Tham khảo tài liệu 'kundu fluid mechanics 2 episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 64 Kinematics Figure Deformation of elements in a parallel shear flow. The element is stretched along the principal axis JCj and compressed along the principal axis X2. Ỵ and that of BC is zero giving y 2 as the overall angular velocity half the vorticity . The average value does not depend on which two mutually perpendicular elements in the Xi X2-plane are chosen to compute it. In contrast the components of sttain rate do depend on the orientation of the element. From Eq. the strain rate tensor of an element such as ABCD with the sides parallel to the Xi X2-axes is 0 e 0 0 0 0 0 0 which shows that there are only off-diagonal elements of e. Therefore the element ABCD undergoes shear but no normal strain. As discussed in Chapter 2 Section 12 and Example a symmetric tensor with zero diagonal elements can be diagonalized by rotating the coordinate system through 45 . It is shown there that along these principal axes denoted by an overbar in Figure the strain rate tensor is 0 0 0 so that there is a linear extension rate of ẽu y 2 a linear compression rate of 22 y 2 and no shear. This can be understood physically by examining the deformation of an element PQRS oriented at 45 which deforms to P Q R S . It is clear that the side PS elongates and the side PQ contracts but the angles between the sides of the element remain 90 . In a small time interval a small spherical element in this flow would become an ellipsoid oriented at 45 to the Xi X2-coordinate system. Summarizing the element ABCD in a parallel shear flow undergoes only shear but no normal strain whereas the element PQRS undergoes only normal but no shear strain. Both of these elements rotate at the same angular velocity. 11. Kinematic CMnideralúiiu of Vortex I iouiH 65 11. Kinematic Considerations of Vortex Flows Hows in circular paths arc called vortex flows some basic forms of which are described in what follows. Solid-Body Rotation Consider first the case in which the velocity is .
