Tetsuro Tsuji*, Koichiro Takita, Satoshi Taguchi
Kyoto University
The 34th International Symposium on Rarefied Gas Dynamics (RGD34)
@The University of Queensland, Brisbane, Australia
July 13-17, 2026 (presentation July 14)
Model of Thermal Slip Under
Near-Wall Fluid-Solid Interaction Potentials
Tsuji, Takita & Taguchi, Physical Review Fluids 11, 114202 (2025)
Introduction
thermo-osmotic flow
What is thermo-osmotic flows?
advection
decoupled
flow
NS system
heat
energy eq.
1
small
scale
gas
or
liquid
characteristic length
surface effect
is important
small scale
e.g. buoyancy force
/ 9
Introduction
molecular-scale effects in near-continuum limit
2
fluid
/ 9
Phys. Rev. Fluids (2025)
Introduction
molecular-scale effects in near-continuum limit
wall
slip boundary condition
slip
“NS + slip b.c.” strategy can be applied to various systems
cf. thermophoresis
Boltzmann eq
MD sims
NS＋slip bc
computational costs to
include molecular-scale effects
〇
×
2
/ 9
Phys. Rev. Fluids (2025)
Slip boundary condition
generalized slip-flow theory
Sone (1969, 2002, 2007)
inter-molecular �interaction model
(HS, inverse power law, etc)
molecular scattering
(diffuse reflection, etc)
flow velocity
unit tangent
slip coefficient
temperature
experimentally difficult to obtain
local excess enthalpy
slip coefficient
distance from the boundary surface
cf.
viscosity
linear response theory
Derjaguin, et al., Surface Forces (1987)
Anzini, et al (2019, 2022, 2025)
Kinefuchi, et al., (2017)
Derjaguin’s model
〇
×
experimentally accessible
intermolecular potential�fluid-solid interaction, volume excl.
〇
×
dense systems
comparison with
experiments
Fu, et al. (2017)
Ganti, et al. (2017)
MD
3
/ 9
Phys. Rev. Fluids (2025)
Slip boundary condition
generalized slip-flow theory
inter-molecular �interaction model
(HS, inverse power law, etc)
molecular scattering
(diffuse reflection, etc)
flow velocity
unit tangent
slip coefficient
temperature
experimentally difficult to obtain
local excess enthalpy
slip coefficient
distance from the boundary surface
cf.
viscosity
linear response theory
Kinefuchi, et al., (2017)
Derjaguin’s model
extend?
Enskog-Vlasov system
Van der Waals fluids
e.g. Takata, et al (2021)
〇
×
experimentally accessible
intermolecular potential�fluid-solid interaction, volume excl.
〇
×
dense systems
comparison with
experiments
Fu, et al. (2017)
Ganti, et al. (2017)
MD
Sone (1969, 2002, 2007)
3
/ 9
Phys. Rev. Fluids (2025)
Slip boundary condition
〇
×
experimentally accessible
inter-molecular potential�fluid-solid interaction, volume excl.
generalized slip-flow theory
inter-molecular �interaction model
(HS, inverse power law, etc)
molecular scattering
(diffuse reflection, etc)
flow velocity
unit tangent
slip coefficient
temperature
experimentally difficult to obtain
local excess enthalpy
slip coefficient
distance from the boundary surface
cf.
viscosity
linear response theory
Kinefuchi, et al., (2017)
Derjaguin’s model
this research
“interactivity” between fluids and solids
sign reversal in MD
Tsuji, et al. (2017); Fu, et al. (2017)�Liu, et al. (2023)
〇
×
dense systems
comparison with
experiments
Fu, et al. (2017)
Ganti, et al. (2017)
MD
3
Sone (1969, 2002, 2007)
/ 9
Phys. Rev. Fluids (2025)
Slip boundary condition
〇
×
experimentally accessible
inter-molecular potential�fluid-solid interaction, volume excl.
