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Viscous stress tensor

The viscous stress tensor, denoted as \tau_{ij}, is a second-order tensor in continuum mechanics that quantifies the internal frictional stresses within a viscous fluid arising from velocity gradients and molecular diffusion, representing the momentum transfer due to viscosity.^[1]^[2] It forms the viscous component of the total Cauchy stress tensor \sigma_{ij}, where \sigma_{ij} = -P \delta_{ij} + \tau_{ij}, with P being the thermodynamic pressure and \delta_{ij} the Kronecker delta, distinguishing it from pressure which acts isotropically even in static fluids.^[3]^[2] In Newtonian fluids, which obey a linear relationship between stress and strain rate, the viscous stress tensor is expressed as \tau_{ij} = 2\mu e_{ij} + \lambda e_{kk} \delta_{ij}, where \mu is the dynamic shear viscosity, \lambda is the second viscosity coefficient (related to bulk viscosity), and e_{ij} is the symmetric rate-of-strain tensor given by e_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right), with u_i as velocity components.^[1]^[3] The diagonal components (i=j) capture normal viscous stresses associated with extension or compression, while off-diagonal components (i \neq j) describe shear stresses that resist relative motion between fluid layers.^[1] For incompressible flows where the divergence of velocity \nabla \cdot \mathbf{u} = 0, the expression simplifies to \tau_{ij} = \mu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right), highlighting the direct proportionality to the velocity gradient tensor.^[2] The viscous stress tensor plays a central role in the Navier-Stokes equations, governing the momentum balance in viscous flows by linking viscous forces to the fluid's deformation rate, which is essential for predicting phenomena like boundary layer development, drag in pipes, and energy dissipation in laminar regimes.^[1]^[2] Its effects dominate at low Reynolds numbers, where viscosity significantly influences flow behavior, but become negligible in high-Reynolds-number turbulent flows.^[2] Viscosity coefficients \mu and \lambda depend on fluid properties such as temperature and density, and for many liquids and gases, \lambda is small compared to \mu, often approximated using Stokes' hypothesis as \lambda = -\frac{2}{3}\mu.^[3]

Fundamentals

Viscous versus Elastic Stress

Elastic stress arises from reversible deformations in solids, where the material returns to its original shape upon removal of the applied force, and is directly proportional to the magnitude of the strain. This relationship is encapsulated in Hooke's law, originally proposed by Robert Hooke in 1678, which states that the stress \sigma is proportional to the strain \epsilon, expressed as \sigma = E \epsilon, where E is Young's modulus representing the material's stiffness.^[4]^[5] In contrast, viscous stress occurs in fluids and is dissipative, involving irreversible energy loss as heat due to the rate of deformation rather than the deformation itself. It is proportional to the strain rate, as seen in simple shear flow where the shear stress \tau is given by \tau = \mu \frac{du}{dy}, with \mu as the dynamic viscosity and \frac{du}{dy} as the velocity gradient.^[6]^[7] This behavior was first recognized by Isaac Newton in his 1687 Philosophiæ Naturalis Principia Mathematica, where he described fluid resistance as proportional to the velocity gradient across layers.^[8] The fundamental distinction highlights elastic stress as reversible and strain-dependent, storing potential energy, whereas viscous stress is irreversible and strain-rate-dependent, dissipating energy through friction between fluid layers.^[9] For instance, compressing and releasing a spring results in elastic rebound to its undeformed state, governed by Hooke's law, while applying shear to honey causes permanent flow without recovery, exemplifying viscous dissipation.^[5]^[7]

Physical Causes of Viscous Stress

Viscous stress arises macroscopically as an internal frictional resistance within a fluid, resulting from relative motion between adjacent fluid layers that exhibit velocity gradients. This friction manifests as a transfer of momentum perpendicular to the flow direction, effectively diffusing momentum across the fluid and opposing the shear.^[10] In this view, the stress is proportional to the rate of strain, representing the fluid's resistance to deformation under shear.^[11] At the microscopic level, the origins of viscous stress differ between gases and liquids. In gases, viscosity stems primarily from intermolecular collisions that transport momentum between layers moving at different speeds, as described by the kinetic theory of gases. According to this theory, the shear viscosity μ is approximated by the expression

