When the fluid is subjected to a macroscopic perturbation, volume variations occur at the macroscopic scale and the fluid can be considered, at first order, as not compressible at the scale of the duct cross-section: \[ \boldsymbol{\nabla}\cdot\boldsymbol{v} = 0 \]

A harmonic pressure gradient of the form $\exp(j\omega t)$ and of small amplitude, is applied in the direction of the cylinder axis ($0z$) such that the fluid flow is assumed to be laminar (i.e. the corresponding Reynolds number is low).

From the symmetry of the problem and considering that the length of the cylinder is large enough so that the acoustic velocity is invariant in the $z$-direction, $\boldsymbol{v}$ is found to depend only on variable $r$: $\boldsymbol{v} = (0,0,v_{z}(r,z))$ and the Navier-Stokes equation is re-written:

\[ \frac{\eta}{r} \frac{\partial}{\partial r}\left(r\frac{\partial v_{z}}{\partial r}\right) -\frac{\partial p}{\partial z} = j\omega\rho_{0}v_{z} \]

This last equation is a cylindrical Bessel equation. Its solution with conditions $v_{z}=0$ at $r=R$ and $v_{z}$ finite at $r=0$ ($\forall z$) is:

\[ v_{z}(r,z,\omega) = -\frac{\pi_{c}(r,\omega)}{\eta}\frac{\partial p}{\partial z} \]

with

\[ \pi_{c}(r,\omega) = -j\delta_{v}^{2} \left(1 - \frac{J_{1}\left(\displaystyle\frac{r}{\delta_{v}}\sqrt{-j}\right)} {J_{0}\left(\displaystyle\frac{R}{\delta_{v}}\sqrt{-j}\right)}\right) \]

where $J_{i}$ is the Bessel function of the first kind and of order $i$ and $\delta_{v}$ the viscous boundary layer for a circular duct geometry. $\delta_{v}$ denotes a characteristic size (or thickness) in which the fluid velocity distribution close to the surface of the wall is considerably perturbed by the viscous forces generated between the motionless wall and the fluid flow:

\[ \delta_{v} = \sqrt{\frac{\eta}{\rho_{0}\omega}} \]

Some authors prefer to define the acoustic Reylnolds number:

\[ \beta = \frac{R}{\delta_{v}} = \sqrt{\frac{\omega\rho_{0}}{\eta}}R \]

which is the ratio of two stresses caused separately by sound pressure and viscosity.

The function $\pi_{c}(r,\omega)$, plotted in the figure below for three values of the ratio $\beta = R/\delta_{v}$, gives the profile of the velocity field in the cylinder over a cross-section:

• • At high frequencies ($\delta_{v} \ll R$), the effects of the fluid viscosity are mainly located near the cylinder wall. For the rest of the cylinder cross-section, the regime of the flow is purely inertial: \[ v_{z}(r,z,\omega) = \frac{1}{j\omega\rho_{0}}\frac{\partial p}{\partial z} \]
• • At low frequencies ($\delta_{v} \gg R$), the viscous effects dominate and change the fluid flow in the whole cross-section.

The mean velocity $\langle\boldsymbol{v}\rangle$ over the cross-section is equal to:

\[ \langle\boldsymbol{v}\rangle = \frac{\displaystyle\int_{0}^{R}2\boldsymbol{v}.\boldsymbol{z}\pi r dr}{\pi R^{2}}.\boldsymbol{z} \]

Making use of

\[ \int_{0}^{a} r J_{0}(r)dr = a J_{1}(a) \]

yields

\[ \langle\boldsymbol{v}\rangle = -\frac{\Pi_{c}(\omega)}{\eta}\frac{\partial p}{\partial z}.\boldsymbol{z} \label{ms.eq.generalized_darcy} \]

with

\[ \Pi_{c}(\omega) = -j\delta_{v}^{2}\left(1 - 2\frac{\delta_{v}}{R\sqrt{-j}} \frac{J_{1}\left(\displaystyle\frac{R}{\delta_{v}}\sqrt{-j}\right)} {J_{0}\left(\displaystyle\frac{R}{\delta_{v}}\sqrt{-j}\right)}\right) \ . \]

At low frequencies (i.e. $\delta_v \gg R$), the fluid flow in the cylinder is governed by Darcy's law, from the macroscopic point of view:

\[ \langle\boldsymbol{v}\rangle = -\frac{\Pi_{c}(0)}{\eta}\frac{\partial p}{\partial z}.\boldsymbol{z} \]

where $\Pi_{c}(0) = R^2/8$ can be defined as a permeability and expressed in darcy or m$^2$ (1 darcy $\simeq$ $10^{-12}$ m$^2$). Note that $\Pi_{c}(0)$ does not depend on the fluid properties but only on the geometry of the duct.

Above a flat plate

The derivation of the boundary viscous layer above a plate can be obtained considering the shear flow generated in a fluid above a flat and infinite plate oscillating in the plane of the plate with a harmonic motion.

Due to friction, the motion of the plate induces a motion in the fluid. From the continuity condition of displacements at the plate-fluid interface, this motion is the same for the plate particles and the fluid particles at the interface. The fluid velocity decreases with the distance $y$ above the plate

Denoting $u_0 \cos(\omega t)$ the harmonic velocity of the flat plate in the $x$-direction (i.e. in the plane of the plate). With the $y$-direction chosen normal to the plate, the rate momentum flux (i. e. the shear stress) in the $y$-direction is $\tau(y) = −\eta \partial u/\partial y$ per unit area so that the net force (in the $x$-direction) per unit area on a fluid element of thickness $dy$ is $\tau(y) − \tau(y + dy) = −(\partial\tau/\partial y)dy$. The equation for the $x$-component of the fluid velocity is then:

\[ \rho \frac{\partial u}{\partial t} = \eta \frac{\partial^2 u}{\partial y^2} \]

The corresponding equation for the complex velocity amplitude $u(\omega)$ is: \[ \frac{\partial^2 u}{\partial y^2} + (j\rho_0\omega)/\eta = 0 \]

with the solution (see e.g. the page on the solution to the undamped 1D harmonic oscillator on how to solve such a second order, linear, homogeneous differential equation):

\[ u = u_0 e^{jky} = u_0 e^{-y/\delta_v}e^{jy/\delta_v} \]

with

\[ k = +(1+j)\sqrt{\rho_0\omega/(2\eta)} \]

and

\[ \delta_v = \displaystyle\sqrt{2\eta/(\rho_0\omega)} \]

NB: only the positive root of $k^2 = \displaystyle\frac{j\rho_0\omega}{\eta}$ is a physical solution