generalized slip-flow theory
inter-molecular �interaction model
(HS, inverse power law, etc)
molecular scattering
(diffuse reflection, etc)
flow velocity
unit tangent
slip coefficient
temperature
Kinefuchi, et al., (2017)
this research
“interactivity” between fluids and solids
sign reversal in MD
Tsuji, et al. (2017); Fu, et al. (2017)�Liu, et al. (2023)
〇
×
dense systems
comparison with
experiments
Fu, et al. (2017)
Ganti, et al. (2017)
MD
3
Sone (1969, 2002, 2007)
Fu, et al. Phys. Rev. Lett. (2017)
/ 9
Phys. Rev. Fluids (2025)
Generalized slip-flow theory for gases
kinetic eq.
molecular velocity
inter-mol. coll.
parameter ε ≈ Kn ≪ 1
scaled
asymptotic theory for Kn ≪ 1
(i.e., frequent inter-mol. collision)
velocity distribution function
characteristic length
mean free path
Kn =
ℓ
L
rarefied gas
(or molecular gas)
large Kn → collision with surfaces > collision with gas molecules
e.g. mass flux
macroscopic quantities
are obtained from f
Sone (2002,2007)
4
/ 9
Phys. Rev. Fluids (2025)
Generalized slip-flow theory for gases
f = f0 + ε f1 + ε2 f2 + ・・・
correction
kinetic eq.
molecular velocity
inter-mol. coll.
parameter ε ≈ Kn ≪ 1
scaled
asymptotic theory for Kn ≪ 1
(i.e., frequent inter-mol. collision)
fluid-dynamic
equations
slip boundary
condition
analysis of Knudsen layer
＋
bulk
boundary-layer
correction�(Knudsen-layer)
＋
sol. =
(kinetic) b.c. cannot be satisfied
×
velocity slip and temperature jump appear in the boundary-layer eqs.
💡
fluid-dynamic type system governs overall behavior
velocity distribution function
application to more general boundary-value problems
Sone (2002,2007)
4
/ 9
Phys. Rev. Fluids (2025)
Generalized slip-flow theory for gases
f = f0 + ε f1 + ε2 f2 + ・・・
correction
kinetic eq.
molecular velocity
inter-mol. coll.
parameter ε ≈ Kn ≪ 1
scaled
asymptotic theory for Kn ≪ 1
(i.e., frequent inter-mol. collision)
fluid-dynamic
equations
slip boundary
condition
analysis of Knudsen layer
＋
bulk
boundary-layer
correction�(Knudsen-layer)
＋
sol. =
velocity slip and temperature jump appear in the boundary-layer eqs.
💡
fluid-dynamic type system governs overall behavior
velocity distribution function
application to more general boundary-value problems
this study
hydrophobic
hydrophilic
U
near-wall
interaction
potential
“interactivity”
wall
4
/ 9
Phys. Rev. Fluids (2025)
Model
local equilibrium
governing equation (BGK type, steady)
kinetic bc (diffuse) ＠X1 = ±D/2
mol. vel.
mol. vel. dist. func.
the moment of
Sone (1966), Niimi (1968), Loyalka (1969), ・・・, Ohwada et al., (1989),
Loyalka & Hickey (1991), ・・・, Takata & Funagane (2013), ・・・
this study
range of interaction：δD (D is the gap size)
5
short-range interaction
long-range interaction
/ 9
Phys. Rev. Fluids (2025)
Model
local equilibrium
governing equation (BGK type, steady)
kinetic bc (diffuse) ＠X1 = ±D/2
mol. vel.
mol. vel. dist. func.
