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522 chapter 19RiversurfaceRiversurfaceyLxyxLHbottombottomFigure 19.4 Cross-sectional view of the flow of a stream around a submerged beam (left)and two parallel plates (right). Both beam and plates have length L along the direction offlow. The flow is seen to be symmetric about the centerline and to be unaffected at thebottom and surface by the submerged object.in a moving fluid [F&W 80]:DvDtdef= (v · ⃗∇)v + ∂v∂t . (19.31)This derivative gives the rate of change, as viewed from a stationary frame, ofthe velocity of material in an element of fluid and so incorporates changes due tothe motion of the fluid (first term) as well as any explicit time dependence of thevelocity. Of particular interest is that Dv/Dt is second order in the velocity, andso its occurrence reflects nonlinearities into the theory. You may think of thesenonlinearities as related to the fictitious (inertial) forces that would occur if wetried to describe the motion in the fluid’s rest frame (an accelerating frame).The material derivative is the leading term in the Navier–Stokes equation,DvDt = ν∇2 v − 1 ρ ⃗ ∇P (ρ, T, x), (19.32)∂v xz∑∂t + ∂v xv j = ν∂x jj=x∂v yz∑∂t + ∂v yv j = ν∂x jj=x∂v zz∑∂t + ∂v zv j = ν∂x jj=xz∑j=xz∑j=xz∑j=x∂ 2 v x∂x 2 j∂ 2 v y∂x 2 j∂ 2 v z∂x 2 j− 1 ρ− 1 ρ∂P∂x ,∂P∂y ,− 1 ∂Pρ ∂z . (19.33)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 522