Collins Mechanical Filters

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73) Hence this yields the following expressions for the history of the relative right-Cauchy-Green tensor (according to equation ( 1 . 74) It is noteworthy that besides the constant velocity component V normal to the wall and the local velocity component u(y) parallel to the wall, only the velocity component u(y + Vs) enters, which a material point possessed at the time in the past s. One can easily satisfy oneself from this fact that in the present case the determinant of the tensor Because the determinant always has the value unity.

Before that we shall look at applications of the important class of steady shear flows in detail. If the plane shear flow sketched in Fig. 1 is steady, then the velocity field and hence the rate of strain of each layer is constant with time. Therefore every fluid particle experiences, apart from a rotation, at all times a constant shear rate y(y). Hence it looks back to a peculiar monotonous deformation history, in which there were no variations at all, so that the memory of the past only contains one bit of information, namely the quantity for the particle concerned.

By the introduction If one puts = curiae satisfied. 1103 > ), then the continuity equation According to these relations the field with two components is reduced to a single Furthermore the stream function ψ has an obvious meaning. The curves Ψ (x,y,t) = constant at a certain time are of course the stream lines, because if one proceeds along such a curve 0 = d * = ^ dx + ^ dy = - vdx + udy 3x 3y applies. 104) The relationship dy/dx = v/u which follows from that is the differen­ tial equation of the stream lines (equation ( 1 .

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