the moment of
Sone (1966), Niimi (1968), Loyalka (1969), ・・・, Ohwada et al., (1989),
Loyalka & Hickey (1991), ・・・, Takata & Funagane (2013), ・・・
goal: analyze this system using slip-flow theory to obtain thermal-slip coefficient
5
mean free path
k ∝
potential range
δ ∝
important
dimensionless
parameters
potential depth
U ∝
/ 9
Phys. Rev. Fluids (2025)
Preliminary
result of numerical analysis
δ
no potential
flow ∝ k
slip-flow theory works
mean free path
6
Q. when slip-theory works?
mean free path
k ∝
potential range
δ ∝
potential depth
U ∝
⇒ let’s do numerical inspection
/ 9
Phys. Rev. Fluids (2025)
Preliminary
δ
result of numerical analysis
δ = 0.01, 0.02, …, 0.09, 0.10
k = 0.01, 0.02, …, 0.09, 0.10
with potential
flow ∝ k
×
6
Q. when slip-theory works?
mean free path
k ∝
potential range
δ ∝
potential depth
U ∝
⇒ let’s do numerical inspection
/ 9
Phys. Rev. Fluids (2025)
Preliminary
result of numerical analysis
δ
δ = 0.01, 0.02, …, 0.09, 0.10
k = 0.01, 0.02, …, 0.09, 0.10
with potential
flow ∝ k
〇
6
Q. when slip-theory works?
mean free path
k ∝
potential range
δ ∝
potential depth
U ∝
⇒ let’s do numerical inspection
/ 9
Phys. Rev. Fluids (2025)
Main result: Thermal slip
assumptions:
nσ3 = O(1)
the molecular-volume effect is NOT negligible
ℓ0 ≈ (nσ2)-1
≈ σ
number density
molecular size
generalized slip-flow theory
thermal-slip boundary condition
thermal speed
“mean free path”
reference temperature
slip coefficient
non-dimensional slip coef. = O(1)
unit tangent
7
/ 9
Phys. Rev. Fluids (2025)
Main result: Thermal slip
slip coef. b2(1) (non-dimensional slip coef) is determined by
solving the Knudsen-layer problem
unknown function
(non-dimensional)
potential
thermal-slip boundary condition
thermal speed
“mean free path”
reference temperature
slip coefficient
non-dimensional slip coef. = O(1)
unit tangent
b.c.
thermal-slip coef.
inhomogeneous term
+ Stokes eqs & heat eq for flow, temperature, and pressure fields
gaussian
boundary-layer correction
Bardos, et al., Comm. Pure Appl. Math. (1986)
Coron et al., Comm. Pure Appl. Math. (1988)
(≈ Kn)
7
/ 9
Phys. Rev. Fluids (2025)
Analysis of thermo-osmosis
u2
slip-flow theory
channel center
wall
u2 = 0
no potential
8
/ 9
Phys. Rev. Fluids (2025)
Analysis of thermo-osmosis
u2
channel center
wall
slip-flow theory
attractive potential
u2 = 0
8
/ 9
Phys. Rev. Fluids (2025)
Analysis of thermo-osmosis
u2
channel center
wall
slip-flow theory
repulsive potential
u2 = 0
8
/ 9
Phys. Rev. Fluids (2025)
Analysis of thermo-osmosis
u2
channel center
wall
slip-flow theory
attractive potential (U > 0)
flow enhance
repulsive potential (U < 0)
flow reversal
with potential (U ≠ 0)
same flow structure, but…
8
repulsive potential
/ 9
Phys. Rev. Fluids (2025)
Analysis of thermo-osmosis
u2
slip-flow theory
attractive potential (U > 0)
flow enhance
repulsive potential (U < 0)
flow reversal
with potential (U ≠ 0)
same flow structure, but…
effect of is not very monotone
mild
drastic
8
(δ /k =)
/ 9
Phys. Rev. Fluids (2025)
Summary
Thank you for your attention!
