1 Introduction
Over the years, various motivations have prompted researchers to investigate the dynamic response of fluid-submerged oscillating objects. A prominent example is the theoretical work of Stokes (Reference Stokes1851) in the mid-nineteenth century on the vibration of a sphere in an unbounded fluid inspired by the motion of a simple pendulum performing indefinitely small oscillations in air. Since then, there have been a number of theoretical investigations focusing on the hydrodynamics of vibrating impermeable particles of different geometries (Ray Reference Ray1936; Kanwal Reference Kanwal1955, Reference Kanwal1964, Reference Kanwal1970; Williams Reference Williams1966; Tuck Reference Tuck1969; Lai & Mockros Reference Lai and Mockros1972; Lawrence & Weinbaum Reference Lawrence and Weinbaum1986, Reference Lawrence and Weinbaum1988; Pozrikidis Reference Pozrikidis1989; Loewenberg Reference Loewenberg1993; Avudainayagam & Geetha Reference Avudainayagam and Geetha1994; Zhang & Stone Reference Zhang and Stone1998; Shu & Chwang Reference Shu and Chwang2001; Shatz Reference Shatz2004, Reference Shatz2005; Barta Reference Barta2011). These studies are motivated by applications ranging from the Brownian motion of microparticles and locomotion of microorganisms to frequency response of cantilever beams used in atomic force microscopes and design of bio-inspired swimming and flying robots.
A closer inspection of the literature indicates that oscillating motions of permeable objects, compared to those of their impermeable counterparts, remain largely unexplored. This is despite their importance in many conventional and emerging applications, such as the design of stabilizers for offshore structures (Molin Reference Molin2001), enhanced oil recovery (Graham & Higdon Reference Graham and Higdon2002), oscillatory rheology of porous-particle suspensions (Looker & Carnie Reference Looker and Carnie2004) and aerodynamics of insect flight (Santhanakrishnan et al. Reference Santhanakrishnan, Robinson, Jones, Low, Gadi, Hedrick and Miller2014). The existing studies, which almost all belong to the current millennium, have only focused on the heaving motion of perforated disks (Molin Reference Molin2001, Reference Molin2011; Liu et al. Reference Liu, Li, Li and He2011; An & Faltinsen Reference An and Faltinsen2013) and translational vibrations of porous spheres and spherical shells (Looker & Carnie Reference Looker and Carnie2004; Vainshtein & Shapiro Reference Vainshtein and Shapiro2009; Tsai & Hsu Reference Tsai and Hsu2010; Ollila, Ala-Nissila & Denniston Reference Ollila, Ala-Nissila and Denniston2012; Prakash, Raja Sekhar & Kohr Reference Prakash, Raja Sekhar and Kohr2012). A missing elementary geometry in these investigations is a cylinder (be it circular or otherwise). Identifying the hydrodynamic characteristics of vibrating cylinders, however, has been shown to be the first step in understanding the behaviour of two-dimensional-like flow structures that appear frequently in systems of current interest (Sader Reference Sader1998; Phan, Aureli & Porfiri Reference Phan, Aureli and Porfiri2013; Ahsan & Aureli Reference Ahsan and Aureli2015).
Here, we examine the hydrodynamics of porous cylinders vibrating in an infinite fluid medium. We consider both transverse and rotational oscillations for circular and elliptical cross-sections. Assuming small-amplitude oscillations, the unsteady Brinkman–Debye–Bueche (BDB) and linearized Navier–Stokes (NS) equations are solved for the fluid flow within and around cylinders, respectively. For circular cylinders, analytical solutions are obtained using the method of fundamental solutions; while for cylinders with elliptical cross-sections, a Fourier-pseudospectral method is devised to numerically calculate the flow field. We present the results in terms of the total forces and torques exerted on the oscillating cylinder. In particular, we scrutinize the influence of oscillation frequency and permeability on the effective mass and damping coefficients that are closely associated with the conservative and dissipative components of the hydrodynamic load, respectively. The latter is proportional to the power required to drive the vibrations. Our calculations reveal that, when the oscillation frequency is high enough, there exists a permeability at which the power (or damping coefficient) is maximum. Perhaps surprisingly, the maximum power can be several times greater than the power expended during the oscillations of an otherwise impermeable cylinder.
Below, we first describe the problem statement and modelling strategy. Details of the analytical and numerical approaches are explained next, followed by a discussion of the results, including the applicability range of the small-amplitude assumption. A short summary and concluding remarks are given in the end and supplemental information is provided in appendices A and B.
2 Problem formulation and solutions
2.1 Governing equations
We consider an infinitely long porous cylinder of characteristic (cross-sectional) length scale
$a$
and permeability
$\unicode[STIX]{x1D705}$
that undergoes sinusoidal oscillations with constant amplitude
$A$
and period
$T$
in an unbounded incompressible fluid of density
$\unicode[STIX]{x1D70C}$
and viscosity
$\unicode[STIX]{x1D707}$
. We assume that the cylinder is neutrally buoyant and the amplitude of oscillations compared to the characteristic length of the cylinder is very small, i.e.
$\unicode[STIX]{x1D706}=A/a\ll 1$
. Also, following Brinkman (Reference Brinkman1947, Reference Brinkman1948) and Debye & Bueche (Reference Debye and Bueche1948), we model the flow through the porous cylinder by adding a term proportional to the relative velocity of the cylinder to the momentum equation. This approach leads to a set of unsteady BDB and linearized NS equations that can be written succinctly in the dimensionless form as (see e.g. Vainshtein & Shapiro Reference Vainshtein and Shapiro2009; Ollila et al.
Reference Ollila, Ala-Nissila and Denniston2012)
$$\begin{eqnarray}\displaystyle {\mathcal{R}}\frac{\unicode[STIX]{x2202}\tilde{\boldsymbol{u}}}{\unicode[STIX]{x2202}\tilde{t}} & = & \displaystyle \unicode[STIX]{x1D6FB}^{2}\tilde{\boldsymbol{u}}-\unicode[STIX]{x1D735}\tilde{p}-\unicode[STIX]{x1D712}\unicode[STIX]{x1D6FD}^{2}(\tilde{\boldsymbol{u}}-\tilde{\boldsymbol{u}}_{c}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\tilde{\boldsymbol{u}} & = & \displaystyle 0,\end{eqnarray}$$
$\tilde{t}$
denotes the time,
$\tilde{p}$
represents the pressure field,
$\tilde{\boldsymbol{u}}$
and
$\tilde{\boldsymbol{u}}_{c}$
are the fluid and cylinder velocities, respectively, and
$\unicode[STIX]{x1D712}$
is a binary mask function that is one inside the cylinder and zero in its exterior. As intended,
$\unicode[STIX]{x1D712}=1$
and
$\unicode[STIX]{x1D712}=0$
give, respectively, the unsteady BDB and unsteady Stokes equations that are coupled through the continuity of velocity and traction vector at the surface of the cylinder. The fluid velocity within the porous region is an averaged quantity over a domain that is large compared to the size of a single pore, but small compared to the cylinder itself. Furthermore,
$\tilde{\boldsymbol{u}}$
is assumed to vanish at infinity. In (2.1), 
$$\begin{eqnarray}\displaystyle {\mathcal{R}}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D71B}a^{2}/\unicode[STIX]{x1D707}\quad \text{and}\quad \unicode[STIX]{x1D6FD}^{2}=a^{2}/\unicode[STIX]{x1D705} & & \displaystyle\end{eqnarray}$$
denote, respectively, the dimensionless frequency and permeability, with
$\unicode[STIX]{x1D71B}=2\unicode[STIX]{x03C0}/T$
being the angular frequency of the oscillations. The parameter
${\mathcal{R}}$
is also known as the oscillatory Reynolds number
$Re_{\unicode[STIX]{x1D71B}}$
. As we will discuss later in § 3, the values of
${\mathcal{R}}$
and
$\unicode[STIX]{x1D6FD}$
determine how fast the oscillations of the surrounding fluid decay as the distance from the cylinder increases and how deep the fluid flow penetrates into the porous cylinder, respectively. The ratio
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FD}/\sqrt{{\mathcal{R}}} & & \displaystyle\end{eqnarray}$$
will be shown to be an important factor in characterizing the hydrodynamic response of the vibrating cylinder. Throughout the article, the length, time, velocity, stress, force per unit length and torque per unit length are non-dimensionalized by
$a$
,
$1/\unicode[STIX]{x1D71B}$
,
$A\unicode[STIX]{x1D71B}$
,
$\unicode[STIX]{x1D707}A\unicode[STIX]{x1D71B}/a$
,
$\unicode[STIX]{x1D707}A\unicode[STIX]{x1D71B}$
and
$\unicode[STIX]{x1D707}aA\unicode[STIX]{x1D71B}$
, respectively.
Given the form of the cylinder motion, it is mathematically convenient to express the velocities and pressure as real parts of complex quantities, i.e.
