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ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
AUTOMODEL DECISIONS OF ONE UNLINEAR EQUATION OF
FILTRATION Jalilova R.K. Email: [email protected]
Jalilova Rahima Kurbanovna - PhD in Physical and Mathematical Sciences, Doсent, Teacher,
DEPARTMENT OF COMPUTATIONAL MATHEMATICS AND INFORMATICS, AZERBAIJAN STATE PEDAGOGICAL UNIVERSITY, BAKU, REPUBLIC OF AZERBAIJAN
Abstract: the paper studies the localization of perturbations caused by the LS-regime with peaking, during the filtration of the gas-liquid mixture. The motion of a liquid with a gas dispersed in it in a porous medium is considered in the following cases: a) in the presence of a source (drain), b) with a variable permeability, c) with a uniformly distributed production rate. During the research it was proved that localization of perturbations caused by LS - regime with exacerbation in a gas-liquid mixture in a porous medium occurs. The velocity increases in the peaking mode near the center of symmetry, and outside this region tends to a constant velocity distribution.
Keywords: equation of Relay, localization of disturbance, the wave processes in two-phase systems.
АВТОМОДЕЛЬНЫЕ РЕШЕНИЯ ОДНОГО НЕЛИНЕЙНОГО УРАВНЕНИЯ ФИЛЬТРАЦИИ Джалилова Р.К.
Джалилова Рахима Курбановна - доктор философии по физико-математическим наукам, доцент,
преподаватель, кафедра вычислительной математики и информатики, Азербайджанский государственный педагогический университет, г. Баку, Азербайджанская Республика
Аннотация: в статье проведено исследование процесса локализации возмущений, вызванной LS-режимом с обострением, при фильтрации газожидкостной смеси. Рассматривается движение жидкости с мелкодиспергированным в ней газом в пористой среде в следующих случаях: а) при наличии источника (стока), б) при переменной проницаемости, в) при равномерно распределенном дебите. В процессе исследований доказано, что происходит локализация возмущений, вызванная LS - режимом с обострением в газожидкостной смеси в пористой среде. Скорость увеличивается в режиме с обострением вблизи центра симметрии, а вне этой области стремится к постоянному распределению скорости.
Ключевые слова: уравнение Реле, локализация возмущений, волновые процессы в двухфазных системах.
УДК 51-73
Submit the action of liquid with small-dispersion in its gas in the porous environmentThe equation of the action of single-measure stream of gas-oil mixture with reckoning of the inertial members has the following view [3, 43]:
1 da 1 ( d (a\\ 1 dp 1 p
--+ — a—1 — 1 =---—--— a (1)
m dt m ^ dx ^ m)) p dx p k
where t-time, x-co-ordinate, w-speed of the action of mixture , p -compactness of mixture ,P-pressure, /Л - viscosity, k-pervious, m-porosity
It is possible to neglect the disturbance with the interaction among the bubbles and submit the action of each bubble independently from other bubbles for condition that the distance between the
bubbles of gas bigger of their radius and essential smaller of the length of wave. On the foundation of the equation of Relay and elementary direct of homogeny model, which connects radius of bubble R with compactness p -[2,283] ,the equation is written in view:
c-n 2c4 v dSp R0 d2 dp
dP = aldp + - 7-.--— + -^--f- (2)
3 (1 -Wo Wo dt 3(1 -w Jw dt
where R0 - balance radius of gas bubble , v - kinetic viscosity, 4 3
(p0 = — TtR Np - the actually volume gas-substance, N-numerous of bubbles in one mass of mixture
a2 = P0 (l — Wo Wop - expression for low-frequency approximate of the speed of sound in two-phase space. The porous space is little compressibility:
m0 2 PP0 m ≈a = — + fta; —0 (3)
po po
where ^((1 -coefficient of compressibility of porous space.
Using the methods of not linear wave dynamic, and so the row of transformations, connecting compactness p with speed of mixture w, we get one equation concerning w, showing the action of gas-liquid mixture in the porous space [3].
dw w dw d2 w d3 w au
— +---r—T + X—T+ w = 0 (4)
dt ap0 dx dx dx 2k
4 v 1 R02a0 where 2ц = —,-г— ;2x = —
3 i1 -PoК' 3 (l V0
In this case the coefficient has the meaning of compactness viscosity, appearing with reckoning of disciplinal losses on the boundary of separation phases. The member from the third derivative describes the influence of the dispersion effects for the action of two-phase mixture in whole. It is famous that in the process of the spreading of disturbances, dissipation balances the un linear effects and assists for the installation of the stationary forms of the wave.
