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On Shearing, Stretching and Spin

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Abstract.

An analysis is presented of stretching, shearing and spin of material line elements in a continuous medium. It is shown how to determine all pairs of material line elements at a point x, at time t, which instantaneously are not subject to shearing. For a given pair not subject to shearing, a formula is presented for the determination of a third material line element such that all three form a triad not subject to shearing, instantaneously. It is seen that there is an infinity of such triads not subject to shearing.

A new decomposition of the velocity gradient L is introduced. In place of the classical decomposition of Cauchy and Stokes, L=d+w, where d is the stretching tensor and w is the spin tensor, the new decomposition is L=?+, where ?, called the ldquo;modified” stretching tensor, is not symmetric, and , called the “modified” spin tensor, is skew-symmetric – the tensor ? being chosen so that it has three linearly independent real right (and left) eigenvectors. The physical interpretation of this decomposition is that the material line elements along the three linearly independent right eigenvectors of ? instantaneously form a triad not subject to shearing. They spin as a rigid body with angular velocity μ (say) associated with . Also, for each decomposition L=?+, there is a decomposition L=? T+\widetilde{\pmb{\mathcal M}}, where \widetilde{\pmb{\mathcal M}} is also skew-symmetric. The triad of material line elements along the right eigenvectors of ? T (the set reciprocal to the right eigenvectors of ?) is also instantaneously not subject to shearing and rotates with angular velocity \widetilde{\boldsymbol{\mu}} (say) associated with \widetilde{\pmb{\mathcal M}}. It is seen that the vorticity vector ω is the mean of the two angular velocities μ and \widetilde{\boldsymbol{\mu}}, ω =(μ+\widetilde{\boldsymbol{\mu}})/2. For irrotational motion, ω =0, so that μ=-\widetilde{\boldsymbol{\mu}}; any triad of material line elements suffering no shearing rotates with angular velocity equal and opposite to that of the reciprocal triad of material line elements.

It is proved that provided d is not spherical, there is an infinity of choices for ? and in the decomposition L=?+.

Two special types of decompositions are introduced.

The first type is called “CCS-decomposition” (where CCS is an abbreviation for Central Circular Section). It is associated with the infinite family of triads (not subject to shearing) with a common edge along the normal to one plane of central circular section of an ellipsoid ? associated with the stretching tensor, and the two other edges arbitrary in the other plane of central circular section of ?. There are two such CCS-decompositions.

The second type is called “triangular decomposition”, because, in a rectangular cartesian coordinate system, ? has three off-diagonal zero elements. There are six such decompositions.

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Received 14 November 2000 and accepted 2 August 2001

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Boulanger, P., Hayes, M. On Shearing, Stretching and Spin. Theoret Comput Fluid Dynamics 15, 199–229 (2002). https://doi.org/10.1007/s001620100050

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Keywords

  • Vorticity
  • Angular Velocity
  • Triad
  • Rigid Body
  • Velocity Gradient