By Bernard LeMehaute
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9 Express the components of acceleration A,, A 6 , A= in a cylindrical system of coordinates. , the axes 0 X and 0 Y are defined from a given point along and perpendicularly, respectively, to a streamline. Give the components of acceleration. In a second case, where it is assumed that paths and streamlines are different, the axis will then be defined at a given point with respect to paths. Then, give also the components of acceleration (R is the radius of curvature of the path). 11 Demonstrate that in a spherical system of coordinates (r, e, ) related to the earth, the operator djdt is: d o o UE o Us o W-+---+--=-+ r o
This is a positive convective acceleration. The fluid tends to resist this acceleration by convective inertia. In a divergent conduit, the velocity decreases and the fluid tends to continue its motion with the same velocity because of its inertia. The applied forces cause a negative convective acceleration. Expansion or contraction of a compressible fluid is the sum oflinear deformations and also results in corresponding inertia forces. 2 It has been seen that the linear deformation velocity components are those given in Equation 4-1.
8 The equations for an average viscous flow through a porous medium are defined by op = Ku ox op = Kv dy where p is the hydrostatic pressure. Demonstrate that such a flow is irrotational. The equations for an average fully turbulent flow through rocks are op = Ku2 ox op = Kv2 ay Demonstrate that such a flow is in general rotational. , ! (av _ au) = ! (ov' _ ou') 2 OX oy 2 ox' oy' u' and v' being the velocity components along the x' axis and the y' axis, respectively. 1 The Continuity in a Pipe The Continuity Principle The principle of continuity expresses the conservation of mass in a given space occupied by a fluid.
An Introduction to Hydrodynamics and Water Waves by Bernard LeMehaute