2. Formulation of the problem and solution approach

Let us consider a flat plate which bounds a viscous fluid and oscillates in its own plane with velocity $U_0^{*}\cos (\omega ^{*}t^{*})$, where $U^{*}_0$ and $\omega ^{*}$ denote the amplitude and the angular frequency of the velocity oscillations, and $t^{*}$ is time. Moreover, let us introduce the viscous length $\delta ^{*}=\sqrt {{2 \nu ^{*}}/{\omega }}$, where $\nu ^{*}$ is the kinematic viscosity of the fluid.

Let us introduce a coordinate system $(x^{*},y^{*},z^{*})$ with the $x^{*}$- and $y^{*}$-axes lying on the plate, the $x^{*}$-axis being in the direction of the plate oscillations, and the $z^{*}$-axis being vertical and pointing upwards.

The hydrodynamic problem is posed by continuity and Navier–Stokes equations along with the no-slip condition at the plate surface and the boundary condition that enforces the vanishing of the velocity far from the plate. Let us introduce dimensionless variables such that the spatial coordinates are scaled with the viscous length $\delta ^{*}$ and time is scaled using the angular frequency of the fluid oscillations,

(2.1a,b)\begin{equation} (x, y, z) = \frac{(x^{*},y^{*},z^{*})}{\delta^{*}},\quad t = \omega^{*}t^{*}. \end{equation}

Moreover, let us denote with ($u,v,w$) and $p$ the dimensionless velocity components and the pressure field, respectively, defined by

(2.2a,b)\begin{equation} (u,v,w) = \frac{(u^{*},v^{*},w^{*})}{U^{*}_0},\quad p=\frac{p^{*}}{\rho^{*}U^{*}_0\delta^{*}\omega^{*}}, \end{equation}

where $\rho ^{*}$ is the density of the fluid. Then, the hydrodynamic problem is posed by

(2.3)\begin{equation} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0 \end{equation}

and

(2.4)$$\begin{gather} \frac{\partial u}{\partial t}+ \frac{R_\delta}{2} \left(u\frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)={-}\frac{\partial p_m}{\partial x} +\frac{1}{2}\left( \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+ \frac{\partial^{2} u}{\partial z^{2}}\right), \end{gather}$$

(2.5)$$\begin{gather}\frac{\partial v}{\partial t} + \frac{R_\delta}{2} \left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\right) ={-}\frac{\partial p_m}{\partial y} +\frac{1}{2}\left( \frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}+\frac{\partial^{2} v}{\partial z^{2}}\right), \end{gather}$$

(2.6)$$\begin{gather}\frac{\partial w}{\partial t} + \frac{R_\delta}{2} \left(u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\right) ={-}\frac{\partial p_m}{\partial z} +\frac{1}{2}\left( \frac{\partial^{2} w}{\partial x^{2}}+\frac{\partial^{2} w}{\partial y^{2}}+\frac{\partial^{2} w}{\partial z^{2}}\right), \end{gather}$$

where the force per unit mass is assumed to be conservative and the modified pressure $p^{*}_m$ is introduced such that $p^{*}_m= p^{*}-\rho ^{*} \varphi ^{*}$, $\varphi ^{*}=-g^{*}z^{*}$ being the dimensional potential function of the body force and $g^{*}$ being the gravitational acceleration.

It can be easily verified that flow field induced by the plate oscillations is described by

(2.7a–d)\begin{equation} u_0=\tfrac{1}{2}\textrm{e}^{-(1+i)z}\ \textrm{e}^{\textrm{i}t}+\mbox{c.c.},\quad v_0=0,\quad w_0=0,\quad p_{m,0}=0, \end{equation}

where $p_{m,0}$ is the leading-order term of the modified pressure, i.e. the actual pressure minus the hydrostatic contribution.

