Volume viscosity

Volume viscosity (also called second coefficient of viscosity or dilatational viscosity or bulk viscosity) represents the irreversible resistance, over and above the reversible resistance caused by isentropic bulk modulus, to a change of volume of fluid . The volume viscosity becomes important only for such effects where fluid compressibility is essential. Volume viscosity is mainly related to the rotational and vibrational energy of the molecules. It is zero for monatomic gases at low density, but can be large for fluids with larger molecules. The volume viscosity is important in describing sound attenuation (as in e.g. Stokes's law), and the absorption of sound energy into the fluid depends on the sound frequency i.e. the rate of fluid compression and expansion. Volume viscosity is also important in describing the fluid dynamics of liquids containing gas bubbles. For an incompressible liquid the volume viscosity is superfluous, and does not appear in the equation of motion. Volume viscosity (also called second coefficient of viscosity or dilatational viscosity or bulk viscosity) represents the irreversible resistance, over and above the reversible resistance caused by isentropic bulk modulus, to a change of volume of fluid . The volume viscosity becomes important only for such effects where fluid compressibility is essential. Volume viscosity is mainly related to the rotational and vibrational energy of the molecules. It is zero for monatomic gases at low density, but can be large for fluids with larger molecules. The volume viscosity is important in describing sound attenuation (as in e.g. Stokes's law), and the absorption of sound energy into the fluid depends on the sound frequency i.e. the rate of fluid compression and expansion. Volume viscosity is also important in describing the fluid dynamics of liquids containing gas bubbles. For an incompressible liquid the volume viscosity is superfluous, and does not appear in the equation of motion. The negative-one-third of the trace of the Cauchy stress tensor at the equilibrium is often identified with the thermodynamic pressure, which only depends upon the equilibrium state potentials like temperature and density (equation of state). In general, the trace of the stress tensor is the sum of thermodynamic pressure contribution and another contribution which is proportional to the divergence of the velocity field. This coefficient of proportionality is called volume viscosity. Common symbols for volume viscosity are ζ {displaystyle zeta } and μ v {displaystyle mu _{v}} . Volume viscosity appears in the classic Navier–Stokes equations if it is written for compressible fluid, as described in the most books on general hydrodynamics and acoustics. where μ {displaystyle mu } is the shear viscosity coefficient and ζ {displaystyle zeta } is the volume viscosity coefficient. The parameters μ {displaystyle mu } and ζ {displaystyle zeta } were originally called the first and second viscosity coefficients, respectively. The operator D v / D t {displaystyle Dmathbf {v} /Dt} is the material derivative. By introducing the tensors S {displaystyle mathbf {S} } , S 0 {displaystyle mathbf {S} _{0}} and C {displaystyle mathbf {C} } , which describe crude shear flow, pure shear flow and compression flow, respectively,