Heat conduction in semi-infinite solid in contact with linearly increasing mass of fluid

C. C. Chao, J. H. Weiner
1956 Quarterly of Applied Mathematics  
The boundary conditions implied by conditions (t) and (it) are: For Eq. (4): (a) ue = 0, ur finite, when 0 = 0. (b) ur = ue = 0, when 6 = a. For Eq. (5): (c) us = 0, ur finite, dur/dd finite, when 6 = 0.* These yield respectively the equations: Cj cos2 a + c2 cos a + c3 = 0, 2ci + c2 = 0. These have ct = c2 = c3 = 0 as their only solution. As Squire [5] has shown, the general solution of Eq. (4) with = c2 = c3 = 0 is: where a is an arbitrary constant. Referring again to the boundary condition
more » ... (a) = ue(a) = 0, we see that there is no finite value of a{a = yields a satisfactory but trivial solution) which satisfies this boundary condition, no matter what value of a is chosen. Thus we have shown that there is no non-trivial solution of the form \p = rf(8) that is compatible with the Navier-Stokes equations and the boundary conditions (i) and (it). The author wishes to thank Prof. Garrett Birkhoff for suggesting the problem and for his helpful advice. *It is assumed that when these boundary conditions are substituted in Eqs. (4) and (5) that the limit d -> 0 is taken, since 6 = 0 is a singular point in the spherical polar coordinate system.
doi:10.1090/qam/79967 fatcat:dvl3d2gvcratrdob6m346nlvwa