Simultaneous effects of slip conditions and wall properties on MHD peristaltic flow of a Maxwell fluid with heat transfer Текст научной статьи по специальности «Физика»

УДК 512.54

Simultaneous Effects of Slip Conditions and Wall Properties on MHD Peristaltic Flow of a Maxwell Fluid with Heat Transfer

Kalidas Das*

Department of Mathematics, Kalyani Government Engineering College, Kalyani, Nadia, West Bengal, Pin:74125, India

Received 07.09.2011, received in revised form 23.12.2011, accepted 05.03.2012 The effects of both wall slip conditions and heat transfer on the magnetohydrodynamics (MHD) peristaltic flow of a Maxwell fluid in a porous planar channel with elastic wall properties have been studied. Mathematical formulation is based upon the modified Darcy’s law. The analytical solution has been derived for the stream function and temperature under the assumptions of small wave number. The results obtained in the analysis have been discussed numerically and explained graphically.

Keywords: peristalsis, Maxwell fluid, modified Darcy’s law, Brinkman number, Knudsen number, Heat transfer coefficient.

Introduction

Peristaltic flows are generated by the propagation of waves along the length of a distensible tube. It plays an indispensable role in transporting many physiological fluids in the body in various situations such as urine transport from kidney to bladder, movement of ovum in the fallopian tubes, the movement of enzyme in the gastrointestinal tract, swallowing of food through oesophagus and the vasomotion of small blood vessels. Some biomedical instruments, like the blood pumps in dialysis and the heart lung machine use the principle of peristaltic pumping to transport fluids without internal moving parts. The mechanism of peristaltic transport has been exploited for industrial applications like sanitary fluid transport, transport of corrosive fluids where the contact of the fluid with the machinery parts is prohibited and transport of a toxic liquid used in nuclear industry to avoid contamination of the outside environment.

The problem of the mechanism of peristaltic transport has attracted the attention of many investigators since the first investigation of Latham [1]. A number of analytical, numerical and experimental [2-15] studies have been conducted to understand peristaltic action for different kinds of fluids under different conditions with reference to physiological and mechanical situations. However the interaction of peristalsis and heat transfer has not received much attention which may become highly relevant and significant in several industrial processes. Also thermodynamical aspects of blood may become significant in processes like oxygenation and hemodialysis [16-18] when blood is drawn out of the body. Recently the combined effects of magnetohydrodynamics and heat transfer on the peristaltic transport of viscous fluid in a channel with compliant walls have been discussed by Mekheimer and Abd elmaboud and co-workers [19,20]. Recently, Hayat et al. [21], Nadeem and Akram [22] developed the problem by considering slip conditions on the boundary of the channel

The study of fluid flows and heat transfer through porous medium has attracted much

* [email protected] © Siberian Federal University. All rights reserved

attention recently. It is well known that flow through a porous medium has practical applications especially in geophysical fluid dynamics. Examples of natural porous media are beach sand, sandstone, limestone, wood, the human lung, bile duct, gall bladder with stones and small blood vessels. In some pathological situations, the distribution of fatty cholesterol and artery clogging blood clots in the lumen of coronary artery can be considered as equivalent to a porous medium.Hayat et al. [23] have analyzed hall effects on peristaltic flow of a Maxwell fluid in a porous medium.Hayat et al. [24] have examined the effect of heat transfer on the peristaltic flow of an electrically conducting fluid in a porous space. Very recently, Hayat et al. [25] have investigated the influence of heat and mass transfer on MHD peristaltic flow of a Maxwell fluid with complaint walls which have not been discussed so far when no-slip condition is no longer valid. The present paper concentrate on this concept.

The main purpose of the present study is to highlight the importance of slip conditions and heat transfer on MHD peristaltic flow of a Maxwell fluid through a porous medium in planar channel with elastic wall properties. The perturbation method has been used for the analytic solution. The features of flow characteristics are analysed by plotting graphs. The significance of the present model over the existing models has been pointed out by comparing the results with other theories. The paper has been organized as follows. In section 2, the problem is first modeled and the non-dimensional governing equations are formulated. Section 3 includes the solutions of the problem.Numerical results and discussion are presented in section 4. The conclusions have been summarized in section 5.

