Langmuir Micro Separated Flow Analysis using Langmuir Slip Boundary Condition

Langmuir Micro Separated Flow Analysis using Langmuir Slip Boundary Condition 200212

Langmuir Micro Separated Flow Analysis using Langmuir Slip Boundary Condition.

i ii NOMENCLATURE..iii LIST OF FIGURES...v LIST OF TABLES vi 1.1 2...4 2.1 4 2.2 5 2.3..9 3...10 3.1 10 3.2 T...17 4...30...31 ABSTRACT.34...35

Langmuir, T. MaxwellLangmuir. Langmuir Beskok, Maxwell, DSMC.,. T 10~60.,. Langmuir.

NOMENCLATURE Roman Symbols A Ec D e F r G r Kn L Ma Pr Re R S T T T 0 w mean area of a site Eckert number Potential parameter Convective flux vector Viscous flux vector Knudsen number Channel length Mach number Prandtl number Reynolds number Gas constant Control surface length Temperature Reference temperature Wall temperature c p Specific heat ratio h o k n r p q B Specific total enthalpy Boltzmann constant Unit vector normal to the control surface Static pressure Heat diffusion u, v Cartesian velocity components u g u w Slip velocity Wall velocity

u 0 u λ Reference velocity Tangential gas velocity x, y Dimensional coordinates Greek Symbols α The fraction of covered surface β Pressure relaxation factor γ Blending factor used in the approximation of convective flux λ Mean free path µ Molecular viscosity ρ Fluid density σ v Accommodation coefficient σ T Thermal-accommodation coefficient τ Viscous stress τ Viscous dissipation in energy equation

List of Figures Fig. 1 Fig. 2 Schematic of the backward-facing step geometry. Unstructured grid after the expansion of the backward-facing step geometry. Fig. 3 Pressure and streamwise velocity along the backward-facing step channel. Comparison of DSMC and Navier-Stokes utilizing the Langmuir slip condition are presented. Fig. 4 Velocity and temperature distribution before, and after the expansion. Various slip models are compared to the DSMC solutions. Fig. 5 Shear stress distributions at top and bottom walls using Beskok, Maxwell and Langmuir slip condition. Fig. 6 Fig. 7 Fig. 8 Schematic of the flow split in a T-shaped planner branch Unstructured grid at a part of main and side branch Mach number contours obtained by the Maxwell (under figure) and Langmuir (up figure) slip models. Fig. 9 Velocity, pressure, and Mach number distribution at a horizontal and cross section. (Re=10) Fig. 10 Velocity, pressure, and Mach number distribution at a horizontal and cross section. (Re=50) Fig. 11 Velocity and Mach number distributions utilizing the Langmuir slip model of increasing Reynolds number. Fig. 12 Distribution of pressure and streamwise velocity using each boundary condition(re=30).

Fig. 13 Pressure and velocity distributions along main branch using each boundary conditions (x = 0.5, 0.95). Fig. 14 Comparison of vorticity for incompressible and compressible rarefied gas flows along lower wall of side branch List of Figures Table 1 Notation and location for the streamwise cuts presented in the step geometry locations TW and BW correspond to the DSMC cell adjacent to the walls. Table 2 Description and location for the streamwise cuts presented in the T-shaped micro-manifold geometry locations.

MEMS(Micro-Electro-Mechanical-Systems), (biomedical system), Bio-chip [1]. MEMS. MEMS,, Navier-Stokes.,. Kn,,. Kn,,. (slip flow), (rarefaction), (compressibility), (intermolecular forces)[1-2]. MEMS [1-10] DSMC(Direct Simulation Monte Carlo) [3]Navier-Stokes [4]. Bird[3] DSMC MEMS,., Navier-Stokes

, DSMC. Kn. Kn 0.01, (no-slip boundary condition) Navier-Stokes,. 0.01 Kn 0.1, (slip boundary condition) Navier-Stokes, (slip flow regime). 0.1 Kn 10, Navier-Stokes (transition flow regime), Kn > 10, (free molecular flow). Boltzmann.,,, MEMS [5]. 1 (the first-order slip boundary condition)maxwell [6-8] (the high-order slip boundary condition)([4],[9])., Myong([2],[5],[10]) [11] Langmuir. Langmuir Navier-Stokes [12]., MEMS,

,,., [9],, DSMC.

