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1 Kinematics 29 steady flow v ¼ vðxÞ the streamlines coincide with the particle trajectories, called the pathlines. For a given velocity field the streamlines are determined from the differential equations: dr  vðx; tÞ ¼ 0 , dx1 dx2 dx3 ¼ ¼ v1 v2 v3 at constant time t ð3:1:17Þ The vorticity vector or for short the vorticity cðx; tÞ of the velocity field is defined as: c ¼ r  v  rot v  curl v ð3:1:18Þ The significance of this concept will be discussed in Sect. 1. e. cðx; tÞ ¼ 0; in a region of the flow.

1. The relevant constitutive equations for the flow given by Eq. 1) are: 52 sRz ¼szR 3 Basic Equations in Fluid Mechanics ! sy dv dv when 6¼ 0 ; jsRz j ¼ jszR j ¼ lþ dR jdv=dRj dR sy when dv ¼0 dR sRR ¼shh ¼ szz ¼ sRh ¼ shR ¼ 0 ð3:8:17Þ sy is a yield shear stress. 5) it follows that: jszR j sy rp ¼ when R 2sy jcj ð3:8:18Þ The Eqs. 17) show that inside a cylindrical surface of radius rp the material flows like solid plug. 5). Because c\0 and dv=dR 0; we obtain: dv sy jcj ¼À Rþ dR 2l l ð3:8:19Þ Integration and the boundary condition v(d/2) = 0 yield: "  2 # !

9 Reduction of Drag in Turbulent Flow Fig. 2) Fig. 8 Flos through a contraction. a Newtonain fluid. b non-Newtonain fluid 23 (b) 10 −1 Turbulent flow. Polymer in water Concentration in ppm: 5, 10 −2 0 450 Newtonian, pure water 16 Laminar f = Re 10 −3 510 ⋅ −4 10 2 10 3 10 4 10 5 10 6 Reynolds number Re =ρvD/ μ the fluid mixture is changed only slightly. For the present example the viscosity l is only increased by 1 % relative to that of water. The reason why very small amounts of polymer additives to a Newtonian fluid like water have such a large effect on drag, is not completely understood.

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