This example is merely given as an instance of discontinuous boundary-conditions. The values of 4, can be at once found, if required, by integrating (30). 85. A very general formula for the functions 4, may be obtained as follows. It may be shewn that if a function ƒ (z) be finite, continuous, and single-valued, and have its first derivative finite, at all points of a space included between two concentric circles about the origin, its value at any point of the space can be expanded in the convergent form If the above conditions be satisfied at all points within a circle having the origin as centre, we retain only the ascending series; if at all points without such a circle, the descending series, with the addition of the constant A,, is sufficient. If the conditions be fulfilled for all points of the plane xy without exception, ƒ (2) can be no other than a constant A.. Putting f(2) = 4+i, introducing polar co-ordinates, and writing the complex constants A, B, in the forms P+iQ, R+iS, respectively, we obtain -n $ = P ̧ + Σr” (P2 cos në – Q„ sin në) + Σ° r ̄” (R2 cos nė + S„ sin n☺) += Q。 + Σïr” (Qn cos nə + P2 sin në) +Σr ̄” (S2 cos nė – R„sin nė) Š 0 This value may be expanded for each boundary in a series of sines and cosines of multiples of 0, by Fourier's theorem. The series thus found must be equivalent to those obtained from (32), whence, equating separately coefficients of sin n↔ and cos no, we obtain four systems of linear equations to determine Pn, Qn, Rn, Sn. 86. Example 6. An infinitely long circular cylinder of radius a is moving with velocity V perpendicular to its length, in an infinite mass of liquid which is at rest at infinity; to find the motion of the fluid supposing it to have been started from rest. The motion will evidently be in two dimensions. Let the origin be taken in the axis of the cylinder, and the axes of x, y in a plane perpendicular to its length. Further let the axis of a be in the direction of the velocity V. The motion having originated from rest will necessarily be irrotational, and will be singlevalued. Also, since ds, taken round the section of the cylinder гаф dn J do ds, is zero, is also single-valued (see Art. 69), so that the formulæ the problem is determinate, by Art. 71. Since the region occupied by the fluid extends to infinity we must in (32) omit the coefficients P, Q. The condition (33) then gives -n-1 V cos 0 = − Σ na ̄”-1 (R1 cos no + S„ sin n0), Va2, and all the The complete solution is therefore given by which can be satisfied only by making R other coefficients zero. These formulæ coincide with those of Art. 80 (e). As this case is one which is readily comparable with experiment, we will calculate the effect of the pressure of the fluid on the surface of the cylinder. The formula (4) of Art. 25 gives where we have omitted the term due to the external impressed forces, the effect of which can be calculated by the ordinary rules of Hydrostatics. The term in (35) expresses the rate at which аф is increasing at a fixed point of space, whereas the value of in (34) is referred to an origin which is in motion with the velocity The pressure at any point of the cylindrical surface is there The resultant pressure on a length of the cylinder is evidently parallel to ; to find its amount per unit length we must multiply (36) by ade. cos and integrate with respect to between the limits 0 and 2π. The only term which gives a result different from zero is the second, which gives if M' be the mass in unit length of the fluid displaced by the cylinder. Compare Art. 105. If in the above example we impress on the fluid and the cylinder a velocity - V in the direction of x, we have the case of a current flowing with velocity V past a fixed cylindrical obstacle. Adding to and the terms - Ve and Va and Vy, respectively, we get a 2 $ = − V (r + 2) cos 0, y = - V (r – a2) sin 0. φ If no external forces act, and if V be constant, we find for the resultant pressure on the cylinder the value zero. 87. To render the formula (31) capable of representing any case of irrotational motion in the space between two concentric circles, we must add to the right-hand side the term If A = P + ¿Q, the corresponding terms in 4, & are Plogr-Q0, P0 + Q log r, respectively. The meaning of these terms will appear from Example 2 above. 27P is the cyclic constant of y, i.e. (Art. 69) the total flux across the inner (or outer) circle; and -2πQ, the cyclic constant of ø, is the circulation in any circuit embracing the origin. The formula (31), as amended by the addition of the term (38), may readily be generalized so as to apply to any case of irrotational motion in a region with circular boundaries, one of which encloses all the rest. In fact, corresponding to each internal boundary we have a series of the form 2' where c, = a + ib say, refers to the centre, and the coefficients ▲, A1, Д,, &c. are in general complex quantities. The difficulty however of determining these coefficients so as to satisfy given boundary conditions is now so great as to render this method of very little utility. Indeed the determination of the irrotational motion of a liquid subject to given boundary conditions is a problem whose exact solution can be effected by direct processes in only a very few cases. Most of the cases for which we know the solution have been obtained by an inverse process; viz. instead of trying to find a solution of the equation (5a) or (7) satisfying given boundary conditions, we take some known solution of the differential equations and enquire what boundary conditions it can be made to satisfy. In this way we may obtain some interesting results in the following two important cases of the general problems in two dimensions. 88. Case I. The boundary of the fluid consists of a rigid cylindrical surface which is in motion with velocity V in a direction perpendicular to its length. Let us take as axis of x the direction of this velocity V, and let ds be an element of the section of the surface by the plane xy. Then at all points of this section velocity of the fluid in the direction of the normal, If we take any possible form of , the equation (39) is the equation of a system of curves each of which would by its motion parallel to a produce the set of stream-lines defined by const. We give a few examples. = (a) If we choose for the form Vy+const., then (39) is satisfied identically for all forms of the boundary. Hence the fluid contained within a cylinder of any shape which has a motion of translation only may move as a solid body. If, further, the cylindrical space occupied by the fluid be simply-connected, this is the only kind of motion possible. This is otherwise evident from Art. 49; for the motion of the fluid and the solid as one mass evidently satisfies the boundary conditions, and is therefore the only solution which the problem admits of. In this system of curves is included a circle of radius a, provided Hence the motion produced in an infinite mass of liquid by a circular cylinder moving through it with velocity V perpendicular to its length, is given by which agrees with (34). (c) With the same notation as in Example 3 let us assume |