82-3 Differential Equation of Fluid Equilibrium 1. Eulers equilibrium equation Euler put forward the equation in 1755 As shown in Figure2-3, Select a infinitesimal wedge-shaped ABCDE in the balance fluid whose length of sides are dx, dy and dz. Assume that the density is p and the pressure is p at point a E Figure 2-3 infinitesimal hexahedron 21
21 §2-3 Differential Equation of Fluid Equilibrium 1. Euler’s equilibrium equation As shown in Figure2—3， Select a infinitesimal wedge-shaped ABCDE in the balance fluid whose length of sides are , and . Assume that the density is and the pressure is at point A . dx dz dy p D z A B E o x y Figure 2—3 infinitesimal hexahedron Euler put forward the equation in 1755. C
流其功学 §2-3流体平衡的微分方程式 一、欧拉平衡方程式 欧拉于1755年提出 如图23,在平衡流体中任取边长为cy、c的一个微 元六面体 ABCDE,设A点的密度为尸,压强为P。 E C B O 图2-3微元六面体 22
22 §2-3流体平衡的微分方程式 一、欧拉平衡方程式 如图2—3，在平衡流体中任取边长为 、 、 的一个微 元六面体ABCDE，设A点的密度为 ，压强为 。 dx dy dz p D z A B E o x y 图2—3微元六面体 欧拉于1755年提出。 C
Obtain from(1-17), the mass force of fluid is dFm=dm(fi+f j+f k) pdxdydE( i +f j+fk)(2-8) Obtain from (2-2) and (2-3) the surface force of fluid is dP==pd (pA-pBdydzi +(pa-pcaxdi+(p-pp )dxdyk In this formula (2-9) Pa The pressure at a point on surfaces DC, BD and BC PB Pc\ Pd--The pressures at a point on surfaces BE, CE and DE
23 Obtain from (1—17）, the mass force of fluid is dF dm( f i f j f k ) m x y z = + + dxdydz( f i f j f k ) x y z = + + （2—8） Obtain from（2—2）and（2—3）,the surface force of fluid is dP pdA = − p p dydzi p p dxdzj p p dxdyk A B A C A D = ( − ) +( − ) +( − ) （2—9） In this formula pA — The pressure at a point on surfaces DC, BD and BC pB 、pC 、pD—The pressures at a point on surfaces BE, CE and DE
流其功学 由(1-17)式可得流体的质量力为: d F =dm(x' f1+∫,j+fk) pdxdydE( i +f j+fk)(2-8) 由(22)、(23)式得流体的表面力为: dP==pdA (pA-PBdydzi+(pa-pcdxdij+(P-pp )dxdyk (2-9) 式中 p4DC、BD、BC面上一点的压强; pB、P、P-BE、CE、DE面上一点的压强。 24
24 由（1—17）式可得流体的质量力为： dF dm( f i f j f k ) m x y z = + + dxdydz( f i f j f k ) x y z = + + （2—8） 由（2—2）、（2—3）式得流体的表面力为： dP pdA = − p p dydzi p p dxdzj p p dxdyk A B A C A D = ( − ) +( − ) +( − ) （2—9） 式中 pA —DC、BD、BC面上一点的压强； pB 、pC 、pD — BE、CE、DE面上一点的压强
Pa=p is given. The pressure is the continuous function of coordinates in the balance fluid. namely=p(x,y, =) Expand it according to the taylor 's formula for multi dimensions continuous functions and omit the infinitesimal above second order the result is D dx p -+ ap dy (2-10) PD=p+ d z Get from(2-9) and(2-10) dP=dxdydz( k) (2—11) ax a az 25
25 is given . The pressure is the continuous function of coordinates in the balance fluid. namely . Expand it according to the taylor’s formula for multi dimensions continuous functions and omit the infinitesimal above second order, the result is pA = p p = p(x, y,z) dz z p p p dy y p p p dx x p p p D C B = + = + = + （2—10） Get from（2—9）and（2—10） ( k ) z p j y p i x p dP dxdydz − − = − （2—11）