HEAT 587 MATHEMATICAL APPENDIX. Let v be the temperature at any point P specified by |, 77, accord ing to any system of three sets of plane or curved orthogonal surfaces used for co-ordinates. Let d, pdri, Wbe the lengths of the edges of the infinitesimal rectangular parallelepiped having P for its centre, and its sides parts of the six surfaces - d%, | + i^|, The rates of variation of temperature per unit of length in the directions corresponding to the variations of |, rj, are respectively 1 dv 1 dv 1 dv ~~ dj ~~jl chj ~ 7 dC Hence the fluxes across three infinitesimal rectangles having their edges parallel to the three pairs of sides of the parallelepiped, and each having its centre at P, are respectively k dv , I*-,!. k Hence the excess of the quantities of heat conducted in to the parallelepiped above those conducted out across the three pairs of faces is - A ) + /*&* T) fj. dti) d v dv J ) ( dl- The effect of this gain of heat is to warm the matter of the parallelepiped at a rate per unit of time equal to the rate of gain of heat divided by e^vd^drid^, the thermal capacity of the matter. Hence dv 1 I d uv dv dv dv d ,
dv ) ..
" This, for the case of the uniform motion of heat (dvjdt = 0), was first given by Lame, to whom the generalized system of curvilinear coordinates for a point is due ("Memoire sur les Lois de 1 JJquilibre des Fluides Etheres," Journal de I ficole Poly technique, vol. iii., cahier xxiii. ). He deduced it from Fourier s equation [(3) below] in terms of plane rectangular coordinates by a laborious transformation. Equation (2) was first given, proved as above, as the direct expression of Fourier s fundamental law of conduction, by W. Thomson (Cam bridge Mathematical Journal, Nov. 1843). For plane rectangular coordinates we have =ju v=l, and if we put x, y, z for |, 77, in this case, (2) becomes dt c ( dx dx J dy dyj dz dz which is Fourier s celebrated fundamental equation. From it we may deduce by transformation the proper forms of the corresponding equation for polar coordinates ; but they are more easily got direct from the equation (2) for generalized coordinates. Thus for ordinary polar coordinates r, 6, </> we have, if we take these for , 77, respectively, A = l, fi = r, v = r sin0 . Hence (2) becomes dt cr 2 (dr dr) siu0 dO dd / sin 2 d^ If k be constant, and we put &/C=K, this becomes dv K ( d ( dv 1 dl . dv 1 T- ~r i -H r 7- + - -H sinfl H at r 2 ( dr dr J sm(? d0 dO J sm a ( or du d*u 1 df. du j_ i sin ft dr* sine d8 d6 dt~" ( dr* sine de-"""de J " sin% d& i ) (6). where u = If again we take for the coordinates r, $, z (polar coordinates in the plane perpendicular to z being denoted by r, <p), we have A = l, H = r, j/ = l, and so find dt c ( rdr dr J rd<^ For the case of k constant we may take it outside the brackets in each of these equations, as we have already done in (4) ; thus (2) becomes (8); or, with K for k/c, the diffusivity ( 81, 82), dv __ f d^v d?v d?v dt dx^ dy^ L It is this restricted form which, with the further restriction that c be constant, is most generally recognized as Fourier s equation of conduction, and it is for it, with these restrictions, that his brilliant solutions were given. These solutions are available for practical use by limiting the range of temperature within which any one solution is continuously applied to a range of temperature within which the values of& and c are each nearly enough constant. "We may expect or 20 C. on each side of the mean temperature to be practically not too wide a range for any case, judging from copper and iron ( 80), the only substances for which hitherto we have any infor mation as to variations of both k and c with temperature. Each of the following expressions I XVII. for v satisfies Fou (9) or its equivalent (6), as the reader will readily verify for himself, sola The special condition corresponding to the peculiar character of " the particular solution is specially noted in each case. I. Instantaneous simple point-source ; a quantity Q of heat sud denly generated at the point (0, 0, 0) at time < = 0, and left to diffuse through an infinite homogeneous solid. EVERY OTHER SOLUTION is OBTAINABLE FROM THIS BY SUMMATION. where . . (10). Verify that f f f vdxdydz = 4ir I vr*drQ; and that
- unless also x=0, y = Q, 2 = 0.
) = when Remark that (11). = when < = .... II. Constant simple point-source, rate q : [-vC*-S^]-&. (12) - The formula within the brackets shows how this obvious solution is derivable from (11). III. Continued point-source ; rate per unit of time, at time t, an arbitrary function, f(t) : K)~^ ( 13 ) IV. Time-periodic simple point-source, rate per unit of time at time t, q sin 2nt : Verify that v satisfies (6); also that - 47rr 2 = g sin2?i< where ?- = 0. Cl/T V. Instantaneous spherical surface source ; a quantity Q suddenly generated over a spherical surface of radius a, and left to diffuse outwards and inwards : r=i Qe-(>-) /4c<- e -(f+a)W ^ _ (15) _ To prove this most easily, verify that it satisfies (6) ; and farther verify that 47T /"T 2 dr = Q; Jo and that v=Q when t = 0, unless also r=a. Remark that (15) becomes identical with (10) when a = ; remark farther that (15) is obtainable from (10) by integration over the spherical surface. VI. Constant spherical surface source ; rate per unit of time from the whole surface, q : (16).
- [ = q l* dt ~ ~ a}<l ^~fT 4< T i
J = q/4irr , where and = qf^ira , where The formula within the brackets shows how this obvious solution is derivable from (13). VII. Time-periodic spherical surface source ; rate per unit of time, at time t, from whole surface, q sin 2nt : _ (r _a)2/4/c X _ e -(
- )- sin [2n- ( K n)WB] , where r>,
where r-*=.a where A B, C, D are constants determined by the conditions that and drj r<a when the two values of r exceed a and fall short of a by infinitely small differences. Verify that v satisfies (6). Also that v is finite when r = 0. VIII. Fourier s " Linear Motion of Heat " ; instantaneous plane source ; quantity per unit surface, a : (18). Verify that this satisfies (9) for the case of v independent of y and
z, and that
