610 



velocity components v and w at infinity. The 

 integral relation (6) now takes the form 



hdnd5 = 1, 



(28) 



The mathematical problem is now completely specified. 



5. THE METHOD OF SOLUTION 



(L - 1) en= -hn.- 



"nn 



'K 



= -Sn- 



(31) 

 (32) 



with the boundary conditions (i) and (ii) stated 

 before, which we need not repeat here. 

 The solution of Eq. (30) is 



ho = Ce 



(n^ + C^) 



and application of the integral condition (28) on 

 hg gives the value l/ir for C, so that 



The mathematical problem just formulated can be 

 solved numerically once X is known. But consider- 

 able effort is required for this solution, since 

 there are three second-order partial differential 

 equations to be solved, two of which are nonlinear. 

 It is true that computers can deal with nonlinear- 

 ities, but the domain is infinite, and some estimate 

 has to be made of how far to go in the numerical 

 computation. Furthermore the integral condition 

 (28) can only be imposed after the computations 

 are done for h, and this makes the computation very 

 cumbersome. 



For arbitrarily large values of A an analytical 

 solution is extremely difficult because the non- 

 linearities present formidable difficulties. We 

 shall attempt a power-series solution of the form 



h = hg + Ahj + A^h2 + . . . , "^ 

 C = 5g + ACj + X^E.^ + . . . , 

 4' = y„ + A*, + A^if, + . . . . 



(29) 



The success or failure of this approach depends 

 not only on the value of A, but also on the magni- 

 tudes of h,/hj,, h2/h, , etc. Thus we need to make 

 an estimate of the range of A, and we have to find 

 out how fast hjj, Su, and fjj decrease as n increases. 

 Furthermore, even the estimate of A cannot be made 

 without knowing the magnitudes of fo- It turns 

 out that a reasonable estimate of A is 



where 



1 -r^ 



— e 



n2 + c2. 



(33) 



Then the solution of Eq. (31) is 



2 -r^ 



^0 = - 37 ^^ 



2 „ -r-^ 

 ■z;— cos 6 • re 



3TT 



where 



6 = tan 



■1 ? 



Given 5q, Eq. (32) can be easily integrated by 

 separation of the variables r and 9. The result is 



H*. 



6TTr 



(1 



). 



(34) 



The isotherms given by Eq. (32) are just concen- 

 tric circles. But the streamlines given by Eq. 

 (34) are already interesting. They are shown in 

 Figure 1, which shows two very prominent vortices, 

 with the vorticity pointing in the x direction. 

 Thus the first approximation already shows the 

 prominent features of the flow pattern in any plane 

 normal to the x axis. Note that both the flow 

 pattern and the temperature field are symmetric 



30 < A < 50. 



Using Eq. (29) , we shall show in the following 

 sections that hj/hg, Cj/Sof ^nd ^i/'^n are all of 

 the order of 10~2. Thus, if A = 30, stopping at 

 the second approximation, that is, at the terms 

 with the first power in A, would introduce an error 

 of about 10?, if we assume, as we evidently can, 

 that the ratio 10"2 would apply to (hn+j)/hji etc. 

 for n equal and greater than 1. If A = 50 this 

 error would be about 25 to 30 percent, and it would 

 be necessary to go to at least the A^ terms to 

 reduce the error to less than 15?. 



We shall delay the presentation of the estimate 

 of A until later and shall proceed with the solution 

 according to the approach in Eq. (29) . In awaiting 

 the experimental determination of A , we shall carry 

 out the solution to the second approximation. 



6. THE FIRST APPROXIMATION 



The first approximation is governed by the equations 



Lhg = 0, 



(30) 



FIGURE 1. Flow pattern from the first approximation. 

 The horizontal axis is the t\ axis, the vertical axis 

 the ? axis, and the arrow indicates the direction of 

 the gravitational acceleration. The value of SittQ is 

 zero on the C axis. It increases toward the left and 

 decreases toward the right. The increments (or decre- 

 ments) are all 0.1. 