2. 2.1 2.,,. nds G nds = S F 0 (1) S F, G.. r F = f i + g j 1 r ; G = ( fv i + gv j). (1a) Re f ρu 2 ρu + p = ρuv ρuh o ; ρv ρuv g = + ; (1b) 2 ρv p ρvh o f v 0 τ xx = τ xy τ x qx ; g v 0 τ xy = τ yy τ y qy., ρ, p, u v x y h. o

Re. τ, q.,.. p = ρrt. (1c). x * = x / L ; y * = y / L ; u * = u / u ; v * = v / u ; ref ref p * 2 = p /( ρ ) ; T * = T / T ; ref ρ * = ρ / ρ ; ref ref u ref h = h ; Re = ( ρ ref u ref L) / µ ref * o 2 o / uref (1a)(1b) *. 2.2,.,, MEMS [1]. MEMS Maxwell-Smoluchowski

[13]., [14] 1879Maxwell, [1]. u g u w 2 σ v = σ v u λ y w (2), λ σ (accommodation coefficient),, v ug, uw., Von Smoluchowskitemperature-jump,. u * g u * w 2 σ v u = Kn σ v y 2 3 ( γ 1) Kn Re T + Ec 2π γ x * * w * * w (3) T 2 σ Kn T * * * T g Tw = * σ T ( γ + 1) Pr y 2γ w (4), *, γ σ, T (thermal-accommodation coefficient), Ec Eckert. Ec u c T T T 2 = 0 = ( γ 1) 0 2 Ma (5) p, u0 ( gas 0 (reference velocity), T = T T ), T0 (reference temperature).,

Maxwell. u σ * * * 2 v u g uw = Kn * σ v y w (6),,,.,., (6) Beskok[4] 2. u g 1 = ) 2 [ u + (1 σ u σ u ] λ v λ +, uλ (tangential gas velocity)., v Maxwell., Navier-Stokes Kn 1, Beskok Kn 2. Myong Langmuir. (adsorption). Langmuir 2 (the fraction of covered surface) α., w (7)

α., (8) [10]. βp α = (8) 1 + βp (9). βp α = (9) 1 + βp, β, (10). Aλ / Kn D e β = exp (10) kbtw kbtw, k B Boltzmann, A D, λ, e. β, D e. N 2 (9)., (9)(10) α. T = αtw + ( 1 α) T (11) 0 u = αuw + ( 1 α) u (12) 0, u0, T0.. Maxwell, Beskok DSMC.

2.3.. - SIMPLE[15-17]. -. - (edge) [18-19]. - 2 1 (piecewise linear reconstruction)[18]. - DemirdzicMuzaferija[15]..

3. 3.1 MEMS,. Langmuir DSMC, Maxwell Beskok2 [9].. T- [20].. Fig. 1,. h, S, / h = 0. 467 S, x = 5. 6., N 2., x / h = 0.86, 15869. Re = 80, Pr = 0. 7, 300K, 2.32 Langmuir, DSMC, Maxwell, Beskok. Fig. 3 5

Fig. 1 Schematic of the backward-facing step geometry. Fig. 2 Unstructured grid after the expansion of the backward-facing step geometry.

DSMC. Table 1.,,. 1.2 < x / h < 1. 6,,, x / h 2.0...,.,. x / h 3.0., 0. Fig. 4 ( x / h = 1. 7 ) ( x / h = 2. 1),. u, v, y, / h = 0. 25. Maxwell,. u, v,, u.,

P 7 6.5 6 5.5 5 Bottom Wall (DSMC) Bottom Wall (Langmuir) Bottom Center (DSMC) Bottom Center (Langmuir) Center (DSMC) Center (Langmuir) Center of Entrance (DSMC) Center of Entrance (Langmuir) Top Wall (DSMC) Top Wall (Langmuir) 4.5 4 3.5 3 1 2 3 4 5 x/h 1.1 1 0.9 0.8 0.7 Filled symbols - DSMC Lines - Langmuir Center of Entrance Center 0.6 U/c 0.5 0.4 Bottom Center 0.3 0.2 Top Wall 0.1 0-0.1 Bottom Wall 1 2 3 4 5 x/h Fig. 3 Pressure and streamwise velocity along the backward-facing step channel. Comparison of DSMC and Navier-Stokes utilizing the Langmuir slip condition are presented.