$\tilde{\boldsymbol{u}}_{c}=\text{Re}(\boldsymbol{u}_{c}\text{e}^{\text{i}\tilde{t}})$
,
$\tilde{\boldsymbol{u}}=\text{Re}(\boldsymbol{u}\text{e}^{\text{i}\tilde{t}})$
and
$\tilde{p}=\text{Re}(p\text{e}^{\text{i}\tilde{t}})$
, with
$\text{i}^{2}=-1$
. Substituting in (2.1), we then have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}-\unicode[STIX]{x1D735}p & = & \displaystyle \unicode[STIX]{x1D709}_{i}^{2}[\boldsymbol{u}-(\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D709}_{i})^{2}\boldsymbol{u}_{c}]\quad \text{when }\unicode[STIX]{x1D712}=1,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}-\unicode[STIX]{x1D735}p & = & \displaystyle \unicode[STIX]{x1D709}_{o}^{2}\boldsymbol{u}\quad \text{when }\unicode[STIX]{x1D712}=0,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u} & = & \displaystyle 0,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}_{i}^{2}=\text{i}{\mathcal{R}}+\unicode[STIX]{x1D6FD}^{2}\quad \text{and}\quad \unicode[STIX]{x1D709}_{o}^{2}=\text{i}{\mathcal{R}}. & & \displaystyle\end{eqnarray}$$
Here and throughout the article, we recognize the applicability domain of parameters and variables, being inside or outside the cylinder, by subscripts
$i$
and
$o$
, respectively. The time-independent equations (2.4a
) and (2.4b
) are in the form of the original BDB equations with complex dimensionless permeabilities. In solving (2.4), we treat circular and non-circular cross-sections separately, as only the former is amenable to analytical solutions.
2.2 Analytical solutions for circular cylinders
A convectional approach to analytically calculate
$\boldsymbol{u}$
for simple geometries, such as spheres and circular cylinders, is to write (2.4) in terms of the streamfunction and solve the corresponding equations by the method of separation of variables. This method is used by Vainshtein & Shapiro (Reference Vainshtein and Shapiro2009) and Ollila et al. (Reference Ollila, Ala-Nissila and Denniston2012) to obtain the velocity and pressure fields for an oscillating sphere. Alternatively, here, we solve (2.4) for a circular cylinder of radius
$a$
(see figure 1) via the method of fundamental solutions. In this approach,
$\boldsymbol{u}-(\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D709}_{i})^{2}\boldsymbol{u}_{c}$
when
$\unicode[STIX]{x1D712}=1$
and
$\boldsymbol{u}$
when
$\unicode[STIX]{x1D712}=0$
are represented, respectively, by the solutions of the homogeneous part of (2.4a
) and (2.4b
) that are in turn singular at infinity and the centre of the cylinder. The fundamental singular solutions necessary for representing the flow within and around the cylinder are analogues of stokeson, potential doubleton and roton for internal, and stokeslet, potential doublet and rotlet for external Stokes flows. The terminologies are adopted from those of Chwang & Wu (Reference Chwang and Wu1975) and Pozrikidis (Reference Pozrikidis1992).
Figure 1. Schematic of a vibrating cylinder (depicted as a dark grey circle of dimensionless radius one) immersed in a boundless fluid (shown in light grey). The dashed enclosing contour highlights the lack of an outer boundary, and
$\unicode[STIX]{x1D712}=0$
and
$\unicode[STIX]{x1D712}=1$
denote areas inside and outside of the cylinder, respectively. Also, double-headed arrows represent transverse and rotational vibrations.
Let the subscripts
$SN$
,
$DN$
and
$RN$
denote the field variables for the equivalents of stokeson, potential doubleton and roton, respectively. Also, suppose analogues of stokeslet, potential doublet and rotlet are represented, respectively, by the subscripts
$S$
,
$D$
and
$R$
. Following this convention, we write the fundamental singularities of the BDB equations for internal and external flows as
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{SN}(\boldsymbol{r};\unicode[STIX]{x1D736}) & = & \displaystyle \left[\text{I}_{0}(\unicode[STIX]{x1D70C}_{i})-\frac{\text{I}_{1}(\unicode[STIX]{x1D70C}_{i})}{\unicode[STIX]{x1D70C}_{i}}\right]\unicode[STIX]{x1D736}-\left[\text{I}_{0}(\unicode[STIX]{x1D70C}_{i})-\frac{2\text{I}_{1}(\unicode[STIX]{x1D70C}_{i})}{\unicode[STIX]{x1D70C}_{i}}\right]\frac{(\unicode[STIX]{x1D736}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{r}}{r^{2}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{SN}(\boldsymbol{r};\unicode[STIX]{x1D736}) & = & \displaystyle 0,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{DN}(\boldsymbol{r};\boldsymbol{\unicode[STIX]{x1D702}}) & = & \displaystyle \boldsymbol{\unicode[STIX]{x1D702}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{DN}(\boldsymbol{r};\boldsymbol{\unicode[STIX]{x1D702}}) & = & \displaystyle -\unicode[STIX]{x1D709}_{i}^{2}(\boldsymbol{\unicode[STIX]{x1D702}}\boldsymbol{\cdot }\boldsymbol{r}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{RN}(\boldsymbol{r};\unicode[STIX]{x1D738}) & = & \displaystyle \frac{\text{I}_{1}(\unicode[STIX]{x1D70C}_{i})}{\unicode[STIX]{x1D70C}_{i}}(\unicode[STIX]{x1D738}\times \boldsymbol{r}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{RN}(\boldsymbol{r};\unicode[STIX]{x1D738}) & = & \displaystyle 0,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{S}(\boldsymbol{r};\unicode[STIX]{x1D736}) & = & \displaystyle \left[\text{K}_{0}(\unicode[STIX]{x1D70C}_{o})+\frac{\text{K}_{1}(\unicode[STIX]{x1D70C}_{o})}{\unicode[STIX]{x1D70C}_{o}}-\frac{1}{\unicode[STIX]{x1D70C}_{o}^{2}}\right]\unicode[STIX]{x1D736}-\left[\text{K}_{0}(\unicode[STIX]{x1D70C}_{o})+\frac{2\text{K}_{1}(\unicode[STIX]{x1D70C}_{o})}{\unicode[STIX]{x1D70C}_{o}}-\frac{2}{\unicode[STIX]{x1D70C}_{o}^{2}}\right]\frac{(\unicode[STIX]{x1D736}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{r}}{r^{2}},\qquad\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{S}(\boldsymbol{r};\unicode[STIX]{x1D736}) & = & \displaystyle \frac{\unicode[STIX]{x1D736}\boldsymbol{\cdot }\boldsymbol{r}}{r^{2}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{D}(\boldsymbol{r};\boldsymbol{\unicode[STIX]{x1D702}}) & = & \displaystyle -\frac{\unicode[STIX]{x1D709}_{o}^{2}}{2}\left\{\left[\text{K}_{0}(\unicode[STIX]{x1D70C}_{o})+\frac{\text{K}_{1}(\unicode[STIX]{x1D70C}_{o})}{\unicode[STIX]{x1D70C}_{o}}\right]\boldsymbol{\unicode[STIX]{x1D702}}-\left[\text{K}_{0}(\unicode[STIX]{x1D70C}_{o})+\frac{2\text{K}_{1}(\unicode[STIX]{x1D70C}_{o})}{\unicode[STIX]{x1D70C}_{o}}\right]\frac{(\boldsymbol{\unicode[STIX]{x1D702}}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{r}}{r^{2}}\right\},\qquad\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{D}(\boldsymbol{r};\boldsymbol{\unicode[STIX]{x1D702}}) & = & \displaystyle 0,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{R}(\boldsymbol{r};\unicode[STIX]{x1D738}) & = & \displaystyle \unicode[STIX]{x1D709}_{o}^{2}\frac{\text{K}_{1}(\unicode[STIX]{x1D70C}_{o})}{\unicode[STIX]{x1D70C}_{o}}(\unicode[STIX]{x1D738}\times \boldsymbol{r}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{R}(\boldsymbol{r};\unicode[STIX]{x1D738}) & = & \displaystyle 0,\end{eqnarray}$$
