《流体力学 Fluid Mechanics》课程PPT教学课件（双语版）第六章 不可压缩理想流体的无旋运动

Rotational nosy unimpressible ideal-fluid 、 Translating motion Velocity of each point concludes ux,u,, known from figure 6-1, ux, uy is translating velocity 2、 Deformation motion (1) Linear deformation Taking AB as example. due to velocity of angle point B is Ou faster than angle point A along direction x(or slower) Ox u so through dt, protraction of side AB along x(or condense)isa- dx- dt x Linear deformation of unit length in unit time called linear deformation velocity, and written, E, i=x,y, z)then dx·d/(dx.at ou similarly E,m,= (6-1) Oy

11 Velocity of each point concludes ,known from figure 6—1, is translating velocity. ux uy , ux uy , （1）Linear deformation Taking AB as example.Due to velocity of angle point B is faster than angle point A along direction x (or slower) , so through , protraction of side AB along x (or condense)is .Linear deformation of unit length in unit time called linear deformation velocity ,and written， then dx x ux         dt dx dt x ux          (i x y z) ij  = , , 1、Translating motion 2、Deformation motion ( ) x u dx dt dx dt x ux x xx     =    = y uy yy    = z uz zz    =  similarly （6—1）

不可忽流的元族运动 1、平移运动 各点的速度中均包含有x2,由图6—1可见,vx,2 是平移速度。 2、变形运动 (1)线变形 以AB为例。因为角点B沿x方向的速度比角点4快(或 du 慢),所以经过h时段后,AB边在x方向的伸长( 或缩短)量为(md。单位时间单位长度的线变形称为线 变形速度,并记为n(=x,y,2),则 Er=dx dt/dxdt) ax 同理 (61) 12 az

12 各点的速度中均包含有 ，由图6—1可见， 是平移速度。 ux uy , ux uy , dx x ux         （1）线变形 以AB为例。因为角点B沿 x 方向的速度比角点A快（或 慢） ，所以经过 时段后，AB边在 x 方向的伸长（ 或缩短）量为 。单位时间单位长度的线变形称为线 变形速度，并记为 ，则 dt dx dt x ux          (i x y z) ij  = , , 1、平移运动 2、变形运动 ( ) x u dx dt dx dt x ux x xx     =    = y uy yy    = z uz zz    =  同理 （6—1）

Rotational flowy impressible ideal-fluid (2) Shear deformation Taking half angular deformation velocity in plane defined as shear known from figure 6-1, angle changing of point A 1s J=x, y, z but i* deformation velocity of liquid micelle, written as &, i ∠CAB-∠C'B′=6+8B and that dxd/dx+-xdkx:d|≈ dt (1) SB= dt/ dy+dy (2) Oy SO Sa+oB t OV According to definition of shear deformation velocity on liquid micelle, obtaining Sa+oB P乙x3=3 2 Ox ay ) E、=E (6 6-2) 2(O E 13 2 az ax

13 Taking half angular deformation velocity in plane defined as shear deformation velocity of liquid micelle,written as known from figure 6-1,angle changing of point A is (i j x y z i j)  i j , = , , but  According to definition of shear deformation velocity on liquid micelle,obtaining CAB−CAB =   +          +   = =           +   = =           +   = + = = x u z u z u y u y u x u dt x z z x xz z y yz z y y x xy yx 2 1 2 1 2 1 2 1           （6—2） and that dt y u x u dt y u dy dt y u dy dt dy y u dt x u dx dt x u dx dt dx x u y x x y x y x y           +   + =               +   =              +   =         so （1） （2） （2）Shear deformation

不可压包滤的运动 (2)剪切变形 将平面上角变形速度之半定义为流体微团的剪切变形速 ,记为(/=xy:但1≠/)由61町知,4点的角度变化 ∠CAB-∠C'B'=6+8B 而 dxd/dx+-xdkx:d|≈ dt (1) SB= dt/ dy+dy oux dt (2) Oy 所以 Sa+oB= t OV 根据流体微团剪切变形速度的定义得 E Sa+oB 2 dt 2 Ox ay ) E、=E (62) 2(O E 2 az ax

14 将平面上角变形速度之半定义为流体微团的剪切变形速 度，记为 由图6-1可知，A点的角度变化 为 (i j x y z i j)  i j , = , , 但  根据流体微团剪切变形速度的定义得 CAB−CAB =   +          +   = =           +   = =           +   = + = = x u z u z u y u y u x u dt x z z x xz z y yz z y y x xy yx 2 1 2 1 2 1 2 1           （6—2） dt y u x u dt y u dy dt y u dy dt dy y u dt x u dx dt x u dx dt dx x u y x x y x y x y           +   + =               +   =              +   =       而   所以 （1） （2） （2）剪切变形

Rotational flowy impressible ideal-fluid 3、 Rotation Taking average value of rotational angular velocity on two straight-line in liquid micell defined as rotational angular velocity of liquid micelle, written as, o, (i=x,y, z), supposing straight-line rotational angular velocity along counter-clockwise is positive, then from formula(1 )(2), rotational angular of side AB in unit time is au, /ox, rotational angular of side AC in unit time is -Ou/ay according to definition of rotational angular velocity on liquid micelle, obtaining 2( oz (63) 2 Ox = 2 Ox Definition O If all liquid micelle translating or deforming motion when liquid flowing, no rotating viz.o =0 =0.=0, then called this flow is irrotational flow(potential flow) If has rotation in liquid micelle, viz. @r, @,, a- at least one is notO of three then called rotational motion vortex motion) 15

15 Taking average value of rotational angular velocity on two straight-line in liquid micelle defined as rotational angular velocity of liquid micelle,written as , ,supposing straight-line rotational angular velocity along counter-clockwise is positive, then from formula(1)(2), rotational angular of side AB in unit time is , rotational angular of side AC in unit time is ,according to definition of rotational angular velocity on liquid micelle, obtaining (i x y z) i  = , , u x  y  u y − x  If all liquid micelle translating or deforming motion when liquid flowing, no rotating,viz. ,then called this flow is irrotational flow (potential flow). = = = 0 x  y z           −   =         −   =           −   = y u x u x u z u z u y u y x z x z y x y x 2 1 2 1 2 1    （6—3）  3、 Rotation Definition： If has rotation in liquid micelle,viz. at least one is not 0 of three, then called rotational motion(vortex motion). x  y z ,