Table 1 Notation and location for the streamwise cuts presented in the step geometry locations TW and BW correspond to the DSMC cell adjacent to the walls. Abbreviation Description Location BW Bottom wall y/h = 0.01675 BC Bottom center y/h = 0.25 C Center y/h = 0.48325 CE Center of entrance y/h = 0.75 TW Top wall y/h = 0.9875 Maxwell, Langmuir, Beskok, DSMC,. Maxwell,. u, v,., 4,. Fig. 5, Beskok, Maxwell Langmuir.. x / h < 2. 8, h x /, 2.8 2.9

350 0-1 300-2 250-3 -4 U[m/s] 200 150 V[m/s] -5-6 -7 100 50 DSMC Maxwell Beskok Langmuir -8-9 -10 0 0.5 0.6 0.7 0.8 0.9 1 y/h -11 0.5 0.6 0.7 0.8 0.9 1 y/h 302 300 298 296 350 280 DSMC Maxwell Beskok Langmuir T[K] 294 292 290 U [m/s] 210 140 288 286 70 284 0 282 0.5 0.6 0.7 0.8 0.9 1 y/h 0 0.25 0.5 0.75 1 y/h 15 310 0 300-15 V [m/s] -30 T[K] 290-45 280-60 0 0.25 0.5 0.75 1 y/h 270 0 0.25 0.5 0.75 1 y/h Fig. 4 Velocity and temperature distribution before, and after the expansion. Various slip models are compared to the DSMC solutions.

0.2 0.15 Top Wall Beskok Maxwell Langmuir 0.1 τ 0.05 0 Bottom Wall -0.05 2 3 4 5 x/h Fig. 5 Shear stress distributions at top and bottom walls using Beskok, Maxwell and Langmuir slip condition.. Beskok,. Langmuir,. T Langmuir. DSMC,.,, Langmuir.

3.2 T 3.1 T. T Maxwell. T Fig. 6, Fig. 7. D 1µm, 10~60.,,. N 2, 14514. Kn 0.04,. Fig. 8Re 30, Mach contour., Mach,., Mach.. Maxwell,. Fig. 9Re=10, x y,, Mach., Maxwell

Fig. 6 Schematic of the flow split in a T-shaped planner branch Fig. 7 Unstructured grid at a part of main and side branch

mach 0.695988 0.649614 0.603239 0.556865 0.510491 0.464117 0.417743 0.371369 0.324994 0.27862 0.232246 0.185872 0.139498 0.0931236 0.0467494 mach 0.686516 0.640752 0.594988 0.549225 0.503461 0.457697 0.411933 0.366169 0.320406 0.274642 0.228878 0.183114 0.137351 0.0915869 0.0458231 Fig. 8 Mach number contours obtained by the Maxwell (under figure) and Langmuir (up figure) slip models.

Y 2.9 2.8 2.7 2.6 2.5 2.4 No-Slip Langmuir Maxwell V 100 90 80 70 60 50 2.3 2.2 2.1 40 30 20 10 20 30 40 U 0.25 0.5 0.75 x 28.5 2.9 2.8 28.25 2.7 2.6 P * 28 Y 2.5 2.4 2.3 27.75 2.2 2.1 0.2 0.4 0.6 0.8 1 x 27 27.5 P * 28 28.5 0.25 2.9 2.8 0.2 2.7 2.6 Ma 0.15 Y 2.5 2.4 0.1 2.3 2.2 0.05 0.2 0.4 0.6 0.8 x 2.1 0.05 0.1 0.15 Ma Fig. 9 Velocity, pressure, and Mach number distribution at a horizontal and cross section. (Re=10)

. Re=10,..., v u,, p * 2 = p /( ρ ). ref u in,..,,.. Maxwell Langmuir. x, y.. Fig. 10 50, x y,, Mach., Mach, Fig. 9. Fig. 11 Langmuir, Mach., Mach, Mach 10 20