1. Mathematical Formulation of the Problem

Consider the flow of an electrically conducting incompressible Maxwell fluid through a porous channel of uniform thickness in presence of a constant transverse magnetic field B0 (see Fig. 1).

Fig. 1. Schematic diagram of the physical model

The induced magnetic field is assumed negligible for small magnetic Reynolds number. The walls of the channel are assumed to be flexible and are taken as a stretched membrane, on which travelling sinusoidal waves of moderate amplitude are imposed. The geometry of the channel wall is given by

y = n(x, t) = d + a sin— (X — at) (1)

A

where d is the mean half width of the channel, a is the amplitude, A is the wave length, t is the time, X is the direction of wave propagation, c is the phase speed of the wave.

The equations governing the motion of the present problem are [25]

du dv dx + dy

P

du du du

dt + UdX + V~3y

dp + dSxx + dSxy

dx dx dy

aB°u + Rx,

dv dv dv

dt + udX + vdy_

dp dS;

= - dt/ +

xy

dx

+

dS,,

yy

dy

+ R

y

pCp

dT dT dT

dt + udt + vdy

(d2T d2T\

= K{dd2 +

S du S dv S

о 9 / + Sxx ГЛ + Syy^ + Sxy

dy2 J dx dy

du, dv

dy + dx

P

(3)

(4)

(5)

where u, v are the velocities in the x and y directions respectively, p is the pressure, p is the density, p is the coefficient of viscosity of fluid, a is the electrical conductivity of the fluid, к is the thermal conductivity, Cp is the specific heat at constant pressure, Sxx, Sxy and Syy are the components of extra stress tensor S, Rx and Ry are the component of Darcy’s resistance R and T is the temperature of the fluid. For a Maxwell fluid, the extra stress tensor S satisfies the following equation:

S + Л1

dS

dt

— LS — SLT

pAi

(6)

Here Л1(> 0) is the relaxation time. The espressions for velocity gradient L and first Rivlin-Erickson tensor A1 are

L = grad V, (7)

A1 = (gradV) + (gradV )T (8)

Since we are considering the slip on the wall, therefore, the corresponding boundary conditions for the present problem can be written as

du u = TP^~ at y = ± n, dy (9)

dT T = To T at y = ± n dy (10)

where T0 is the temperature at the walls, в and 7 are the dimensional slip parameters.

The governing equation of motion of the flexible wall may be expressed as :

L*(n)= p - po (11)

where L* is an operator, which is used to represent the motion of stretched membrane with viscosity damping force, flexural rigidity of the plate etc such that

L*

dx2

d2

+ mdj2 + d

d

1 dt

+B

d4 dx4

+ H.

(12)

The continuity of stress at y = ± n and using x-momentum equation yield

d_

dx

L*(n)

dp

dx

dSxx dSxy ___^9 du,

~ + ~W - 0u + Rx - P~t a y = ± n

(13)

Here p0 is the pressure on the outside surface of the wall due to the tension in the muscles, т is the elastic tension in the membrane, m is the mass per unit area, d1 is the coefficient of viscous damping forces, B is the flexural rigidity of the plate and H is the spring stiffness. Here we assumed po = 0.

The Darcy’s resistance R in Maxwell fluid can be obtained from the equation

(l + A

d 1 dt

R

-Pv

(14)

where k is the permeability parameter, Ai is the relaxation parameter and V is the velocity of the fluid.

д'ф дф

Introducing ф such u = d—, v = —— and the following non-dimensional quantities :

dy dx

ф ху П a S d d2 p u cdP a

— ; х = cd d; у= d; П = d; £ = d; S = A ; p = c/xA ; Re = ; в x

Y; Р = d; r PvCP а к ; dSij а ; A1 -c^ A1c = ~d ; A = ct A ; M = J-B0d: x

k Td3 mcd3 E3 = d1 d3 ; E4 Bd3

d2 ; E1 = A3^c; E2 a3^ ; A2 ^ = A3xc; E5

T-To

n

-; А = У

в;

d;

Hd;

Ap,c ’

CpT(

(15)

in (3)-(5) and using (14), we finally get (after dropping bars)

8,8 , d \

SRe

i +SAi ( dt + фудх — ф ay)