\mu \approx \frac{1}{3} \rho \lambda \bar{v},

where ρ is the fluid density, λ is the mean free path of the molecules, and \bar{v} is the average thermal velocity; this relation was first derived by James Clerk Maxwell in 1860.^[12] In liquids, viscous stress originates from stronger molecular interactions, including short-range repulsive forces and shear-induced alignments of molecules, which hinder relative motion more effectively than in dilute gases. These effects are captured in models like the Lennard-Jones potential, where viscosity emerges from correlated atomic fluctuations and caging dynamics that dissipate energy during flow.^[13] Unlike elastic stress, which allows reversible recovery of stored energy, viscous stress is inherently irreversible, converting mechanical work into thermal energy through dissipation and thereby generating entropy in the system. This entropy production distinguishes viscous processes as non-equilibrium phenomena, governed by principles of irreversible thermodynamics where frictional losses increase the overall disorder.^[10]^[14] In the continuum approximation of fluid mechanics, viscous stress is modeled under the assumption that the fluid behaves as a continuous medium, valid when flow scales greatly exceed molecular dimensions—a condition known as the continuum hypothesis. This averaging over microscopic scales also relies on the no-slip boundary condition at solid-fluid interfaces, where viscous interactions ensure the fluid velocity matches the wall velocity, preventing slip due to adhesive forces.^[15]^[16]

Mathematical Definition

The Viscous Stress Tensor

In continuum mechanics, the viscous stress tensor represents the contribution to the internal forces in a fluid arising from viscous effects, such as friction between fluid layers during motion. The total Cauchy stress tensor \sigma_{ij}, which describes the state of stress at a point in the material, is decomposed into an isotropic pressure term and the viscous part:

\sigma_{ij} = -p \delta_{ij} + \tau_{ij},

where p is the hydrostatic pressure, \delta_{ij} is the Kronecker delta, and \tau_{ij} denotes the components of the viscous stress tensor.^[17]^[18] This decomposition separates the reversible, thermodynamic pressure effects from the dissipative viscous stresses that depend on the fluid's deformation rate.^[3] In Cartesian coordinates, the viscous stress tensor \tau_{ij} is a second-rank tensor expressed as a $3 \times 3 symmetric matrix, with diagonal elements corresponding to normal viscous stresses and off-diagonal elements to shear viscous stresses. These components capture the momentum transfer due to viscosity in three dimensions, influencing both extension/compression and shearing of fluid elements.^[18]^[3] Within continuum mechanics, the viscous stress tensor \tau_{ij} generally relates to the velocity field \mathbf{v} of the fluid through constitutive relations that link stress to kinematics, with specific forms depending on the fluid model (as explored in subsequent sections). Common notation uses \tau_{ij} for the viscous contribution, though some texts employ \varepsilon_{ij} to distinguish it from other stress components. For incompressible flows, where the fluid volume is conserved, the viscous stress tensor is traceless (\tau_{kk} = 0), ensuring it is purely deviatoric and focused on shape change without volumetric effects.^[17]^[3]

Symmetry of the Tensor

In standard continuum mechanics for fluids, the viscous stress tensor \tau is symmetric, satisfying \tau_{ij} = \tau_{ji}. This property follows directly from the conservation of angular momentum in the absence of distributed body torques or internal couples.^[17] Consider a small cubic fluid element; the net torque arising from the off-diagonal components of the stress tensor, such as (\tau_{yx} - \tau_{xy}) \Delta x \Delta y \Delta z, must equal the rate of change of the element's angular momentum, I \dot{\theta}, where I is the moment of inertia. In the limit as the element size approaches zero and assuming no volumetric torques, this balance requires \tau_{yx} = \tau_{xy}, with analogous relations for other components.^[17] More generally, the local angular momentum equation in the absence of body couples implies \epsilon_{ikj} \frac{\partial \tau_{jl}}{\partial x_k} = 0, which, for sufficiently smooth fields, enforces the symmetry \tau_{ij} = \tau_{ji}.^[3] This symmetric form is the standard assumption in the Navier-Stokes equations for most simple and complex fluids, where microscopic effects like particle spin are negligible.^[19] However, symmetry breaks down in generalized continuum theories, such as micropolar fluid models that incorporate microstructure and internal rotations, as in liquid crystals or suspensions; here, couple stresses introduce an asymmetric component to \tau.^[20] Asymmetry can also emerge in non-equilibrium thermodynamics, where shear stresses generate moments of internal forces that do not align with standard symmetry assumptions.^[21] In contexts involving rotational degrees of freedom, such as chiral active fluids or molecular liquids, a rotational viscosity \kappa parameterizes the antisymmetric part: \tau_{ij} - \tau_{ji} = 2\kappa \omega_{ij}, where \omega_{ij} = \frac{1}{2} (\partial_i u_j - \partial_j u_i) is the vorticity tensor.^[22] Such effects are relevant under external influences like magnetic fields in plasmas, where kinetic theory yields anisotropic transport that can support antisymmetric contributions, or in rotating frames via Coriolis terms that couple to non-standard viscous responses.^[23] The symmetry of \tau reduces its independent components from nine to six, which streamlines the specification of constitutive relations and simplifies numerical implementations in fluid dynamics simulations.^[17]