To investigate the stability of this basic flow, let us consider a perturbation of small amplitude superimposed on the basic flow and let us determine its time development. Since, according to Squire's theorem, two-dimensional perturbations of the flow field are more unstable than three-dimensional perturbations, let us assume

(2.8)\begin{equation} (u,v,w,p_m)=(u_0,0,0,p_{m,0})+\epsilon (u_1,0,w_1,p_{m,1}), \end{equation}

where $\epsilon$ is assumed to be much smaller than unity. Neglecting terms of order $\epsilon ^{2}$, continuity and momentum equations read

(2.9)$$\begin{gather} \frac{\partial u_1}{\partial x} +\frac{\partial w_1}{\partial z}=0, \end{gather}$$

(2.10)$$\begin{gather}\frac{\partial u_1}{\partial t}+ \frac{R_\delta}{2} \left[ u_0\frac{\partial u_1}{\partial x} +w_1\frac{\partial u_0}{\partial z} \right] ={-}\frac{\partial p_{m,1}}{\partial x}+\frac{1}{2}\left[\frac{\partial^{2} u_1}{\partial x^{2}}+\frac{\partial^{2} u_1}{\partial z^{2}}\right], \end{gather}$$

(2.11)$$\begin{gather}\frac{\partial w_1}{\partial t}+ \frac{R_\delta}{2}\left[ u_0\frac{\partial w_1}{\partial x} \right] ={-}\frac{\partial p_{m,1}}{\partial z}+\frac{1}{2}\left[ \frac{\partial^{2} w_1}{\partial x^{2}}+\frac{\partial^{2} w_1}{\partial z^{2}}\right]. \end{gather}$$

The continuity equation is satisfied if a stream function $\psi$ is introduced such that

(2.12)\begin{equation} (u_1,w_1)=\left(\frac{\partial \psi}{\partial z},-\frac{\partial \psi}{\partial x}\right). \end{equation}

Then, eliminating the pressure field from (2.10) and (2.11), the linearized vorticity equation is obtained as

(2.13)\begin{align} & \frac{\partial^{3} \psi}{\partial t \partial x^{2}}+\frac{\partial^{3} \psi}{\partial t \partial z^{2}} + \frac{R_\delta}{2}\left[ u_0 \left( \frac{\partial^{3} \psi}{\partial x^{3}}+ \frac{\partial^{3} \psi}{\partial z^{2} \partial x} \right) -\frac{\partial^{2} u_0}{\partial z^{2}} \frac{\partial \psi}{\partial x} \right] \nonumber\\ &\quad = \frac{1}{2}\left( \frac{\partial^{4} \psi}{\partial x^{4}} +2 \frac{\partial^{4} \psi}{\partial x^{2} \partial z^{2}} +\frac{\partial^{4} \psi}{\partial z^{4}} \right). \end{align}

Because of the linearity of (2.13), it is possible to consider the generic streamwise harmonic components of the perturbation. Moreover, because the growth of perturbations of the basic laminar flow is observed when the Reynolds number is much larger than unity, the amplitude of the flow perturbations can be assumed to grow on a temporal scale $\tau$ much faster than that characterizing the time development of the basic flow. Hence, a ‘momentary’ criterion for the instability of the basic flow can be used (Shen Reference Shen1961). Indeed, the Reynolds number can be thought to be the ratio between the characteristic temporal scale $1/\omega ^{*}$ of the fluid oscillations and the convective temporal scale $\delta ^{*}/U^{*}_0$ of the time development of the perturbation. Hence, let us consider a solution of (2.13) in the form

(2.14)\begin{equation} \psi(x,z,t) = \mbox{Real}\left\{ f(z,t) \exp \left[ \textrm{i} \alpha \left( x - \frac{R_\delta}{2} \int c(\tau)\,{\textrm{d}}\tau \right) \right] \right\} + \mbox{h.o.t.}, \end{equation}

where h.o.t. indicates higher-order terms. In (2.14), $\alpha$ is the wavenumber of the generic Fourier component of the flow perturbation in the streamwise direction, the real part $c_r$ of the complex quantity $c$ is its wave speed and the imaginary part $c_i$ controls its growth/decay rate.

At the leading order of approximation, when (2.14) is plugged into (2.13), the following homogeneous differential equation is obtained:

(2.15)\begin{equation} \left[ u_0(z,t) - c(t) \right] {\mathcal{N}}^{2} f(z,t) - \frac{\partial^{2} u_0(z,t)}{\partial z^{2}} f(z,t) = \frac{1}{i \alpha R_\delta} {\mathcal{N}}^{4} f(z,t), \end{equation}

where the operator ${\mathcal {N}}^{2}$ is defined by ${\mathcal {N}}^{2}={\partial ^{2}}/{\partial z^{2}}-\alpha ^{2}$. The boundary conditions force the vanishing of the velocity at the wall and far from it

