Boundary Value Problems in Transport with Mixed or Oblique Derivative Boundary Conditions-i Formulation of Equivalent Integral Equations

When an mtemally heated body IS cooled along Its boundary by a penpherally flowmg flmd that IS contmually replemshed from an external source, a dlfferentml energy balance on the boundary leads to unfamiliar boundary condltlons Such boundary condltlons Involve mued second denuatrues with respect to spatial vmables, which under addltlonal assumptions (such as an mhmte heat transfer coefficient) lead to oblrque denvatrve boundary condltlons, I e at the boundary an obhque denvative of the temperature IS specdied ClassIcal attempts at solution of elhptlc partml differential equauons with oblique denvatlve boundary condltlons have been through the estabhshment of equtvalent stngular mtegral equations, using complex analytic continuation The theory of singular integral equations IS comphcated, however Usmg appropriate Green’s functions, the boundary value problems of Interest have been reduced to equivalent integral equations m this work Whde oblique denvatlve boundary value problems are shown to lead to smgular integral equations, the mlxed denvahve boundary value problem IS shown to yield Fredholm Integral equations directly This surpnsmg finding IS mathematIcally slgndicant, because Fredholm mtegral equations are solved more easdy, and physically slgndicant because the mlxed denvafive boundary con&on IS the more reahstlc condluon in the present context A method of solution of Fredholm integral equations 1s discussed More compbcated boundary condltlons In whrch axud conductIon in the coolant fhud IS important have also been shown to lead to Fredholm integral equations Fmally a transient problem has been formulated 1 lNTROlKJCTlON Steady state problems m the transport of energy or mass 111 stationary media lead to boundary value problems mvolvmg Laplace’s equation (wlthout volumetnc sources) or Poisson’s equation (mth volumetnc sources), which must be satisfied m some domam of Interest together with pertment boundary condrtlons The claswcal treatment of these boundary value problems[l] has employed the DYlchlet boundary condltlon, which stlpulates the value of the unknown function at the boundary, the Neumann boundary condlhon, which specties the normal denvatlve on the boundary, or the mwed (Robin) boundary condition that lays down the boundary value of a hnear combmation of the function and Its normal derivative Indeed many practrcal problems m energy or mass transfer fit neatly into the above scheme From a mathematical vlewpornt too, the boundary value problems Just mentloned possess the advantage of compnsmg a partml ddferentml operator with symmetIlc domams, which when coupled with the boon of separability of the operator, frequently yKlds the bonus of an analytical sefles solution tBy such drfferentlatlon, it must be admltted that some mformataon IS “lost” and dependmg on the other boundary condrtins. some nsk of non-umqueness of solution IS introduced The posslbdlty of non-umqueness of solution can be seen physIcally m that the Inlet temperature of the coolant does not appear in eqn (3)’ An Integral verSlOn of thus boundary condrtion. whach appears later as eqn (7),1s more approprmte m thus regard It IS, however, easy to run into sltuatlons of energy and mass transfer, where the boundary condrtlons are not of the types Just described In fact, a reahstic formulatlon of most problems of transport m statronary media whrch are bounded by a flowmg fluld, whl mvanably produce boundary conditions that are outside the above category Thus, for example, consider a slab, which IS heated by a volumetnc source term, such as by means of an electnc current or by the slab bemg of nuclear matenal The slab IS allowed to cool by a penpherally flowmg fluid as shown m Fag 1 The temperature of the slab, T(x, y) will clearly satisfy the dtierentral equation