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Slip boundary conditions for the moving contact line in molecular dynamics and continuum simulations
- 1. Slip boundary conditions for the moving contact line in molecular dynamics and continuum simulations Anoosheh Niavarani and Nikolai V. Priezjev Department of Mechanical Engineering Michigan State University Movies and preprints @ http://www.egr.msu.edu/~niavaran • A. Niavarani and N. V. Priezjev, “Modeling the combined effect of surface roughness and shear rate on slip flow of simple fluids”, Physical Review E 81, 011606 (2010). • A. Niavarani and N. V. Priezjev, “The effective slip length and vortex formation in laminar flow over a rough surface”, Physics of Fluids 21, 052105 (2009).
- 2. Introduction Question: How to model the moving contact line problem in a shear flow using molecular dynamics and continuum methods? Equilibrium: sθ : Contact angle γ : Surface tension γFluid 1 Fluid 2 sθ1γ 2γ 2 1cos sγ θ γ γ= − sd θθ ≠ Fluid 1 Fluid 2 γ 2γ1γ U • No-slip boundary condition leads to divergence of energy dissipation (unphysical) • Contact line singularity is regularized by introduction of the slip region near the contact line. Moving contact line: 2 ( cos sin )r c d r θ µ τ θ θ= − Huh and Scriven, J. Colloid Interface Sci. 35, 85 (1971) Department of Mechanical Engineering Michigan State University Young’s equation
- 3. • The Navier model describes the slip boundary condition at the solid/liquid interface: Linear relation outside contact line: U Contact line • The friction coefficients can be estimated from the molecular dynamics simulation away from the contact line. (cos cos )CL contact suβ γ θ θ= −• At the contact line • Our goal is to use molecular dynamics simulations to estimate the stress tensors, friction coefficient, and flow profiles and determine the correct boundary condition for continuum modeling. What is the boundary condition near the moving contact line? ,slipuτ β= 0/ Lβ µ= L0 h Solid wall U x z uslip u(z) Ren and E, Physics of Fluids 19, 022101 (2007) Qian, Wang, and Sheng, Phys. Rev. E 68, 016306 (2003) β ,slipu β ,contact CLu β Department of Mechanical Engineering Michigan State University
- 4. 1−=δ 12 ww εε = w2γ 90=sθ Fluid 1 Fluid 2 Fluid 1 1 2γ γ= 12 2 ww εε = Fluid 1 Fluid 2 Fluid 1 1 2γ γ> 130≈sθ γFluid 1 Fluid 2 sθ1γ 2γ 2 1cos sγ θ γ γ= − Immiscible fluids − = 612 4)( σ δ σ ε rr rVLJ ij i i i i j i V my m y f y≠ ∂ + Γ =− + ∂ ∑ 1 τ − Γ = BT=1.1 kε ( ) ( ) 2 ( )i i Bf t f t mk T t tδ′ ′= Γ − if 2 1/2 ( )mτ σ ε= σ LJ molecular length scale ε LJ energy scale LJ time scale Lennard-Jones potential: Langevin Thermostat Friction coefficient Equation of motion: Gaussian random force 3 81.0 − = σρFluid density Details of the molecular dynamics simulations x z sθ sθ
- 5. 2 || )2.07.3()]()([ − ⊥ ±=−= ∫ εσττγ drrrSurface tension xxτ yyτ zzτ Extracting normal stresses from molecular dynamics σ/x σ/z U = 0Equilibrium 12 ww εε =1 2γ γ= x z ,xxτ yyτ zzτ • To calculate the surface tension, the normal stresses are estimated accurately using a modified Irving-Kirkwood relation. • The surface tension from molecular dynamics simulations is then used in continuum simulations.
