DA ROSA, C. E.; ROSSI, F. D.; JACOBI, L. F.; KNACKFUSS, R. F.

doi:10.5540/tema.2020.021.03.0559

C. E. DA ROSA

Instituto Federal Farroupilha - Campus São Vicente do Sul, Universidade Federal de Santa Maria, Santa Maria-RS, Brasil - E-mail: cinara.rosa@iffarroupilha.edu.br

http://orcid.org/0000-0002-9077-5031

F. D. ROSSI

Instituto Federal Farroupilha - Campus Alegrete, Alegrete-RS, Brasil- E-mail: fabio.rossi@iffarroupilha.edu.br

http://orcid.org/0000-0002-2450-1024

L. F. JACOBI

Universidade Federal de Santa Maria, Santa Maria-RS, Brasil - E-mail: lucianefj8@gmail.com

http://orcid.org/0000-0003-4622-9292

R. F. KNACKFUSS

Universidade Federal de Santa Maria, Santa Maria-RS, Brasil - E-mail: rfknackfuss@gmail.com

http://orcid.org/0000-0001-6997-3474

*Autor correspondente: Cinara Ewerling da Rosa - E-mail: cinara.rosa@iffarroupilha.edu.br

Modelo Cinético	β	ϖ	ε _p	ε _t
Modelo BGK	0	0	1	1
Modelo S	1/3	0	1	3/2
Modelo GJ	5/9	1/3	3/2	9/4
Modelo MRS	$1 - (16 / 15) 2^{1 / 2}$	$1 - (8 / 5) 2^{1 / 2}$	$(5 / 16) 2^{1 / 2}$	$(15 / 32) 2^{1 / 2}$

Caso	α _t1	α _n1	α _t2	α _n2	u _w1	u _w2	2a ¹
“A”	0.25	0.50	0.75	0.25	1.0	-2.0	1.0(X)
“B”	0.50	0.75	0.25	0.25	0.0	1.0	2.0(X)

η	BGK	S	GJ	MRS	ELB ¹⁴14 M. Garcia & E. Siewert . The linearized Boltzmann equation with Cercignani-Lampis boundary conditions: Basic flow problems in a plane channel. Europen Journal of Mechanics B/Fluids, 28 (2009), 387-396.
0.0	1.2476(-1)	1.6431(-1)	1.6755(-1)	1.6094(-1)	9.5650(-2)
0.2	1.8637(-1)	2.3631(-1)	2.5250(-1)	2.1695(-1)	1.5862(-1)
0.4	1.9851(-1)	2.5121(-1)	2.7000(-1)	2.2957(-1)	1.7148(-1)
0.6	1.8763(-1)	2.3752(-1)	2.5262(-1)	2.2168(-1)	1.6197(-1)
0.8	1.5069(-1)	1.9228(-1)	1.9668(-1)	1.9157(-1)	1.2678(-1)
1.0	4.1982(-2)	6.8671(-2)	4.6411(-2)	1.0616(-1)	1.9727(-2)

η	BGK	S	GJ	MRS	ELB ¹⁴14 M. Garcia & E. Siewert . The linearized Boltzmann equation with Cercignani-Lampis boundary conditions: Basic flow problems in a plane channel. Europen Journal of Mechanics B/Fluids, 28 (2009), 387-396.
0.0	2.7700(-1)	3.0219(-1)	3.3321(-1)	2.5602(-1)	-1.9451(-1)
0.2	2.7914(-1)	3.0327(-1)	3.3317(-1)	2.5784(-1)	1.6753(-1)
0.4	2.7155(-1)	2.9214(-1)	3.1799(-1)	2.5176(-1)	2.1371(-1)
0.6	2.5203(-1)	2.6589(-1)	2.8389(-1)	2.3592(-1)	2.1962(-1)
0.8	2.1376(-1)	2.1587(-1)	2.1989(-1)	2.0459(-1)	1.8979(-1)
1.0	1.1234(-1)	8.9549(-2)	6.0825(-2)	1.2324(-1)	5.2614(-2)

	BGK	S		GJ		MRS		ELB ¹⁴14 M. Garcia & E. Siewert . The linearized Boltzmann equation with Cercignani-Lampis boundary conditions: Basic flow problems in a plane channel. Europen Journal of Mechanics B/Fluids, 28 (2009), 387-396.
	$ε_{p} = ε_{t}$	ε _p	ε _t	ε _p	ε _t	ε _p	ε _t
Caso “A”	35	76	40	82	58	66	28	11
Caso “B”	36	77	45	85	52	63	27	11

