2.1. Airflow viscous sublayer

We present the fluid-dynamic functions of the viscous sublayer, for a subsonic flow with order-one Mach number, in the form of asymptotic expansions:

(2.6) \begin{equation} \left.\begin{gathered} u = Re^{{-}1/8}U_{*}(t_{*},x_{*},Y_{*})+\cdots, \\ v= Re^{{-}3/8}V_{*}(t_{*},x_{*},Y_{*})+ \cdots , \\ p = Re^{{-}1/4}P_{*}(t_{*},x_{*},Y_{*})+\cdots, \\ \rho=\rho_{w}+O(Re^{{-}1/8})+\cdots, \\ \mu = \mu_{w}+O(Re^{{-}1/8})+ \cdots, \end{gathered}\right\} \end{equation}

where the independent variables are defined as

(2.7a–c)\begin{equation} t_{*}=\sigma_{\mu}\,Re^{1/4} t, \quad x_{*}=Re^{3/8}(x-1),\quad Y_{*}=Re^{5/8}y. \end{equation}

Note that $\sigma _{\mu }$ first appears in the asymptotic solution for the film. Namely, the asymptotic solution for the vertical component of the film flow which is obtained by considering the shear-stress continuity between the two fluids. Consequently, $\sigma _{\mu }$ re-appears in the time scale by assuming that the kinematic condition holds at the interface. The substitution of (2.6)–(2.7a–c) into the Navier–Stokes equations (2.3) yields

(2.8a)$$\begin{gather} \rho_{w}\left(U_{*} \frac{\partial U_{*}}{\partial x_{*}}+V_{*}\frac{\partial U_{*}}{\partial Y_{*}}\right) ={-} \frac{{\rm d} P_{*}}{{\rm d}x_{*}}+\mu_{w}\frac{\partial^2 U_{*}}{\partial Y_{*}^2}, \end{gather}$$

(2.8b)$$\begin{gather}\frac{\partial U_{*}}{\partial x_{*}}+\frac{\partial V_{*}}{\partial Y_{*}}=0. \end{gather}$$

We assume that the interface between the air and liquid film is situated underneath of the lower tier in the triple-deck structure, and is represented by the equation

(2.9)\begin{equation} Y_{*}=H_{*}(t_{*},x_{*}). \end{equation}

We write the boundary conditions for (2.8) as

(2.10a)$$\begin{gather} U_{*}=V_{*} =0 \quad \text{at} \ Y_{*}=H_{*}(t_{*},x_{*}), \end{gather}$$

(2.10b)$$\begin{gather}U_{*}=\lambda(Y_{*}-H_{0}^{*})+\cdots \quad \text{as} \ x_{*} \rightarrow-\infty, \end{gather}$$

(2.10c)$$\begin{gather}U_{*}= \lambda Y_{*}+A_{*}(t_{*},x_{*})+\cdots \quad \text{as} \ Y_{*} \rightarrow \infty, \end{gather}$$

where $H_{0}^{*}$ is a constant representing the film thickness before the interaction region and $\lambda$ represents the dimensionless skin friction in the airflow. To find the value of $\lambda$ for a compressible boundary layer we used numerical methods, namely, the fourth-order Runge–Kutta method with Newton's iteration.

2.2. Upper tier

In the upper tier, the fluid-dynamic functions are represented by the asymptotic expansions

(2.11)\begin{equation} \left.\begin{gathered} u = 1+Re^{{-}1/4}u_{*}(t_{*},x_{*},y_{*})+\cdots, \\ v=Re^{{-}1/4}v_{*}(t_{*},x_{*},y_{*})+ \cdots , \\ p = Re^{{-}1/4}p_{*}(t_{*},x_{*},y_{*})+\cdots, \\ \rho=1+Re^{{-}1/4}\rho_{*}(t_{*},x_{*},y_{*})+\cdots, \\ h=\frac{1}{(\gamma -1)M_{\infty}^2}+Re^{{-}1/4}h_{*}(t_{*},x_{*},y_{*})+ \cdots, \end{gathered}\right\} \end{equation}

with the independent variables are

(2.12a–c)\begin{equation} t_{*}=\sigma_{\mu}\,Re^{1/4} t, \quad x_{*}=Re^{3/8}x,\quad y_{*}=Re^{3/8}y. \end{equation}

Substituting (2.11) and (2.12a–c) into the Navier–Stokes equations (2.3) yields the following leading-order terms which describe the pressure $p_{*}$ in the inviscid upper tier:

(2.13a)\begin{equation} (1-M_{\infty}^2 ) \frac{\partial^2 p_{*}}{\partial x_{*}^2}+\frac{\partial^2 p_{*}}{\partial y_{*}^2}=0. \end{equation}

To solve (2.13a) we define the boundary conditions as

(2.13b)$$\begin{gather} \frac{\partial p_{*}}{\partial y_{*}}=\frac{1}{\lambda}\frac{\partial^2 A_{*}}{\partial x_{*}^2} \quad \text{at} \ y_{*}=0, \end{gather}$$

(2.13c)$$\begin{gather}p_{*} \rightarrow 0 \quad \text{as} \ x_{*}^2+y_{*}^2 \rightarrow \infty. \end{gather}$$

The condition (2.13b) is deduced by matching with the solution in the middle tier of the airflow.