Tsuji, Takita & Taguchi
Physical Review Fluids 11, 114202 (2025)
thermal-slip coefficient
potential strength
sign reversal
9
/ 9
Phys. Rev. Fluids (2025)
EOF
Main result: Thermal slip
step 2
u2
u2
flow velocity, temperature, pressure, etc.
u2G
u2K
slip coef b2(1)
Stokes eqs & heat eq
fluid-dynamic
equations
slip boundary
condition
analysis of Knudsen layer
＋
bulk
boundary-layer
correction�(Knudsen-layer)
＋
sol. =
KL problem
(=Y2(1))
step 1
7
/ 9
Phys. Rev. Fluids (2025)
Main result: Thermal slip
u2G
u2K
slip coef b2(1)
Stokes eqs & heat eq
fluid-dynamic
equations
slip boundary
condition
analysis of Knudsen layer
＋
bulk
boundary-layer
correction�(Knudsen-layer)
＋
sol. =
KL problem
(=Y2(1))
step 1
step 2
remark
u2
u2
flow velocity, temperature, pressure, etc.
7
/ 9
Phys. Rev. Fluids (2025)
Similarity with molecular simulation
Wang, et al. Nano Lett. (2020)
Fan, et al.
Int. J. Heat Mass Trans. (2024)
Fu, et al. Phys. Rev. Lett. (2017)
Qi, et al. Phys. Fluids (2024)
10
/ 9
Phys. Rev. Fluids (2025)
Similarity with molecular simulation
Fu, et al. Phys. Rev. Lett. (2017)
Ganti, et al., Phys. Rev. Lett. (2017)
Qi, et al., Phys. Fluids (2024)
M12 = KTST0
present definition of
thermo-osmotic coefficient
thermal-slip coefficient
thermal speed
“mean free path”
reference temperature
dimensionless
slip coef. = O(1)
M12 = 10-9 –10-6 m2/s
similar order of magnitude
PRESENT STUDY
11
M12
/ 9
Phys. Rev. Fluids (2025)
Contents
speaker: Tetsuro Tsuji (Kyoto University)
title: Thermally-induced flows in microfluidic systems:
from optothermal fluidic experiments to non-equilibrium gaseous modeling
Phys. Rev. Appl. (2023)
Phys. Rev. Fluids (2025)
Introduction
What is thermo-osmosis?
advection
decoupled
flow
NS system
heat
energy eq.
1
small
scale
gas
or
liquid
characteristic length
surface effect
is important
small scale
/ 9
Introduction
What is thermo-osmosis?
characteristic length
mean free path
Kn =
ℓ
L
= O(1)
Knudsen number (Kn) is a good indicator of
the significance of surface properties
case of gases
gas-surface interaction
molecular gas
(or rarefied gas)
2
large Kn → collision with surfaces > collision with gas molecules
cf. Boltzmann equation
/ 9
Introduction
vacuum
What is thermo-osmosis?
no-slip b.c. for ordinary viscous fluids
flow over a boundary with
a temperature gradient
thermal creep for rarefied gases
Maxwell (1879)
thermal-stress slip flow
nonlinear-thermal-stress flow
thermal edge flow
other types of
thermal flows
Sone (2007)
3
/ 9
Introduction
vacuum
What is thermo-osmosis?
no-slip b.c. for ordinary viscous fluids
thermal creep for rarefied gases
Maxwell (1879)
thermal-stress slip flow
nonlinear-thermal-stress flow
thermal edge flow
other types of
thermal flows
Sone (2007)
3
applications ?
flow over a boundary with
a temperature gradient
/ 9
Introduction
Sugimoto & Sone (2005)
Taguchi & Tsuji (2022)
(thermally-driven pump without mechanically moving parts)
X. Wang, et al. Knudsen Pumps: a Review. Microsystems Nanoeng. (2020)
application (ex. 1)
What is thermo-osmosis?