The Strain Rate Tensor

The strain rate tensor, also referred to as the rate-of-deformation tensor, is a fundamental kinematic quantity in fluid mechanics that quantifies the rate at which a fluid element deforms due to velocity variations. It is defined as the symmetric portion of the velocity gradient tensor, with components given by

e_{ij} = \frac{1}{2} \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right),

where \mathbf{v} = (v_1, v_2, v_3) is the velocity field and x_i are the spatial coordinates.^[18]^[24]^[25] This tensor arises from the linear approximation of the relative velocity between two nearby fluid particles, capturing the deformative motion while excluding rigid-body rotation.^[25] The velocity gradient tensor decomposes into this symmetric strain rate tensor and an antisymmetric vorticity tensor.^[24] The components of the strain rate tensor describe specific aspects of fluid deformation. The diagonal elements, such as e_{11} = \frac{\partial v_1}{\partial x_1}, represent the rates of linear extension or compression along the respective coordinate directions.^[18]^[3] The off-diagonal elements, for instance e_{12} = \frac{1}{2} \left( \frac{\partial v_1}{\partial x_2} + \frac{\partial v_2}{\partial x_1} \right), quantify the rates of shearing deformation in the corresponding planes.^[24] The trace of the tensor, e_{kk} = \frac{\partial v_k}{\partial x_k} = \nabla \cdot \mathbf{v}, corresponds to the dilatation rate, or the volumetric expansion rate per unit volume of the fluid element.^[24]^[26] Due to its symmetry, e_{ij} = e_{ji}, the tensor has at most six independent components in three dimensions.^[18] In practical flows, the strain rate tensor links directly to the kinematics of the velocity field. For example, in a simple shear flow with velocity \mathbf{v} = (u(y), 0, 0), the tensor simplifies such that the only non-zero off-diagonal component is e_{xy} = e_{yx} = \frac{1}{2} \frac{du}{dy}, representing the shear rate.^[18] This definition presupposes a differentiable velocity field \mathbf{v}(\mathbf{x}, t) derived from the Navier-Stokes equations governing the fluid motion.^[24] The formulation relies on the infinitesimal strain approximation, which holds for flows at low Mach numbers where compressibility effects are negligible and linearizations of velocity differences are valid.^[25]^[27]

Newtonian Fluids

General Newtonian Media

In general Newtonian media, the viscous stress tensor \tau_{ij} is related to the strain rate tensor e_{kl} through a linear constitutive equation that does not presuppose isotropy:

\tau_{ij} = \mu_{ijkl} e_{kl},

where \mu_{ijkl} is a fourth-rank viscosity tensor with up to 81 components in its most general form. This relation assumes a linear dependence between the viscous stresses and the rate of strain, as well as local thermodynamic equilibrium, meaning the constitutive law depends only on the local state variables without memory effects or nonlocal influences.^[28]^[3] Due to the symmetry of both the viscous stress tensor (\tau_{ij} = \tau_{ji}) and the strain rate tensor (e_{kl} = e_{lk}), the number of independent components in \mu_{ijkl} reduces from 81 to 36. If the viscosity tensor additionally satisfies major symmetry (\mu_{ijkl} = \mu_{klij}), which aligns with the existence of a strain energy potential in analogous elastic contexts, the number further decreases to 21 independent components. This 21-component form applies to the lowest symmetry crystal classes, such as triclinic, while higher symmetries like cubic reduce the independent components to as few as three, reflecting the material's directional properties.^[29]^[28] This general tensorial framework extends the foundational work of George Gabriel Stokes, who in 1845 proposed a linear relation for the viscous stresses in isotropic fluids, thereby generalizing the concept to anisotropic media like liquid crystals or polycrystalline materials where directional variations in viscosity arise from molecular orientation or structural alignment.^[30]

Isotropic Newtonian Case

In isotropic Newtonian fluids, the material response to deformation is independent of direction, leading to a simplified form of the viscous stress tensor that depends on only two scalar viscosity coefficients. This reduction occurs because isotropy and tensor symmetry constrain the general fourth-order viscosity tensor, which could have up to 21 independent components in anisotropic cases, to just these scalars: the shear viscosity \mu and the second viscosity coefficient \lambda. The resulting expression for the viscous stress tensor \tau_{ij} is

\tau_{ij} = 2\mu \, e_{ij} + \lambda \, (e_{kk}) \, \delta_{ij},

where e_{ij} is the symmetric strain rate tensor, e_{kk} = \nabla \cdot \mathbf{v} is its trace (the velocity divergence), and \delta_{ij} is the Kronecker delta.