$\boldsymbol{r}$
is the position vector in the
$x$
–
$y$
plane with
$|\boldsymbol{r}|=r=1$
representing the surface of the cylinder, and
$\text{I}_{n}$
and
$\text{K}_{n}$
are modified Bessel functions of the first and second kinds and of order
$n$
, respectively (Abramowitz & Stegun Reference Abramowitz and Stegun1972). Also,
$\unicode[STIX]{x1D70C}_{i}=\unicode[STIX]{x1D709}_{i}r$
,
$\unicode[STIX]{x1D70C}_{o}=\unicode[STIX]{x1D709}_{o}r$
and
$\unicode[STIX]{x1D736}$
,
$\unicode[STIX]{x1D737}$
and
$\unicode[STIX]{x1D738}$
are constant vectors reflecting the strength of the singular solutions. We note that
$\unicode[STIX]{x1D736}$
and
$\unicode[STIX]{x1D737}$
lie in the
$x$
–
$y$
plane, to which
$\unicode[STIX]{x1D738}$
is perpendicular. Equations (2.6a
)–(2.11b
) are derived in a similar fashion to their Stokes flow analogues (see e.g. Chwang & Wu Reference Chwang and Wu1975; Pozrikidis Reference Pozrikidis1992). We first consider the translational oscillations where, without loss of generality, we set
$\boldsymbol{u}_{c}=\boldsymbol{e}_{y}$
with
$\boldsymbol{e}_{y}$
being the unit vector in the
$y$
direction. In this case, we express the flow field as
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{i} & = & \displaystyle \boldsymbol{u}_{SN}(\boldsymbol{r};\unicode[STIX]{x1D6FC}_{i}\boldsymbol{e}_{y})+\boldsymbol{u}_{DN}(\boldsymbol{r};\unicode[STIX]{x1D702}_{i}\boldsymbol{e}_{y})+(\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D709}_{i})^{2}\boldsymbol{e}_{y},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{i} & = & \displaystyle p_{SN}(\boldsymbol{r};\unicode[STIX]{x1D6FC}_{i}\boldsymbol{e}_{y})+p_{DN}(\boldsymbol{r};\unicode[STIX]{x1D702}_{i}\boldsymbol{e}_{y}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{o} & = & \displaystyle \boldsymbol{u}_{S}(\boldsymbol{r};\unicode[STIX]{x1D6FC}_{o}\boldsymbol{e}_{y})+\boldsymbol{u}_{D}(\boldsymbol{r};\unicode[STIX]{x1D702}_{o}\boldsymbol{e}_{y}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{o} & = & \displaystyle p_{S}(\boldsymbol{r};\unicode[STIX]{x1D6FC}_{o}\boldsymbol{e}_{y})+p_{D}(\boldsymbol{r};\unicode[STIX]{x1D702}_{o}\boldsymbol{e}_{y}),\end{eqnarray}$$
$\unicode[STIX]{x1D6FC}_{i}$
,
$\unicode[STIX]{x1D6FC}_{o}$
,
$\unicode[STIX]{x1D702}_{i}$
and
$\unicode[STIX]{x1D702}_{o}$
are determined by applying, at
$r=1$
, 
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{i}=\boldsymbol{u}_{o}\quad \text{and}\quad \boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{i}=\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D748}_{o}, & & \displaystyle\end{eqnarray}$$
with
$\unicode[STIX]{x1D748}$
denoting the dimensionless (complex) stress tensor and
$\boldsymbol{n}=\boldsymbol{e}_{r}=\boldsymbol{r}/r$
. Given the first condition in (2.14) and recalling that
$\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0$
everywhere, the continuity of the surface traction condition can be replaced by the continuity of pressure and vorticity
$\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$
(see e.g. Pozrikidis Reference Pozrikidis2011, p. 241), i.e.
$$\begin{eqnarray}\displaystyle p_{i}=p_{o}\quad \text{and}\quad \unicode[STIX]{x1D74E}_{i}=\unicode[STIX]{x1D74E}_{o}. & & \displaystyle\end{eqnarray}$$
Note that, in two dimensions, the vorticity vector is effectively a scalar since it has only one non-zero component, which is normal to the
$x$
–
$y$
plane.
Substitution of (2.11) and (2.12) into the left-hand equation of (2.14) and the two equations of (2.15), yields, after some manipulations,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{i}=\frac{A_{3}}{A_{2}},\quad \unicode[STIX]{x1D702}_{i}=-\frac{A_{1}}{\unicode[STIX]{x1D709}_{i}^{2}A_{2}},\quad \unicode[STIX]{x1D6FC}_{o}=\frac{A_{1}}{A_{2}},\quad \unicode[STIX]{x1D702}_{o}=\frac{2(A_{1}+A_{4})}{\unicode[STIX]{x1D709}_{o}^{2}A_{2}}, & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle A_{1} & = & \displaystyle [2\unicode[STIX]{x1D709}_{o}(\unicode[STIX]{x1D709}_{i}^{2}-\unicode[STIX]{x1D709}_{o}^{2})\text{K}_{1}(\unicode[STIX]{x1D709}_{o})+\unicode[STIX]{x1D709}_{i}^{2}\unicode[STIX]{x1D709}_{o}^{2}\text{K}_{0}(\unicode[STIX]{x1D709}_{o})]\text{I}_{1}(\unicode[STIX]{x1D709}_{i})+\unicode[STIX]{x1D709}_{i}\unicode[STIX]{x1D709}_{o}^{3}\text{K}_{1}(\unicode[STIX]{x1D709}_{o})\text{I}_{0}(\unicode[STIX]{x1D709}_{i}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle A_{2} & = & \displaystyle \unicode[STIX]{x1D6FD}^{-2}\{[2\unicode[STIX]{x1D709}_{o}(\unicode[STIX]{x1D709}_{i}^{2}-\unicode[STIX]{x1D709}_{o}^{2})\text{K}_{1}(\unicode[STIX]{x1D709}_{o})+\unicode[STIX]{x1D709}_{i}^{2}(\unicode[STIX]{x1D709}_{i}^{2}+\unicode[STIX]{x1D709}_{o}^{2})\text{K}_{0}(\unicode[STIX]{x1D709}_{o})]\text{I}_{1}(\unicode[STIX]{x1D709}_{i})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D709}_{i}\unicode[STIX]{x1D709}_{o}(\unicode[STIX]{x1D709}_{i}^{2}+\unicode[STIX]{x1D709}_{o}^{2})\text{K}_{1}(\unicode[STIX]{x1D709}_{o})\text{I}_{0}(\unicode[STIX]{x1D709}_{i})\!\},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle A_{3} & = & \displaystyle -2\unicode[STIX]{x1D709}_{i}\unicode[STIX]{x1D709}_{o}\text{K}_{1}(\unicode[STIX]{x1D709}_{o}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle A_{4} & = & \displaystyle -2\unicode[STIX]{x1D709}_{i}^{2}\text{I}_{1}(\unicode[STIX]{x1D709}_{i}).\end{eqnarray}$$
$\boldsymbol{u}$
and
$p$
in the entire domain, the (complex) total force per unit length of the cylinder is calculated as (see e.g. Felderhof Reference Felderhof2014) 
$$\begin{eqnarray}\displaystyle \boldsymbol{F}=\unicode[STIX]{x1D6FD}^{2}\!\int _{S}(\boldsymbol{u}_{i}-\boldsymbol{u}_{c})\,\text{d}S=\frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FD}^{2}}{\unicode[STIX]{x1D709}_{i}^{2}}[\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D709}_{i}\text{I}_{1}(\unicode[STIX]{x1D709}_{i})+\unicode[STIX]{x1D702}_{i}\unicode[STIX]{x1D709}_{i}^{2}-\text{i}{\mathcal{R}}]\boldsymbol{e}_{y}. & & \displaystyle\end{eqnarray}$$
The effective mass and damping coefficients are then obtained according to
$$\begin{eqnarray}\displaystyle C_{e}^{t}=-\text{Im}[F/(\unicode[STIX]{x03C0}{\mathcal{R}})]\quad \text{and}\quad C_{d}^{t}=-\text{Re}[F/(\unicode[STIX]{x03C0}{\mathcal{R}})], & & \displaystyle\end{eqnarray}$$
respectively, where
$\boldsymbol{F}=F\boldsymbol{e}_{y}$
and the superscript
$t$
underscores transverse oscillations. We note that
$C_{e}^{t}$
represents both the inertia of the fluid mass within the porous cylinder and that of the surrounding fluid.
