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The NI transition is a first-order phase transition, albeit it is very weak. The order parameter is the Q {\displaystyle \mathbf {Q} } tensor, which is symmetric, traceless, second-order tensor and vanishes in the isotropic liquid phase.
A second-order fluid is a fluid where the stress tensor is the sum of all tensors that can be formed from the ... truncated at the second-order. For an isotropic ...
A scalar function that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density, or Helmholtz free energy, of a nonlinear material possessing isotropic symmetry.
In physics, tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase. [1] The Q {\displaystyle \mathbf {Q} } tensor is a second-order, traceless, symmetric tensor and is defined by [ 2 ] [ 3 ] [ 4 ]
For a second order isotropic tensor (i.e. a tensor having the same components in any coordinate system) like the Lagrangian strain tensor have the invariants where is the trace operator, and {,,, …}.
For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense, [29] and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces. [30] A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more
It can be shown that the stress tensor is a contravariant second order tensor, which is a statement of how it transforms under a change of the coordinate system. From an x i -system to an x i ' -system, the components σ ij in the initial system are transformed into the components σ ij ' in the new system according to the tensor transformation ...
In order to apply this to the Navier–Stokes equations, three assumptions were made by Stokes: The stress tensor is a linear function of the strain rate tensor or equivalently the velocity gradient. The fluid is isotropic. For a fluid at rest, ∇ ⋅ τ must be zero (so that hydrostatic pressure results).