η	BGK	S		GJ		MRS		ELB ¹⁴14 M. Garcia & E. Siewert . The linearized Boltzmann equation with Cercignani-Lampis boundary conditions: Basic flow problems in a plane channel. Europen Journal of Mechanics B/Fluids, 28 (2009), 387-396.
	$ε_{p} = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$
0.0	-0.6862	-0.8715	-0.6685	-0.8490	-0.6481	-0.9176	-0.4956	-0.4956
0.2	-0.8291	-1.0585	-0.8326	-1.0640	-0.8419	-1.0706	-0.8304	-0.6539
0.4	-0.8600	-1.1010	-0.8687	-1.1138	-0.8858	-1.1063	-0.8581	-0.6870
0.6	-0.8425	-1.0754	-0.8489	-1.0845	-0.8641	-1.0882	-0.8432	-0.6733
0.8	-0.7708	-0.9749	-0.7661	-0.9683	-0.7684	-1.0122	-0.7810	-0.6057
1.0	-0.5475	-0.6858	-0.5069	-0.6330	-0.4658	-0.7917	-0.5876	-0.3590

η	BGK	S		GJ		MRS		ELB ¹⁴14 M. Garcia & E. Siewert . The linearized Boltzmann equation with Cercignani-Lampis boundary conditions: Basic flow problems in a plane channel. Europen Journal of Mechanics B/Fluids, 28 (2009), 387-396.
	$ε_{p} = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$
0.0	13.134	13.985	14.144	16.427	16.315	9.784	10.117	8.101
0.2	4.113	4.793	4.488	5.761	5.237	3.339	3.155	2.567
0.4	0.348	0.163	0.449	0.292	0.633	0.155	0.262	0.400
0.6	-2.980	-4.112	-3.161	-4.817	-3.455	-2.716	-2.270	-1.421
0.8	-7.410	-9.494	-8.063	-1.135	-9.251	-6.273	-5.596	-4.294
1.0	-19.226	-21.501	-21.243	-25.888	-25.133	-14.153	-14.399	-12.114

η	BGK	S		GJ		MRS		ELB ¹⁴14 M. Garcia & E. Siewert . The linearized Boltzmann equation with Cercignani-Lampis boundary conditions: Basic flow problems in a plane channel. Europen Journal of Mechanics B/Fluids, 28 (2009), 387-396.
	$ε_{p} = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$
0.0	-1.0296	-1.0286	-0.9065	-1.0120	-0.8929	-1.0623	-0.9371	-0.7703
0.2	-1.1532	-1.1522	-1.0661	-1.1468	-1.0641	-1.1671	-1.0766	-0.9737
0.4	-1.2496	-1.2490	-1.1906	-1.2453	-1.1887	-1.2582	-1.1975	-1.1261
0.6	-1.3420	-1.3420	-1.3103	-1.3388	-1.3073	-1.3472	-1.3156	-1.2724
0.8	-1.4376	-1.4380	-1.4338	-1.4361	-1.4304	-1.4380	-1.4362	-1.4233
1.0	-1.5588	-1.5592	-1.5895	-1.5670	-1.5957	-1.5420	-1.5743	-1.6204

η	BGK	S		GJ		MRS		ELB ¹⁴14 M. Garcia & E. Siewert . The linearized Boltzmann equation with Cercignani-Lampis boundary conditions: Basic flow problems in a plane channel. Europen Journal of Mechanics B/Fluids, 28 (2009), 387-396.
	$ε = ε_{p} = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$
0.0	0.2224	0.2222	0.2025	0.2204	0.2013	0.2269	0.2060	0.1821
0.2	0.2763	0.2761	0.2707	0.2768	0.2718	0.2761	0.2698	0.2660
0.4	0.3197	0.3198	0.3265	0.3199	0.3266	0.3195	0.3263	0.3345
0.6	0.3623	0.3626	0.3812	0.3621	0.3804	0.3624	0.3821	0.4019
0.8	0.4070	0.4076	0.4384	0.4068	0.4368	0.4066	0.4395	0.4713
1.0	0.4665	0.4671	0.5133	0.4705	0.5156	0.4587	0.5069	0.5625

