Modeling the combined effect of surface roughness and shear rate on slip flow of simple fluids

Size: px

Start display at page:

Download "Modeling the combined effect of surface roughness and shear rate on slip flow of simple fluids"

Pamela Webb
5 years ago
Views:

1 Modeling the combined effect of surface roughness and shear rate on slip flow of simple fluids Anoosheh Niavarani and Nikolai Priezjev November 2009 A. Niavarani and N.V. Priezjev, Modeling the combined effect of surface roughness and shear rate of slip flow of simple fluids, Phys. Rev. E 81, (2010). Department of Mechanical Engineering Michigan State University

2 Introduction The no-slip boundary condition works well for macro- flows, however, in micro and nanoflows molecular dynamics simulations and experiments report the existence of a boundary slip. MD simulations have shown that at the interface between simple 1 and polymeric 2 liquids and flat walls the slip length is a function of shear rate. 1 P. A. Thompson and S. M. Troian, Nature (1997) 2 N. V. Priezjev and S. M. Troian, Phys. Rev. Lett. (2004) z x h L 0 L 0 ( γ ) U u(z) u s Solid wall Navier slip law u u s = s = L L 0 ( 0 γ ) γ γ du ( z) γ = du ( z) γ dz dz Surface roughness reduces effective slip length. For flow of simple 1 and polymeric 2 liquids past periodically corrugated surface there is an excellent agreement between MD and continuum results for the effective slip length (if ). λ 30σ Constant local slip length from MD was used as a boundary condition for continuum simulation. 1 N. V. Priezjev and S. M. Troian, J. Fluid Mech. (2006) 2 A. Niavarani and N. V. Priezjev, J. Chem. Phys. (2008)

3 Introduction Question: How to model the combined effect of surface roughness and shear rate? z Fluid flow x Rough surface u s ut ( n) ( x) = Ls ( γ ) n Solution 1: Perform full MD simulation of the flow past rough surface (very expensive computationally!!!). Solution 2: From MD simulation extract only slip length as a function of shear rate L s (γ ) for flat walls. And then, use L s (γ ) as a local boundary condition for the flow over rough surface in the Navier-Stokes equation. (Is it possible?) In this study we compare Solution 1 and Solution 2 Department of Mechanical Engineering Michigan State University

Details of molecular dynamics (MD) simulations Department of Mechanical Engineering ρ w = 0.67 σ 3 ρ = 0.81 σ 3 ρ w = 3.1 σ 3 L = λ 42σ = 0.

4 Details of molecular dynamics (MD) simulations Department of Mechanical Engineering ρ w = 0.67 σ 3 ρ = 0.81 σ 3 ρ w = 3.1 σ 3 L = λ 42σ = z 2πa ka = λ Equation of motion: V my m y f f i ij i + Γ i = + i i j yi Gaussian random force f () t f ( t ) = 2 mk TΓδ ( t t ) i i B Γ=τ 1 Langevin Thermostat T=1.1ε k B 3 ρ = 0.81σ Lennard-Jones potential: r r V LJ (r) = 4ε σ σ τ = ( mσ ε) Friction coefficient Fluid density 2 1/ σ LJ molecular length scale ε LJ energy scale LJ time scale Michigan State University

5 The effective slip length L eff and intrinsic slip length L 0 Shear flow over a rough surface z Local velocity profile u(z) x u s L s R(x) z( x) = asin(2 π x/ λ) L eff is the effective slip length, which characterizes the flow over macroscopically rough surface Department of Mechanical Engineering u s ( x) = L s u( z) n 1 L s 1 = L L s : local slip length L 0 : intrinsic slip length extracted from MD simulation 1 R( 0 x R(x) : local radius of curvature Michigan State University )

6 Details of continuum simulations (Finite Element Method) Equations of motion (penalty formulation): Bilinear quadrilateral elements Boundary condition: us = L0 ( γ )[( n ) us + us / R( x)] R(x): local radius of curvature (+) concave, ( ) convex L 0 : intrinsic slip length as a function of Shear rate: γ = ( n. ) u u / R( x) s + s shape function Galerkin formulation: Department of Mechanical Engineering Michigan State University

7 MD simulations: Velocity profiles and slip length for flat walls Velocity profiles Slip length vs. shear rate U γ L 0 /σ h L 0 u s u(z) u s Velocity profiles are linear across the channel Shear rate is extracted from a linear fit to velocity profiles No-slip at the upper wall (commensurable wall/fluid densities) Department of Mechanical Engineering Slip length is a highly nonlinear function of shear rate u s = L 0 ( ) γ γ du ( z) γ = dz Michigan State University

