Explore chapters and articles related to this topic
Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
Tensors encountered in fluid mechanics are of second order, i.e., they are characterized by an ordered pair of coordinate directions. Tensors are often denoted by uppercase, boldface type or lower case, boldface Greek letters, such as τ for the stress tensor. A tensor field is a tensor-valued function that associates a tensor with each point of a given region in space. Tensor addition and multiplication of a tensor by a scalar are commutative and associative. If R, S, and T are tensors of the same type, and m and n are scalars, then
Tensors
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
The idea behind a tensor field is analogous to that of a scalar or vector field, i.e., it associates a tensor of a given type to each point in the base space. Naturally, since scalars and vectors are different types of tensors, scalar and vector fields are the most natural examples of tensor fields. Just as we have used the tangent and dual bases to describe different vectors, a tensor field can be described by expressing its components in the coordinate basis as functions on the base space. For example, a type (2,0) tensor field T would be written as () T(p)=Tab(p)eab(p),
Tensors in Rectangular Rtesian Coordinate Systems
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
which shows that the quantities bi indeed form components of a vector field. Similarly, the contraction with respect to the indices i, k yields components of a vector field bj = ∂aij/∂xi. The quantities ∂aij/∂xi and ∂aij/∂xj are called the components of the divergence of the tensor field aij with respect to the indices i and j, respectively. In the general case, we define the divergence of a tensor field with respect to any one of its indices as the contraction of the corresponding tensor gradient with respect to that index. It follows from Theorem 1 above and Theorem 1 of Sec. 5.7 that the divergence of a tensor field of order r yields a tensor field of order r - 1.
Numerical simulation of the 2D lid-driven cavity flow of chiral liquid crystals
Published in Liquid Crystals, 2023
In the LdG theory, the microstructure and orientation of LCs are described by a second-order tensor , known as the tensor order parameter. All tensors at each spatial point form a tensor field. The tensor can be described as follows [4,13]: