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

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

ρ − 4 2 2 2γ ⎛ u 2

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 find 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 obtain: ( ~ 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 longitudinal variable x . As a consequence of (13), p ~ depends only on the coordinate x , and T ~ is a linear function of the variable y with its coefficients depending on x . By using 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 longitudinal 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 integrated 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 longitudinal velocity have known explicit expressions [1,2]. With a temperature gradient along the walls, due to the expression (15), the flow velocity and the slipping along the wall will depend on this gradient: The dynamic and thermal effects are coupled. In order to obtain

p~ , it is necessary to perform a numerical integration 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 profiles are different. On Fig. 3, the associated velocity profiles 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 profiles along the micro-channel FIGURE 3. Velocity profiles 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 inside 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 inside the channel. The values P i , T i , , P o , T o are read on the simulation results (see Table 1). The following characteristic scales are introduced: 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 using these values as data for the solution of (15) and (16), one obtains 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 profiles along the micro-channel axis are represented: The continuous lines 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 longitudinal 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 nonlinear evolution of the pressure (continuous lines compared to the dashed lines on Fig. 4). The second result set corresponds to the four simulations n°6 to n°9 where the temperature decreases linearly 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 profiles and the longitudinal velocity profiles 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 continuous lines to the asymptotic solutions. As previously, the two results, asymptotic and DSMC, are in good agreement. Depending on the simulation conditions, the pressure gradient inside 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 increases 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 underline the influence of a temperature gradient along the walls. In particular, on the expression o

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