(These changes comprise the vorticity of the flow, which is the curl (rotational) ∇ × v of the velocity which is also the antisymmetric part of the velocity gradient ∇ v.) Įither way, the strain rate tensor E( p, t) expresses the rate at which the mean velocity changes in the medium as one moves away from the point p – except for the changes due to rotation of the medium about p as a rigid body, which do not change the relative distances of the particles and only contribute to the rotational part of the viscous stress via the rotation of the individual particles themselves. In any chosen coordinate system with axes numbered 1, 2, 3, this viscous stress tensor can be represented as a 3 × 3 matrix of real numbers: Like the total and elastic stresses, the viscous stress around a certain point in the material, at any time, can be modeled by a stress tensor, a linear relationship between the normal direction vector of an ideal plane through the point and the local stress density on that plane at that point. These stresses generally include an elastic ("static") stress component, that is related to the current amount of deformation and acts to restore the material to its rest state and a viscous stress component, that depends on the rate at which the deformation is changing with time and opposes that change. Internal mechanical stresses in a continuous medium are generally related to deformation of the material from some "relaxed" (unstressed) state. In non-Newtonian fluids, on the other hand, the relation between ε and E can be extremely non-linear, and ε may even depend on other features of the flow besides E.ĭefinition Viscous versus elastic stress : 304 In the absence of such a coupling, the viscous stress tensor will have only two independent parameters and will be symmetric. If the fluid is isotropic as well as Newtonian, the viscosity tensor μ will have only three independent real parameters: a bulk viscosity coefficient, that defines the resistance of the medium to gradual uniform compression a dynamic viscosity coefficient that expresses its resistance to gradual shearing, and a rotational viscosity coefficient which results from a coupling between the fluid flow and the rotation of the individual particles. In a Newtonian fluid, by definition, the relation between ε and E is perfectly linear, and the viscosity tensor μ is independent of the state of motion or stress in the fluid. The tensor μ has four indices and consists of 3 × 3 × 3 × 3 real numbers (of which only 21 are independent). In many situations there is an approximately linear relation between those matrices that is, a fourth-order viscosity tensor μ such that ε = μE. In an arbitrary coordinate system, the viscous stress ε and the strain rate E at a specific point and time can be represented by 3 × 3 matrices of real numbers. For a completely fluid material, the elastic term reduces to the hydrostatic pressure. In viscoelastic materials, whose behavior is intermediate between those of liquids and solids, the total stress tensor comprises both viscous and elastic ("static") components. However, elastic stress is due to the amount of deformation ( strain), while viscous stress is due to the rate of change of deformation over time (strain rate). Both tensors map the normal vector of a surface element to the density and direction of the stress acting on that surface element. The viscous stress tensor is formally similar to the elastic stress tensor (Cauchy tensor) that describes internal forces in an elastic material due to its deformation. The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point.
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