no-slip b.c. for ordinary viscous fluids
thermal creep for rarefied gases
Maxwell (1879)
one-way flow
thermal-stress slip flow
nonlinear-thermal-stress flow
thermal edge flow
other types of
thermal flows
Sone (2007)
thermal-edge flow
one-way flow
3
flow over a boundary with
a temperature gradient
/ 9
Comparison with numerical analysis
attractive potential
11
/14
Comparison with numerical analysis
attractive potential
magnification near the wall
11
/14
Comparison with numerical analysis
repulsive potential
magnification near the wall
11
/14
Comparison with numerical analysis
slip-flow theory
numerical analysis
U = 1
mean free path
range of potential
12
/14
Comparison with numerical analysis
slip-flow theory
numerical analysis
U = 1
ratio
quantitative agreement
between slip-flow theory
and numerical analysis
12
/14
Slip-flow theory under the potential
step 1
decomposition into bulk and Knudsen layer (KL)
fluid-dynamic
equations
slip boundary
condition
Knudsen-layer analysis
＋
bulk
boundary-layer�(Knudsen-layer)
＋
solution =
asymptotic analysis
for small k
local
equilibrium
density
velocity
temperature
bulk
KL
step 2
G
K
bulk is NOT affected by potential
Boltzmann eq
MD sims
fluid eqs
＋slip bc
〇
×
1
/14
Slip-flow theory under the potential
remainder
step 2
asymptotic analysis for small k for bulk part
bulk
KL
1
rewrite
O(δ/k)
O(U/δ)
the effective range of is confined in near-wall region with thickness δ
magnified
near wall
without potential
2
/14
Slip-flow theory under the potential
step 2
asymptotic analysis for small k for bulk part
bulk
KL
rewrite
remainder
O(δ/k)
O(U/δ)
without potential
2
/14
Slip-flow theory under the potential
step 2
asymptotic analysis for small k for bulk part
bulk
KL
rewrite
remainder
(Hilbert expansion)
O(δ/k)
O(U/δ)
without potential
2
/14
Slip-flow theory under the potential
step 2
asymptotic analysis for small k for bulk part
bulk
KL
rewrite
remainder
velocity distribution functions
Stokes equations
same as conventional results
O(δ/k)
O(1)
O(U/δ)
2
eq of state
/14
Slip-flow theory under the potential
step 2
asymptotic analysis for small k for bulk part
bulk
KL
1
rewrite
remainder
velocity distribution functions
corrections are necessary…
O(δ/k)
O(1)
O(U/δ)
〇
no slip b.c.
〇
×
×
2
/14
Slip-flow theory under the potential
fluid-dynamic
equations
slip boundary
condition
Knudsen-layer analysis
＋
bulk
boundary-layer�(Knudsen-layer)
＋
solution =
asymptotic analysis
for small k
step 3
Knudsen-layer analysis
bulk
KL
correction
step 3
(near x1 = 1/2)
(NOT moderately varying)
remainder
neglected terms in bulk analysis
appears as inhomogeneous terms
without potential
3
/14
Slip-flow theory under the potential
remainder from bulk
step 3
Knudsen-layer analysis
value on the boundary
leading order
e.g.
local equilibrium
bulk
KL
correction
potential near x1=1/2
4
/14
Slip-flow theory under the potential
step 3
Knudsen-layer analysis
leading order
remainder from bulk
b.c.
(correction must vanish at infinity)
bulk
KL
correction
leading-order solution
4
/14
Slip-flow theory under the potential
step 3
Knudsen-layer analysis
first order
bulk
KL
correction
temperature jump
shear slip
thermal slip
no flux across the wall
5
/14
Slip-flow theory under the potential
first order
temperature jump
shear slip
thermal slip
no flux across the wall
b.c.
boundary value of bulk solution at the first order
slip
jump
5
/14
Slip-flow theory under the potential
first order
no flux across the wall
b.c.
boundary value of bulk solution at the first order
slip
jump
decomposition
thanks to the linearity of the system,
we arrive at three b.v. problems
5
/14
Slip-flow theory under the potential
first order
no flux across the wall
b.c.
