ACTA MECHANICA SINICA, Vol.9, No.1 February 1993 Science Press, Beijing, China Allerton Press, INC., New York, U.S.A.

ISSN 0567-7718

GENERAL EXACT SOLUTION OF INCOMPRESSIBLE POTENTIAL FLOWS AROUND TWO CIRCLES

Wang Qianxi ( ~ 1 ~ ) Zhuang Lixian ( ~ [ , ~ ) Tong Binggang ( ~ )

(University of Science and Technology of China, Hefei, Anhui 230026, China)

ABSTRACT: Three exact solutions are obtained for 2-D incompressible potential flows around two moving circles in three cases: (i) expansion (or contraction) of themselves, (ii) approaching (or departing from) each other, (iii) moving perpendicularly to the line connecting the centres in opposite directions. Meanwhile, an- other set of two exact solutions is obtained for 2-D incompressible potential flows between two moving eccen- tric circles in two cases: moving parallelly or perpendicularly to the line connecting the centres.

KEY WORDS: incompressible potential flow, unsteady flow, conformal mapping

I. I N T R O D U C T I O N

According to the superposition principle , the incompressible potential flow around two circles moving and expanding in a fluid can be divided into four elementary parts induced respectively by two circles fixed in a uniform s t ream, expanding (or contract ing) , approaching (or departing f rom) each o ther , moving perpendicularly to the line connecting the centres i n opposite direct ions, as sketched in Fig. 1. Here the potential flow induced by two circles means the 2-D potential flow induced b y two parallel circular cylinders of infinite length .

~V_ I.____ U -

~ V c

~ W t �9

(a) (b) (c) (d)

Fig. 1 The four cases of two circles moving or expanding in a fluid : (a) fixed in a uniform stream, (b) expansion of themselves, (c) approaching (or departing from) each other, (d moving perpendicularly to the line connecting the centres in opposite directions

The incompressible potential flow between two moving eccentric circles can be divided into two elementary parts induced by the movements parallel and perpendicular to the line connecting the centres in opposite direct ions, as sketched in F ig .2 .

Lagally (1929) obtained the exact solution for the steady potential flow of two circles fixed in a uniform stream TM. The other problems mentioned above are unsteady ones and remain unsolved up to n o w . In

@@ (a) (b)

this paper , the region outside (or between ) the two Fig.2 The movements of two circles parallel(a) circles is mapped into an annulus. Then all the and perpendicular (b) to the line connecting remaining problems are solved analytically in the the centers in opposite directions

Received 7 December 1991