Next, we consider the rotational vibrations of a porous circular cylinder about its longitudinal axis. We set
$A=a\unicode[STIX]{x1D6E9}$
with
$\unicode[STIX]{x1D6E9}$
being the angular amplitude of oscillations. Consequently,
$\boldsymbol{u}_{c}=\boldsymbol{e}_{z}\times \boldsymbol{r}$
, where
$\boldsymbol{e}_{z}$
is the unit vector in the
$z$
direction of the Cartesian coordinate system
$(x,y,z)$
. Here, the internal and external flows take the forms of
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{i} & = & \displaystyle \boldsymbol{u}_{RN}(\boldsymbol{r};\unicode[STIX]{x1D6FE}_{i}\boldsymbol{e}_{z})+(\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D709}_{i})^{2}(\boldsymbol{e}_{z}\times \boldsymbol{r}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{i} & = & \displaystyle p_{RN}(\boldsymbol{r};\unicode[STIX]{x1D6FE}_{i}\boldsymbol{e}_{z}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{o} & = & \displaystyle \boldsymbol{u}_{R}(\boldsymbol{r};\unicode[STIX]{x1D6FE}_{o}\boldsymbol{e}_{z}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{o} & = & \displaystyle p_{R}(\boldsymbol{r};\unicode[STIX]{x1D6FE}_{o}\boldsymbol{e}_{z}),\end{eqnarray}$$
$\unicode[STIX]{x1D6FE}_{i}$
and
$\unicode[STIX]{x1D6FE}_{o}$
are complex constants. Following the procedure that led to (2.16), the application of the continuity boundary conditions yields 
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{i} & = & \displaystyle -\frac{\unicode[STIX]{x1D709}_{o}\unicode[STIX]{x1D6FD}^{2}\text{K}_{2}(\unicode[STIX]{x1D709}_{o})}{\unicode[STIX]{x1D709}_{i}[\unicode[STIX]{x1D709}_{i}\text{I}_{0}(\unicode[STIX]{x1D709}_{i})\text{K}_{1}(\unicode[STIX]{x1D709}_{o})+\unicode[STIX]{x1D709}_{o}\text{I}_{1}(\unicode[STIX]{x1D709}_{i})\text{K}_{0}(\unicode[STIX]{x1D709}_{o})]},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{o} & = & \displaystyle \frac{\unicode[STIX]{x1D6FD}^{2}\text{I}_{2}(\unicode[STIX]{x1D709}_{i})}{\unicode[STIX]{x1D709}_{i}\unicode[STIX]{x1D709}_{o}[\unicode[STIX]{x1D709}_{i}\text{I}_{0}(\unicode[STIX]{x1D709}_{i})\text{K}_{1}(\unicode[STIX]{x1D709}_{o})+\unicode[STIX]{x1D709}_{o}\text{I}_{1}(\unicode[STIX]{x1D709}_{i})\text{K}_{0}(\unicode[STIX]{x1D709}_{o})]},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{T}=\unicode[STIX]{x1D6FD}^{2}\int _{S}[\boldsymbol{r}\times (\boldsymbol{u}_{i}-\boldsymbol{u}_{c})]\,\text{d}S=\frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FD}^{2}}{2\unicode[STIX]{x1D709}_{i}^{2}}[4\unicode[STIX]{x1D6FE}_{i}\text{I}_{2}(\unicode[STIX]{x1D709}_{i})-\text{i}{\mathcal{R}}]\boldsymbol{e}_{z}. & & \displaystyle\end{eqnarray}$$
The effective mass and damping coefficients are derived from the complex torque as
$$\begin{eqnarray}\displaystyle C_{e}^{r}=-\text{Im}[T/(\unicode[STIX]{x03C0}{\mathcal{R}}R_{g}^{2})]\quad \text{and}\quad C_{d}^{r}=-\text{Re}[T/(\unicode[STIX]{x03C0}{\mathcal{R}}R_{g}^{2})], & & \displaystyle\end{eqnarray}$$
respectively, where
$R_{g}$
is the radius of gyration of the cross-section, defined as
$R_{g}^{2}=\int _{S}r^{2}\,\text{d}S/\!\int _{S}\,\text{d}S$
,
$\boldsymbol{T}=T\boldsymbol{e}_{z}$
, and the superscript
$r$
denotes rotational oscillations.
2.3 Numerical solutions for non-circular cylinders
Figure 2. Schematic illustrating a periodic domain of length
$L$
(depicted as a thick black square) that is uniformly discretized into an array of square elements. The dark grey regions represent identical cylinders of non-circular cross-section that are spaced a domain length apart from one another.
Obtaining closed-form solutions for
$\boldsymbol{u}$
and
$p$
for cylinders of non-circular cross-sections is very challenging, if not impossible. Thus, we resort to numerics to study their dynamic response under transverse and rotational oscillations. To this end, we approximate the fluid flow within and around an isolated vibrating cylinder with that through and surrounding each member of a periodic array of oscillating cylinders (see figure 2). The latter is known to converge to the former as the spacing between the cylinders in the array becomes very large. We then employ a Fourier-pseudospectral method to calculate the velocity field and the resulting hydrodynamic loads.
We begin by taking the Fourier transform of (2.4), which, after rearranging, leads to
$$\begin{eqnarray}\displaystyle (k^{2}+\unicode[STIX]{x1D709}_{o}^{2})\hat{\boldsymbol{u}}+\text{i}\boldsymbol{k}\hat{p}=-\unicode[STIX]{x1D6FD}^{2}(\widehat{\unicode[STIX]{x1D712}\boldsymbol{u}}-\widehat{\unicode[STIX]{x1D712}\boldsymbol{u}}_{c})\quad \text{and}\quad \text{i}\boldsymbol{k}\boldsymbol{\cdot }\hat{\boldsymbol{u}}=0, & & \displaystyle\end{eqnarray}$$
where
$\boldsymbol{k}$
is the two-dimensional wavevector with
$k=|\boldsymbol{k}|$
and the hat overbar denotes the transformed variables and terms. Taking the dot product of each side of the momentum equation in (2.25) by
$\text{i}\boldsymbol{k}$
(i.e. taking the divergence), we find
$$\begin{eqnarray}\displaystyle \hat{p}=\unicode[STIX]{x1D6FD}^{2}\frac{\text{i}\boldsymbol{k}}{k^{2}}\boldsymbol{\cdot }(\widehat{\unicode[STIX]{x1D712}\boldsymbol{u}}-\widehat{\unicode[STIX]{x1D712}\boldsymbol{u}}_{c}), & & \displaystyle\end{eqnarray}$$
which, after substitution back into the momentum equation, results in
$$\begin{eqnarray}\displaystyle \boldsymbol{u}+\mathfrak{F}^{-1}\left\{\unicode[STIX]{x1D701}\left[\widehat{\unicode[STIX]{x1D712}\boldsymbol{u}}-\frac{\boldsymbol{k}}{k^{2}}(\boldsymbol{k}\boldsymbol{\cdot }\widehat{\unicode[STIX]{x1D712}\boldsymbol{u}})\right]\right\}=\mathfrak{F}^{-1}\left\{\unicode[STIX]{x1D701}\left[\widehat{\unicode[STIX]{x1D712}\boldsymbol{u}}_{c}-\frac{\boldsymbol{k}}{k^{2}}(\boldsymbol{k}\boldsymbol{\cdot }\widehat{\unicode[STIX]{x1D712}\boldsymbol{u}}_{c})\right]\right\}, & & \displaystyle\end{eqnarray}$$
where
$\mathfrak{F}^{-1}$
represents the inverse Fourier transform and
$\unicode[STIX]{x1D701}=\unicode[STIX]{x1D6FD}^{2}/(k^{2}+\unicode[STIX]{x1D709}_{o}^{2})$
.
To solve this equation numerically, we first discretize the periodic domain (see figure 2) into a uniform distribution of
$N$
square elements whose centres lie inside the domain boundary. Then, we enforce (2.27) at the centre of the elements (also known as the collocation points) while using a fast Fourier transform (FFT) to calculate the direct and inverse transforms. This gives rise to a set of linear equations for
$\boldsymbol{u}$
at the centre of the grid cells, which can be solved iteratively via the generalized minimal residual method (GMRES). Note that in this approach the continuity boundary conditions at the surface of the cylinder are automatically satisfied since a single velocity field represents the entire flow. Finally, we calculate the total force and torque exerted on the cylinder using
$$\begin{eqnarray}\displaystyle \boldsymbol{F} & = & \displaystyle \unicode[STIX]{x1D6FD}^{2}\mathop{\sum }_{j=1}^{N}S_{j}\unicode[STIX]{x1D712}_{j}(\boldsymbol{u}_{j}-\boldsymbol{u}_{c}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{T} & = & \displaystyle \unicode[STIX]{x1D6FD}^{2}\mathop{\sum }_{j=1}^{N}S_{j}\unicode[STIX]{x1D712}_{j}[\boldsymbol{r}_{j}\times (\boldsymbol{u}_{j}-\boldsymbol{u}_{c})],\end{eqnarray}$$
$S_{j}$
,
$\unicode[STIX]{x1D712}_{j}$
,
$\boldsymbol{u}_{j}$
and
$\boldsymbol{r}_{j}$
denote the surface area, value of the mask function, velocity and location of the centre of the
$j$
th element, respectively. Throughout the calculations, we set the length of the periodic domain
$L$
and number of grid points such that always
$20\leqslant L/a$
and the size of the largest grid cell is smaller than
$0.01a$
. Recall that
$a$
is the characteristic length scale of the cross-section, e.g. the radius for circular and semi-major axis for elliptical cylinders. Further, the mask function is smoothed over five nodes by a Gaussian filter to avoid spurious Gibbs oscillations that usually appear when taking the FFT of discontinuous functions (see e.g. Kolomenskiy & Schneider Reference Kolomenskiy and Schneider2009). We verify the accuracy of our numerical approach by comparing its results for circular cylinders with the analytical formulae of § 2.2, and find that the relative errors of the effective mass and damping coefficients for
$0<\unicode[STIX]{x1D6FD}\leqslant 10^{2}$
and
$10^{-1}\leqslant {\mathcal{R}}\leqslant 10^{3}$
are less than
$2.2\,\%$
. Extending the upper limits of
$\unicode[STIX]{x1D6FD}$
and
${\mathcal{R}}$
in numerical simulations requires finer grid resolutions, and reducing the lower limit of
${\mathcal{R}}$
demands larger periodic domains. It is worth noting that the limit of
${\mathcal{R}}\rightarrow 0$
is logarithmically singular for the case of transverse oscillations (see e.g. table 1 in appendix A), i.e. no steady-state solution exists for
${\mathcal{R}}=0$
.