2.9 300 2.8 2.7 250 Y 2.6 2.5 2.4 No-Slip Langmuir Maxwell V 200 150 2.3 2.2 100 2.1 40 60 80 100 120 140 U 0.2 0.4 0.6 0.8 x 2.9 0.925 2.8 0.9 2.7 0.875 2.6 P * 0.85 Y 2.5 0.825 2.4 0.8 2.3 0.775 2.2 0.75 0.2 0.4 0.6 0.8 x 2.1 0.6 0.7 0.8 P * 0.9 1 2.9 0.8 2.8 0.7 2.7 0.6 2.6 Ma 0.5 Y 2.5 2.4 0.4 0.3 2.3 2.2 0.2 0.2 0.4 0.6 0.8 x 2.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Ma Fig. 10 Velocity, pressure, and Mach number distribution at a horizontal and cross section. (Re=50)

2.9 300 2.8 2.7 250 2.6 200 Y 2.5 2.4 V 150 2.3 2.2 2.1 Re=10 Re=20 Re=30 Re=40 Re=50 Re=60 50 100 150 U 100 50 0.2 0.4 0.6 0.8 x 0.9 Re = 10 Re = 20 Re = 30 0.8 Re = 40 Re = 50 Re = 60 0.7 0.6 2.9 2.8 2.7 2.6 Re = 10 Re = 20 Re = 30 Re = 40 Re = 50 Re = 60 Ma 0.5 0.4 Y 2.5 2.4 0.3 2.3 0.2 2.2 0.1 0.2 0.4 0.6 0.8 x 2.1 0.1 0.2 0.3 0.4 0.5 Ma Fig. 11 Velocity and Mach number distributions utilizing the Langmuir slip model of increasing Reynolds number. Table 2 Description and location for the streamwise cuts presented in the T- shaped micro-manifold geometry locations. Description Location Top wall y = 2.95 Center y = 2.5 Bottom wall y = 2.05

3 2.8 2.6 Top wall Langmuir condition Maxwell condition No-Slip condition P * 2.4 Center 2.2 2 Bottom wall 1.8 0 1 2 3 4 x 140 100 Center Langmuir condition Maxwell condition No-Slip condition U Top Wall 60 20 Bottom Wall 0 1 2 3 4 x Fig. 12 Distribution of pressure and streamwise velocity using each boundary condition(re=30).

,. 20,. Fig. 123, 3. Table 2, 30. (top wall), x1.8. (bottom wall). (center),.,. (bottom wall) x1. 3.1,. 0,,., 3, MEMS,.,.,.

3.4 Langmuir condition Maxwell condition 3.2 No-Slip condition 3 2.8 x=0.5 P * 2.6 2.4 2.2 2 1.8 x=0.95 1 2 3 4 5 Distance along main branch 250 200 main center (x=0.5) V 150 100 main side (x=0.95) 50 1 2 3 4 5 Distance along main branch Fig. 13 Pressure and velocity distributions along main branch using each boundary conditions (x = 0.5, 0.95).

0.07 0.06 0.05 0.04 Re = 10 Re = 50 Re = 100 Re = 200 Vorticity 0.03 0.02 0.01 0-0.01-0.02-0.03 1 2 3 x 0.12 0.1 0.08 Re = 10 Re = 30 Re = 50 Re = 60 0.06 Vorticity 0.04 0.02 0-0.02-0.04 1 2 3 4 x Fig. 14 Comparison of vorticity for incompressible and compressible rarefied gas flows along lower wall of side branch

,,.,.. Fig. 13 30, = 0. 5 x, x = 0. 95. (center) (side).,.,...,,.,..,. Fig. 14 (vorticity)., Langmuir.

,..,. T..,,. 3.1 Langmuir.

4. Langmuir. T, Langmuir. MEMS. Langmuir Maxwell1, Beskok2, DSMC,,. T,.,.,,,., MaxwellLangmuir., 1. MEMS Langmuir.

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Abstract The current study analyzes Langmuir slip boundary condition theoretically and it is tested in the practical numerical analysis for separation-associated flow. Slip phenomenon at the channel wall is properly implemented by various numerical slip boundary conditions including Langmuir slip model. Compressible backward-facing step flow is compared to other analysis results with the purpose of Langmuir slip model validation. The numerical solutions of pressure and velocity distributions where separation occurs, are in good agreement with other numerical results. Numerical analysis is conducted for Reynolds numbers from 10 to 60 for a prediction of separation at T-shaped micro manifold. Reattachment length of flows shows nonlinear distribution at the wall of side branch. The Langmuir slip model predicts fairly the physics in terms of slip effect and separation.

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