\) The second coefficient \(\lambda

relates to the bulk viscosity \zeta, which governs resistance to uniform expansion or compression, via \lambda = \zeta - \frac{2}{3} \mu.() For incompressible flows, where \nabla \cdot \mathbf{v} = 0, the trace term vanishes, yielding \tau_{ij} = 2\mu \, e_{ij}. In this limit, the tensor becomes traceless, as the bulk contribution is absent, and only the shear viscosity \mu determines the stress response.() Common isotropic Newtonian fluids exhibit characteristic shear viscosities that illustrate the scale of viscous effects. For water at 20°C, \mu \approx 10^{-3} Pa·s, reflecting its relatively high resistance to shear compared to gases.

\) In contrast, dry air at [standard temperature and pressure](/page/Standard_temperature_and_pressure) (approximately 20°C and 1 atm) has \(\mu \approx 1.8 \times 10^{-5}

Pa·s, orders of magnitude lower, which facilitates rapid momentum transfer in atmospheric flows.()

Shear and Bulk Viscous Stress

In Newtonian fluids, the viscous stress tensor \tau_{ij} can be decomposed into a deviatoric (shear) component \tau_{ij}^{(d)} and a volumetric (bulk) component \tau_{ij}^{(v)}, such that \tau_{ij} = \tau_{ij}^{(d)} + \tau_{ij}^{(v)}. The deviatoric part is given by \tau_{ij}^{(d)} = 2\mu e_{ij}^{(d)}, where \mu is the shear viscosity coefficient and e_{ij}^{(d)} is the traceless deviatoric strain rate tensor, representing the rate of shape deformation without volume change. The volumetric part is \tau_{ij}^{(v)} = \zeta (\nabla \cdot \mathbf{v}) \delta_{ij}, where \zeta is the bulk viscosity coefficient and \nabla \cdot \mathbf{v} is the divergence of the velocity field, capturing resistance to uniform expansion or compression.^[31] Physically, the shear component \tau_{ij}^{(d)} arises from the tangential forces that cause relative sliding between adjacent fluid layers, leading to momentum transfer and flow resistance in shear-dominated motions such as pipe flow or boundary layers. In contrast, the bulk component \tau_{ij}^{(v)} opposes isotropic volumetric changes, such as those during rapid compression or rarefaction, by generating normal stresses proportional to the rate of volume variation. In monatomic gases, bulk viscosity is often negligible (\zeta = 0) under the Stokes hypothesis, which assumes no internal relaxation processes beyond shear, simplifying models for dilute ideal gases.^[32]^[33] This decomposition aligns with the isotropic Newtonian constitutive equation, where the full viscous stress combines both components to relate strain rate to stress linearly. The shear viscosity \mu governs the magnitude of tangential stresses in applications involving directional flow, while bulk viscosity \zeta influences dilatational effects, though \mu remains constant in true Newtonian behavior. Bulk viscosity is rarely significant in simple fluids like water or air but plays a critical role in complex systems such as polymers, where it contributes to energy dissipation during deformation. It is typically measured through acoustic methods, such as analyzing sound attenuation in solutions, where excess damping beyond classical predictions reveals \zeta values.^[31]^[34]

Extensions

Non-Newtonian Fluids

Non-Newtonian fluids exhibit a viscous stress tensor that deviates from the linear proportionality to the rate-of-strain tensor characteristic of Newtonian fluids, often manifesting as nonlinear dependencies on shear rate, stress history, or deformation magnitude. In these materials, the constitutive relation generalizes to forms where the apparent viscosity μ varies with the magnitude of the rate-of-strain tensor ė, its invariants, or temporal evolution, leading to complex flow behaviors in engineering and biological systems.^[35] A foundational model for such nonlinearity is the power-law fluid, where the viscous stress tensor is given by

\boldsymbol{\tau} = K |\dot{\mathbf{e}}|^{n-1} \dot{\mathbf{e}},

with K as the consistency index and n as the flow behavior index; when n \neq 1, the fluid displays shear-rate-dependent viscosity. For n < 1, the fluid is shear-thinning (pseudoplastic), where viscosity decreases under increasing shear, as observed in blood and polymer solutions, facilitating easier flow in high-shear regions like capillaries. Conversely, for n > 1, shear-thickening (dilatant) behavior occurs, with viscosity increasing under stress, exemplified by cornstarch-water suspensions used in impact-resistant materials.^[36]^[37]^[38] Viscoelastic non-Newtonian fluids incorporate elastic memory effects, where the stress tensor depends on both the current deformation rate and past history, often modeled by differential equations balancing viscous and elastic components. The Maxwell model, a seminal viscoelastic constitutive relation, is expressed as