The methodology explained above can be applied to any cross-sectional geometry (circular or otherwise), but here we focus only on ellipses due to their practical significance. Furthermore, this numerical approach can be extended to solve the coupled unsteady BDB–NS equations, including the nonlinear term we neglected so far thanks to the small-amplitude assumption. In its most simple form, the vorticity–streamfunction formulation of the equations are first transformed into the Fourier space. Then, the nonlinear advective and permeability-dependent terms are treated explicitly (e.g. using a second-order Adams–Bashforth method), while the linear diffusive term is integrated exactly via the integrating factor technique (see e.g. Kolomenskiy & Schneider Reference Kolomenskiy and Schneider2009). Note that the mask function is, in general, time-dependent and varies according to the position of the object, unless the equations are written in a non-inertial reference frame attached to the oscillating object. Here, we use the extended approach to solve the full BDB–NS equations to find out (for a few representative cases) how broadly applicable the predictions coming from the solution of the linearized equations are.
3 Results and discussion
We begin by analysing the results for transverse oscillations of first circular and then elliptical cylinders. Before we proceed, it is useful to remember that the magnitude of the (real) force experienced by the cylinder can be written as
$$\begin{eqnarray}\displaystyle \tilde{F} & = & \displaystyle |\tilde{\boldsymbol{F}}|=|\text{Re}(\boldsymbol{F}\text{e}^{\text{i}\tilde{t}})|=|F|\sin (\tilde{t}-\unicode[STIX]{x1D719})\nonumber\\ \displaystyle & = & \displaystyle |F|\cos \unicode[STIX]{x1D719}\sin \tilde{t}-|F|\sin \unicode[STIX]{x1D719}\cos \tilde{t}=\unicode[STIX]{x03C0}{\mathcal{R}}(C_{e}^{t}\sin \tilde{t}-C_{d}^{t}\cos \tilde{t}),\end{eqnarray}$$
where
$\unicode[STIX]{x1D719}$
is the phase lag between the force and prescribed displacement
${\tilde{y}}=\sin \tilde{t}$
. The average power required to sustain the vibrations in one period can then be calculated as
$$\begin{eqnarray}\displaystyle \tilde{{\mathcal{P}}}=\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}\tilde{\boldsymbol{F}}\boldsymbol{\cdot }\tilde{\boldsymbol{u}}_{c}\,\text{d}\tilde{t}=-\frac{\unicode[STIX]{x03C0}}{2}{\mathcal{R}}C_{d}^{t}. & & \displaystyle\end{eqnarray}$$
Equation (3.2) highlights the fact that the average power consumption is directly proportional to the damping coefficient.
Figure 3(a,b) shows the variations of the effective mass and damping coefficients of a circular cylinder as a function of
$\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FD}/\sqrt{{\mathcal{R}}}$
for different values of the dimensionless frequency
${\mathcal{R}}$
. The parameter
$\unicode[STIX]{x1D6FF}$
represents the ratio of two length scales, namely the oscillation penetration length that scales with
$1/\sqrt{{\mathcal{R}}}$
and the characteristic permeation length that scales with
$1/\unicode[STIX]{x1D6FD}$
(see e.g. Vainshtein & Shapiro Reference Vainshtein and Shapiro2009). Dashed lines in the insets are plots of the asymptotic expressions for the corresponding coefficients in the limits of high/low oscillation frequency and permeability (see table 1 in appendix A). The behaviour of the dimensionless force magnitude and phase shift between the force and prescribed displacement (see (3.1)) associated with the plots of
$C_{e}^{t}$
and
$C_{d}^{t}$
are shown in figures 3(c) and 3(d), respectively.
Figure 3. Dimensionless (a) effective mass and (b) damping coefficients of a transversely oscillating porous circular cylinder as a function of
$\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FD}/\sqrt{{\mathcal{R}}}$
for different values of the dimensionless frequency
${\mathcal{R}}$
. Dashed lines in the insets are plots of the asymptotic expressions in the limits of high/low oscillation frequency and permeability (see table 1 in appendix A). Dimensionless (c) force magnitude and (d) phase shift between the force and prescribed displacement associated with the plots in (a) and (b) (see (3.1)).
Inspecting figure 3(a), we see that the curves for the effective mass coefficient all start as a nearly horizontal line when the permeability is very low (
$\unicode[STIX]{x1D6FF}\gg 1$
). With an increase in the permeability, as one might intuitively expect,
$C_{e}^{t}$
begins to decay. This trend continues as
$C_{e}^{t}$
asymptotically approaches zero with a rate that is proportional to
$\unicode[STIX]{x1D6FF}^{4}$
(see table 1 in appendix A). The rapid decay in the magnitude of the effective mass coefficient is due to two factors (see (3.1)): (i) the monotonic decrease of the force magnitude with increasing permeability, as depicted in figure 3(c); and (ii) the rise of the phase lag to
$\unicode[STIX]{x03C0}/2$
at high permeabilities (see figure 3
d). Each of these factors has a
$\unicode[STIX]{x1D6FF}^{2}$
contribution to the overall decay rate of
$C_{e}^{t}$
.
Figure 4. Dimensionless (a,c) effective mass and (b,d) damping coefficients of transversely oscillating porous elliptical cylinders of various aspect ratio
$\unicode[STIX]{x1D700}$
. The dimensionless frequency is
${\mathcal{R}}=10^{-1}$
in (a,b), and
${\mathcal{R}}=10^{3}$
in the primary plots and
${\mathcal{R}}=10$
in the insets of (c,d). The results are plotted versus the modified
$\unicode[STIX]{x1D6FF}$
, i.e.
$\unicode[STIX]{x1D6FF}$
multiplied by
$\sqrt{\unicode[STIX]{x1D700}}$
.
Similar to the effective mass plots, the curves for the damping coefficient plateau in the zero-permeability limit (see figure 3
b). However, unlike those plots, when the oscillation frequency is not too low,
$C_{d}^{t}$
initially increases on increasing the permeability from nil, reaches a maximum and then approaches zero as the permeability tends to infinity (see figure 3
b). In this limit, all curves converge to a single line of slope
$\unicode[STIX]{x1D6FF}^{2}$
(see also table 1 in appendix A). The existence of a maximum in
$C_{d}^{t}$
at moderate to high frequencies suggests that, in these regimes, the increase in
$\sin \unicode[STIX]{x1D719}$
initially outweighs the drop in
$|F|$
and, by doing so, yields a greater damping coefficient (see (3.1)). This trend reverses, of course, as the permeability continues to increase. Evidently, the maximum
$C_{d}^{t}$
seems always to occur at
$\unicode[STIX]{x1D6FF}\sim O(1)$
. Perhaps even more intriguing, the maximum
$C_{d}^{t}$
, depending on the frequency, may be many times greater than the damping coefficient of the impermeable cylinder
$C_{d,\infty }^{t}$
. For instance, at
${\mathcal{R}}=10^{3}$
, the ratio
$C_{d,max}^{t}/C_{d,\infty }^{t}$
is approximately
$11.2$
, whereas it is close to
$3.8$
for
${\mathcal{R}}=10^{2}$
(see figure 3
b). Interestingly, the reduction in
$C_{e}^{t}$
, on the other hand, is not typically as pronounced as the increase in
$C_{d}^{e}$
at low permeabilities. For example, at
${\mathcal{R}}=10^{3}$
and
$\unicode[STIX]{x1D6FF}=3$
,
$C_{d}^{t}/C_{d,\infty }^{t}=5.66$
and
$C_{e}^{t}/C_{e,\infty }^{t}=0.92$
or, at
${\mathcal{R}}=10^{2}$
and
$\unicode[STIX]{x1D6FF}=3$
,
$C_{d}^{t}/C_{d,\infty }^{t}=2.39$
and
$C_{e}^{t}/C_{e,\infty }^{t}=0.85$
.
Next, we consider transverse oscillations of elliptical cylinders in the direction of their minor axes. The results are presented in figure 4. Here, the cylinder’s semi-major axis is chosen as its characteristic length
$a$
and the ratio of the cylinder’s minor to major axis (its aspect ratio) is denoted by
$\unicode[STIX]{x1D700}$
. More importantly, the coefficients are plotted versus
$\unicode[STIX]{x1D6FF}\sqrt{\unicode[STIX]{x1D700}}$
(the modified
$\unicode[STIX]{x1D6FF}$
). Multiplying
$\unicode[STIX]{x1D6FF}$
with
$\sqrt{\unicode[STIX]{x1D700}}$
is equivalent to redefining the dimensionless permeability as
$\unicode[STIX]{x1D6FD}^{2}=ab/\unicode[STIX]{x1D705}$
(see Masoud, Stone & Shelley Reference Masoud, Stone and Shelley2013), where
$a$
and
$b$
represent the semi-major and semi-minor axes of the cross-section, respectively. Remember that the original definition is
$\unicode[STIX]{x1D6FD}^{2}=a^{2}/\unicode[STIX]{x1D705}$
. We find that, when the effect of inertia is weak (
${\mathcal{R}}\ll 1$
), the plots for various aspect ratios nearly collapse onto a single master curve and, hence, are very well represented by the analytical results for a circular cylinder (i.e. solid black lines in figure 4(a,b)). As the frequency of the oscillations increases, deviations from a perfect collapse of data begin to appear (see figure 4(c,d) and their insets). Mismatches in the
$C_{d}^{t}$
plots are mainly limited to the range
$0.5\leqslant \unicode[STIX]{x1D6FF}\leqslant 10$
, whereas the deviations in the
$C_{e}^{t}$
curves are more widespread, particularly at very large
${\mathcal{R}}$
. Overall, from figure 4, we learn that, if adjusted properly, the closed-form expressions derived in § 2.2 for the effective mass and damping coefficients of transversely oscillating circular cylinders can be used to obtain very good estimates for the coefficients of similarly vibrating elliptical cylinders over a broad span of conditions.