д д d ai + ф дх — фх ду

1

' 2 дуф + дуф\ +

K У дх2 + ду2 ) +

д2 д2 Ф

+Sa-a- (Sxx — Syy) — M2

д д д

1 + SAi ( + фу дх — фх ду

S2 дХФ+дУФ)

x2 y2

2 2

аур — S2ш) Sxy + <16>

x y

„ (а д д\„ 2а2в д2в

ШеРг{т + ф дх — фх ду)в = SiX + ду2

+ (дф — s2?i) Sx

S(Sxx Syy) дхду +

\ ду2 дх2 J

Jxy

(17)

in which the components of extra stress tensor can be obtained through Eqs. (6)-(8) and are given by

Sxx + A1

Syy + A1

/ 8 д d \

S ( ду + фу дх фx ду) Sxx 2(SSxxфxy + Sxy фуу )

' / a 8 d \

S f дЦ + фу дх фx д j Syy + 2(S Sxy фxx + SSyy фxy )

2SфXy 7

= — 2Sф

xy

Sxy + A1

0 0 0

S ( дЦ + Фу дх — ^ ду ) Sxy + (S Sxx^x — Syy Фуу )

фуу S 'dxxi

(18)

(19)

(20)

where Re is tha Reynolds number, S is the wave number, M is the Hartman number, Pr is the Prandtl number, Ec is the Eckert number, Br (= EcPr )is the Brinkman number, E1, E2, E3, E4 and E5 are non-dimensional elasticity parameters.

Also the boundary conditions reduce to

ф

д2ф

ду = T вдуу, at у = ± п = ± [1 + £ sin 2п(х — t)], дв

в = т Yду,аt у = ± П7

д д д

1 + SAd х + фу дх — ф ду

д3

E1 дх3 +E2 дхдЬ2

3 2

+E3

д5

тдх + 4 дх5 + 5 дх

( 8

д

1 + SAW ж + фух — фx- )

Л)

ду)

Sx

S

х

+

Sx

( 8

д

у

— M 2фу — ReS ——+ Фу^~ —

t

х

(21)

(22)

(23)

Ху ) Фу

— кфу,at у = ± п

2

c

c

п

It is worth pointing out that the present model can be further developed in future using the following geometry of the channal wall:

1. Sinusoidal wave:

2. Triangular wave:

n(x,t) = d + bX + a sin

2n

T

(X - ct).

n(x, t) = d + bX + a

8

n3

E

n= 1

(-1)n-1 2n- 1

cos

2n(2n

Л

-(X - ct)} .

3. Square wave:

n(x,t)

d + bX + a

4

n

E

n=1

(-i)n-1

2n — 1

sin

2n(2n

Л

-(X - ct)} .

2. Solution of the Problem

As the Eqs. (16)-(23) are highly non-linear differential equations,analytical solutions valid for all arbitrary parameters involving in these equations, seems to be impossible to find.We have used perturbation method in this section to find analytical solution. For perturbation solution , we expand flow quantities in a power series of S as follows

/ = О + xy + S2/ 2 + ....,

в = во + se 1 + S2e 2 + ,

Sxx — S0xx + SS1xx + S2S2xx +

Sxy = S0xy + SS1xy + S2S2xy +

„ Syy = S0yy + SS1yy + S2S2yy +

(24)

If we substitute (24) into (16)-(23) and seperate the terms of different orders in S, we obtain the following system of partial differential equations for stream function and temperature together with boundary conditions

2.1. Zeroth Order System

d2So 2 dy2 N ^°yy = 0 (25)

d2во + B / S 0 dy2 + /0yy S0xy (26)

S0xx 2Л1 ^0yy S0xy (27)

S0xy Л1 ^0yy S0yy = ^0yy ? (28)

S0yy (29)

d/0 Rd2/0 , О = T & * 2 ,at У = ± П dy dy2 (30)

дв0 в0 = T l^~,at y = ± n, dy (31)

F d3 + F д 3 + F _0!_ F dx3 + F dxdt2 + F dtdx

1

+ F d + F д + F4 WT + Fw

dx5 dx

dS°- N2 фоу ,at у = ±n ду

where N2 = M2 + — K

(32)

2.2. First Order System

/д d . д \ . , д , д

R‘[at + ф0уdx -ф0-d;) ф0уу = 4dt + фоуdx -Фо"ду

д2 д2 Si

+dxdy(Soxx - Soyy) - ^ф1уу + dy2

д 2S0xy ду2

- M2фоуу

(33)