\boldsymbol{\tau} + \lambda \frac{D \boldsymbol{\tau}}{Dt} = 2 \mu \dot{\mathbf{e}},

with \lambda as the relaxation time and \frac{D}{Dt} denoting an objective time derivative (typically upper-convected for polymeric flows); this captures phenomena like stress relaxation in polymer melts. An extension, the Oldroyd-B model, combines solvent viscosity \eta_s with a polymeric contribution, yielding

\boldsymbol{\tau} + \lambda_1 \stackrel{\nabla}{\boldsymbol{\tau}} = 2 \eta_p \dot{\mathbf{e}} + 2 \lambda_2 \eta_p \stackrel{\nabla}{\dot{\mathbf{e}}},

where \stackrel{\nabla}{}{\cdot} is the upper-convected derivative, \lambda_1 and \lambda_2 are relaxation and retardation times, and \eta_p is the polymeric viscosity; this model better predicts instabilities in dilute polymer solutions.^[39]^[40] Yield-stress fluids, such as Bingham plastics, introduce a threshold stress below which the material behaves rigidly, with the viscous stress tensor activating only when the stress magnitude exceeds \tau_0:

\boldsymbol{\tau} = \tau_0 \frac{\dot{\mathbf{e}}}{|\dot{\mathbf{e}}|} + \mu \dot{\mathbf{e}} \quad \text{for} \quad |\boldsymbol{\tau}| > \tau_0,

and \boldsymbol{\tau} = 0 otherwise; this describes toothpaste or drilling muds, where flow initiates only after yielding. In rheology, these models enable analysis of non-Newtonian flows in applications like food processing, biomedical devices, and enhanced oil recovery, where the generalized stress tensor informs predictions of wall shear stress and flow instabilities.^[41]^[42]

Anisotropic and Advanced Cases

In anisotropic fluids, the viscous stress tensor exhibits directional dependence due to the material's internal structure, such as in liquid crystals where molecular orientation influences flow resistance. Unlike isotropic cases, the fourth-rank viscosity tensor \mu_{ijkl} couples velocity gradients to stress in a way that varies with the director field \mathbf{n}, leading to up to five independent scalar viscosity coefficients in nematic phases as described by the Ericksen-Leslie-Parodi theory.^[43] For example, in nematic liquid crystals, these coefficients account for phenomena like tumbling or flow alignment, where the stress tensor \tau_{ij} = \mu_{ijkl} \partial_k v_l incorporates both symmetric strain-rate contributions and antisymmetric parts tied to rotation.^[44] This anisotropy arises from the elongated molecular shape, enabling applications in display technologies and soft robotics, but complicating flow predictions due to orientation-dependent dissipation.^[45] In relativistic contexts, the viscous stress generalizes to a four-tensor \pi^{\mu\nu} in the energy-momentum tensor T^{\mu\nu} = (\epsilon + p) u^\mu u^\nu + p g^{\mu\nu} + \pi^{\mu\nu}, where it represents deviations from perfect fluid behavior in high-speed or high-density regimes like heavy-ion collisions.^[46] The Eckart theory provides a first-order approximation, expressing \pi^{\mu\nu} = -2\eta \sigma^{\mu\nu} - \zeta \theta \Delta^{\mu\nu} with shear viscosity \eta, bulk viscosity \zeta, shear tensor \sigma^{\mu\nu}, expansion \theta, and projector \Delta^{\mu\nu}, but it suffers from acausality and instabilities at short wavelengths.^[47] To address these, the Israel-Stewart theory introduces second-order dynamics, treating \pi^{\mu\nu} as an independent variable evolving via relaxation equations like \tau_\pi \Delta^{\mu\alpha} \Delta^{\nu\beta} u^\lambda \nabla_\lambda \pi_{\alpha\beta} + \pi^{\mu\nu} = -2\eta \sigma^{\mu\nu}, ensuring hyperbolic propagation and stability for numerical simulations in relativistic hydrodynamics.^[48] This framework has been pivotal in modeling quark-gluon plasma, where viscous effects influence collective flow observables.^[49] Beyond these, turbulent flows introduce effective stresses analogous to viscous ones through the Reynolds stress tensor -\rho \overline{u_i' u_j'}, which mimics molecular viscosity in the Navier-Stokes equations but originates from momentum transport by velocity fluctuations rather than intermolecular collisions.^[50] This analogy allows closure models like Boussinesq's eddy viscosity hypothesis, \tau_{ij}^{\rm turb} = - \nu_t (\partial_i \overline{u}_j + \partial_j \overline{u}_i), but it does not represent true microscopic viscosity and requires turbulence modeling for accuracy in engineering applications.^[51] In micropolar fluids, which model suspensions with microstructure like blood or polymeric solutions, the stress tensor becomes asymmetric to account for micro-rotations \omega_k, yielding \tau_{ij} = \lambda \epsilon_{kk} \delta_{ij} + \mu (\epsilon_{ij} + \epsilon_{ji}) + \kappa (\epsilon_{ji} - \epsilon_{ij}) + higher-order terms, where \kappa captures the antisymmetric part linked to spin inertia.^[20] This asymmetry distinguishes micropolar theory from classical continua, enabling predictions of couple stresses in flows with suspended particles.^[52] Recent advances in computational fluid dynamics (CFD) post-2000 have enhanced modeling of anisotropic viscosity through coupled orientation-transport equations and adaptive meshing, particularly for nematic liquid crystals and composite materials. For instance, finite-volume methods incorporating the full Leslie-Ericksen viscosities simulate defect dynamics and flow instabilities with improved accuracy, as seen in simulations of microfluidic devices where anisotropy amplifies shear banding.^[45] Anisotropic mesh adaptation techniques, evolving since the early 2000s, refine grids along principal strain directions to capture viscosity gradients efficiently, reducing computational cost by orders of magnitude in high-fidelity viscous flow predictions for aerospace and geophysics. These developments, often integrated with machine learning for parameter estimation, have enabled real-time optimization in anisotropic media simulations. As of 2025, further integrations of artificial intelligence, including large language model-based agents for OpenFOAM and reinforcement learning for anisotropic p-adaptation, have advanced real-time simulations and error estimation in anisotropic viscous flows.^[53]^[54]^[55]^[56]

