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Where p(u. t ) is the density (see Figure 2. I. I ). The principle of conservation of mass states that: t1( ,LI( V,) ). rlt = 0 or M ( V,) = M ( V,,), where C',, is the reference configuration. siorl). 1) D(p,,(X))lDt= 0 = (DpjDt)J + pDJIDt. Using Euler's formula and the condition J # 0, we obtain, DpjDt = - p ( V - v ) . :~rlerirrrl i~c~rsiotl). 3) are known as the continuity equation or mass balance. ,? = jl DG , 11 Dr d r> ~ i h c r eG is a n y scalar, vector. 2. Principle of conservation of linear momentum /rttcgrul cersion (or Euler's uxiom) This principle states that for a material volume V, linear m o m e n t u m is conserved.
Velocity, acceleration, Lagrangian and Eulerian viewpoints The velocity is defined as - and it is the velocity of the particle X. t')I~ = a(X, t). , the property of the particle X that happens to be at the spatial location x at time t. Thus, v(X, t ) is the Lagrangian velocity and v(x, t14 is the Eulerian velocity. In most classical problems in fluid mechanics it is enough to obtain the spatial description. t)lx, and represents the change at a fixed position x. u,)e, (see Appendix). The expression allows the computation of the acceleration at (x, t ) without computing the motion first.
V xv=o. , same relationship as streamlines and velocity field). Obviously, vortex lines cannot cross for that would imply that a given fluid element has two different rates of rotation. Similarly, physical arguments indicate that a vortcx line cannot end somewhere in the fluid. 7).