1 Ху

д2ел

д

ду2 ~ RePr ( дt + ф0у д{( ф0х дУ )90 Вг [ф0ууS1 ху + ф1ууS0ху + ф0ху (S0xx S0уу )]? (34)

д

д

: ду'

S i xx + Ai

д д д

dt + ф0у д^ ф0х ду J S0xx 2(^0ххф0ху + S0ху ф1уу + S1 ху ф0уу )

S1 уу + А1

д

д

д

dt + ф0у дx ф0х ду J S0yy + 2^0уу ф0ху

^1ху + А1

д д д

dt + ф0у dx — ф0х ду ) ^ху — Соууф1уу + ^ууф0уу)

- 2ф0ху >

= ф1уу

= 2фоху, (35) (36)

хг = те

ду

д2 ф1 ду2

, at у = ±п,

а д^1

91 = Т7"ду, at у = ±п,

. , д д д \

А4 dt + фоу ax - ф0х ау)

дSоxy

д + ^ - N 2ф

dx ду

1у

д3 F2 д3 + F3

dx3 + dxdt2

+ фоу д 0 1 д ^ ["

dx дуУ

at у = ±n

д2 + F _д^ + F _д_

дtдx 4 dx5 5 dx

П =

A1

dS,

оху

ду

- (A1M2 + йв)фоу

(37)

(38)

(39)

(40)

П

2.3. Zeroth Order Solution

The solutions of Equs.(25),(26), satisfying conditions(30)-(32) straight forward can be written as

( sinh Ny

фо = L

Br L2

-у

8(cosh Ny + eN sinh Ny)2

{ N (cosh Ny + eN sinh Ny)

{(cosh 2Ny + 2yN sinh 2Ny - cosh 2Ny) +

+2N2 (у2 - n2 - 2yn)}

where L =

8en3

^ sin 2n(x - t) - ^F1 + F2 - 4n2F4 - 4^2^

cos 2n(x -1)

The non-dimensional heat transfer coefficient at the wall is given by

BrL yx /олг2

Zo = Пх^Оу (n) =

4(cosh Ny + eN sinh Ny)2

(2N2y - N sinh 2Ny)

(41)

(42)

(43)

9

о

2.4. First Order Solution

Invoking the value of ф0 in Equs.(35)-(37), the solution of Equs.(33), (34)satisfying conditions (38)-(40) can be written as

Ф1 = A3y + A4 sinh Ny + A5y cosh Ny + A6y2 sinh Ny + A7 sinh 2Ny

(44)

9\ = L12(y2 — y2 — 2y7) + L13(y4 — y4 — 4y37) + L14(cosh Ny — cosh Ny — Ny sinh Ny)+ +L15(cosh 2Ny—cosh 2Ny — 2N7 sinh 2Ny) + L16(cosh 3Ny —cosh 3Ny — 3Ny sinh 3Ny)—

2

— ^ (L6L7 +4L4)(y sinh Ny—y sinh Ny—7 sinh Ny — N77 cosh Ny) + L17(y sinh2Ny— (45) —y sinh 2Ny — 7 sinh 2Ny — 2N 7y cosh 2Ny) + 2L4 (y 2cosh Ny — y2cosh Ny — 2y7 coshN y—

— Ny72 sinh Ny) + 4-Nl (y2 cosh2Ny — y2 cosh2Ny — 2y7 cosh2Ny — 2N7y2 sinh2Ny)

The non-dimensional heat transfer coefficient at the wall is given by

Zi = #0x(y) + Vxdiy (y) =

r [(cosh Ny + pN sinh Ny) {(sinh 2Ny + 27N cosh2Ny—

Br L2Nyx

4(cosh Ny + eN sinh Ny)3

— 2N (7 + y)} — 2N7(sinh Ny + ^N cosh Ny)(sinh2Ny — 2Ny)] +

+yx[2yL12 + 4y3L13 + NL14 sinh Ny + 2NL15 sinh2Ny + 3NL16 sinh3Ny—

2

— N (L6L7 + 4L4)(sinh Ny + Ny cosh Ny) + 2L4(2y7 cosh Ny + Ny2 sinh Ny)+

+ Co (2y7cosh2Ny + 2Ny2 sinh2Ny) + L17(sinh2Ny + 2Ny cosh2Ny)] 4N2

(46)

where A1 =

A2 =

N

(cosh Ny + eN sinh Ny)2 LLxN2

[(Lt — LLx)(cosh Ny + ^N sinh Ny)+

+LN (sinh Ny + ^N cosh Ny)(Lyx — yt)],

cosh Ny + eN sinh Ny ’

