# Page:EB1911 - Volume 14.djvu/140

128
[HYDRODYNAMICS
HYDROMECHANICS

direction 0 along a “plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.

It is convenient to introduce the function S2 =1<>§ I = l°g=<=o;and u=co, ¢=oo across the end jg' of the jet, bounded by the curved lines AP], A'P"I', . over which the skin velocity is Q. The stream lines xBA], xAjI' are given by ul/=0, m; so that if c denotes the ultimate breadth ]' of the jet, where the velocity may be supposed uniform and equal to the skin velocity Q, m = Qc, c = m/Q.

If there are more B corners than one, either on xA or x'A', the expression for Q' is the product of corresponding factors, such as in (5).

Restricting the attention to a single corner B, " - '. - b- .fl (Cog 713-Q-1 Sin ch nf! =ch lcg “cos n6+i sh log “sin 116 = %<f~+@~»> = <1> sh nS2=Sh log cos n0+i ch log "sin 110, V =%<r~-¢-> = ' <8> °o>a>b>o>a'> -OO: -(9). and then § 2 l / b-a.b-a' ¢§ u n1 du " 21l(u"b){ (u-a.u-a ' du " ru' U?) the formulas by which the conformal representation is obtained.

For the (Z polygon has a right angle at u =a, a', and a zero angle at uh=b, where H changes from o to sir/n and (2 increases by surgz; so t at ' d9 A /(b-a.b-a') d?¢'(u-bN(u-a.u-al)'whereA' zn »' (H) And the w polygon has a zero angle at u=o, ob, where gb changes from 0 to m and back again, so that w changes by im, and gl; =§ , where B = -2. ' (12) Along the stream line xBAP], , 1 ¢=o, u=a¢-*f¢/"'; (13) and over the jet surface JPA, where the skin velocity is Q, 5% = -q = - Qy u 5ae1Y.l'Q/7" =a¢vrs/c, (14) denoting the arc AP by s, starting at u=a; ' 4 j g ch nQ=cosn8¥J§ -E€;/-ki, A (15) shjnQ=i sinn0=i/E1 g, (16) <=° >u=¢w*"'> fr, (17) and this gives the intrinsic equation of the jet, and then the radius of curvature ds xd¢ idw idw dn "= '35 =QTo='Q'J§ 2=Q'8T¢/322 c u-b~/- u-a.u-a') 2 a-b.b—a) not requiring the integration of (II) and (12) n =§ ' ""T/( If 0 =a across the end jj' of the jet, where u = so, q=Q, I chnSl=cosna= ggv, sh n§ z=isin na=i f%, (19)

Then A a-b.b-a', . .2 a-a COS 21l|."C0s 2M-2 -isln 21la-5:5 V x/(a bb a')/ Si" 2“=2 (20) =sin Znav*-"°'**"(a';lizZ a')i 2n§ b /(a-b.b-a') Tp' (I+u-b)w/(u-a.u-a') (21) =a-a'-l-(a+a') cos zna-[a+a'+(a-a') cos 2na]cos 2710 (a-a') sin' zna Xcos zna-cos 2710 sin 2110 Along the wall AB, cos n6=o, sin n0=I, a>u> b, (22) <24> £ f § fL>i i fQ. A du~°°d4> dt * -1rqu 1r qu ' (25) QE f “Q Q 7' c "' b q u Pl(f1-b)~/(1¢-0')~l-/(b-¢')~/(affu) '/“rg 6) " ' v(a-a')»/(u-lf) u' (2 Along the wall Bx, cos n8= 1, sin n0=o, b> u>o (27) n f ' ¢h-v=¢h1°g(%) =/f';v'11/#Ei <28> sh nQ=sh log n = {¥~§ 1/%. (29) Atxwhere ¢=oo, u=o, and q=q0, A Q " @;'1'/2 /'I-b/:E (qt) r /a -E' z>+ 517' Q - 430)

In crossing to the line of flow x'A'P']', nl/ changes from 0 to m, so thatwith g ==Q across j]', while across xx' the velocity is go, so that 1 m=sz°-xx'=Q-JJ' 1 (31) 1 '.Q. Nb-a'/<2 / G-1»/;@]=/» § “g\$“ Z b' Q-a' b » (32) giving the contraction of the jet compared with the initial breadth of the stream.

Along the line of, flow x'A'P' I', ¢=m, u=a'e~'f¢/"', and from x' to A', cos n0=1, sin n&=o, G. U Q»¢sq- chnil-chlog - a af b u, (33) it Q "=Jll-bJ1l"'0.I A sh 710 sh log E17 B-:I-, <>>~>f1'- on Along the jet surface A']', q =Q, gn ' 1 g A ch nn=cos no=/f% %7/5;-ji, (35) sh n9=isinn0=i'!l%:§ /9l: -3, (37) al>u =ale, r/|c> my giving the intrinsic equation.

41. The first problem of this kind, worked out by H. v. Helmholtz, of the efflux of a jet between two edges A and A1 in an infinite wall, is obtained by the symmetrical duplication of the above, with n=I, b¢0, a'=-°o, asinfig.5, V chS2=?, shS2= ff; ' (1) and along the jet APJ, oo > u =ae"'/°> a, Shn=-isana=¢ § =¢¢-4"-'°, (2) -me -mf 8) 0

PM-= T°sin6ds= ei /ds=§ ¢i /=§ sin0, (3) 