(2.16)\begin{equation} \left.\begin{gathered} \frac{\partial f}{\partial z} = 0,\quad f=0\quad \mbox{at}\ z=0,\\ \frac{\partial f}{\partial z} = \rightarrow 0,\quad f \rightarrow 0\ \mbox{for}\ z \rightarrow \infty. \end{gathered}\right\} \end{equation}

At this stage, it is worth pointing out that the viscous term is retained in (2.15) even if it is of order $1/R_\delta$, because it turns out to be significant, at the leading order of approximation, within a viscous layer close to the wall and within critical layers. As pointed out by Blondeaux & Vittori (Reference Blondeaux and Vittori1994), a formal approach would require the solution of the inviscid version of (2.15) and, then, the matching of the inviscid solution with the solution which holds in the viscous and critical layers. However, such a procedure would involve much tedious and heavy algebra which can be avoided by the direct solution of (2.15). Blondeaux & Vittori (Reference Blondeaux and Vittori1994) showed that the direct solution of (2.15) coincides, to the required order of approximation, with the solution obtained on the basis of the rigorous matched asymptotic approach, previously briefly outlined.

A non-vanishing solution of the hydrodynamic homogeneous problem posed by (2.15) and its boundary conditions can be found by forcing an eigenrelation which provides the value of $c$ as a function of $t$, $\alpha$ and $R_\delta$.

To find the eigenvalues $c$ and the corresponding eigenfunctions $f$, the approach employed by Blondeaux & Seminara (Reference Blondeaux and Seminara1979) is employed. The stream function $\psi$ is written in the form

(2.17)\begin{align} f &= \sum_{n=0}^{\infty} \sum_{m=0}^{n} a_{n,m}(t)\exp({[-\alpha -n(1+i)+2mi]z}) \nonumber\\ &\quad +b_{n,m}(t)\exp({[-\sigma -n(1+i)+2mi]z}, \end{align}

where $\sigma =\sqrt {\alpha ^{2}-2 \ \textrm {i} \alpha R_\delta c}$ and the coefficients $a_{n,m}$ and $b_{n,m}$ can be written as functions of $a_{0,0}$ and $b_{0,0}$ by means of the recursive relationships

(2.18)\begin{align} & \left(a_{n,m},b_{n,m}\right) = \textrm{i}\alpha R_\delta \left\{\frac{\mu}{2}\textrm{e}^{\textrm{i}t} \left[\left(-\left(\alpha,\sigma\right) - {\mathcal{C}}_{n,m} \right)^{2} -\alpha^{2} -2\textrm{i} \right]\left(a_{n-1,m},b_{n-1,m}\right) \vphantom{\frac{\zeta}{2}\textrm{e}^{-\textrm{i}t} \left[\left(-\left(\alpha,\sigma\right) - {\mathcal{C}}_{n,m}-2\textrm{i}\right)^{2} -\alpha^{2} +2\textrm{i}\right]\left(a_{n-1,m-1},b_{n-1,m-1}\right)}\right. \nonumber\\ &\quad \left.+\frac{\zeta}{2}\textrm{e}^{-\textrm{i}t} \left[\left(-\left(\alpha,\sigma\right) - {\mathcal{C}}_{n,m}-2\textrm{i}\right)^{2} -\alpha^{2} +2\textrm{i}\right]\left(a_{n-1,m-1},b_{n-1,m-1}\right) \right\} \nonumber\\ &\quad \times\left\{ \left[\left(-\left(\alpha,\sigma\right) - {\mathcal{D}}_{n,m} \right)^{2} -\alpha^{2} \right]^{2} + \textrm{i} \alpha R_\delta c \left[\left(-\left(\alpha,\sigma\right) - {\mathcal{D}}_{n,m} \right)^{2} -\alpha^{2} \right] \right\}^{{-}1}, \end{align}

where

(2.19a,b)\begin{equation} {\mathcal{C}}_{n,m}=(n-1)(1+i)-2mi,\quad {\mathcal{D}}_{n,m}={\mathcal{C}}_{n,m}-1-i. \end{equation}

Moreover, $\mu =0$ when $m=n$, whereas $\mu =1$ when $0\le m \le n-1$. Similarly, $\zeta =0$ when $m=0$, and $\zeta =1$ when $1\le m \le n$.