- 6. Distribution of the shear stress along the lower wall in equilibrium (U=0) Snapshot of the atoms near the contact line Tangential stress along the lower wall x z 10σ • The tangential stresses along the lower wall is calculated from LJ forces per unit area between wall atoms and fluids molecules. • The negative and positive stresses, within 5σ from the contact line, are due to a reduced density in the fluid/fluid interfacial region. /x σ xzτ Department of Mechanical Engineering Michigan State University
- 7. sd θθ ≠ dθ U Dynamic contact angle and shape of interface in steady-state shear flow Fluid 1 Fluid 2 γ 2γ1γ τσ /05.0=U 0.1 /U σ τ= U = 0 σ/x • As the wall speed (Capillary number) increases, the contact angle becomes larger. • The dynamic contact angle is .90dθ > Dynamic contact angle vs wall speed U x z dθ dθ dθ Department of Mechanical Engineering Michigan State University
- 8. 240σ/x0 σ/z 25 σ/x σ/z 0 240 25 U U τσ /1.0=U • The flow velocities are computed from the time averaging of instantaneous molecule speeds in small spatial bins over a long period of time. Extracting macroscopic velocities from molecular dynamics Velocity profiles in the first fluid layer First fluid layer σ/x slipu x z • In each fluid phase the slip velocity of the first fluid layer increases near the contact line (symbols), but the overall slip velocity is smaller than wall speed at the contact line (dashed line). Fluid 1 Fluid 2
- 9. U 0 240 25 σ/x σ/x Velocity profile in the first fluid layer slipu Slip velocity and the friction coefficient from molecular dynamics /z σ U First fluid layer Velocity profile along the z direction ( )u z σ/x τσ /01.0=U 0.05 /U σ τ= 0.1 /U σ τ= τσ /01.0=U 0.05 /U σ τ= 0.1 /U σ τ= • The slip velocity near the contact line becomes larger with increasing the wall speed. • The velocity profiles are linear and the slip length is calculated from a linear fit to the profiles. 0 2.2L σ= 0/ 0.9Lβ µ= = Department of Mechanical Engineering Michigan State University
- 10. τ3000Movie length ~ Motion of the contact line at high shear rates Department of Mechanical Engineering Michigan State University • At higher capillary numbers (higher U) the contact line undergoes an unsteady motion. • High stresses at the contact line lead to a pronounced curvature of the fluid-fluid interface and a breakup of a continuous fluid phase. 0.2 /U σ τ= 2 1w wε ε=http://www.egr.msu.edu/~niavaran
- 11. 0 u x ∂ = ∂ 0v = 2 ( ( ) ) u u u p u f t ρ µ ∂ + ⋅∇ = −∇ + ∇ + ∂ 0u∇⋅ = ( )k k k k f n x x Sγκ δ=− − ∆∑ 1 ( )t t t k k kx x t u+ = + ∆ ( )k ij k ij u u x xδ= −∑ Interface location predicted by marker points Details of the continuum modeling of the moving contact line Navier-Stokes equation applied on the fixed grids Boundary conditions U U • Away from the contact point (single phase fluid): Navier Slip B.C.: xz slipuτ β= , , CLγ β β extracted from molecular dynamics simulations • At the contact point (the marker point): (cos cos )CL contact suβ γ θ θ= − dim 1 ( ( ) )1 ( ) (1 cos ) 2 m m k k k m x x x x if x x d d d π δ = − −= + − ≤∏ • Near the contact point: a distribution function interpolates the velocities between the contact point and single phase fluid slipu slipu contactu The system size and the flow properties are the same as in the molecular dynamics method x z
- 12. 0.01U = 0.05U = 0.1U = 0.01U = 0.05U = 0.1U = Molecular dynamics results=dashed lines Dynamic contact angle and flow profiles near the moving contact line 0.05U = 0.1U = 0.01U = x z dθ dθ dθ σ/x slipu Velocity profiles in the first fluid layer • The slip velocity and the contact angle increase at higher wall speeds. • The continuum results agree well with molecular dynamics simulations. Molecular dynamics results=dashed line Dynamic contact angle vs wall speed U dθ
- 13. • The slip boundary conditions near the moving contact line extracted from MD simulations were used in the continuum solution of the Navier-Stokes equation in the same geometry to reproduce velocity profiles and the shape of the fluid-fluid interface. • The MD results show that both dynamic contact angle and slip velocity near the contact line increase with increasing the capillary number (Ca). • At higher capillary numbers (higher U) the contact line undergoes an unsteady motion. High stresses at the contact line lead to a pronounced curvature of the fluid-fluid interface and a breakup of a continuous fluid phase. Important conclusions Department of Mechanical Engineering Michigan State University