1.试对柱坐标形式的微六面体,建立运动方程 解:系统总动量变化率≡控制体内动量 变化率+经控制面净流出的动量 控制体内动量变化率 (puroroeo at 48 x 经控制面净流出的动量 r方向(mmn).-( pu, urSA=) Lou, ur+ (pu uirOr1680z-[pu, ur-(pu urSrS8sz .(pu urSr580z 0方向 (pupuors80z 06 Z方向c()roO0
1. 试对柱坐标形式的微六面体,建立运动方程. r dr dz x z d 解: 系统总动量变化率= 控制体内动量 变化率 + 经控制面净流出的动量 控制体内动量变化率: 经控制面净流出的动量 r 方向 方向 Z 方向 ( ) ( ) [ ( ) ] [ ( ) ] ( ) r r r r r r r r r r u ur z u ur z u ur u ur r z u ur u ur r z r r u ur r z r + − = + − − = ( ) u u r z ( ) z u ru r z z ( ) ur r z t
系统总动量变化率 (pir)+ (pu ir)+ (pueui)+.(pu ruord80z fril ap 1 a +-(pu,r)+-(pun)+-(pu.)+pr Oi+ncin+lt如+n2186 t r ar r06 ar r a0 az Du =p-rOro86z Dt 柱坐标中=l,en+le+e, 06 于是 Du a a, ue a L-+ 06 Que +ugee +ue) +u Or ragaz De +. ue oue+u z +0) ar r a8 +u au u ou,+u c)e t 06 az
系统总动量变化率 [ ( ) ( ) ( ) ( )] 1 1 { [ ( ) ( ) ( )] [ ]} r z r z r z ur u ur u u u ru r z t r z u u u u u ru u r u u r u u r z t r r r z t r r z Du r r z Dt + + + = + + + + + + + = 柱坐标中 , , r r r z z r e e u u e u e u e e e = + + = = − 2 ( )( ) ( ) ( ) ( ) r z r r z z r r r r r z r r r z z z z z r z z Du u u u u e u e u e Dt t r r z u u u u u u u u e t r r z r u u u u u u u u u e t r r z r u u u u u u u e t r r z = + + + + + = + + + − + + + + + + + + + 于是
微元体所受的表面力 or(rp,)+agta(rp )]dededr l 06 (ro2)+=+on+-(ro0)le+ 06 ()+2n+2()E}mk 06 微元体所受的重力 FB=pgrord00z=(pg, e,+pgee+pge ror800z
[ ( ) ( )] {[ ( ) ( )] [ ( ) ( )] [ ( ) ( )] } S r z r rr zr r r r z z rz zz z p F rp rp d dzdr r z r r e r z r r e r z r r e drd dz z z = + + = + − + + + + + + + + 微元体所受的表面力 微元体所受的重力 ( ) F gr r z g e g e g e r r z B r r z z = = + +
依据动量定理 ①D。,O m08++[(m)+aaV lr t a0- az r Pg +-(ro)+ 00-0w+y(ro 06 +u dug u, ue 06 P8a+-[(ro)+ 00+0+(rod 06 az L + 06 n8:+0 rO.)+ +—(r 00 az
依据动量定理 1 [ ( ) ( )] r z Du p g rp rp Dt r r z = + + + 2 1 [ ( ) ( )] 1 [ ( ) ( )] 1 [ ( ) r r r r r z r x rr zr r r z r r z z z z z r z z rz u u u u u u u u t r r z r g r r r r z u u u u u u u u u t r r z r g r r r r z u u u u u u u t r r z g r r z + + + − = + + − + + + + + = + + + + + + + = + + ( )] z zz r z +
2.教科书(1.6) 在下边习题求解中会用到以下公式,请参阅有关资料。 柱坐标与直角坐标 =rose y=rsin 6 6= arc tg COS u= cos ou+sin e y ar r ae a cos0 a ue =-sin e u+ cos81 =sin e-+ r a0 ⅩA(r,O,z) u =w p=p(r,O,=)=p(x,y,=) r=r(x,y),b=0(x,y),z= ui +v+wk=u, e, +upeo+ue ao ao ar a0 00 cos e ao sin 0 ag ly=ui·n+y·en+wk:e,= cos0 u+ sino v ax ar ax a0 ax A ao ao ar a0 a0 ao cose ag ue=ui.ea+vj.eg+wk.ee=-sin 0 u+cos 0 v =sin0-+ ay a o0 ay ar r a0 u=ui.e+ve+wk.e=w
= = = sin cos z z y r x r = = = + arc tg 2 2 z z x y r x y = = − + = + sin cos cos sin u w u u v u u v z r = + = − = cos sin sin cos z z y r r x r r 2. 教科书(1.6) 在下边习题求解中会用到以下公式,请参阅有关资料。 柱坐标与直角坐标 X A(r, , z) Z r Y ( , , ) ( , , ) ( , ), ( , ), sin cos cos sin r z x y z r r x y x y z z r x r x x r r r y r y y r r z z = = = = = = + = − = + = + = cos sin sin cos r r z z r r r r z z z z ui vj wk u e u e u e u ui e vj e wk e u v u ui e vj e wk e u v u ui e vj e wk e w + + = + + = + + = + = + + = − + = + + =