The boundary conditions at the wall lead to

(2.20)\begin{equation} \left.\begin{gathered} A_{11} a_{0,0} + A_{12} b_{0,0} = 0,\\ A_{21} a_{0,0} + A_{22} b_{0,0} = 0, \end{gathered}\right\} \end{equation}

where

(2.21)\begin{equation} \left.\begin{gathered} A_{11}=\sum_0^{\infty} \sum_{m=0}^{n} \frac{a_{n,m}}{a_{0,0}},\\ A_{12}=\sum_0^{\infty} \sum_{m=0}^{n} \frac{b_{n,m}}{b_{0,0}},\\ A_{21}=\sum_0^{\infty} \sum_{m=0}^{n} \left(-\alpha -n(1+i)+2 m i \right) \frac{a_{n,m}}{a_{0,0}},\\ A_{22}=\sum_0^{\infty} \sum_{m=0}^{n} \left(-\sigma -n(1+i)+2 m i \right) \frac{b_{n,m}}{b_{0,0}}. \end{gathered}\right\} \end{equation}

A non-trivial solution of the system (2.20) can be found if and only if

(2.22)\begin{equation} A_{11}A_{22}-A_{12}A_{21}=0, \end{equation}

which provides the eigenrelation $c=c(t,\alpha ,R_\delta )$.

This flow field vanishes when $z$ tends to infinity unless $c_r=0$ and $c_i < -({\alpha R_\delta }/{2})$. As discussed in Blondeaux & Seminara (Reference Blondeaux and Seminara1979), when $c_r=0$ and $c_i < -({\alpha R_\delta }/{2})$, a continuous spectrum of eigenvalues exists but the eigenfunctions do no decay moving far away from the oscillating wall. Indeed, for such values of $c$ the solution of the eigenvalue problem becomes

(2.23)\begin{align} f &= a \ \textrm{e}^{-\alpha z}+b \cos\left( \sqrt{\alpha^{2} + \alpha R_\delta c_i} z \right) +c \sin\left( \sqrt{\alpha^{2} + \alpha R_\delta c_i} z \right) \nonumber\\ &\quad + \mbox{exponential decaying terms}. \end{align}

Therefore, the continuous spectrum will be no longer considered in the following.

3. The results

The eigenfunction $f$ and the eigenvalues $c$ are evaluated by truncating the summation (2.17) after $N$ terms, with $N$ large enough that the addition of further terms does not modify the results. The value of $N$ depends on $R_\delta$ and $\alpha$. For example, for $\alpha =0.5$ and $R_\delta =100$, already $N=15$ provides reliable results (five significant digits), but for $\alpha =0.5$ and $R_\delta =1000$, values of $N$ larger than $35$ are required to obtain the same accuracy. However, the evaluation of $A_{11}, A_{12}, A_{21}$ and $A_{22}$ requires a relatively small effort even when the accurate evaluation of $\psi$ asks for a large number of terms in (2.17). It is worth pointing out that increasing values of $N$ demand the evaluation of $a_{n,m}$ and $b_{n,m}$ with more digits. For example, for $\alpha =0.5$, $R_\delta =100$ and $t=0.0$, the single-precision floating-point format is enough, but for $\alpha =0.5$, $R_\delta =1500$ and $t=0.0$, the quadruple-precision is required. Because of the small computational costs required by the numerical procedure used to obtain the eigenvalues $c$ and the eigenfunctions $f$, the results described in the following were obtained by fixing $N=75$ and using the quadruple-precision for all the values of $\alpha$ and $R_\delta$.

Before discussing in detail the quantitative results provided by the solution of the eigenvalue problem described in the previous section, let us point out that for any eigenvalue $c(t)=c_r(t)+\textrm {i}c_i(t)$, there are other eigenvalues equal to minus the complex conjugate of $c$ at phases of the cycle equal to $t \pm {\rm \pi}$ $(c_r(t)+\textrm {i} c_i(t) = -c_r(t \pm {\rm \pi}) + \textrm {i} c_i(t \pm {\rm \pi}) )$.

