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Asymptotic Modeling of Flows in Micro-Channel by Using ...

ρ − 4 2 2 2γ ⎛ u 2 xx + vxy ⎞ γ M 0 ε ( ρ u u + ) + − ⎜ ⎟ x ρ v u y p x M 0 Kn0 u yy + ε = 0 π 3 ⎝ ⎠ (9) 4 2 4 2γ 2 ⎛ v yy + u xy 2 ⎞ γ M 0 ε ( ρ u v + ) + − ⎜ ⎟ x ρ v v y p y M 0 Kn0 ε + ε v xx = 0 π 3 ⎝ ⎠ (10) 2γ 4 2 2 2 ( u T + vT ) + ( γ −1)( u + v ) p − ( γ −1) M Kn ε ( u + v − u v ) 2γ π x y 2 4 2 2 0 ( γ −1) M Kn ( u + ε v + 2 ε u v ) − ⎜T + T ⎟ = 0 0 0 y x y x π y x 0 0 3 2γ 1 Kn π Pr M 0 ⎛ ⎝ x xx y ε 1 2 x yy y ⎞ ⎠ (11) In addition, the dimensionless ideal gas law takes the simple form written with dimensionless variables. We have, for y = ±1 / 2 : p = ρ . The boundary conditions (6) are now 1 ∂ u 3 2γ Kn0 1 ∂ T 2 γ Kn0 1 ∂ T u = m Kn0 + , v = 0 , T = Tw m (12) ρ T ∂ y 4 π M ρ T ∂ x γ + 1 Pr ρ T ∂ y 0 It is important to note that the small parameter ε does not appear **in** (12). If the boundary conditions (6) had been written with second order terms [1,2], then all the new terms would appear with the same order **in** (12). THE ASYMPTOTIC SOLUTION IN THE FIRST APPROXIMATION In order to f**in**d the first approximation **of** the problem (8) to (12), we put: u = u ~ + O ( ε 2 ) , v = v~ + O ( ε 2 ) , ρ = p = p~ + O ( ε 2 ) , T = T ~ + O ( ε 2 ) . The terms O ( ε 2 ) are neglected and we obta**in**: ( ~ p u ~ ) ( ~ p ) = 0 x + v ~ y , p x − 2γ / M 0 Kn0 u ~ yy = 0 , ~ p y = 0 , T ~ yy = 0 (13) u ~ Kn T ~ Kn T ~ u ~ 1 ∂ 3 2γ , v ~ , T ~ T T ~ 0 1 Kn p~ y M p~ ~ ∂ 2 γ 0 1 ∂ = m 0 + = 0 = w m ∂ 4 π T ∂ x + 1 Pr p~ T ~ 0 γ ∂ y (14) In equation (14), the dimensionless temperature T w on the two walls depends only on the longitud**in**al variable x . As a consequence **of** (13), p ~ depends only on the coord**in**ate x , and T ~ is a l**in**ear function **of** the variable y with its coefficients depend**in**g on x . By us**in**g the third condition **in** (14), due to the symmetry **of** the problem, one sees that T ~ = T w . The second equation **in** (13) and the first boundary condition **in** (14) are used to write the expression **of** the longitud**in**al velocity: ~ p y ~ p / Kn dT u ~ ⎛ 2 x 1 ⎞ x 3 2γ π 0 = ⎜ − ⎟ − + w (15) 2γ / π M Kn 2 8 p / M T 4 p M Tw dx 0 0 ⎝ ⎠ 2 2γ π 0 w 0 Now we pay attention to the first equation **in** (13) and to the boundary condition on v ~ **in** (14). The equation is **in**tegrated between y = −1 / 2 and y = 1 / 2 . In addition, at the upstream and downstream ends **of** the microchannel, we prescribe constant pressures: p~ = p~ i for x = 0 and p~ = p~ o for x = 1 . The pressure p~ = ~ p ( x ) is the solution **of** a second order differential equation with boundary conditions: d dx ⎪ ⎧ p~ p~ x p~ x 3 2γ / π Kn ⎨ + − 12 2 2 4 p M ⎪⎩ γ / π M 0 T 0 T w w 0 dT ⎪ ⎫ w ⎬ = 0 dx ⎪⎭ p~ ~ x = p , { } =0 i , { } o p~ ~ x=1 = p (16) In the particular case where T w is constant, the pressure and the longitud**in**al velocity have known explicit expressions [1,2]. With a temperature gradient along the walls, due to the expression (15), the flow velocity and the slipp**in**g along the wall will depend on this gradient: The dynamic and thermal effects are coupled. In order to obta**in**