References

[1]
8: Introduction to Viscous Flows - Engineering LibreTexts
May 1, 2022 · This stress tensor consists of the various stresses that can occur on an element of fluid. ... define the viscous stress tensor. CH-7.1.png Figure ...<|control11|><|separator|>
[2]
Viscous Stress - an overview | ScienceDirect Topics
Viscous stress is defined as the stress caused by molecular diffusion in a fluid, resulting in the transfer of momentum when fluid molecules move into ...
[3]
[PDF] CHAPTER 6 Viscosity & The Stress Tensor
Stress is a force per unit area, with normal and shear components. The stress tensor (σij) is defined as Σi(n) = σij nj. Viscosity is a fluid's resistance to ...
[4]
Hooke's law: applications of a recurring principle
Robert Hooke was a 17th century British physicist. In 1678, he proposed Hooke's Law in his essay “Ut tensio sic vis” (9) by stating that “The power of any ...
[5]
Elastic Region | MATSE 81: Materials In Today's World
Mathematically, this can be written as σ=Eε, and more generally is known as a form of Hooke's law. E is the proportionality constant and is called the modulus ...
[6]
Viscoelastic Material - an overview | ScienceDirect Topics
For an elastic material, the stress is directly proportional to the strain according to Hooke's law (Eq. 2.1). In a viscous fluid (i.e., Newtonian fluid), the ...
[7]
[PDF] Viscosity
Thus the velocity varies linearly across the gap and the shear stress τ required to maintain an upper plate velocity of U for a gap of h is given by μU/h. We ...
[8]
Revisiting viscosity from macroscopic to nanoscale regimes
### Summary of Newton's Description of Viscosity in Principia 1687
[9]
[PDF] Thermodynamics of viscoelastic fluids: the temperature equation.
We discussed both the irreversible (dissipative) part and the reversible (elastic) part of the temperature equation. The reversible part is closely related to.
[10]
Liquid Viscosity - Galileo
Viscosity is, essentially, fluid friction. Like friction between moving solids, viscosity transforms kinetic energy of (macroscopic) motion into heat energy.
[11]
Viscous Fluid - an overview | ScienceDirect Topics
A viscous fluid resists flow, exhibiting different velocities among layers due to shear forces, with internal friction opposing relative motion.
[12]
New Foundations Laid by Maxwell and Boltzmann (1860 - 1872)
Aug 24, 2001 · Maxwell showed that if the kinetic theory of gases is correct, both expectations will be wrong, because the mechanism that produces viscosity is ...
[13]
Microscopic Origins of the Viscosity of a Lennard-Jones Liquid
Aug 11, 2022 · Abstract. Unlike crystalline solids or ideal gases, transport properties remain difficult to describe from a microscopic point of view in ...
[14]
[PDF] On the Irreversible Production of Entropy - CORE
The causes contributing to the irreversible production of entropy include the dissipation of mechanical energy by friction and by viscous forces, the transfer ...<|separator|>
[15]
[PDF] The No-Slip Boundary Condition in Fluid Mechanics
A moving fluid in contact with a solid body cannot have velocity relative to the body. Even though the question whether there is slip has been ...
[16]
No Slip Condition - an overview | ScienceDirect Topics
The no-slip condition is defined as the requirement that the velocity of a viscous fluid at the interface with a solid body matches the velocity of the ...
[17]
[PDF] Equation of Motion for Viscous Fluids - MIT
A solution yields the velocity components and pressure at the boundaries, from which one obtains the stress tensor components via equation (42) [or. (43)] and ...
[18]
[PDF] Chapter 3 The Stress Tensor for a Fluid and the Navier Stokes ...
The stress tensor for a fluid can be put in diagonal form, and for a fluid at rest, it must have only diagonal terms, which is the static pressure.