Figure 5. Dimensionless (a) effective mass and (b) damping coefficients of a rotationally oscillating porous circular cylinder as a function of
$\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FD}/\sqrt{{\mathcal{R}}}$
for different values of the dimensionless frequency
${\mathcal{R}}$
. Dashed lines in the insets are plots of the asymptotic expressions in the limits of high/low oscillation frequency and permeability (see table 1 in appendix A). (c) Dimensionless torque magnitude and (d) phase shift between the torque and prescribed angular displacement associated with the plots in (a) and (b) (see (3.3)).
Figure 6. Dimensionless (a,c) effective mass and (b,d) damping coefficients of rotationally oscillating porous elliptical cylinders of various aspect ratio
$\unicode[STIX]{x1D700}$
. The dimensionless frequency is
${\mathcal{R}}=10^{-1}$
in (a,b), and
${\mathcal{R}}=10^{3}$
in the primary plots and
${\mathcal{R}}=10$
in the insets of (c,d).
Having discussed the transverse oscillations, we now examine rotational vibrations. Again, it is constructive to recall that
$$\begin{eqnarray}\displaystyle \tilde{T} & = & \displaystyle |\tilde{\boldsymbol{T}}|=|\text{Re}(\boldsymbol{T}\text{e}^{\text{i}\tilde{t}})|=|T|\sin (\tilde{t}-\unicode[STIX]{x1D719})\nonumber\\ \displaystyle & = & \displaystyle |T|\cos \unicode[STIX]{x1D719}\sin \tilde{t}-|T|\sin \unicode[STIX]{x1D719}\cos \tilde{t}=\unicode[STIX]{x03C0}{\mathcal{R}}R_{g}^{2}(C_{e}^{r}\sin \tilde{t}-C_{d}^{r}\cos \tilde{t}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \tilde{{\mathcal{P}}}=\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}\tilde{\boldsymbol{T}}\boldsymbol{\cdot }\tilde{\unicode[STIX]{x1D734}}_{c}\,\text{d}\tilde{t}=-\frac{\unicode[STIX]{x03C0}}{2}{\mathcal{R}}R_{g}^{2}C_{d}^{r}, & & \displaystyle\end{eqnarray}$$
where
$\tilde{\unicode[STIX]{x1D734}}_{c}=\cos \tilde{t}\,\boldsymbol{e}_{z}$
is the angular velocity of the cylinder. The results for circular and elliptical cylinders are presented, respectively, in figures 5 and 6, whose formats are identical to those of figures 3 and 4, in that order. As can be seen in these figures, the plots for
$C_{e}^{r}$
and
$C_{d}^{r}$
exhibit many of the same features as those for
$C_{e}^{t}$
and
$C_{d}^{t}$
, chief among which are the existence of a maximum in the
$C_{d}^{r}$
curves when inertial effects are dominant and the overall decay of the coefficients with increasing permeability. Despite close qualitative similarities, there are of course quantitative differences. Most notably, at the same
${\mathcal{R}}$
, the ratio
$C_{d,max}^{r}/C_{d,\infty }^{r}$
is almost half of
$C_{d,max}^{t}/C_{d,\infty }^{t}$
, and
$\unicode[STIX]{x1D6FF}$
corresponding to
$C_{d,max}^{t}$
is shifted towards one.
A natural question that may present itself at this point is: How do the behaviour of the effective mass and damping coefficients for vibrating cylinders compare with those for an oscillating sphere? To answer this question, we have analytically solved for
$C_{e}^{t}$
,
$C_{d}^{t}$
,
$C_{e}^{r}$
and
$C_{d}^{r}$
of a sphere using the method of § 2.2. The derivation details along with the plots of the coefficients versus
$\unicode[STIX]{x1D6FF}$
(see figure 8) are presented in appendix B. To the best of our knowledge, these results have not been reported correctly or at all elsewhere. The comparison of figures 3 and 5 against figure 8 reveals that there is a strong resemblance between the behaviour of the effective mass and damping coefficients of cylinders and spheres. It also indicates that the critical dimensionless frequency
${\mathcal{R}}_{c}$
, beyond which the damping coefficient maximizes at a finite
$\unicode[STIX]{x1D6FF}$
, is higher for a sphere than it is for a circular cylinder. In addition, we observe that, at a given
${\mathcal{R}}>{\mathcal{R}}_{c}$
, the ratios
$C_{d,max}^{t}/C_{d,\infty }^{t}$
and
$C_{d,max}^{r}/C_{d,\infty }^{r}$
for a sphere are roughly 35 % and 20 % less than their counterparts for a circular cylinder, respectively.
Figure 7. Percentage difference between the results of full and linearized NS calculations for the damping coefficients of porous cylinders oscillating (a) transversely and (b) rotationally with finite amplitudes. The primary plots belong to circular cylinders, whereas those in the insets are for elliptical cylinders of aspect ratio
$\unicode[STIX]{x1D700}=0.1$
. The dimensionless frequency is
${\mathcal{R}}=10^{3}$
in all plots.
In concluding this section, we examine the validity of the small-amplitude assumption under which the presented results were obtained. Figure 7 shows the percentage difference between the results of full and linearized NS calculations for the damping coefficient of finite-amplitude transverse and rotational oscillations. The results are shown for two representative cross-sections, circular and elliptical with aspect ratio
$\unicode[STIX]{x1D700}=0.1$
, and for the highest frequency considered in this study, i.e.
${\mathcal{R}}=10^{3}$
, where the largest deviations from the small-amplitude theory are expected to occur. When the cross-section is circular (
$\unicode[STIX]{x1D700}=1$
), we see that the difference stays below 5 % for the dimensionless amplitude as high as
$\unicode[STIX]{x1D706}=0.3$
, which indicates how wide the applicability range of the small-amplitude assumption can be. For the same ratio of the amplitude to aspect ratio (i.e.
$\unicode[STIX]{x1D706}/\unicode[STIX]{x1D700}=0.3$
), we see that the difference is similarly low (less than 6 %) for the elliptical cross-section of
$\unicode[STIX]{x1D700}=0.1$
. That an error of similar magnitude is reached at a lower amplitude for
$\unicode[STIX]{x1D700}=0.1$
, compared to
$\unicode[STIX]{x1D700}=1$
, is due to the more vigorous vortex shedding from the sharp edges as the cross-section gets thinner. This nonlinear phenomenon is not captured by the linearized model.
4 Conclusion
We systematically investigated the dynamic response of fluid-submerged porous (circular and elliptical) cylinders under periodic (transverse and rotational) oscillations of small amplitude. Using analytical and numerical modelling, we demonstrated how the oscillation frequency and permeability affect the hydrodynamic performance of cylinders characterized by their effective mass and damping coefficients. Perhaps surprisingly, we discovered that, at high enough oscillation frequencies, permeable cylinders can significantly outperform their impermeable counterparts when used as dampers to dissipate oscillatory mechanical energy. What is intriguing is that, in those cases, a drastic change in the damping characteristics of impermeable cylinders can be achieved by making them slightly permeable. Our calculations indicate that the optimal damping occurs when the ratio of the characteristic length of oscillation penetration and the characteristic permeation length is of order one. This and other findings of our study can serve as engineering guidelines for designing porous damping structures with superior hydrodynamic performance, which may be used, for example, as stabilizers in deep-water offshore platforms to dampen unwanted environmental disturbances. We also hope that the insights gained by our analyses guide and/or motivate future theoretical and experimental investigations on the subject. Potentially interesting directions to pursue are considering other geometries, large-amplitude oscillations and confinement effects. Performing pore-resolved numerical simulations would be well worth while, too.
Acknowledgements
We thank M. Aureli for stimulating discussions. This research was carried out in part using the computational resources provided by the Superior high-performance computing facility at Michigan Technological University.
