca 3/11/2021 Paris, July 2003 6

Flow near the tip: inflow or stagnation point? Ca log 10( L) 3/11/2021 Paris, July 2003 7

Anatomy of a cusp Hydrodynamic picture: the interface of zero thickness terminates in a cusp singularity Shikhmurzaev’s picture: there is an interfacial layer (1) of unspecified nature; its properties change starting from equilibrium values; the cusp widens into a transition zone (2), and a surface-tension-relaxation tail (3) becomes a gradually disappearing internal interface. Diffuse interface picture: A narrow interfacial zone (1) is close to local equilibrium; vapor condenses near the cusp (2); bulk fluid relaxes to equilibrium in the diffusion zone (3) 3/11/2021 Paris, July 2003 8

Equilibrium density functional theory Free energy of inhomogeneous fluid e. g. hard-core potential: Euler – Lagrange equation: chemical potential static equation of state 3/11/2021 Paris, July 2003 9

Dynamic diffuse interface theory Cahn – Hilliard equation (in Galilean frame) dynamic equation of state coupling to hydrodynamics: G – mobility Stokes equation: continuity equation: compressible flow in boundary region: 3/11/2021 Paris, July 2003 10

Multiscale perturbation scheme Find an equilibrium density profile: liquid at z – at z Perturbations: • Weak gradient of chemical potential • Propagation of interphase boundary • Disjoining/conjoining potential • Interfacial curvature Expand in scale ratio Use solvability condition Match to outer solution 3/11/2021 Paris, July 2003 11

Equilibrium density profile 1 D static density functional equation Q(z) = Density profile for a van der Waals fluid h– 3 tail Gibbs surface (defines the nominal thickness) 3/11/2021 Paris, July 2003 12

Perturbation scheme: inner solution Inner equation for chemical potential: Inner chemical potential: • Material balance across the layer: • Dynamic shift of chemical potential: 3/11/2021 Paris, July 2003 13

Interaction of interfaces liquid vapor liquid h Change of surface tension Shift of chemical potential: equilibrium …valid at h>>d …significant at h~d 3/11/2021 shifted equilibrium Paris, July 2003 14

1 d solutions of the nonlocal equations h/d 3/11/2021 Paris, July 2003 15

Variable surface tension vs. gap width h<d h>d Ca h/d Cusp appears at a finite Ca Ca( ) Ca(d) it is lower than Ca in the hydrodynamic solution with the same tip curvature log 10( L or L/d) 3/11/2021 Paris, July 2003 16

Effect of variable surface tension: velocity d = 10– 8 �=d velocity � Ca ����� � blow-up near the tip d = 10– 8 Ca ����� velocity q d = 10– 6 q 3/11/2021 Paris, July 2003 17

Larger Ca? Higher-order singularity – 1/3 > a > – 8/19 0 > b > – 1/95 close together tail: close together a=– 1/3 b=0 a=– 8/19 b=– 1/95 tip: diverge 3/11/2021 Paris, July 2003 18

Outer solution – gas phase longitudinal vapor flux liquid condensation zone gas flow liquid longitudinal air flux longitudinal condensation flux local condensation flux condensation starts when the gap narrows to provide necessary driving force gap h at the entrance of condensation zone 3/11/2021 ~10– 7 m Paris, July 2003 19

Outer solution – liquid phase density depletion convective diffusion equation ( u – velocity along the cusp) equilibrium shifted equilibrium potential difference driving the condensation flux Weak but slowly decaying density depletion downstream h/d 3/11/2021 Paris, July 2003 20

Conclusions Cusp singularity is resolved trough molecular interactions on nanoscale Effect of variable surface tension: Cusp is formed at finite capillary number • Weak Marangoni flow near the cusp • Effect of condensation: Air is removed from the cusp by diffusion • Weak depletion of liquid density near the cusp • “Relaxation” effect: • 3/11/2021 Weak depletion tail downstream from the cusp Paris, July 2003 21