1. Finite Element Method for Thermocapillary Flow
Zekang Cheng, Jie Li
BP Institute & Engineering Department, University of Cambridge, Cambrige, UK
Introduction
Crucial fluid properties such as density and surface tension are temperature
dependent.
Temperature variation leads to non-uniform surface tension and hence
induces thermocapillary flow.
Thermocapillary flow is ubiquitous in industrial applications including
thermal induced inkjet printing. It can also be used to manipulate micro
droplets in microfluidics.
(a) (b)
Figure 1 : Thermocapillary flow in: (a) thermal inkjet printer; (b) microdrop manipulation
with microheaters
Governing Equations
Continuity:
· u = 0
Navier-Stokes Equation:
ρ
du
dt
= − p + · µ (( u)] + ( u)T
) + f
Heat Transfer Equation (Φ: viscous dissipation):
ρcp
dT
dt
= (k T) + Φ
Interfacial Boundary Condition (n: normal vector; κ: curvature):
−pI + µ ( u) + ( u)T
· n
+
−
= σκn + σ
Temperature Dependence:
ρ = ρ0 − α(T − T0)
σ = σ0 − β(T − T0)
Finite Element Method
Taylor-Hood Element
Figure 2 : Taylor-Hood Element: velocity and
temperature is defined at ◦ point (P2) and
pressure is defined at • point (P1).
u =
5
i=0
uiφi v =
5
i=0
viφi
T =
5
i=0
Tiφi p =
2
i=0
uiψi
φi(xj) = δij
ψi(xj) = δij
Garlerkin Method
Multiply N-S equation and heat transfer equation with φ and continuity
equation with ψ, then integrate them over the triangle (weak formulation).
Use discretized u, v, p, T and obtain a set of linear equations.
Deal with the material derivatives with method of characteristics.
Isoparametric P2 Method
Figure 3 : Fitting curved boundary
with second-order segments
x(s) =
2
i=0
xiφi(s)
y(s) =
2
i=0
yiφi(s)
φi(sj) = δij
Adaptive Moving Mesh
Moving mesh for interface tracking
Adaptive mesh for a good-quality mesh
Boundary at tn
Boundary at tn+1
Figure 4 : Interfacial nodes move in a
Lagrangian way, while most interior nodes stay
fixed.
Edge Swap
Vertex Splitting
Edge Contraction
Vertex Contraction
Edge Splitting
Figure 5 : Mesh refinement
Drop/Bubble Migration with Thermocapillarity
Stokes solution for a spherical bubble under a vertical T∞
Theoretical terminal speed UYGB for a spherical droplet (1 and 2 denotes
the fluid outside and inside the drop respectively; ν = µ/ρ; κ = k/(ρcp)):
UYGB =
2U
(2 + χ)(2 + 3λ)
U = β T∞R/µ1, λ = ν2/ν1, , χ = κ2/κ1
Comparison with analytical solution (a free-surface bubble: χ = λ = 0)
and convergence study
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
1 10 100 1000
(a) (b)
Figure 6 : (a) −, → and → represent the free surface, numerical solution and analytical
solution respectively: (b): Relative error with analytical solution shows second-order
convergence (N is node number on the interface, •, • denote results from straight-edge
element and isoparametric P2 element while − denotes 5/N2
and − denotes 0.45/N2
)
N-S Solver (still working on it)
Non-dimensional number:
Re =
UR
ν1
, Ma =
UR
κ1
, Ca =
ρUµ1
σ0
Terminal temperature and velocity field for different Ma
(Re = 1.0, Ca = 0.2, χ = 1, λ = 1):
0.0 0.5 1.0 1.5 2.0−1.0
−0.5
0.0
0.5
1.0
1.5
2.0 Ma = 1
0.0 0.5 1.0 1.5 2.0
Ma = 10
0.0 0.5 1.0 1.5 2.0
Ma = 100
−0.8
0.0
0.8
1.6
Figure 7 : Velocity in the reference frame moving with the drop.
Future Work
Validation of the N-S solver on moving mesh
Adaptive moving mesh with isoparametric P2 triangles
FEM for Thermocapillary Flow <cz295@cam.ac.uk