A3 = ът2 tx [{(1 — Де)К + A1} (NA1 — A2) cosh Ny + A2Ny {(1 — Rey)K + A1 y} sinh Ny] +

N 2 К +LLx — Lt +

A4 =

L

(cosh Ny + eN sinh Ny)3 1

{Lx (cosh Ny + eN sinh Ny) —

—LNyx(sinh Ny + ^Ncosh Ny)} ,

A5

N (cosh Ny + eN sinh Ny) 1

{A3 + (A5 + A6Ny2) cosh Ny + A2Ny + 2A6(y + в)+

+2A5^N + A6N2y2^} , A1LLx

{(ДеК — A1)(A1N — A2) + N2KA1(2NA1 — A2)} —

4N 4K

A6 = tnA|k {ДеК + (N2К — 1)A1} A1L

cosh Ny + eN sinh Ny

4N 3K

A7 =

L1

3N(cosh Ny+eNsinh Ny)3

4(cosh Ny+eN sinh Ny)3 L2 = cosh 2Ny + 27N sinh 2Ny — 2N 2y(y + 27)

Pr ReBr L2Nyt

{Lx(cosh Ny+eNsinh Ny) — LNyx(sinh Ny+^Ncosh Ny)} , {Lt(cosh Ny+eNsinh Ny) —Lyt(sinh Ny + ^N cosh Ny)} ,

L3 =

4(cosh Ny + eN sinh Ny)2

{sinh2Ny + 27Ncosh2Ny — 2N(y + 7)} ,

Pr ReBrL

L4

L5

Le

L7

Le

L9 = L10 L11 L12 L13 L14

L15

L1e

L17

Pr ReBr L2

4(cosh Nn + pN sinh Nn)4 Pr ReBr L3Nnx

4 cosh Nn + pN sinh Nn)3 Pr Re

N (cosh Nn+pN sinh Nn)2

Pr ReBr L2N

{Lx(cosh Nn + в sinh Nn) — Lnx(sinh Nn + eN cosh Nn)} , {sinh2Nn + 2yNcosh2Nn — 2N(n + 7)} ,

{Lx(cosh Nn+eN sinh Nn) — Lnx(sinh Nn+eN cosh Nn)} ,

(A4 N2 + 2NA5 + 2Ae), (NA5 +4Ae),

4(cosh Nn + eN sinh Nn)2 ’

Br LN

2(cosh Nn + eN sinh Nn)

Br LN2

2(cosh Nn + eN sinh Nn)

Br LN 3Ae

2(cosh Nn + eN sinh Nn) ’

2Br LN 3A7

2(cosh Nn + eN sinh Nn) ’

2 {L3 — Lg + L1L2 — (L5 + L2L4)(cosh Nn + eN sinh Nn)} ,

C {2LxL7 N — L10 + 2N2L1 — 2L4N2 (cosh Nn + eN sinh Nn)} 1

(L5 + 23L4 + L2L4 + 7LeL7 — L1I),

1

2N2

1

16N 6 1

18N2 1

[4N4 {L4(cosh Nn+eN sinh Nn) — L1 +L8}+4N3(LxL7 — L9) + (4N2 — 1)L10],

(2L11 + L6L7 — L4),

4N 3

{N (L9 — LxL7) — 2L1O}

3. Numerical Results and Discussion

This section aims to analyze the behaviours of the streamlines, temperature and heat transfer coefficient graphically for embedded flow parameters in the present problem.

An interesting phenomenon of peristaltic motion in the wave frame is trapping which is basically the formation of an internally circulating bolus of fluid by closed streamlines. This trapped bolus is pushed ahead with the peristaltic wave. The trapping phenomena for different values of M, K, e, E1, E2, E3, E4 and E5 are shown in Fig. 2. It is observed from Figs. 2a, 2b that the trapped bolus which are moving as whole decreases in size with the increase in M. The effect of porosity parameter K on the trapping is illustrated in Figs. 2b, 2c and observed that the size of trapped bolus gradually increases with increasing K. Figs. 2b, 2d depict that the size of trapped bolus increases with increasing e. The variation of compliant wall parameters is studied in Figs. 2e-2i. It is observed that the volume of the trapped bolus increases with increasing E1 and E2 but the effect is reverse for E3, E4 and E5.

