Studies on the Motion of Viscous Flows— III 



If now the velocity measuring instruments are such that they cannot meas- 

 ure accurately any quantities of the order of 0.0004 u^ I\f2 or smaller, then 

 within this accuracy the boundary condition at infinity is met at a finite distance 

 of 50a. Such a distance would here be called a physically infinite distance. In 

 fact, we may assert that the flow becomes uniform at a finite distance of 50a. 

 That it is 50a we concluded after looking at Table 1 and the accuracy of the in- 

 struments. Moreover, we made use of the solution obtained by the consideration 

 of the condition at infinity. 



We shall now show that for the same physical process described by the pre- 

 vious solution there is another mathematical procedure consistent with the physi- 

 cal ideas to determine physically infinite distance. 



In this procedure we first make the assertion that there is a finite distance 

 h at and beyond which the flow can be regarded as uniform. We may express 

 this by the condition 



u (h,6') = -u^ cos + e^cos^ 



VQ(h,(9) = +u^ sinfi* + e^ six\6 (A) 



■ . ^ .:■ •- Po(i^.^) = P» + Cp . ' - --. 



where u^ and p,^ are constant magnitudes of velocity and pressure recorded by 

 the instruments and which may be regarded as known completely. The errors 

 e^, e^ in velocity and "e^ in pressure are to be regarded as unknown, but less 

 than the error bounds of the instruments, which may be regraded as known. 



Using the method of separation of variables, we obtain as a general solution 

 of the harmonic equation 



-/-(,(?, 5) = (c^ sin\5+ C2 cos>v5)(c3 r^+ c^r"-*^) + Cg^ log^ r + c^e + c, log^ ^ + Cg , 



where c^, c^,..., Cg and K are arbitrary constants. This satisfies ^^"1'^= 0. 



We may take Cg = because 41 ^ is to be determined within an arbitrary 

 constant. For periodicity in d we must have k an integer, say n and Cj, Cg 

 equal to zero. Superposition for all n gives 



V^o(r,5) - c^ log^? + J] (Cj„sinn5+C2^cosn5)(c3„?"+c^^?-") . 



n= 1 



By differentiation we get 



Gq(?,0) = - ^:- = + 2_, n (Cj^ cos n0 - c^^ sin n^)(c3^r"-'+c^^ ?-"-!) 

 r 30 n= 1 



513 