Give some he first limited spreading of speed:
w(x,0) = W0 (x) (5)
Auto model decisions of the mission (4)-(5) are investigated:
w(x, t) = g (t 0(g), where g = x / p(t) (6)
Putting (6) in (4) determinations the functions:
( \ 2/3 /- \1/3
g(t) = [l " ^J P(t) = [l " ^J (7)
where Г -arbitrary parameter of devising of variable. The mission has the auto model decision:
f 4-2/3
w(x,t) = ll-tJ 0(g), g = x(1 -^)-1/3 (8)
where 0(g) -the decision of equation
d 6 1 ndd 1 sdO 2
X-r +-6-+ — £-+ — 6 = 0 (9)
d£ ap0 d£ 3t d£ 3r
( t Y'3
So x = £l 1--I then 0 < t < tf , half width of area of spreading disturbance is shorten.
Decision w(x, t) is the decision of the regime with aggravation. In this case half width of the first division w0 (x) bigger than half width of decision w(x, t) when
0 < t < tf .We see the localization of disturbances. When £ ^ ro then decision of equation (9) has asymptotic [3, 44]:
6 ^ c£- (10)
From (10) and (6) we get that the main member of asymptotic decomposition of the speed with
x ^ ro
w(x, t) ^ cx ~2 (11) does not depend on time .It shows on the localization of disturbances; speed increases in the regime with aggravation in shortening area near the centre of symmetry, but out of that area it aspires to the constant spreading of speed, it means to definite of the following expression (11).
Analogous investigations of the process of localization disturbances which causing of LS-the regime with aggravation in gas-liquid mixture are concluded with reckoning of the action of source.
The member which takes into consideration the action of source in the equation of inseparable is inserted:
^-¿Pl + rtw) (12)
at ox
The equation which descriptions the action of gas-liquid mixture in porous spare is written in the following view:
0w w 0w 02w 03w au q(w) dq(w) ^
— +---r——+ x—v+ w + ^-J--ria ' = 0 (13)
0t ap0 0x 0x 0x 2k 2 0(x)
The auto model decisions of missions (13),(5) in view (6) are investigated. Submit the case when the source with the degree mode depends on w:
q(w) = q0wx +&0w
Putting (6) in equation (13) when r = 0 determinates the function g(t) and (p(t) with the formulas (7) where 6(£) -the decision of equation.
d0 1 nd6 1 ^d6 2 „ 05/2 x—r+—6—+— £—+—6+q0 — = o (14) d£ ap0 d£ 3r d£ 3r 2
We will look for the equation answering the following boundary conditions:
d 36 d6
-00 > 0,6 > 0,6P > 0, P > 1
d£ £ d£ £ £
when £ ^ ro then decision of equation has asymptotic(10).Therefore the main member of asymptotic decomposition of the speed does not depend on time [4, 64].
Now submit the auto model decision of the mission with variable of pervious. A law of inflexion of pervious on layer applies middle-aged:
f x }J
k(x) = k0 I — I where k0 -coefficient of pervious for x=h, h-capacity of layer. It is not
^ h J
difficult to persuade if j=3 then the decision is auto model and has the view(8).Function satisfies to ordinary differential equation:
d36 1 nd0 1 ¡.¿G f 2 ^ h3 V 65/2 „
dg3 ap0 dg 3t" dg ^3r 2£0 gJ
We see 0(g) has asymptotic if g ^ ® it means goes on the localization of disturbances[4,65].
Submit the mission (13),(5)by setting debit which distributions follow the square of deposit ;except the general action of mixture from each element of volume layer it is possible to productive the selection of mixture for setting intensive.
In that case the compactness of debit determinations by formula q/x or wx / x .It is not difficult to persuade for y = 2 the decision is auto model. The equation concerning 0(g) has the following view:
d30 1 „ d0 1 „d0 ( 2 a^ h3 \ 1 02
-6 — +—g — +
6 +--= 0
dg3 ap0 dg 3т dg {3т 2 £0 g3 J g 2
Function 0(g) has asymptotic if g —> да ,The decision w(x, t) determinates by formula (8) and has asymptotic if X — да(11) [4,63]. In all submitting chances:
a) by presence source
b) by variable pervious
c) by the investigation of the process of localization disturbances are passed, which causing of LS-the regime with aggravation in gas-liquid mixture.
References / Список литературы
1. Barenblatt G.I. Similarity, auto model, inermediate symptotic.Gidrometeo, 1978.
2. The wave processes in two-phase systems/Book science art. by red V.E. Neporanov. Novosibirsk, 1980. Р. 283.
3. JalilovaR.G. The wave processes in two-phase systems // Dan Azerb.1985. XLI, № 7. Р. 42-47.
4. Jalilova R.G. Automodel decisions of one not linear equation of filtration of gas-liquid mixture.// Material VI Repub. Conf. by mathematics and mechanics, initiated to 40 year of Victory. Baku. «Science», 1985. Р. 62-65.