Appendix A. Asymptotic expressions for the effective mass and damping coefficients of circular cylinders
The expressions derived for the effective mass and damping coefficients of circular cylinders in § 2.2 involve modified Bessel functions of complex arguments. To make the intricate calculations more accessible and to gain additional physical insights, in table 1 we present simplified formulae for the effective mass and damping coefficients of circular cylinders in the asymptotic limits of high/low oscillation frequency and permeability. Several considerations are made in deriving the asymptotic expressions. First, in the two limits involving
$\unicode[STIX]{x1D6FF}\rightarrow 0$
, only the largest non-zero terms are retained (see the top two rows of table 1). Second, in the limit
$\unicode[STIX]{x1D6FF}\rightarrow \infty$
and
${\mathcal{R}}\rightarrow 0$
, we demand that
$\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FF}\sqrt{{\mathcal{R}}}\rightarrow \infty$
. This additional restriction is not needed in the opposite limit
$\unicode[STIX]{x1D6FF}\rightarrow 0$
and
${\mathcal{R}}\rightarrow \infty$
, because, in that situation, the magnitude of
$\unicode[STIX]{x1D709}_{i}$
goes to infinity irrespective of whether
$\unicode[STIX]{x1D6FF}\sqrt{{\mathcal{R}}}$
is finite, decays to zero or blows up (see (2.5)). Third, when
$\unicode[STIX]{x1D6FF}\rightarrow \infty$
and
${\mathcal{R}}\rightarrow \infty$
, it is assumed that
$\unicode[STIX]{x1D6FF}$
and
$\sqrt{{\mathcal{R}}}$
increase at about the same rate. Lastly, in cases where
$\unicode[STIX]{x1D6FF}\rightarrow \infty$
, we keep the largest terms involving
$\unicode[STIX]{x1D6FF}$
(i.e. the leading-order corrections due to the permeability of the cylinder) and
$\unicode[STIX]{x1D6FF}$
-independent terms of the same or greater order of magnitude (see the bottom two rows of table 1).
Table 1. Asymptotic expressions for the effective mass and damping coefficients of transversely and rotationally oscillating porous circular cylinders. The new parameters
$\unicode[STIX]{x1D6EC}$
and
$\unicode[STIX]{x1D6E4}$
are defined as
$\unicode[STIX]{x1D6EC}=\ln (\sqrt{{\mathcal{R}}}/2)+\unicode[STIX]{x1D6F6}$
and
$\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D6EC}^{2}+\unicode[STIX]{x03C0}^{2}/16$
, where
$\unicode[STIX]{x1D6F6}$
is the Euler constant.
Appendix B. Small-amplitude oscillations of a porous sphere
The solution approach adopted in § 2.2 to solve for the flow field within and around a vibrating porous circular cylinder can also be used to derive closed-form expressions for the velocity distribution in the interior and exterior of an oscillating porous sphere of radius
$a$
when the amplitude of oscillations is small relative to
$a$
. To proceed with the derivations, we need to first determine the fundamental singularities of the BDB equations for three-dimensional internal and external flows. The three-dimensional equivalents of stokeson, potential doubleton and roton, and of stokeslet, potential doublet and rotlet (see § 2.2 for the name convention) can be shown to be, respectively (see also Pozrikidis Reference Pozrikidis1989),
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{SN}(\boldsymbol{r};\unicode[STIX]{x1D736}) & = & \displaystyle \frac{({\unicode[STIX]{x1D70C}_{i}}^{2}+1)\sinh \unicode[STIX]{x1D70C}_{i}\!-\!\unicode[STIX]{x1D70C}_{i}\cosh \unicode[STIX]{x1D70C}_{i}}{\unicode[STIX]{x1D70C}_{i}^{3}}\!\left[\unicode[STIX]{x1D736}-\frac{3\unicode[STIX]{x1D709}_{i}^{2}(\unicode[STIX]{x1D736}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{r}}{\unicode[STIX]{x1D70C}_{i}^{2}}\right]\!+\frac{2\unicode[STIX]{x1D709}_{i}^{2}\sinh \unicode[STIX]{x1D70C}_{i}(\unicode[STIX]{x1D736}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{r}}{\unicode[STIX]{x1D70C}_{i}^{3}},\qquad\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{SN}(\boldsymbol{r};\unicode[STIX]{x1D736}) & = & \displaystyle 0,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{DN}(\boldsymbol{r};\boldsymbol{\unicode[STIX]{x1D702}}) & = & \displaystyle \boldsymbol{\unicode[STIX]{x1D702}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{DN}(\boldsymbol{r};\boldsymbol{\unicode[STIX]{x1D702}}) & = & \displaystyle -\unicode[STIX]{x1D709}_{i}^{2}(\boldsymbol{\unicode[STIX]{x1D702}}\boldsymbol{\cdot }\boldsymbol{r}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{RN}(\boldsymbol{r};\unicode[STIX]{x1D738}) & = & \displaystyle \frac{\unicode[STIX]{x1D70C}_{i}\cosh \unicode[STIX]{x1D70C}_{i}-\sinh \unicode[STIX]{x1D70C}_{i}}{\unicode[STIX]{x1D70C}_{i}^{3}}(\unicode[STIX]{x1D738}\times \boldsymbol{r}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{RN}(\boldsymbol{r};\unicode[STIX]{x1D738}) & = & \displaystyle 0,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{S}(\boldsymbol{r};\unicode[STIX]{x1D736}) & = & \displaystyle 2\left[\text{e}^{-\unicode[STIX]{x1D70C}_{o}}\left(1+\frac{1}{\unicode[STIX]{x1D70C}_{o}}+\frac{1}{\unicode[STIX]{x1D70C}_{o}^{2}}\right)-\frac{1}{\unicode[STIX]{x1D70C}_{o}^{2}}\right]\frac{\unicode[STIX]{x1D736}}{r}\nonumber\\ \displaystyle & & \displaystyle -\,2\left[\text{e}^{-\unicode[STIX]{x1D70C}_{o}}\left(1+\frac{3}{\unicode[STIX]{x1D70C}_{o}}+\frac{3}{\unicode[STIX]{x1D70C}_{o}^{2}}\right)-\frac{3}{\unicode[STIX]{x1D70C}_{o}^{2}}\right]\frac{(\unicode[STIX]{x1D736}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{r}}{r^{3}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{S}(\boldsymbol{r};\unicode[STIX]{x1D736}) & = & \displaystyle \frac{2(\unicode[STIX]{x1D736}\boldsymbol{\cdot }\boldsymbol{r})}{r^{3}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{D}(\boldsymbol{r};\boldsymbol{\unicode[STIX]{x1D702}}) & = & \displaystyle -\text{e}^{-\unicode[STIX]{x1D70C}_{o}}(1+\unicode[STIX]{x1D70C}_{o}+\unicode[STIX]{x1D70C}_{o}^{2})\left[\frac{\boldsymbol{\unicode[STIX]{x1D702}}}{r^{3}}-\frac{3(\boldsymbol{\unicode[STIX]{x1D702}}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{r}}{r^{5}}\right]-\frac{2\text{e}^{-\unicode[STIX]{x1D70C}_{o}}\unicode[STIX]{x1D70C}_{o}^{2}(\boldsymbol{\unicode[STIX]{x1D702}}\boldsymbol{\cdot }\boldsymbol{r})\boldsymbol{r}}{r^{5}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{D}(\boldsymbol{r};\boldsymbol{\unicode[STIX]{x1D702}}) & = & \displaystyle 0,\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{R}(\boldsymbol{r};\unicode[STIX]{x1D738}) & = & \displaystyle \frac{2\text{e}^{-\unicode[STIX]{x1D70C}_{o}}(\unicode[STIX]{x1D70C}_{o}+1)}{r^{3}}(\unicode[STIX]{x1D738}\times \boldsymbol{r}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{R}(\boldsymbol{r};\unicode[STIX]{x1D738}) & = & \displaystyle 0.\end{eqnarray}$$
Figure 8. Dimensionless (a,c) effective mass and (b,d) damping coefficients of (a,b) transversely and (c,d) rotationally oscillating porous spheres as a function of
$\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FD}/\sqrt{{\mathcal{R}}}$
for different values of the dimensionless frequency
${\mathcal{R}}$
. Dashed lines in the insets are plots of the asymptotic expressions in the limits of high/low oscillation frequency and permeability (see table 2).
