Tetsuro Tsuji (Kyoto University, Japan)
joint work with Koichiro Takita and Satoshi Taguchi (Kyoto University)
Slip-flow Theory for Thermo-osmosis
The 26th RIMS workshop on
Mathematical Analysis in Fluid and Gas Dynamics�organized by Masashi Aiki and Masahiro Suzuki
July 2-4 (2025) @ RIMS, Kyoto University (presentation on July 3, 15:00-15:50)
Tsuji, Takita, Taguchi, arXiv 2506.20229 (2025)
Table of contents
(10 min)
(10 min)
(20 min)
speaker: Tetsuro Tsuji (Kyoto University)
title: Slip-flow Theory for Thermo-osmosis
Tsuji, Takita, Taguchi, arXiv 2506.20229 (2025)
Introduction
What is thermo-osmosis ?
advection
decoupled
flow
NS system
heat
energy eq.
1
characteristic length
surface effect
is important
small scale
small
scale
molecular simulation
Fu, et al. (2017)
experiments
Tsuji, et al. (2023)
molecular-scale effects are important
gas
or
liquid
initial configuration
heating starts at 0 sec
thermo-osmosis
7 µm
500 nm
/ 3
Introduction
molecular effects in fluids
2
bulk ≫ molecular scale
/ 3
Introduction
molecular effects in fluids
2
wall
slip boundary condition
slip
/ 3
Introduction
molecular effects in fluids
2
wall
slip boundary condition
slip
“fluid eqs + slip b.c.” strategy can be applied to various systems
cf. thermophoresis
slip-flow theory
Boltzmann eq
MD sims
fluid eqs＋slip bc
costs for molecular-scale effects in fluids
〇
×
Tsuji, et al. (2023)
/ 3
Introduction
parameter ε ≈ Kn ≪ 1
inter-mol. coll.
kinetic eq.
molecular velocity
scaled
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
small Kn → frequent collision between gas molecules
e.g. mass flux
macroscopic quantities
are obtained from f
what is slip-flow theory?
3
/ 3
Introduction
parameter ε ≈ Kn ≪ 1
inter-mol. coll.
kinetic eq.
molecular velocity
scaled
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
small Kn → frequent collision between gas molecules
e.g. mass flux
macroscopic quantities
are obtained from f
what is slip-flow theory?
asymptotic theory for Kn ≪ 1
(i.e., frequent inter-mol. collision)
Sone (2007)
3
/ 3
Introduction
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
Sone (2007)
what is slip-flow theory?
3
〇
×
systematic
×
×
inter-molecular potential�fluid-solid interaction
volume excl.
flow velocity
unit tangent
slip coeff
temperature
e.g. thermal slip
Boltzmann
MD
fluid eqs
+ slip
〇
×
costs
/ 3
Introduction
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. =
fluid-dynamic type system governs overall behavior
velocity distribution function
this study
hydrophobic
hydrophilic
U
near-wall
interaction
potential
model of fluid-solid interaction
wall
what is slip-flow theory?
3
Boltzmann
MD
fluid eqs
+ slip
〇
×
costs
velocity slip and temperature jump appear in the boundary-layer eqs.
💡
〇
×
systematic
×
×
inter-molecular potential�fluid-solid interaction
volume excl.
flow velocity
unit tangent
slip coeff
temperature
e.g. thermal slip
/ 3
Table of contents
(10 min)
(10 min)
(20 min)
speaker: Tetsuro Tsuji (Kyoto University)
title: Slip-flow Theory for Thermo-osmosis
Tsuji, Takita, Taguchi, arXiv 2506.20229 (2025)
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
/ 3
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
/ 3
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
/ 3
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
/ 3
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
/ 3
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
/ 3
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
/ 3
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
/ 3
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
/ 3
Description of the problem
result of numerical analysis
additional scaling assumption
δ
no potential
flow ∝ k
slip-flow theory works
mean free path
range of potential
mean free path
3
/ 3
Description of the problem
additional scaling assumption
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
×
3
/ 3
Description of the problem
additional scaling assumption
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
3
/ 3
Table of contents
(10 min)
(10 min)
(20 min)
speaker: Tetsuro Tsuji (Kyoto University)
title: Slip-flow Theory for Thermo-osmosis
Tsuji, Takita, Taguchi, arXiv 2506.20229 (2025)
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
Table of contents
(10 min)
(10 min)
(20 min)
speaker: Tetsuro Tsuji (Kyoto University)
title: Slip-flow Theory for Thermo-osmosis
Tsuji, Takita, Taguchi, arXiv 2506.20229 (2025)
Analysis of thermo-osmosis
Stokes eq
slip b.c.
new unknown
macroscopic quantities
goal: analyze this system using
compare
thermal-slip coef.
slip coef + KL correcton
PDE for and
8
/14
Analysis of thermo-osmosis
u2
slip-flow theory
channel center
wall
u2 = 0
no potential
9
/14
Analysis of thermo-osmosis
u2
channel center
wall
slip-flow theory
attractive potential
u2 = 0
9
/14
Analysis of thermo-osmosis
u2
channel center
wall
slip-flow theory
repulsive potential
u2 = 0
9
/14
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…
9
repulsive potential
/14
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
10
/14
Table of contents
(10 min)
(10 min)
(20 min)
speaker: Tetsuro Tsuji (Kyoto University)
title: Slip-flow Theory for Thermo-osmosis
Tsuji, Takita, Taguchi, arXiv 2506.20229 (2025)
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
Table of contents
(10 min)
(10 min)
(20 min)
speaker: Tetsuro Tsuji (Kyoto University)
title: Slip-flow Theory for Thermo-osmosis
Tsuji, Takita, Taguchi, arXiv 2506.20229 (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)
13
/14
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
13
M12
/14
Conclusion
Tsuji, Takita, Taguchi, arXiv 2506.20229 (2025)