Figure 1 shows the real and imaginary parts of $c$ as a function of the phase $t$ of the cycle for $\alpha =0.5$ and two different values of the Reynolds number, namely $R_\delta =80$ and $R_\delta =90$, which are just below and above the critical value determined by Blondeaux & Seminara (Reference Blondeaux and Seminara1979), namely $R_{\delta ,c1}=85$. For the smaller value of $R_\delta$, the growth rate of the perturbation, i.e. the imaginary part of $c$, is always negative and the perturbation always decays. On the other hand, for the larger value of $R_\delta$, small parts of the cycle exist such that the perturbation grows.

Figure 1. Real (a) and imaginary (b) parts of the growth rate $c$ for $\alpha =0.5$ and $R_\delta =80$ (dashed–dotted line) and $R_\delta =90$ (continuous line). The labels (1), (2), (3), (4) appearing near the curves allow the association of any group of values of $c_r$ with the corresponding group of values of $c_i$.

The size of the unstable parts of the cycle increases as the Reynolds number is increased, as it appears looking at figure 2 where the eigenvalue $c_i$ is plotted versus $t$ for $\alpha =0.5$ and $R_\delta =150$ and $R_\delta =300$.

Figure 2. Imaginary part of the growth rate $c$ for $\alpha =0.5$ and $R_\delta =150$ (continuous line) and $R_\delta =300$ (dashed–dotted line).

Figure 3 shows the eigenfunction $f$ (see (2.17)) for $R_\delta =300, \alpha =0.5$, $t={3{\rm \pi} }/{2}$ and $t=1.8396$. At $t={3{\rm \pi} }/{2}$, the eigenfunction is significant in a layer, the thickness of which is similar to that of the Stokes boundary layer, i.e. the $|f|$ is significant up to values $z$ of order $1$. At $t=1.8396$, the discrete spectrum of eigenvalues is going to merge into the continuous spectrum and the eigenfunction tends to pervade the whole fluid domain since for smaller values of $t$ no mode exists that decays moving away from the oscillating plate. By analysing at the velocity profiles of the basic flow at different phases of the cycle, it appears that the instability of the Stokes flow is predicted when an inflection point is present in a region of the flow where both the velocity and its derivative are large and the Reynolds number is large enough to lead to a significant interaction between the perturbations and the basic flow.

Figure 3. Real (continuous line) and imaginary (dashed–dotted line) parts of the eigenfunction $f(z)$ (see (2.17)) for $R_\delta =300, \alpha =0.5$. (a) For $t={3{\rm \pi} }/{2}$ (point $B$ in this figure), at this phase the perturbation grows. (b) For $t=1.8396$ (point $A$ in this figure), at this phase the perturbation decays and the discrete spectrum of the eigenvalues is going to merge into the continuous spectrum.

An investigation of the parameter space showed that the Stokes boundary layer is ‘momentarily’ unstable during the oscillatory cycle for values of $R_\delta$ larger than $85$ and the most unstable wavenumber $\alpha _{c,1}$ is equal to $0.5$. Figure 4 shows the maximum value of the imaginary part $c_i$ of $c$ as a function of $\alpha$ and $R_\delta$ irrespective of the phase of the cycle at which the maximum takes place. The line characterized by $c_i=0$ is the marginal stability curve in the plane $(\alpha ,R_\delta )$.

Figure 4. The maximum values of $c_i$ observed during the cycle, irrespective of the phase at which they are observed, plotted versus $\alpha$ and $R_\delta$.

Figure 5 shows the phase $t_{ins}$ of the cycle at which the instability first occurs as a function of the Reynolds number. Close to the critical conditions, the instability is predicted close to $t={{\rm \pi} }/{2}+n {\rm \pi}$ with $n=0,1,2,\ldots$ but the instability takes place earlier as the Reynolds number is increased.

Figure 5. Phase $t_{ins}$ of the cycle, at which instability first occurs, plotted versus the Reynolds number for $\alpha =0.5$. Similar results are obtained for different values of $\alpha$. Needless to say that the instability is predicted also at $t_{ins}+n{\rm \pi}$ with $n=0,1,2,\ldots$.

