An element of a flowing liquid or gas will endure forces from the surrounding fluid, including
viscous stress forces that cause it to gradually deform over time. These forces can be mathematically
first order approximated by a
viscous stress tensor, usually denoted by \tau. The deformation of a fluid element, relative to some previous state, can be first order approximated by a
strain tensor that changes with time. The time derivative of that tensor is the
strain rate tensor, that expresses how the element's deformation is changing with time; and is also the
gradient of the velocity
vector field v at that point, often denoted \nabla v. The tensors \tau and \nabla v can be expressed by 3×3
matrices, relative to any chosen
coordinate system. The fluid is said to be Newtonian if these matrices are related by the
equation \boldsymbol{\tau} = \boldsymbol{\mu} (\nabla v) where \mu is a fixed 3×3×3×3 fourth order tensor that does not depend on the velocity or stress state of the fluid.
Incompressible isotropic case For an
incompressible and isotropic Newtonian fluid in
laminar flow only in the direction (i.e. where viscosity is isotropic in the fluid), the shear stress is related to the strain rate by the simple
constitutive equation \tau = \mu\ \frac{ \mathrm{d} u }{\ \mathrm{d} y\ } where • \tau is the
shear stress ("
skin drag") in the fluid, • \mu is a scalar constant of proportionality, the
dynamic viscosity of the fluid • \frac{du}{dy} is the
derivative in the direction , normal to , of the
flow velocity component that is oriented along the direction . In case of a general 2‑D incompressibile flow in the plane , the Newton constitutive equation become: \tau_{xy} = \mu \left( \frac{\partial u}{\partial y} +\frac{\partial v}{\partial x} \right) where: • \tau_{xy} is the
shear stress ("
skin drag") in the fluid, • \frac{\ \partial u}{\partial y} is the
partial derivative in the direction of the
flow velocity component that is oriented along the direction . • \frac{\partial v}{\ \partial x } is the partial derivative in the direction of the flow velocity component that is oriented along the direction . We can now generalize to the case of an
incompressible flow with a general direction in the 3‑D space, the above constitutive equation becomes \tau_{ij} = \mu \left(\frac{\ \partial v_i}{\partial x_j} + \frac{\ \partial v_j}{\partial x_i} \right) where • x_j is the jth spatial coordinate • v_i is the fluid's velocity in the direction of axis i • \tau_{ij} is the j-th component of the stress acting on the faces of the fluid element perpendicular to axis i. It is the ij-th component of the shear stress tensor or written in more compact tensor notation \boldsymbol{\tau} = \mu \left(\ \nabla\mathbf{u} + \nabla\mathbf{u}^\top \right) where \nabla \mathbf{u} is the flow velocity gradient. An alternative way of stating this constitutive equation is: where \boldsymbol{\varepsilon} = \tfrac{1}{2} \left(\ \mathbf{\nabla u} + \mathbf{\nabla u}^\top \right) is the rate-of-
strain tensor. So this decomposition can be made explicit as: This constitutive equation is also called the
Newton law of viscosity. The total
stress tensor \boldsymbol{\sigma} can always be decomposed as the sum of the
isotropic stress tensor and the
deviatoric stress tensor (\boldsymbol \sigma' ): \boldsymbol \sigma = \tfrac{1}{3} \operatorname{tr}\left( \boldsymbol \sigma \right)\ \mathbf I + \boldsymbol \sigma' where \mathbf{I} is the identity tensor. In the incompressible case, the isotropic stress is simply proportional to the thermodynamic
pressure p: p = - \tfrac{1}{3} \operatorname{tr}\left( \boldsymbol \sigma \right) = - \tfrac{1}{3} \sum_k \sigma_{kk}\ , and the deviatoric stress is coincident with the shear stress tensor \boldsymbol \tau: \boldsymbol \sigma' = \boldsymbol \tau = \mu \left(\ \nabla \mathbf{u} + \left( \nabla\mathbf{u} \right)^\top \right) ~. The stress
constitutive equation then becomes \sigma_{ij} = - p\ \delta_{ij} + \mu \left( \frac{\ \partial v_i}{\ \partial x_j} + \frac{\ \partial v_j}{\ \partial x_i} \right) or written in more compact tensor notation \boldsymbol{\sigma} = - p\ \mathbf{I} + \mu \left(\ \nabla\mathbf{u} + \left(\nabla \mathbf{u}\right)^\top \right) ~.