p~ , it is necessary to perform a numerical **in**tegration **of** problem (16). Then, the velocity u ~ is deduced from (15). Three cases for the dimensionless temperature T w ( x ) along the channel walls are chosen: a) = 1 + ( 1− 2x ) / 7 , b) T w = 1 + ( 1− 4x ) 7 for 0 ≤ x ≤ 0. 5 and T w = 6 / 7 for 0. 5 ≤ x ≤ 1 , c) T w = ( 2 / 7 )( −2x + x + 1) + 6 / 7 . In each case p i = 2.998 and p o = 2.306. On Fig. 2, the three pressure solutions **of** equation (16) are given. We observe that these pressure pr**of**iles are different. On Fig. 3, the associated velocity pr**of**iles **in** the micro-channel section x = 0.4 are shown. The effect due to the temperature gradient is obvious. 2 T w 3 2.9 2.8 Pressure on case a) case b) case c) 0.04 0.03 0.02 Velocity **in** x = 0.4 case a) case b) case c) 2.7 2.6 a b c 0.01 0 -0.01 b a 2.5 -0.02 c 2.4 -0.03 2.3 -0.04 2.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -0.05 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 x FIGURE 2. Pressure pr**of**iles along the micro-channel FIGURE 3. Velocity pr**of**iles **in** the section x= 0.4 ASYMPTOTIC SOLUTION AND DSMC SIMULATION COMPARISON The aim **of** this section is to compare, for a number **of** flows **in**side a micro-channel, the results given **by** DSMC simulations and **by** the asymptotic method. For each simulation (n°1 to n°9), the pressures P **in** and P out and the temperatures T **in** and T out take different values, as shown **in** Table 1. The DS2V code is used **in** its steady version [9]. At its last stage, the code yields the steady solution for the flow **in**side the channel. The values P i , T i , , P o , T o are read on the simulation results (see Table 1). The follow**in**g characteristic scales are **in**troduced: T = ( T T ) / 2 , c i + Pc = (3µ / h) 2π rT c , U c = ( 3h / l) 2π r T c . Then p~ i = Pi / Pc and p~ o = Po / Pc . By us**in**g these values as data for the solution **of** (15) and (16), one obta**in**s the solution through the asymptotic method. Two result sets are now shown. First, the particular case where T w is a constant is considered (simulations n°1 to n°5). In Fig. 4, the dimensionless pressure pr**of**iles along the micro-channel axis are represented: The cont**in**uous l**in**es correspond to the asymptotic solutions and the dots to the DSMC simulations. The two results are **in** a very good agreement. In Fig. 5, the difference between the two results, namely 100(1 − PAS / PDSMC ) , is shown. The fluctuations between the two solutions are smaller than 0.5%. For the longitud**in**al velocity, similar comparisons have been made, which show fluctuations between asymptotic and DSMC results less than 2.5%. As **in** many studies [1,2,7,11], we note the nonl**in**ear evolution **of** the pressure (cont**in**uous l**in**es compared to the dashed l**in**es on Fig. 4). The second result set corresponds to the four simulations n°6 to n°9 where the temperature decreases l**in**early from 400 K to 300 K along the micro-channel walls, and where the entrance pressure P **in** varies from 130 kPa to 110 kPa . In Fig. 6 the pressure pr**of**iles and the longitud**in**al velocity pr**of**iles along the channel axis are presented (here, all the quantities are dimensional and X = x − 5µm ). The dots correspond to the DSMC simulations, and the cont**in**uous l**in**es to the asymptotic solutions. As previously, the two results, asymptotic and DSMC, are **in** good agreement. Depend**in**g on the simulation conditions, the pressure gradient **in**side the channel may be negative (n°6, n°7), near zero (n°8) or positive (n°9). The associated velocity is positive (n°6, n°7), near zero (n°8) or negative (n°9). This conclusion is **in** agreement with the formula (15) where a negative pressure gradient **in**creases the velocity and a negative temperature gradient decreases the velocity. This thermal effect is well known **in** the scope **of** rarefied gas dynamics [14]. CONCLUSION The asymptotic analysis **of** the flow **in** a micro-channel leads to solutions **in** very good agreement with DSMC numerical simulations, **in** the flow configurations here considered. The analytical expressions deduced from the asymptotic process underl**in**e the **in**fluence **of** a temperature gradient along the walls. In particular, on the expression o

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