[19]
More considerations about the symmetry of the stress tensor of fluids
Jul 25, 2024 · There are simple gas flows in which the symmetrical stress tensor contradicts the mechanism of gas viscosity. The usual proofs of the symmetry ...
[20]
[PDF] Theory of Micropolar Fluids - DTIC
The theory of microfluids introduced by Eringen ' deals with a class of fluids which exhibit certain microscopic effects arising from the local structure ...
[21]
More considerations about the symmetry of the stress tensor of fluids
Jul 28, 2024 · This moment of internal forces may lead to nonsymmetry of the stress tensor and is, in general, nonzero as long as shear stress exists. A ...
[22]
Rotational viscosity of fluids composed of linear molecules
Jun 10, 2008 · Thus, the rotational viscosity is a transport coefficient that describes the decay of the antisymmetric stress after an infinitely small ...
[23]
[PDF] arXiv:0905.0086v3 [physics.plasm-ph] 3 Nov 2011
Nov 3, 2011 · The model by Braginskii for the viscous stress tensor is used to determine the shear and gyroviscous forces acting within a toroidally confined ...
[24]
None
### Summary of Rate of Strain Tensor from Kinematics of Fluid Motion
[25]
[PDF] MIT OpenCourseWare http://ocw.mit.edu Electromechanical ...
We can arrive at our definition of the strain-rate tensor in a more precise fashion by considering the relative velocity of fluid at two adjacent points in ...<|control11|><|separator|>
[26]
[PDF] 22.581 Module 7: Stress, Viscosity, and The Navier-Stokes Equations
Volumetric strain rate = Rate of change of the volume per unit volume. The infinitesimal volume δV = δxδyδz. Volumetric Strain Rate = 1. δV. D. Dt. (δV) = 1.
[27]
[PDF] A Physical Introduction to Fluid Mechanics
Dec 19, 2013 · Viscous flows are flows where viscous stresses exert significant forces on the fluid, and viscous energy dissipation is impor- tant. Here ...
[28]
[PDF] arXiv:nlin/0312006v1 [nlin.CD] 2 Dec 2003
Dec 2, 2003 · ijmn + ηA ijmn . (9). So, generically, the “viscosity tensor” has 36 independent components, of which 21 belong to the pair-symmetric part. ηS.
[29]
[PDF] The Shear Viscosity in Anisotropic Phases - arXiv
Nov 5, 2015 · rotational invariance, has 21 independent components (when the field theory lives in 3+1 ... ate components of the viscosity tensor become ...
[30]
[PDF] On the theories of the internal friction of fluids in motion
STOKESs, M.A., Fel- low of Pembroke College. [Read April 14, 1845.] THE equations of Fluid Motion commonly employed depend upon the fundamental hypothesis.
[31]
[PDF] LECTURE NOTES ON INTERMEDIATE FLUID MECHANICS
Jul 1, 2025 · These are lecture notes for AME 60635, Intermediate Fluid Mechanics, taught in the De- partment of Aerospace and Mechanical Engineering of ...
[32]
Understanding Shear Stress in Fluids and Its Effects
The shear stress of a fluid can be defined as a unit area amount of force acting on the fluid parallel to a very small element of the surface.
[33]
[PDF] A brief introduction to bulk viscosity of fluids - arXiv
Mar 15, 2023 · This hypoth- esis assumes bulk viscosity, ib, to be identically zero. The Stokes' hypothesis is a reasonable approximation for commonly observed ...
[34]
Bulk viscosity of polymer solutions - AIP Publishing
Oct 1, 1977 · Particular application is made to the theory of sound attenuation in polymer solutions as one potential means for measuring the bulk viscosity.
[35]
nonNewtonian Fluids - an overview | ScienceDirect Topics
The viscous stress tensor is such as σxy = η γ ˙ . The viscosity η of these fluids can decrease or increase when γ ˙ is increased; the fluid is then said to ...
[36]
Power Law Model - an overview | ScienceDirect Topics
The power law model is defined as a non-Newtonian relation used in engineering calculations, characterized by the equation η = m γ ˙ n − 1, where m is the ...