boundary value of bulk solution at the first order
slip
jump
decomposition
our problem (thermo-osmosis)
thanks to the linearity of the system,
we arrive at three b.v. problems
5
/14
Slip-flow theory under the potential
decomposition
thermal-slip coefficient
bulk
KL
correction
6
/14
Slip-flow theory under the potential
u2G
u2K
slip coef b2(1)
Stokes equations
fluid-dynamic
equations
slip boundary
condition
＋
bulk
boundary-layer
(Knudsen-layer)
＋
sol. =
KL problem
(=Y2(1))
step 2
u2
u2
flow velocity, temperature, pressure, etc.
step 1
7
/14
Description of the problem
Sone (1966), Niimi (1968), Loyalka (1969), ・・・, Ohwada et al., (1989),
Loyalka & Hickey (1991), ・・・, Takata & Funagane (2013), ・・・
classical problem
1
wall temperature
reference temperature
dimensionless parameter ≪ 1
BGK model
/19
Description of the problem
Sone (1966), Niimi (1968), Loyalka (1969), ・・・, Ohwada et al., (1989),
Loyalka & Hickey (1991), ・・・, Takata & Funagane (2013), ・・・
classical problem
wall temperature
reference temperature
dimensionless parameter ≪ 1
1
BGK model
/19
Description of the problem
Sone (1966), Niimi (1968), Loyalka (1969), ・・・, Ohwada et al., (1989),
Loyalka & Hickey (1991), ・・・, Takata & Funagane (2013), ・・・
classical problem
wall temperature
reference temperature
dimensionless parameter ≪ 1
local equilibrium at rest with
temperature T0 and density ρ0*(X1)
normalization
constant
average
f0*
short-range interaction
long-range interaction
gas const.
1
BGK model
/19
Description of the problem
Sone (1966), Niimi (1968), Loyalka (1969), ・・・, Ohwada et al., (1989),
Loyalka & Hickey (1991), ・・・, Takata & Funagane (2013), ・・・
classical problem
wall temperature
reference temperature
dimensionless parameter ≪ 1
f0*
short-range interaction
long-range interaction
velocity distribution function
reference
perturbation
linearize
1
BGK model
/19
Description of the problem
reference temperature
dimensionless parameter ≪ 1
unknown
position
molecular velocity
potential
at x1 = 1/2
at x1 = -1/2
pseudo-Sutherland-type near-wall potential
parameter ≪ 1
(dimensionless range of the potential)
near the wall at x1 = 1/2
parameter = O(1)
(magnitude of the potential with sign)
reference length = D
2
/19
Description of the problem
reference temperature
dimensionless parameter ≪ 1
unknown
position
molecular velocity
parameter ≈ Knudsen number ≪ 1
reference density
near the wall at x1 = 1/2
local
equilibrium
density
velocity
temperature
gaussian
2
/19
Description of the problem
reference temperature
dimensionless parameter ≪ 1
unknown
position
molecular velocity
near the wall at x1 = 1/2
wall temperature gradient
diffuse
reflection
short-range interaction
long-range interaction
parameter ≪ 1
parameter ≪ 1
parameter = O(1)
2
/19
Description of the problem
unknown
position
molecular velocity
near the wall at x1 = 1/2
wall temperature gradient
diffuse
reflection
short-range interaction
long-range interaction
parameter ≪ 1
parameter ≪ 1
parameter = O(1)
goal: analyze this system using
compare
thermal-slip coef.
slip coefficient
(on the wall)
2
/19
Description of the problem
param.
physical meaning
temperature gradient (∝ perturbation)
mean free path (gas rarefaction)
range of the potential (∝ molecular diameter)
magnitude (& sign) of the potential
order
≪ or ≪1
≪ 1
≪ 1
O(1)
assumptions so far introduced…
additional scaling assumption
mean free path
ℓ0 ≈ (nσ2)-1
number density
molecular diameter
σ
≈
nσ3 = O(1)
the molecular-volume effect may NOT be negligible
3
/19