The effect of various parameters, say K, M, Br, 7 and e on temperature are illustrated in Figs. 3-8.We observed that the temperature increases with increase of 7, Br and K while it decreases with increasing M and e. Further it can be noted that the temperature at the upper wall is minimum and it increases slowly towardes the middle portion of the channel. Fig. 8 is made to see the variation of the temperature for various values of compliant wall parameters E1, E2, E3, E4 and E5. It is observed that the temperature increases with an increase of E3 and E4 while it decreases with increasing E1, E2 and E5.

Variations of the heat transfer coefficient at the wall have been presented in Figs. 9-13 for various values of K, M, Br, 7 and e with fixed values of other parameters. One can observe

Fig. 2a-2d. Streamlines for different values of M, K, в: (a)M=2, K=0.03, в=0.2, (b)M=1, K=0.03, в=0.2, (c)M=1, K=0.05, в=0.2, (d)M=1, K=0.03, в=0.1

Fig. 2e-2h. Streamlines for different values of E1,E2,E3,E4, E5: (e)Ei =0.9, E2=0.5, Ез=0.2, E4=0.02, E5=0.5, (f)Ei=0.7, E2=0.7, Es=0.2, E4=0.02, E5=0.5, (g)Ei=0.7, E2=0.5, Es=0.1, E4=0.02, E5=0.5, (h)Ei=0.7, E2=0.5, Es=0.2, E4=0.01, E5=0.5

Fig. 2i. Streamlines for different values of E1, E2,E3,E4, E5: (i)E1=0.7, E2=0.5, E3=0.2, E4=0.02, E5=0.3

that the absolute value of heat transfer coefficient decreases with increase of M and fi. However it increases with increasing 7, Br and K.

Fig. 3. Variation of temperature with y for different values of K

Fig. 5. Variation of temperature with y for different values of Br

Fig. 7. Variation of temperature with y for different values of 7

Fig. 4. Variation of temperature with y for different values of M

Fig. 6. Variation of temperature with y for different values of в

Fig. 8. Variation of temperature with y for different values of Еь E2, E3, E4 and E5

Fig. 9. Effect of K on heat transfer coefficient

Fig. 10. Effect of M on heat transfer coefficient

Fig. 11. Effect of Br on heat transfer coefficient

Fig. 12. Effect of в on heat transfer coefficient

Fig. 13. Effect of y on heat transfer coefficient

Conclusions

In this work, the combined effects of slip conditions and heat transfer on MHD peristaltic flow of a Maxwell fluid in a porous channel with influence of wall properties are studied. The closed

form analytical solutions of the problem are obtained using perturbation method. The results are discussed through graphs and concluded the following observations :

(i) The volume of the trapped bolus decreases by increasing both M, Ei and E2. Moreover the effect is reverse for K, в, E3, E4 and E5.

(ii) The temperature field decreases with increase in both M and в while with increase in 7, Br and K the temperature field increases.

(iii) The absolute value of heat transfer coefficient decreases with increasing M and в but it increases with the increasing 7, Br and K in the vicinity of the upper wall.

(iv) The analytical resultes obtained in this work are more generalised form of Hayat et al. [25] and can be taken as a limiting case by taking в ^ 0 and 7 ^ 0.

Acknowledgement

I gratefully acknowledge the referees for their constructive comments which improved the quality of the paper.

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Совместное влияние условий проскальзывания и свойств стенки на волнообразное МГД-течение максвелловской жидкости с учетом теплопереноса

Калидас Дас

Было изучено влияние условий проскальзывания и теплопереноса на волнообразное магнитогидродинамическое (МГД) течение максвелловской жидкости в пористом плоском канале с упругими стенками. Математическая формулировка задачи основана на модифицированном уравнении Дарси. Аналитическое решение было получено для функции тока и температуры в предположении малых волновых чисел. Полученные результаты представлены в численной и графической форме.

Ключевые слова: волнообразование, максвелловская жидкость, модифицированный закон Дарси, число Бринкмана, число Кнудсена, коэффициент теплопроводности.