Molecular dynamics
MD and continuum simulations are carried out to investigate the influence of shear rate and surface roughness on slip flow of a Newtonian fluid. For weak wall-fluid interaction energy, the nonlinear shear-rate dependence of the intrinsic slip length in the flow over an atomically flat surface is computed by MD simulations. We describe laminar flow away from a curved boundary by means of the effective slip length defined with respect to the mean height of the surface roughness. Both the magnitude of the effective slip length and the slope of its rate dependence are significantly reduced in the presence of periodic surface roughness. We then numerically solve the Navier-Stokes equation for the flow over the rough surface using the rate-dependent intrinsic slip length as a local boundary condition. Continuum simulations reproduce the behavior of the effective slip length obtained from MD simulations at low shear rates. The slight discrepancy between MD and continuum results at high shear rates is explained by examination of the local velocity profiles and the pressure distribution along the wavy surface. We found that in the region where the curved boundary faces the mainstream flow, the local slip is suppressed due to the increase in pressure. The results of the comparative analysis can potentially lead to the development of an efficient algorithm for modeling rate-dependent slip flows over rough surfaces.

1. Modeling the combined effect of surface roughness
and shear rate on slip flow of simple fluids
Department of Mechanical Engineering Michigan State University
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, 011606 (2010).
Anoosheh Niavarani and Nikolai Priezjev
www.egr.msu.edu/~niavaran
November 2009

2. Introduction
 MD simulations have shown that at the
interface between simple1 and polymeric2
liquids and flat walls the slip length is a
function of shear rate.
h
Solid wall
U
x
z
us
u(z)
1 P. A. Thompson and S. M. Troian, Nature (1997)
2 N. V. Priezjev and S. M. Troian, Phys. Rev. Lett. (2004)
 Surface roughness reduces effective slip length.
 For flow of simple1 and polymeric2 liquids past
periodically corrugated surface there is an
excellent agreement between MD and continuum
results for the effective slip length (if ).
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)
dz
zdu )(
=γ
Navier slip law
L0
γ0Lus = γγ )(0Lus =
)(0 γL
dz
zdu )(
=γ
σλ 30≥
 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.

3. x
z
n
nu
Lxu t
ss
∂
∂
=
)(
)()( γ
Introduction
Question: How to model the combined effect of surface roughness and shear rate?
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
for flat walls. And then, use 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
)(γsL
)(γsL
Fluid flow
Rough surface
Department of Mechanical Engineering Michigan State University

4. Details of molecular dynamics (MD) simulations
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
12 6
LJ
r r
V (r) 4ε
σ σ
− −
    
= −    
     
2 1/2
( )mτ σ ε=
σ LJ molecular length scale
ε LJ energy scale
LJ time scale
Lennard-Jones potential:
Department of Mechanical Engineering Michigan State University
Langevin Thermostat
Friction coefficient
Equation of motion:
Gaussian random force
ρ w = 3.1 σ −3
ρ w = 0.67 σ −3
ρ = 0.81 σ −3
σλ 42≈=zL 27.0
2
==
λ
πa
ka
3
81.0 −
= σρFluid density

5. ( ) sin(2 / )z ax xπ λ=
Shear flow over a rough surface
The effective slip length Leff and intrinsic slip length L0
Leff is the effective slip length, which characterizes
the flow over macroscopically rough surface
Department of Mechanical Engineering Michigan State University
z
R(x) : local radius of curvature
Ls
us
R(x)
u(z)
x
Local velocity profile
Ls : local slip length
L0 : intrinsic slip length extracted
from MD simulation
n
zu
Lxu ss
∂
∂
=
)(
)(
)(
111
0 xRLLs
−=

6. Equations of motion (penalty formulation):
Boundary condition:
Bilinear quadrilateral elements
shape function
Details of continuum simulations (Finite Element Method)
Department of Mechanical Engineering Michigan State University
R(x): local radius of curvature
(+) concave, (–) convex
L0: intrinsic slip length as a function of
Galerkin formulation:
)](/))[((0 xRuunLu sss +∇⋅=

γ
)(/).( xRuun ss +∇=

γShear rate:

7. γ
us
Department of Mechanical Engineering Michigan State University
dz
zdu )(
=γγγ )(0Lus =
 Slip length is a highly nonlinear function of
shear rate
 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)
L0
h
U
us
u(z)
L0/σ
MD simulations: Velocity profiles and slip length for flat walls
Velocity profiles Slip length vs. shear rate

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

9. Detailed analysis of the flow near the curved boundary
Local pressure Local temperature
 Pressure: average normal force on wall atoms from
fluids monomers within the cutoff radius
 For pressure is similar to equilibrium case
and is mostly affected by the wall shape
 For pressure increase when and
τσ /0.1≤U
τσ /0.1>U 12.0/ <λx
54.0/ >λx 5.0/ =λx 3
/36.2 σε== flatPP
 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
First fluid layer

10. Local velocity profiles in four region (I-IV) along the stationary lower wall
τσ /0.1=U
Small upper wall speed
17Re ≈
Local tangential 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 => ut (I) > ut (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
Local normal velocity

11. Local normal velocityNormal velocity Density
Large upper wall speed τσ /0.6=U 100Re ≈
Local tangential velocity
 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
Pressure Contours from Navier-Stokes
min
max

12. Intrinsic slip length and the effect of local pressure along a rough surface
 L0 in MD is calculated from friction coefficient
kf due to uncertainty in estimating local shear rate
 L0 in MD and NS is shear rate dependent
 L0 in NS solution is calculated from 9th order
polynomial fit and is not pressure dependent
 The viscosity is constant within the error bars fk
L
µ
=0
s
t
f
u
P
k = Pt: tangential stress
us: slip velocity
Local intrinsic slip length Local pressure
3
15.015.2 −
±= ετσµ
 Good agreement between L0 from MD and
continuum on the right side of the peak
 The L0 from MD is about 3-4 smaller
than NS due to higher pressure regions
σ

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 L0 is studied
 The bulk density and viscosity are independent of the temperature
 The friction coefficient only depends on the L0
 L0 is reduced about 3 as a function of pressure at higher U
 The discrepancy between Leff from MD and continuum is caused by high pressure region on the
left side of the peak
σ

14. Important conclusions
Department of Mechanical Engineering http://www.egr.msu.edu/~niavaran Michigan State University
n
nu
Lxu t
ss
∂
∂
=
)(
)()( γ
Fluid flow
Rough surface
 The flow over a rough surface with local slip boundary
condition was investigated numerically.
 The local slip boundary condition can not be
determined from continuum analysis. Instead, MD
simulations of flow over flat walls give which
is a highly nonlinear function of shear rate.
 Navier-Stokes equation for flow over rough surface
is solved using as a local boundary condition.
 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.
)(γsL
)(γsL
)(γsL
 The oscillatory pattern of the normal velocity profiles correlates well with the fluid layering
near the wall in Regions I and III.