η	BGK	S		GJ		MRS
	$ε = ε_{p} = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$	$ε = ε_{p}$	$ε = ε_{t}$
q _p	-	x	-	x	-	-	-
u _p	x	-	-	x	-	-	-
q _t	-	x	-	x	-	x	-
u _t	-	x	-	x	x	x	-
q _c	-	-	-	-	-	-	-
u _c	x	x	-	-	-	x	-

R_{l} (v' : v) = \frac{m^{2} v_{n}}{2 π α_{n_{l}} α_{t_{l}} (2 - α_{t_{l}}) (k_{B} T_{w})^{2}} {\hat{I}}_{0} (\frac{(1 - α_{n_{l}})^{1 / 2} m v_{n} v'_{n}}{α_{n_{l}} k_{B} T_{w}}) e x p [- \frac{m [v_{n}^{2} + (1 - α_{n_{l}}) v'_{n}^{2}]}{2 k_{B} T_{w} α_{n_{l}}} - \frac{m [v_{t} - (1 - α_{t_{l}}) v'_{t}]^{2}}{2 k_{B} T_{w} α_{t_{l}} (2 - α_{t_{l}})}]

h (\mp a, \pm c_{x}, c_{y}, c_{z}) = 2 u_{w_{l}} α_{t_{l}} c_{y} + \int_{0}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} h (\mp a, \mp c'_{x}, c'_{y}, c'_{z}) R_{l} (\mp c'_{x}, c'_{y}, c'_{z} : \pm c_{x}, c_{y}, c_{z}) d c'_{x} d c'_{y} d c'_{z}

Ψ (ξ) = π^{- 1 / 2} e^{- ξ^{2}}, H (τ, ξ) = {[\begin{matrix} h_{1} (τ, ξ) h_{2} (τ, ξ) \end{matrix}]}^{T}, K (ξ', ξ) = [\begin{matrix} 1 + (2 β / 5) (ξ^{2} - 1 / 2) (ξ'^{2} - 1 / 2) + 2 ϖ ξ' ξ & (2^{3 / 2} β / 5) (ξ^{2} - 1 / 2) \\ (2^{3 / 2} β / 5) (ξ'^{2} - 1 / 2) & 4 β / 5 \end{matrix}]

H^{p} (τ, ξ) = k_{P} {[\begin{matrix} \begin{matrix} ϵ \frac{(1 - ϖ) τ^{2}}{2} - τ ϵ + \frac{(5 - 4 β) ξ^{2}}{5 ϵ (1 - β)} \end{matrix} & \frac{2^{1 / 2} β}{5 ϵ (1 - β)} \end{matrix}]}^{T} + \frac{k_{T}}{2 ϵ (β - 1)} {[\begin{matrix} ξ^{2} & 2^{1 / 2} \end{matrix}]}^{T} .

u (τ) = A_{1} + B_{1} ϵ τ + + \sum_{j = 2}^{2 N} [A_{j} e^{- \frac{ϵ (a + τ)}{ν_{j}}} + B_{j} e^{- \frac{ϵ (a - τ)}{ν_{j}}}] \sum_{k = 1}^{N} w_{k} Ψ (ξ_{k}) {[\begin{matrix} 1 \\ 0 \end{matrix}]}^{T} [Φ_{+} (ν_{j}) + Φ_{-} (ν_{j})]

q (τ) = \sum_{j = 2}^{2 N} [A_{j} e^{- \frac{ϵ (a + τ)}{ν_{j}}} + B_{j} e^{- \frac{ϵ (a - τ)}{ν_{j}}}] \sum_{k = 1}^{N} w_{k} Ψ (ξ_{k}) {[\begin{matrix} \begin{matrix} ξ_{k}^{2} - \frac{1}{2} \end{matrix} \\ 2^{1 / 2} \end{matrix}]}^{T} [Φ_{+} (ν_{j}) + Φ_{-} (ν_{j})] .

[1] *Autor correspondente: Cinara Ewerling da Rosa - E-mail: cinara.rosa@iffarroupilha.edu.br

Brasil

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Modelo Cinético Integrador para Fluxo de Gás Resultante do Movimento de Transferência de Massa em um Canal de Placas Heterogêneas

ABSTRACT