These results were already obtained by Blondeaux & Seminara (Reference Blondeaux and Seminara1979). However, Blondeaux & Seminara (Reference Blondeaux and Seminara1979) did not look at the behaviour of $c$ for larger values of $R_\delta$. Figure 6 shows the imaginary part of $c$ as a function of $t$ for $\alpha =0.5$ and $R_\delta =500$ and $1500$. For the smallest value of $R_\delta$, there is a time interval during which even the most unstable mode is characterized by a negative value of $c_i$, implying a momentary decay of the perturbation. On the other hand, for $R_\delta =1500$, there is an unstable mode during the whole cycle. In figure 6, the points $D$ indicate the phase of the cycle such that the perturbation which propagates in the positive direction of the $x$-axis has the same growth rate as the perturbation which propagates in the negative direction. To determine the critical value $R_{\delta ,c3}$ which leads to the growth of perturbations at any phases of the cycle, the evaluation of $c_i$ is carried out for many values of $R_\delta$ and $\alpha$. Figure 7 shows the value $c_i$ at the point $D$ (see panel (b) of figure 6) plotted in the plane $(\alpha , R_\delta )$. Looking at figure 6, it can be realized that the instability pervades the whole oscillation cycle when $c_i$ at the point $D$ is zero or positive. It follows that the results plotted in figure 7 allow the determination of the value $R_{\delta ,c3}$ of the Reynolds number such that, for $R_\delta$ larger than $R_{\delta ,c3}$, a growing perturbation is present during the whole cycle. It turns out that $R_{\delta ,c3} = 818$ and the most unstable perturbation is characterized by the wavenumber $\alpha _{c,3} = 0.4$. Hence, it can be inferred that the fully turbulent regime is predicted by the present analysis when the Reynolds number is larger than $818$.

Figure 6. Imaginary part of the growth rate $c$ for $\alpha =0.5$ and $R_\delta =500$ (continuous line) and $R_\delta =1500$ (dashed–dotted line). Panel (b) is an enlargement of panel (a).

Figure 7. Imaginary part $c_i$ of the growth rate at the point $D$ (see figure 6) plotted versus $\alpha$ and $R_\delta$.

In figure 8, the value of $c_i$ is plotted for $\alpha =0.5$ and different values of $R_\delta$. When the Reynolds number is not too far from $R_{\delta ,c1}$, the instability is restricted to small parts of the oscillatory cycle. However, when the Reynolds number is increased beyond $200$, the unstable time interval rapidly increases, and for $R_\delta \simeq 550$ the instability pervades a large part of the decelerating phases and the whole accelerating phases. Indeed, the lower limit $t_1$ of the unstable time intervals does not change significantly when the Reynolds number is increased, whereas the upper limit $t_2$ moves quickly to $2{\rm \pi}$. We remind the reader that $c_i(t-{\rm \pi} )=c_i(t)$ is also an eigenvalue which is not plotted in figure 8 for the sake of clarity. Hence, the instability phases also range between $t_1 \pm n {\rm \pi}$ and $t_2 \pm n {\rm \pi}$.

Figure 8. Imaginary part of the growth rate $c$ for $\alpha =0.5$ and different values of $R_\delta$.

The existence of a further critical value of the Reynolds number $R_{\delta ,c2}$ can be inferred, beyond which, not only a growing perturbation exists, but transition to turbulence is promoted during a large time interval. The different qualitative behaviour of the perturbations, when the Reynolds number moves from $200$ to $550$ and beyond, clearly appears when the values of $t_1$ and $t_2$, i.e. the boundaries of the unstable time intervals, are plotted versus the Reynolds number. Figure 9 clearly shows that, while $t_1$ decreases continuously with $R_\delta$, $t_2$ is characterized by a sudden and rapid growth when $R_\delta$ becomes larger than approximately $200$. The rapid growth of the unstable phases is more evident if $t_2$ and $d t_2/d R_\delta$ are plotted versus the Reynolds number. Indeed figure 10 shows that the value of $d t_2/d R_\delta$ decreases from large values to small values when $R_\delta$ moves from $85$ to $187.5$ but it suddenly jumps to larger values when $R_\delta$ moves from $187.5$ to $200$, and decays again for values of $R_\delta$ larger than $200$.

Figure 9. Value of the boundaries of the unstable time intervals during the oscillation cycle as a function of $R_\delta$ ($\alpha =0.5$). A similar curve does exist with $t$ decreased by ${\rm \pi}$.

Figure 10. Value of $t_2$ (a) and of ${\textrm {d}}t_2/{\textrm {d}}R_\delta$ (b) evaluated by means of a forward finite difference approximation of the data plotted in panel (a) for $\alpha =0.5$.