As explained in § 2.2, for the translational oscillations, the velocity and pressure fields take the forms of
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{i} & = & \displaystyle \boldsymbol{u}_{SN}(\boldsymbol{r};\unicode[STIX]{x1D6FC}_{i}\boldsymbol{e}_{y})+\boldsymbol{u}_{DN}(\boldsymbol{r};\unicode[STIX]{x1D702}_{i}\boldsymbol{e}_{y})+(\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D709}_{i})^{2}\boldsymbol{e}_{y},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{i} & = & \displaystyle p_{SN}(\boldsymbol{r};\unicode[STIX]{x1D6FC}_{i}\boldsymbol{e}_{y})+p_{DN}(\boldsymbol{r};\unicode[STIX]{x1D702}_{i}\boldsymbol{e}_{y}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{o} & = & \displaystyle \boldsymbol{u}_{S}(\boldsymbol{r};\unicode[STIX]{x1D6FC}_{o}\boldsymbol{e}_{y})+\boldsymbol{u}_{D}(\boldsymbol{r};\unicode[STIX]{x1D702}_{o}\boldsymbol{e}_{y}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{o} & = & \displaystyle p_{S}(\boldsymbol{r};\unicode[STIX]{x1D6FC}_{o}\boldsymbol{e}_{y})+p_{D}(\boldsymbol{r};\unicode[STIX]{x1D702}_{o}\boldsymbol{e}_{y}),\end{eqnarray}$$
$r=1$
, the unknown coefficients are found to be 
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{i}=\frac{A_{3}}{A_{2}},\quad \unicode[STIX]{x1D702}_{i}=-\frac{\unicode[STIX]{x1D6FD}^{2}}{\unicode[STIX]{x1D709}_{i}^{2}}\left(1-\frac{A_{1}}{A_{2}}\right),\quad \unicode[STIX]{x1D6FC}_{o}=\frac{\unicode[STIX]{x1D6FD}^{2}}{2}\left(1-\frac{A_{1}}{A_{2}}\right),\quad \unicode[STIX]{x1D702}_{o}=\frac{\unicode[STIX]{x1D6FD}^{2}}{\unicode[STIX]{x1D709}_{o}^{2}}\left(1-\frac{A_{1}+A_{4}}{A_{2}}\right), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle A_{1} & = & \displaystyle 2\unicode[STIX]{x1D709}_{i}^{4}(\unicode[STIX]{x1D709}_{i}\cosh \unicode[STIX]{x1D709}_{i}+\unicode[STIX]{x1D709}_{o}\sinh \unicode[STIX]{x1D709}_{i}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle A_{2} & = & \displaystyle \cosh \unicode[STIX]{x1D709}_{i}[3\unicode[STIX]{x1D6FD}^{2}\unicode[STIX]{x1D709}_{i}(\unicode[STIX]{x1D709}_{o}+1)+\unicode[STIX]{x1D709}_{i}^{3}\unicode[STIX]{x1D709}_{o}^{2}+2\unicode[STIX]{x1D709}_{i}^{5}]\nonumber\\ \displaystyle & & \displaystyle +\,\sinh \unicode[STIX]{x1D709}_{i}[-3\unicode[STIX]{x1D6FD}^{2}(\unicode[STIX]{x1D709}_{o}+1)+\unicode[STIX]{x1D709}_{i}^{2}\unicode[STIX]{x1D709}_{o}^{3}+2\unicode[STIX]{x1D709}_{i}^{4}\unicode[STIX]{x1D709}_{0}],\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle A_{3} & = & \displaystyle -3\unicode[STIX]{x1D6FD}^{2}\unicode[STIX]{x1D709}_{i}^{3}(\unicode[STIX]{x1D709}_{o}+1),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle A_{4} & = & \displaystyle 3\unicode[STIX]{x1D709}_{i}^{2}e^{\unicode[STIX]{x1D709}_{o}}(\unicode[STIX]{x1D709}_{i}\cosh \unicode[STIX]{x1D709}_{i}-\sinh \unicode[STIX]{x1D709}_{i}).\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{F}=\unicode[STIX]{x1D6FD}^{2}\int _{S}(\boldsymbol{u}_{i}-\boldsymbol{u}_{c})\,\text{d}S=\frac{4\unicode[STIX]{x03C0}}{3}\frac{\unicode[STIX]{x1D6FD}^{2}}{\unicode[STIX]{x1D709}_{i}^{2}}\left[\frac{2\unicode[STIX]{x1D6FC}_{i}}{\unicode[STIX]{x1D709}_{i}}(\unicode[STIX]{x1D709}_{i}\cosh \unicode[STIX]{x1D709}_{i}-\sinh \unicode[STIX]{x1D709}_{i})+\unicode[STIX]{x1D702}_{i}\unicode[STIX]{x1D709}_{i}^{2}-\text{i}{\mathcal{R}}\right]\boldsymbol{e}_{y}. & & \displaystyle\end{eqnarray}$$
Again per § 2.2, the flow field for the rotational vibrations is expressed as
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{i} & = & \displaystyle \boldsymbol{u}_{RN}(\boldsymbol{r};\unicode[STIX]{x1D6FE}_{i}\boldsymbol{e}_{z})+(\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D709}_{i})^{2}(\boldsymbol{e}_{z}\times \boldsymbol{r}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{i} & = & \displaystyle p_{RN}(\boldsymbol{r};\unicode[STIX]{x1D6FE}_{i}\boldsymbol{e}_{z}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{o} & = & \displaystyle \boldsymbol{u}_{R}(\boldsymbol{r};\unicode[STIX]{x1D6FE}_{o}\boldsymbol{e}_{z}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p_{o} & = & \displaystyle p_{R}(\boldsymbol{r};\unicode[STIX]{x1D6FE}_{o}\boldsymbol{e}_{z}),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{i} & = & \displaystyle \displaystyle \frac{-\unicode[STIX]{x1D6FD}^{2}\unicode[STIX]{x1D709}_{i}(\unicode[STIX]{x1D709}_{o}^{2}+3\unicode[STIX]{x1D709}_{o}+3)}{\unicode[STIX]{x1D6FD}^{2}\sinh \unicode[STIX]{x1D709}_{i}+\unicode[STIX]{x1D709}_{i}\unicode[STIX]{x1D709}_{o}(\unicode[STIX]{x1D709}_{o}\cosh \unicode[STIX]{x1D709}_{i}+\unicode[STIX]{x1D709}_{i}\sinh \unicode[STIX]{x1D709}_{i})},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{o} & = & \displaystyle \displaystyle \frac{\unicode[STIX]{x1D6FD}^{2}\text{e}^{\unicode[STIX]{x1D709}_{o}}[(\unicode[STIX]{x1D709}_{i}^{2}+3)\sinh \unicode[STIX]{x1D709}_{i}-3\unicode[STIX]{x1D709}_{i}\cosh \unicode[STIX]{x1D709}_{i}]}{2\unicode[STIX]{x1D709}_{i}^{2}[\unicode[STIX]{x1D6FD}^{2}\sinh \unicode[STIX]{x1D709}_{i}+\unicode[STIX]{x1D709}_{i}\unicode[STIX]{x1D709}_{o}(\unicode[STIX]{x1D709}_{o}\cosh \unicode[STIX]{x1D709}_{i}+\unicode[STIX]{x1D709}_{i}\sinh \unicode[STIX]{x1D709}_{i})]}.\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \boldsymbol{T}=\unicode[STIX]{x1D6FD}^{2}\!\!\int _{S}[\boldsymbol{r}\times (\boldsymbol{u}_{i}-\boldsymbol{u}_{c})]\,\text{d}S=\frac{8\unicode[STIX]{x03C0}}{3}\frac{\unicode[STIX]{x1D6FD}^{2}}{\unicode[STIX]{x1D709}_{i}^{2}}\left\{\frac{\unicode[STIX]{x1D6FE}_{i}}{\unicode[STIX]{x1D709}_{i}^{3}}[(\unicode[STIX]{x1D709}_{i}^{2}+3)\sinh \unicode[STIX]{x1D709}_{i}-3\unicode[STIX]{x1D709}_{i}\cosh \unicode[STIX]{x1D709}_{i}]-\frac{\text{i}{\mathcal{R}}}{5}\right\}\boldsymbol{e}_{z}.\qquad & & \displaystyle\end{eqnarray}$$
Table 2. Asymptotic expressions for the effective mass and damping coefficients of transversely and rotationally oscillating porous spheres.
The corresponding effective mass and damping coefficients are obtained by substituting
$F$
and
$T$
from above into
$$\begin{eqnarray}\displaystyle C_{e}^{t}=-\text{Im}[F/(4\unicode[STIX]{x03C0}{\mathcal{R}}/3)]\quad \text{and}\quad C_{d}^{t}=-\text{Re}[F/(4\unicode[STIX]{x03C0}{\mathcal{R}}/3)], & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle C_{e}^{r}=-\text{Im}[T/(4\unicode[STIX]{x03C0}{\mathcal{R}}R_{g}^{2}/3)]\quad \text{and}\quad C_{d}^{r}=-\text{Re}[T/(4\unicode[STIX]{x03C0}{\mathcal{R}}R_{g}^{2}/3)], & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle R_{g}^{2}=\frac{\displaystyle \int _{V}(x^{2}+y^{2})\,\text{d}V}{\displaystyle \int _{V}\,\text{d}V}. & & \displaystyle\end{eqnarray}$$
Figure 8 shows the variations of
$C_{e}^{t}$
,
$C_{d}^{t}$
,
$C_{e}^{r}$
and
$C_{d}^{r}$
versus
$\unicode[STIX]{x1D6FF}$
for various
${\mathcal{R}}$
. The figure is supplemented by table 2, which presents simple expressions justifying the behaviour of the plots in the asymptotic limits of low/high
$\unicode[STIX]{x1D6FF}$
and
${\mathcal{R}}$
. The formulae are obtained subject to the same considerations as in appendix A.