[37]
Viscosity of Newtonian and Non-Newtonian Fluids - RheoSense
Shear thinning fluids, also known as pseudo-plastics, are ubiquitous in industrial and biological processes. Common examples include ketchup, paints and blood.
[38]
Non-Newtonian Models | Materials - SimScale
May 22, 2025 · Non-Newtonian models model fluids with non-linear shear stress/rate relations, where viscosity is not fixed. Examples include ketchup, custard, ...<|separator|>
[39]
The relationship between viscoelasticity and elasticity - PMC
In contrast to elastic solids, viscoelastic liquids such as polymer solutions or emulsions do not possess a permanent reference state. Instead, they exhibit a ...
[40]
[PDF] Oldroyd B, and not A?
Oct 20, 2021 · The Oldroyd-B model has a constant shear viscosity, a quadratic first normal stress difference and zero second normal stress difference,.
[41]
Bingham plastic – Knowledge and References - Taylor & Francis
The Bingham–Plastic model includes both yield stress and a limiting viscosity at finite shear rate (Bingham 1916) where is the shear stress, is the yield point ...
[42]
Rheology - Soft Matter Mechanics Laboratory
Rheology is the study of the deformation and flow of matter, distinguishing between Newtonian and non-Newtonian fluids based on their viscosity behavior.
[43]
Viscosity coefficients for anisotropic, nematic fluids based on ...
The constitutive relation for the dynamic stress tensor T of an incompressible anisotropic fluid with velocity v is given by the following expression linear in ...
[44]
Anisotropic Fluids - an overview | ScienceDirect Topics
In nematic liquid crystals, flow instabilities mostly originate from their anisotropic properties through viscous forces and torques that couple the velocity ...
[45]
[PDF] Computational fluid dynamics for nematic liquid crystals
Nov 13, 2015 · In this paper, we use a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress.
[46]
First-Order General-Relativistic Viscous Fluid Dynamics | Phys. Rev. X
May 24, 2022 · We see that T Eckart μ ν corresponds to the choices where the bulk scalar Π = - ζ ∇ μ u μ , the shear-stress tensor is given by π μ ν = - 2 η σ ...Missing: μν | Show results with:μν
[47]
Relativistic Non-Perfect Fluids - Oxford Academic
S μ ν = π μ ν + Π h μ ν ,. (6.11). where πμν is called the anisotropic stress tensor, Π is the viscous bulk pressure (or “dynamic pressure” or “non-equilibrium ...
[48]
[PDF] General-relativistic viscous fluids - Indico Global
A =heat flow; π = viscous shear stress. p = p(ϱ, n). 5/20. Page 18. From ... For the Israel-Stewart theory: Stability holds (Hiscock-Lindblom, '83; Olson ...
[49]
[2104.14197] Hybrid model with viscous relativistic hydrodynamics
Apr 29, 2021 · Since the applicability of the Israel-Stewart hydrodynamics assumes the perturbative character of the viscous stress tensor, \pi^{\mu\nu}, which ...Missing: Eckart four- π^ μν
[50]
8.3: Calculation of Reynolds Stress - Geosciences LibreTexts
Nov 11, 2024 · Reynolds stress can be calculated from direct measurements, by analogy with molecular viscosity, or from statistical turbulence theory.
[51]
Reynolds Stress Model - an overview | ScienceDirect Topics
Instead of modelling the turbulence viscosity, Reynolds stress models use the Reynolds stress transport equations to specify the Reynolds stress tensor in the ...<|separator|>
[52]
Analytical approach for micropolar fluid flow in a channel with ...
Sep 15, 2023 · There are two asymmetric tensors in the theory of micropolar fluids. The first tensor is the asymmetric stress tensor T = T ij and the ...
[53]
The new paradigm of computational fluid dynamics - AIP Publishing
Aug 5, 2025 · Computational fluid dynamics (CFD) integrated with machine learning (ML) is an emerging and rapidly growing research field.