Ivancevic_Applied-Diff-Geom

80 Applied Differential Geometry: A Modern Introduction2.1.5.2 Forces Acting on a FluidA fluid contained in a finite volume is subject to the action of both volumeforces F i and surface forces S i , which are respectively defined by∫∮F i = ρf i dv, and S i = σ ij da j . (2.39)vHere, f i is a force vector acting on an elementary mass dm, so that theelementary volume force is given bydF i = f i dm = ρf i dv,which is the integrand in the volume integral on l.h.s of (2.39). σ ij =σ ij (x k , t) is the stress tensor–field of the fluid, so that the elementary forceacting on the closed oriented surface a is given bydS i = σ ij da j ,where da j is an oriented element of the surface a; this is the integrand inthe surface integral on the r.h.s of (2.39).On the other hand, the elementary momentum dK i of a fluid particle(with elementary volume dv and elementary mass dm = ρdv) equals theproduct of dm with the particle’s velocity u i , i.e.,dK i = u i dm = ρu i dv,so that the total momentum of the finite fluid volume v is given by thevolume integral∫K i = ρu i dv. (2.40)vNow, the Newtonian–like force law for the fluid states that the timederivative of the fluid momentum equals the resulting force acting on it,˙K i = F i , where the resulting force F i is given by the sum of surface andvolume forces,∮F i = S i + F i =From (2.40), taking the time derivative and usingaa∫σ ij da j + ρf i dv. (2.41)v∫˙K i = ρ ˙u i dv,v˙ ρdv = 0, we get