In the theory of heat conduction, the flux across a

Chapter 0, Problem 10.1

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In the theory of heat conduction, the flux across a surface within a solid body al a point on !hat surt'ace is the quantity of heat nowing in a speci lied direction nonnal 10 the surface per unit lime per unit area al the point. Flux is. therefore. measured in such units as calorics per second per square centimeter. It is denoted here by . and it varies with the normal derivative of the temperature T at the point on the surface: ( I ) dT =-KdN (K > 0). Relation (I) is known as Fourier~11 law and the constam K is called Che thennal collductivity of the material of the solid. \vhich is assumed 10 be homogeneous.~ 'll1e law is named for the Fn.:1x:h lll0>. His hook. c:iled in Appc1xJix I. is a dassic: in the theory of heat rnndul"lion. 365 366 APPLICATIO!'S OF Cm~r:OR!\IAL MAPPli\G CllAP. 10 The points in the solid can be assigned rectangular coordinates in thrccdimcnsional space. and we restrict our attention to those cases in which the temperature T varies with only the .r and y coordinates. Since T does not vary v ... ith the coordinate along the axis perpendicular to the .ry plane. the How of heat is. then. twodimensional and parallel to that plane. We agree. moreover. that the How is in a steady state: that is. T does not vary with time. It is assumed that no rhennal energy is created or destroyed within the solid. Thal is. no heat sources or sinks arc present there. Also. the temperature function T(.r. y) and its panial derivatives of the lirst and second order arc continuous at each point interior to the solid. This statement and expression (I) for the nux of heat arc postulates in the mathematical theory of heat conduction. postulates that also apply at points within a solid containing a continuous distribution of sources or sinks. Consider now an clement of volume that is i ntcrior to the sol id and has the shape of a rectangular prism of unit height perpendicular to the .ry plane. with base ~.r by ~Y in the plane (Fig. I 54 ). The time mtc of now of heat tow