Effect of magnetic field on the half zone Marangoni of a low fluid

Department of Aerospace Engineering, Tokyo Metropolitan University

Toshio Tagawa and Tomohiro Kitayama

2014/7/7 1 TAGAWA LABORATORY Research background 1

• The manufacturing technology in semiconductor single crystals is of great importance in the development of information society. • The larger and higher quality of single crystals have been required.

• Single crystals of silicon are manufactured by ① Czochralski Method（CZ method） ② Floating-Zone Method（FZ method）

Demerits • Difficulty in manufacturing large diameter size • Striation due to oscillatory Marangoni convection Fig. Single crystals of silicon Ref.：http://www.rikuryo.or.jp/worldeye/nederland/episode02.html）

2014/7/7 2 TAGAWA LABORATORY Research background 2

Transition process in FZ method

• For low Prandtl number fluid, there are two critical . The flow becomes 3D-steady first then it becomes oscillatory.

Existance of the two critical Reynolds numbers

Fig.：The difference of transition depending on the Prandtl number [2]

2014/7/7 3 TAGAWA LABORATORY Objective • The Marangoni convection of low Prandtl number fluid can be controlled by using the magnetic field because of the electric conducting melt.

• However, the optimal shape and strength of the applied magnetic field related to the FZ method is still unknown.

1．Reproduction of transition process of low Prandtl number fluid for a Marangoni convection • Especially in the two-step transition which is typical for low Prandtl number fluid. • Verification of the numerical code

2 ． Effect of external magnetic fields on the oscillatory Marangoni convection and flow transition • Three types of external magnetic fields, such as vertical, horizontal and cusp-shaped, are applied

2014/7/7 4 TAGAWA LABORATORY Governing equations

Fluid and heat Nomenclature • Continuity of mass r r ∇ ⋅U = 0 B：magnetic filed r • Momentum equations J：electric current density r P：pressure ∂U r r 2 r 2 r r + ()U ⋅∇ U = −∇P + ∇ U + Ha J × B T：temperature ( ) ∂τ r U：velocity vector Electromagnetic force • Energy equation Greek letters ⎛ ∂T r ⎞ 2 Pr ⎜ + ()U ⋅ ∇ T ⎟ = ∇ T ⎝ ∂τ ⎠ τ：time Ψ：electric potential

Elecromagnetism Parameter • Orm’s law • Conservation of electric charge Pr：Prandtl number r r r r Ha：Hartmann number J = −∇Ψ +U × B ∇ ⋅ J = 0

2014/7/7 5 TAGAWA LABORATORY Dimensionless numbers

• Marangoni number • Reynolds number Nomenclature a ： rarius of cylinder − γ ()θ −θ a γθ (θh −θc )a Ma Ma = θ h c Re = = b ： reference magnetic field μα μν Pr 0 （ applied uniform magnetic field ）

• Prandtl number • Hartmann number Greek letters ν σ α ： thermal diffusivity Pr = Ha = b0a α μ γθ ： temperature coefficient of surface tension ： viscosity • μ ν ： kinematic viscosity

θc ： temperature at cold wall Qconvection Nu = θh ： temperature at hot wall Qconduction σ： electric conductivity ： time τ

2014/7/7 6 TAGAWA LABORATORY Initial and boundary conditions

• Initial condition • B. C.（Velocity） Velocity：No-slip and No-penetrate Top and bottom walls：No-slip Temperature： Heat conduction state Free surface：Marangoni effect Central axis：Ozoe＆Toh’s method Z Hot disk : no-slip ( T=0.5 ) 1 • B. C.（Temperature） Top and bottom walls ：Isothermal U = 0 Free surface Adiabatic ∂V V 1 ∂T ： − = − Re ∂R R R ∂ϕ Central axis ：Averaged from the extraporated values ∂W ∂T = − Re Free Free surface Central axis ∂R ∂Z ∂T = 0 • B. C.（Electric current density） ∂R All the boundaries：Insulated（normal component is R zero 0 Cold disk : no-slip ( T=-0.5 ) 1 ） Central axis：Ozoe＆Toh’s method Fig.：Boundary conditions