+ ul cos a sine a +vl sin e 06 a cos0 a+wa (cosO u+sine v)=+=(sin 0 u+ coSe v)aa 06 +u ar r ae a sin 6 a cos e r06 u= coSu+sin bv a cos0 a sn6-+ l =-sin Ou+cos0 v ar r a0
( ) ( ) sin cos cos sin 1 cos sin sin cos r z u u v w x y z u v w r r r r z u v u v w r r z u u u r r z = + + = − + + + = + + − + + = + + = = − + = + sin cos cos sin u w u u v u u v z r = + = − = cos sin sin cos z z y r r x r r
(V川]=cosb(Vsin(nV cosolu +u +sin|u ra0 cos 6-+sin e e-+sin e 06 6 +u. cos0-+sin 0 u,(cos u+sin v)+er a (cos0 u+sin 0 v)-(sin 8 u+cos 0 v)] +u-(cos0 u+sin 0 v) lo tu ar r- r a sin 6 a ar cos0 u=cosO u+sin 6 v ar sIn 8+ cose sin eu+cos0 v r
( ) cos ( ) sin( ) cos sin cos sin cos sin cos sin cos s r r z r z r z r u u u u u v u u u v v v u u u u u u r r z r r z u v u v u u r r r u v u z z u u r = + = + + + + + = + + + + + = + ( ) 2 in [ (cos sin ) ( sin cos )] (cos sin ) z r r r r z r r r r z u v u v u v r u u v z u u u u u u u r r z u u u u u u u r r z r + + − − + + + = + − + = + + − = = − + = + sin cos cos sin u w u u v u u v z r = + = − = cos sin sin cos z z y r r x r r
球坐标与直角坐标 x=rsin 0 cos o y=rsin sin g 6= arc tg z=rose o=arc tg 2 u, = sin 0 cos u+sin sin v+ cos 0 w uo =cos 6 cos p u+ cos sin v+(sin 0)w u,=-sin o u+ cos ov a. cos 6 cosφa SIn sin b cos ax a8 rsin 0 ao a cos 0 sinφa.cosφo esinφ ar a0 rsn 0 ao =cos0-+ 06
= = = cos sin sin sin cos z r y r x r = + = = + + arc tg z arc tg 2 2 2 2 2 x y x y r x y z = − + = + + − = + + sin cos cos cos cos sin ( sin ) sin cos sin sin cos u u v u u v w ur u v w − + + = sin cos cos sin sin cos x r r r + + = sin cos sin cos sin sin y r r r + − = z r r sin cos 球坐标与直角坐标
3.小球在理想流体中作缓慢匀速直线运动,试给出小球表面流体速度u,ν,W 所必须满足的边界条件 解:取固定坐标系如图,球面方程为 [x-x0()]+y2+2=a 令F=[x-x0()+y2+22-a 物面边界条件为, DE AF OF aFaF +1 + Dt at 2(x-x0)uo+2(x-x0)+2y+2z=0 (x-x0)(-u4)+vy+z=0 式中 dx(t)
3. 小球在理想流体中作缓慢匀速直线运动,试给出小球表面流体速度 所必须满足的边界条件 u, v,w 解: 取固定坐标系如图,球面方程为 2 2 2 2 0 [x − x (t)] + y + z = a 2 2 2 2 0 F = [x − x (t)] + y + z − a = 0 Dt DF = 0 + + + z F w y F v x F u t F − 2(x − x0 )u0 + u2(x − x0 ) + v2y + w2z = 0 dt dx t u ( ) 0 0 = 令 物面边界条件为 , 式中: (x − x0 )(u − u0 ) + v y + wz = 0 0 x x 0 0 dx u dt = o
又解:取运动坐标系Ox'yz固结在小球上 球面方程 +y个22 令F=x2+y2+z12-a 流体相对于动坐标系速度-o,v,W 动坐标系和固定坐标系间关系为,x=x-x0,y=y,z=z 0 l,=(l-l0)2+y+k his VF 2x'i+2y'j+22 VE VE 将以上两式代入物面条件得 (x-x0(l-l0)+yy+w==0
又解: 取运动坐标系 固结在小球上 , o' x' y'z' 2 2 2 2 x' +y' +z' = a 2 2 2 2 F' = x' +y' +z' −a 球面方程 令 流体相对于动坐标系速度 动坐标系和固定坐标系间关系为, , , , u − u0 v w x' = x − x , y' = y,z' = z 0 ur n = 0 u u u i vj wk r = ( − 0 ) + + ' 2 ' 2 ' 2 ' ' ' F x i y j z k n F F + + = = (x − x0 )(u − u0 ) + v y + wz = 0 将以上两式代入物面条件得: y x 0 u