General compressible case The Newton's constitutive law for a compressible flow results from the following assumptions on the Cauchy stress tensor: \boldsymbol \sigma = - \left[ p - \left(\lambda + \tfrac23 \mu\right) \left(\nabla\cdot\mathbf{u}\right) \right] \mathbf I + \mu \left(\nabla\mathbf{u} + \left( \nabla\mathbf{u} \right)^\mathrm{T} - \tfrac23 \left(\nabla\cdot\mathbf{u}\right)\mathbf I\right) Introducing the
bulk viscosity \zeta, \zeta \equiv \lambda + \tfrac23 \mu , we arrive to the linear
constitutive equation in the form usually employed in
thermal hydraulics: \boldsymbol \sigma = -p \mathbf I + \mu \left(\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T}\right) + \left(\zeta - \frac 2 3 \mu \right) (\nabla\cdot\mathbf{u}) \mathbf I. Note that in the compressible case the pressure is no more proportional to the
isotropic stress term, since there is the additional bulk viscosity term: p = - \frac 1 3 \operatorname{tr} (\boldsymbol \sigma) + \zeta (\nabla\cdot\mathbf{u}) and the
deviatoric stress tensor \boldsymbol \sigma' is still coincident with the shear stress tensor \boldsymbol \tau (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which is proportional to the shear viscosity: \boldsymbol \sigma' = \boldsymbol \tau = \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right] Note that the incompressible case correspond to the assumption that the pressure constrains the flow so that the volume of
fluid elements is constant:
isochoric flow resulting in a
solenoidal velocity field with \nabla \cdot \mathbf{u} = 0. So one returns to the expressions for pressure and deviatoric stress seen in the preceding paragraph. Both bulk viscosity \zeta and dynamic viscosity \mu need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these
transport coefficient in the
conservation variables is called an
equation of state. Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the
dispersion. In some cases, the
second viscosity \zeta can be assumed to be constant in which case, the effect of the volume viscosity \zeta is that the mechanical pressure is not equivalent to the thermodynamic
pressure: as demonstrated below. \nabla\cdot(\nabla\cdot \mathbf u)\mathbf I=\nabla (\nabla \cdot \mathbf u), \bar{p} \equiv p - \zeta \, \nabla \cdot \mathbf{u} , However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves, where second viscosity coefficient becomes important) by explicitly assuming \zeta = 0. The assumption of setting \zeta = 0 is called as the
Stokes hypothesis. The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory; for other gases and liquids, Stokes hypothesis is generally incorrect. Finally, note that Stokes hypothesis is less restrictive that the one of incompressible flow. In fact, in the incompressible flow both the bulk viscosity term, and the shear viscosity term in the divergence of the flow velocity term disappears, while in the Stokes hypothesis the first term also disappears but the second one still remains.
For anisotropic fluids More generally, in a non-isotropic Newtonian fluid, the coefficient \mu that relates internal friction stresses to the
spatial derivatives of the velocity field is replaced by a nine-element
viscous stress tensor \mu_{ij}. There is general formula for friction force in a liquid: The vector
differential of friction force is equal the viscosity tensor increased on
vector product differential of the area vector of adjoining a liquid layers and
rotor of velocity: d \mathbf{F} = \mu _ {ij} \, d\mathbf{S} \times\nabla\times \, \mathbf {u} where \mu _ {ij} is the viscosity
tensor. The diagonal components of viscosity tensor is molecular viscosity of a liquid, and not diagonal components –
turbulence eddy viscosity. == Newton's law of viscosity ==