2014/7/7 7 TAGAWA LABORATORY Computational strategy

Half-Zone model（Aspect ratio, As = 1.0） • Coordinate system （Characteristic length：radius of cylinder, a ） 3D cylindrical coordinate system Z

Hot disk • Discretization a Equidistant staggered grids l

• Finite difference scheme Free surface φ Time term：Runge-Kutta method Cold disk R Inertial term：UTOPIA scheme Fig.：Schematic of a Half-Zone model Other therms：2nd order Cenrtal difference Half-Zone model • Algorithm • Neglecting the influence of gas phase • Static interface（Normal flow is zero: u=0） HSMAC method • Gravity is ignored（No buoyancy）

2014/7/7 8 TAGAWA LABORATORY External applied magnetic field

• Three types of magnetic field, vertical, horizontal and cusp-shaped are considered.

（Vertical case） （Horizontal case） （Cusp-shaped case） from φ=0 to φ=π direction

2014/7/7 9 TAGAWA LABORATORY m=2+1

”(m+1) type: a time-dependent disturbance of m=1 is imposed on the basic steady flow of m” m=2+1 indicates a oscillatory flow superposed on the 3D-steady flow.

The right hand side figures the typical such oscillatory flow.

Fig.：Flow at m=2+11]

2014/7/7 10 TAGAWA LABORATORY Effect of the Reynolds number (Ha = 0)

2D steady (m=0)

• As increase in Re, the maximum value of Nu increases. • For Re=6000 and 7000, 3D-steady flow takes place. Oscillatory (m=2+1) • For Re≧8000, oscillatory flow takes place.

3D steady (m=2) The critical Reynolds number for the transition to oscillatory flow is about 7000~8000. Fig.：Transient responses of the average Nusselt number on the heated wall.

2014/7/7 11 TAGAWA LABORATORY Visualization for flow transition

Computational conditions

Pr 0.02 Re 10000 grids (30×36×30) （R×φ×Z）

•The present computational results reproduced both the 3D-steady flow and the oscillatory flow depending the Reynolds number. •In the case of the oscillatory flow, the obtained wavenumber was coincident with that of the previous study done by Imaishi et al. •The present computational code was verified.

2014/7/7 12 TAGAWA LABORATORY Vertical magnetic field (Re = 10000)

Fig.：The transient responses of the average Fig.：Transient responses of the azimuthal Nusselt number on the heated wall. component of velocity.

• As increase in Hartmann number, Nusselt number is reduced because convection is effectively damped.

2014/7/7 13 TAGAWA LABORATORY Vertical magnetic field (Re = 10000)

Ha=10 Ha=20 Ha=30

Fig.：Temperature and velocity fields • As increase in Ha, the convection flow is confined in the vicinity of free surface. • The azimuthal disturbance is not significantly damped because the Lorentz force does not act on the free surface.

2014/7/7 14 TAGAWA LABORATORY Damping effect

1.3 Dependency of Nusselt number

1.25 Vertical Horizontal • For low Hartmann number 1.2 Cusp (Ha ≦ 10), the horizontal 1.15 magnetic field effectively Nu reduces the convection. 1.1

1.05 • For high Hartmann number, the vertical or cusp-shaped 1 0 5 10 15 20 25 30 magnetic field reduces the Ha convection. Fig.：Dependency of Nusselt number as a function of Hartmann number

2014/7/7 15 TAGAWA LABORATORY Conclusions

• The effect of the external magnetic fields applied either in the vertical, horizontal or cusp-shaped direction on a half-zone model of electro-conducting liquid was numerically investigated.

1. The vertical magnetic field damps the convection effectively among the three cases of magnetic field. 2. The horizontal magnetic field is the most effective to suppress the oscillatory flow.

2014/7/7 16 TAGAWA LABORATORY