Volume Viscosity
Definition
$$ \zeta := \lambda + {\frac{ 2 }{ 3 }} \mu $$ In a three-dimensional fluid system, for the two Lamé parameters $\lambda$, $\mu$ the constant $\zeta$ defined as above is called the bulk viscosity.
Explanation
Bulk viscosity arises particularly when dealing with compressible fluids, and in a simplified form of the compressible Navier–Stokes equations it appears as $\xi = \zeta / \rho$, expressed as the bulk viscosity divided by the density, i.e., the bulk kinematic viscosity.
As can be seen from the derivation, in the two-dimensional case it becomes $\zeta = \lambda + \mu$.
Derivation
$\zeta$ is a definition, so it does not need to be derived, but it is worth examining algebraically why it takes such an unusual form. Since it is simply a weighted sum of the two parameters, it does not carry profound physical significance; it should be regarded mainly as a notational simplification.
Cauchy stress tensor and Lamé parameters: In particular, in three-dimensional space consider the velocity field at time $t$ and spatial coordinate $\mathbf{x} = \left( x_{1} , x_{2} , x_{3} \right)$ represented by the velocity vector as above. Let this fluid be a Newtonian fluid with viscosity and compressibility. The isotropic Cauchy stress tensor $\sigma$ can be written in terms of the symmetrized gradient $\varepsilon$ as follows. $$ \sigma = - p I + 2 \mu \varepsilon + \lambda \tr \left( \varepsilon \right) I $$
Viewing the Cauchy stress tensor $\sigma$ from the perspective of pressure $p$ naturally brings out $\zeta$. Since $I$ is the identity matrix, the trace of $\sigma$ is $$ \tr \left( \sigma \right) = - 3 p + 2 \mu \tr \left( \varepsilon \right) + 3 \lambda \tr \left( \varepsilon \right) $$ Rearranging this for $p$ gives $$ \begin{align*} p =& - {\frac{ 1 }{ 3 }} \tr \left( \sigma \right) + \left( {\frac{ 2 }{ 3 }} \mu + \lambda \right) \tr \left( \varepsilon \right) \\ =& - {\frac{ 1 }{ 3 }} \tr \left( \sigma \right) + \zeta \tr \left( \varepsilon \right) \end{align*} $$ As shown, $\zeta$ only appears by grouping the coefficients of $\tr \left( \varepsilon \right)$.
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