The contact problem of two conducting plano-convex solids having different conductivities is considered assuming that steady state heat conducting takes places. The problem is formulated sos as to involved a pair of dual integral equations having Legendre functions with complex index. These equations are reduced to a single integral equation which is then solved iteratively. Lastly, the quantities of physical interest are found out

Machwe, V. G.

Singh, B. M.

English

Defence Science Journal

ON THE MTXED BOUNDARY VALUE PROBLEM OF STEADY STATE REAT CONDUCTION IN A DOMAIN FORMED BY CEMENTING TWO PLANO-CONVEX SOLIDS Samrat Ashok Teohnologioal Institute, Vidisha (M.P.) (Received 17 September 1971; revised 29 January 1972) The oontaot problem of two oonducting plano-convex solids having different conduotivities is considered assuming that steady state heat aonduotion takes plaoe. The problem is formulated so as to involve apair of dualintegral equa- tions having Legendre funotions with complex index. These equations are reduced to a single integral equation whioh is then solved iteratively. Lastly, the quantities of physicalinterest are found out. In reoent years several papers have been published on the dual integral equations. These are important while solving the boundary value problems of Mathematical Physics with mixed boundary conditions. &jority of them have .been considered through the Hankel transformation whose kernels are expressible in aylindrical functions. Also the dual integral equations with kernels expressible in Legendre functions with complex index have recently been investigatedlW4. These equations belong to the class connected with Mehler-Fock integral transformation and are of considerable interest in various problems of Mathematical Physics. References of mixed boundary value and boundary value problems of heat conduction are available=-'. In the prasent paper we have reduced our problem into simultaneous dual integral equations having Legendre functions with complex index, and then they are reduced to Fredholm integral equation of second kind. Finally it is solved iter4tively. To solve the problem we introduce a system of toroidal coordinates (cc, p) related to cylindrical coordinates ( r ,  z )  by the expressions o sinh a ' p= -  cosh a + cos /3 O < a < c o , O <  /3<2r .  a sin 19 Z = cosh a $ 00s /3 t The temperature distributions in two plmo-convex 4: 0 solids ABCOA & ADCOA is here considered. The total region ABCDA is formed by two intersecting spheres I' as shown in the Fig. 1. EC and FA portions of the , cemented surfaces of the solids are perfectly inhlated I' and A and C are rigidly conneoted. In the upper solid r the temperature function is prescribed along ABC. Hence we have the boundary conditions : Fig. 1 -Region ABCDA formed by twointereeoting spheres - Nl = 0, aP = 0, a. < a-< co. (2) (the insulation takes place along the sh&d lines). - Dm. SOI. J., VOL. 23, JANUWY 1973 The cemented portion denoted bet&n EF is perfectly conducting. Since we shall assume ADC of the lower solid. is a sink, at the surface of separation of the two medk we have the dary conditions. ul = U2, 8 = 0 ,  O < a < a o ,  where K1 and K2 are the conductivities of the upper and lower solids rmpedively. As already mentioned the cemented surfso? has EC and FA part insulated and on the lower surfm ADC tempemtdre is taken to be zero. On these lines if we take t5l = - on the lower surface, we can write : where u, and u, are the solutions of Laplace's eqclation < .  < w v2 u1 = 0, , (a) ' (7) V 2 u - 0 .  (b) - R E D U C T I O N  T O  I N T E G R A L  E Q U A T I O N S  For the upper plaao-convex solid pre assume the solution of the Laplace's equation in the form : I 0 . " 0<p<.p1, o < u < o o .  - ,  (8) - * - ' This form satisfies the condition (1). &so A (T), is unknown constant. For the lower solid a suitable tempam- ture function is Here B is unknown constant. This form satisfies the conditiow (6). In satisfying the boundarJr con& tion8 (3), (4), (2) & (5) the following equations arB obtained : B (T) tanh t512 T - A (7) tanh & T tanhn T p- + iT (co~h a )  d ~ =  I v1(.) l / l+cosha  ' 0 i O <  a < a o ,  (10) a,, < a. Y - where ,y= - KI k K2 7 S O M E  U S E F U L  R - E S U L T S  As i -(' We give below some results1, ,whioh we shall now mike use of. c o s ~ t p - ~ + ~ ~  (c03ha)dT= 2(cosha--cosht) f" *- * [ I" H (a  - t), - + .(la) 0 J' \ p- t+ i7  (cosha)~&l lhnrss i41 . sd~'=  *2(cosh8-coda) --I H (8 - a). (16) 0 . He& H (t) is Heavy-side unit function. I" , h e  MehleGFook transform1 is given by ' GO - f ( a ) = I s ( T ) P - t + i T  ( w s h a ) d r ,  . (16) 0 \ a thw , g(7) = T ~ ~ . T T J  J (a)  k i t  i7  (oosh a )da ,  (17) 0 and henoe the following relation can easily be derived : p- + + (co?h a )  sinh a d a C O S T S  = --- 1 / 2  as (~03h s- cosh a)& (18) 0 S O L U T I O N  O F  S I M U L T A N E O U S  D U A L  I N T E G R A L  E Q U A T I O N S  I N V O L V I N G  L E G E N D R E  F U N C T I O N S O F  I M A G I N A R Y  A R G U M E N T  We shall now solve the equations (10) to (13). Let us assume ~ A (T) = qb (t) COB T t dt, -- i" \ - (19) 0 where+(t)is unknown. Equation(l9)oanbe written in the form after integrating it by parts. , ( j - A (7) = + ($1 sh "0 - L J +f (t) t dt . T T (20) 0 DEE, SOI, ;f., V O ~ .  23, JANOARY 19113 , With the help of (20) and then (15) it can be shown that (12) is satisfied identically for a which has a oontinuous derivative. We also have i T A (T) tsnh a T p- + ir ( C O S ~  a) d~ = d (a01 (61 (t) dt d 2  (cosh'u,, - cosh a) t - wsh u ' 0 a : 0 < Ea-c< ag, (21) Now from (11) I T B (7) tanh n r p- + (~osh a) dr = . a d (4 d 2  (~osh Q - C O S ~  a) 0 i8 I O < a < ~ .  (22) HaKing use of (16) & (17) weaan get easrly from (13) & (22) -i a0 a, sinh a P-g + i7 (GO& a) du - &J p- g + (wsh u) sin cosh a, - cash a 0 0 I , l/oosh t - cosha * (23) a I - .* . * On intarchanging the order of integrations*in the sebbid iriZegwl of (23) and then integrating by parts and finally using(l8), we get : I (24) 0 Equation (10) can be written in the form , (cosh a )  d T f 0 $XWGH $ ab~aHW3Z : Hi& Boundary Value Problem of Beat t7onduotio~ If we substitute the, m1ues of 3 (~) l& A "(P) from (24) & (19) in'(26), then using (14) we find that a a, m A - 2 / Z  Vl(4 . . d l + w s h a  6 0 0 ~ 0 s T t P - & + i ~  ' W S  # l d T  + odj$( t )ddl .  co%h & T O O B ~ ~ O  0 Equntion (26) is AbhI type. H~noe t&e soldtion is obtained by using (18) : t sinh a vl (a) d a - 2 4 (t) = . ' 4 1  + msh LX dco4.1 t - cash a n (u - 1) Q - "" a. 3' t eos-T u d T + msh a T cosh n T 0 0 I a0 a0 cosh (7r - A) T ,JT (a - 1) + "/ ' ( u ) ~ u ~  C O S ~ / ~ ~ T C O S ~ W T  C03 t ws T U ~ T ,  0 a . . O < t < a , , .  (27) If 1 vl (a) = T2 , (constant), then (27) om be written in the form sech ($) 4 (0 = (o - I) f (a - 1) 37 f P (4 [ (u, ti + 8; (u $ 0 O < d < a o .  (28) where q ( u , t ) =  - f . m s h ( 7 r - a ) ~  c m ~ t a o s ~ a d ~ ,  cosh T fll  COP^ n T 0 and I (29) cosh (m - p2) T cosh T p2 cosh 7r T COS T t COS 7 u d T , (b) 0 1 1 \ 1 4 1  . /  DEB. SCI. J.,  VOL. %, JAX~JARY 1973 Equation (28) is Fredholm integral equation of second kind having kernel K ,  (u, t) $ K ,  (u,  t). Equation (28) is a standard equation. ~ < - S O L U T I O N  O F  B R E D H O L I  X N T E G B A L  E Q U A T I O N %  Equation (28) can be solved for any suitable parti~ular value of ISl and Here ,we ahall get the itehtive solution of the Fredholm integral equation and obtain the solution of (28) as a power series in a, provided that a, is sufliciently small. fi 37 n' If p, = - and ., = f 2 then the domain ABCDA represents a sphere. Equation (28) reduce8 to i sech (i) a, $ (u) cosb - ZL c ~ s h  - t du 2, - 2  9 ( t )  = n'(a - 1) n' +  k h  u + oosh t B 0 O < t  < a , .  (30) If we take - t := r a,, u = x a,, . 9 (7 a,) = Y (r go) = (r)! . say. - , - then (30) takes the form r a  LEU r a0 s e ~ h  --j- 1 .  cssh .O oosh 22 2 - Y (9) = - + 2%. j Y ( x )  n ( a  - 1) n' cosh r a, + oosh x a, ax, o O < r < l .  (31) I f  a, is very small such that a, < < 1, then we can represent 7 a0 x a $o>h - cosh O- 2 2 1 5 a04 = - - aa? ($2 + r2) + -4! 32 ( x 4 + r 4 + 6 x g r 2 )  + O  (a$)+ ..... (32) cosh rqa, + msh 3 a, 2 16 and 1 - 6 A ~ T )  = $ Y ( r )  cos T r aO dr ;, - 0 (36) Henoe ' ' f . 1 C a0 A(T) = r ( u - l i  + + ( u - ] )  + ~ [ _ 4 ( & - 1 \ -  \ 1  7 2  - - 1 ---i 2 4 r ( u - - 1 )  6 n ( u - I )  + P(o--l)  ' 1 - -j- .......... 8(<u - 1 ) ra ' (37) I 7 ,  S O M E  A P P R O X I M A T E  R E S U L T S  - - P. We shall get the total quaatit; of heat passing per second through the circle 'of radius Q, this I circle situated at the surface of separation of two media from upper solid to lower solid. This quantity of s equal to - 4 8 - - ! f a '  Q1 = - 2n kl uo . - - - F 0 - > -  - 1\ - . . 4 3 .  ? +. - = z d a n h a ,  J' oos l~z  L ~ ~ A ( T ) ~  - ( c3sh.u \ tsnh V T  d~ . 1. ,(B) - *+c  \ 0 0 7 SINGE & MAOHWE : Mixed Boundary Value Problem of Beat Conduotion We shall now get the approximate results for temperature fanotions. Por this purpose, we make use 1 7t oQ(31). Rot  tBe sdid portion A B C Q, if we take vl (3 = and = , equation (81 aan la written 6- as follows : 1 u1 = --=- + 4 C Q B ~  a + oos [ ((a- + (D - jll d2 e62 I1 ) + 1 1 where P )  T P ( w s h a )  t a n h w d ~ ,  " - & + i ~  (42) 0 wsh - T 2 and P ( oosh a ) tanh VT d s  . " --&+ i7 (43) ~ o s h  - T 0 2 \,(42) and (43) are oonverged infinite integrals. Similarly we oan write the expassion for temperaturefunotion assigned to the lower solid portion AOCDA as I 212 = u dmsh a -Pms j? aoS1 ao2 81 [ ( - 1 ) "  + (u-1)"' where sl = f sinh 1 + B (' ) tanh TT p ( oosh a ) d ~ ,  -g+ iv (45) o oosh - T 2 snd tanh n~ p (oosh a ) d ~  " -$+ i 7  (46) o cosh- T 2 Here (46) and (46) are again oonvergent infinite integr&1s. With the help oF.8 the values of these integrals oan be f o d  out. 0 

On The Mixed Boundary Value Problem of Steady State Heat Conduction In A Domain Formed By Cementing Two Plano-Convex Solids

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On The Mixed Boundary Value Problem of Steady State Heat Conduction In A Domain Formed By Cementing Two Plano-Convex Solids

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