Method of calculating the dimensions of greenhouse-type single slope watermaker by taking into account the accumulation of solar energy parnik tipidagi bir nishabli suv chuchutgichi o

Elmurodov R.U. - Guliston davlat universiteti dotsenti, fizika-matematika fanlari nomzodi.
E-mail: [e-mail]
Abdurakibov A.A.- Guliston davlat universieti “ Qayta tiklanuvchi energiya manbalari va barqaror atrof-muhit fizikasi” yo‘nalishi magistranti. E-mail: [e-mail]
УДК 532.546
APPROXIMATE SOLUTION OF ONE PROBLEM OF STEPHAN'S TYPE FOR SMALL
TIME VALUES
VAQTNING KICHIK QIYMATLARIDA STEFAN TIPIDAGI MASALANI TAQRIBIY YECHISH
ПРИБЛИЖЕННОЕ РЕШЕНИЕ ОДНОЙ ЗАДАЧИ ТИПА СТЕФАНА ДЛЯ МАЛЫХ
ЗНАЧЕНИЙ ВРЕМЕНИ

Jamuratov Kengash, Axmidov Xoliqul

Gulistan State University, 120100. Gulistan City, Sirdarya region, 4th District
E-mail: [e-mail]

Abstract. During filling and operation of small canals and reservoirs, the groundwater level (GW) rises and if the initial level ^h_l is located below the critical depth ^y₀, then there will come a ^{point in time} t  t₀, such that the level in them reaches the critical level ^h_kp. With a further rise in the water level in canals (reservoirs), due to the dependence of ^ on the groundwater level, two motion areas with a moving interface x l t( ), l t( )₀  0 appear in the region h(0,t) ( )t  h l t t( ( ), )  h_kp ,there will be evaporation, but in region h_l  h x t( , )  h_kp it is absent.
Within the hydraulic theory of the filter, neither the value of the GW h x t( , ) satisfies the Boussinesq equation
h x t(t, )  k xh x t( , )h x t( , )  , (1)

   x  
where k  filtration coefficient, coefficient of water loss, ^ rate of evaporation determined by the equality
 _ f h( h_kp, ),t h  h_kp
  (2)
0 , h  h_kp
f h( h_kp, )t known function with its arguments.
To simplify the study of the problem, equation (1) is usually considered in a linearized form. In the work, a separate linearization of equation (1) is carried out for each of the areas
0  x  l t( ) (h  h_kp ), l t( )  x   (h  h_kp)
assuming that

h h, h h x t( , )   1 x  hkp h
x h2 h, h  hkp
 x
w here h₁(h_kp,h_m] и h₂ (h h_l , _kp ] some average values of h x t( , ), h_m  max( ).t
t
Given the above assumptions about the dynamics of groundwater near new channels and reservoirs in the presence of evaporation, it is formulated as follows:
Find function U x t( , )  h x t( , )hul t_l ( ), l t( )₀  0 in space _t^0 x x: l t( ) that satisfy the equation:
Ut  a2 ( )x 2xU2  (3)

and conditions
U x t( , ) _{t t 0} ( )x U x t, ( , ) _x0( )t ,U x t( , ) _x 0 (4)
U x t( , )x l t ( ) 0 U x t( , ) x l t ( ) 0 0  hkp hl (t  t0 ) (5)
^h₁ U x l t_{ }( ) 0 ^h₂ U x l t_{ }( ) 0 (t  ^t₀) . (6)
x x
Here

 2( )x  1  1 a

(7)

 2  2 ^ f U( 0( ))t ,U  0; ^ 0,U 0;
			(8)

^a²  ^k h  const, 0  x l t( ), a²  k h  const l t, ( )  x .
 _t^₀ ( , ): 0x t  x , t₀  t T.
Problem (3)-(6) belongs to the class of problems with an unknown interface.
The difference between problem (3)-(6) from the well-known Stefan problem [1] is that the flow (flow) is continuous at an unknown interface, while the flow in Stefan's problems is discontinuous and this gap is proportional to the speed of moving the moving interface ( front).
The requirement l t( )₀  0 and the indicated difference T in the Stefan condition creates additional difficulties in the study of problem (3)-(6).
In this regard, it is also of independent mathematical interest.