8 Effective slip length as a function of shear rate for a rough surface Slip length vs. shear rate rate-dependent slip at high shear rates 9 th order polynomial fit slight discrepancy constant slip length at low shear rate excellent agreement monotonic increase ka = 2πa λ u s u (z) detailed analysis Constant slip length at low shear rate, highly nonlinear function of shear rate at high shear rates Blue line is a polynomial fit used in Navier-Stokes solution Surface roughness reduces the effective slip length and its shear rate dependence Excellent agreement between MD and NS at low shear rates but continuum results slightly overestimate MD at high shear rates u s L s R(x) us = L0( γ ) n γ = us n us + R( x) us + R(x) L 0 ( γ ) : polynomial fit

Detailed analysis of the flow near the curved boundary First fluid layer Local pressure Local temperature Pressure: average normal force on wall atoms from

0σ /τ pressure increase when x / λ < 0. 12 and x / λ > 0.54 x / λ = 0. 5 P = P flat 3 = 2.

9 Detailed analysis of the flow near the curved boundary First fluid layer Local pressure Local temperature Pressure: average normal force on wall atoms from fluids monomers within the cutoff radius For U 1.0σ /τ pressure is similar to equilibrium case and is mostly affected by the wall shape For U >1.0σ /τ pressure increase when x / λ < and x / λ > 0.54 x / λ = 0. 5 P = P flat 3 = 2.36ε / σ Average temperature is calculated in the first fluid layer At small upper wall speeds, temperature is equal to equilibrium temperature With increasing upper wall speed the temperature also increases and eventually becomes non-uniformly distributed

10 Local velocity profiles in four region (I-IV) along the stationary lower wall Small upper wall speed U =1.0σ /τ Re 17 Local tangential velocity Local normal velocity At small U, excellent agreement between NS and MD especially inside the valley (IV) The slip velocity above the peak (II) is larger than other regions Even at low Re, the inertia term in the NS equation breaks the symmetry of the flow with respect to the peak => u t (I) > u t (III) In regions I and III, negative and positive normal velocities are due to high pressure and low pressure regions The oscillation of the MD velocity profiles correlates well with the layering of the fluid density normal to the wall

11 Large upper wall speed U = 6.0σ / τ Re 100 Pressure Contours from Navier-Stokes max min Local tangential velocity Normal Local velocity normal velocity Density For large U, the tangential velocity from NS is larger than from MD MD velocity profiles are curved close to the surface, and, therefore, shear rate is subject to uncertainty The oscillations in the normal velocity profiles are more pronounced in MD simulations The location of maximum in the normal velocity profile correlates with the minimum in the density profile

12 Intrinsic slip length and the effect of local pressure along a rough surface Local intrinsic slip length Local pressure L 0 in MD is calculated from friction coefficient k f due to uncertainty in estimating local shear rate L 0 in MD and NS is shear rate dependent L 0 in NS solution is calculated from 9 th order polynomial fit and is not pressure dependent The viscosity is constant within the error bars Good agreement between L 0 from MD and continuum on the right side of the peak The L 0 from MD is about 3-4 σ smaller than NS due to higher pressure regions µ Pt L0 = k f = P k t : tangential stress f us 3 µ = 2.15 ± 0.15ετσ u s : slip velocity

13 Intrinsic slip length and the effect of local pressure Intrinsic slip length vs. pressure for flat wall Bulk viscosity By increasing the temperature of the Langevin thermostat the effect of pressure on L 0 is studied The bulk density and viscosity are independent of the temperature The friction coefficient only depends on the L 0 L 0 is reduced about 3 σ as a function of pressure at higher U The discrepancy between L eff from MD and continuum is caused by high pressure region on the left side of the peak

14 Important conclusions The flow over a rough surface with local slip boundary condition was investigated numerically. The local slip boundary condition L s (γ ) can not be determined from continuum analysis. Instead, MD simulations of flow over flat walls give L s (γ ) which is a highly nonlinear function of shear rate. Navier-Stokes equation for flow over rough surface is solved using (γ ) as a local boundary condition. L s The continuum solution reproduced MD results for the rate-dependent slip length in the flow over a rough surface. The main cause of discrepancy between MD and continuum at higher shear rates is due to reduction in the local intrinsic slip length in high pressure regions. u Fluid flow Rough surface s ut ( n) ( x) = Ls ( γ ) n The oscillatory pattern of the normal velocity profiles correlates well with the fluid layering near the wall in Regions I and III. Department of Mechanical Engineering Michigan State University

The effective slip length and vortex formation in laminar flow over a rough surface

The effective slip length and vortex formation in laminar flow over a rough surface Anoosheh Niavarani and Nikolai V. Priezjev Movies and preprints @ http://www.egr.msu.edu/~niavaran A. Niavarani and N.V.