2.3. Film flow

The asymptotic expansions of the fluid-dynamic functions in film flow are written as

(2.14) \begin{equation} \left.\begin{gathered} u = \sigma_{\mu}\,Re^{{-}1/8}U'_{*}(t_{*},x_{*},Y_{*})+\cdots, \quad v = \sigma_{\mu}\,Re^{{-}3/8}V'_{*}(t_{*},x_{*},Y_{*})+\cdots, \\ p = Re^{{-}1/4}P'_{*}(t_{*},x_{*},Y_{*})+\cdots, \quad \hat{\rho } = \frac{1}{\sigma _{\rho }} \rho _{\infty } + \cdots , \quad \hat{\mu } = \frac{1}{\sigma _{\mu }} \mu _{\infty } + \cdots , \end{gathered}\right\} \end{equation}

with independent variables defined by (2.7a–c). We substitute (2.14) and (2.7a–c) into the Navier–Stokes equations (2.3) while assuming ${\sigma _{\mu }^2}/{\sigma _{\rho }} \ll 1$. Dominant terms yield

(2.15)\begin{equation} \left.\begin{gathered} \frac{\partial P_{*}'}{\partial x_{*}}= \frac{\partial^2 U_{*}'}{\partial Y_{*}^2}, \\ \frac{\partial P_{*}'}{\partial Y_{*}}=0, \\ \frac{\partial U_{*}'}{\partial x_{*}}+\frac{\partial V_{*}'}{\partial Y_{*}}=0. \end{gathered}\right\} \end{equation}

We allow the body surface to deform in pure vertical motion

(2.16)\begin{equation} Y_{*}=Y_{w}^{*}(t_{*},x_{*}), \end{equation}

then the no-slip conditions on the body surface are written as

(2.17)\begin{equation} U_{*}'=0, \quad V_{*}'=\frac{\partial Y_{w}^*}{\partial t_{*}} \quad \text{at}\quad Y_{*}=Y_{w}^{*}(t_{*},x_{*}). \end{equation}

In addition, we need to formulate the boundary conditions on the interface. First, we consider the tangential stress to be continuous across the interface and the interface is the kinematic (impermeability) condition. When dealing with the normal stress, we remember that the pressure does not change across the tiers in the airflow. At the interface, we need to consider the pressure jump which is due to the surface tension. Thus, we formulate surface tension of the liquid film using the normal stress condition

(2.18)\begin{equation} p_*= P_{*}' +\varrho \frac{\partial ^2 H}{\partial x_*^2}, \end{equation}

where $\varrho$ represents surface tension. By considering equal tangential stress condition at the interface, we define the initial condition as

(2.19)\begin{equation} U_{*}'=\lambda Y_{*} \quad \text{at} \ x_{*}={-}\infty, \ Y_{*} \in [0,H_{0}^{*}). \end{equation}

2.4. Affine transformation

The viscous–inviscid interaction problem involves five parameters: the dimensionless skin friction before the interaction region $\lambda$, the fluid density $\rho _{w}$ and viscosity coefficient $\mu _{w}$ on the body surface, the compressibility parameter $\beta$ and the initial film thickness $H_{0}^{*}$. We begin with the airflow in the lower tier regime which is described by (2.8) subject to boundary conditions (2.10). We define the compressibility parameter as $\beta = \sqrt {1 - M_{\infty }^2}$ for a subsonic flow and introduce the affine transformations as

(2.20)\begin{equation} \left.\begin{gathered} t_{*}=\frac{\mu_{w}^{{-}3/2} }{\lambda^{{-}1} \beta^{1/2}} \breve{T}, \quad x_{*}=\frac{\mu_{w}^{{-}1/4} \rho_{w}^{{-}1/2}}{\lambda^{5/4} \beta^{3/4}}\breve{X},\\ Y_{*}=\frac{\mu_{w}^{1/4} \rho_{w}^{{-}1/2}}{\lambda^{3/4} \beta^{1/4}}\breve{Y}+H_{*}(x_{*}),\quad U_{*}=\frac{\mu_{w}^{1/4} \rho_{w}^{{-}1/2}}{\lambda^{{-}1/4} \beta^{1/4}}\breve{U},\\ V_{*}=\frac{\mu_{w}^{3/4} \rho_{w}^{{-}1/2}}{\lambda^{{-}3/4} \beta^{{-}1/4}}\breve{V}+U_{*}\frac{\partial H_{*}}{\partial x_{*}}, \quad P_{*}=\frac{\mu_{w}^{1/2}}{\lambda^{{-}1/2} \beta^{1/2}}\breve{P},\\ A_{*}=\frac{\mu_{w}^{1/4} \rho_{w}^{{-}1/2}}{\lambda^{{-}1/4} \beta^{1/4}}\breve{A}-\lambda H_{*}(x_{*}),\quad H_{*}=\frac{\mu_{w}^{1/4} \rho_{w}^{{-}1/2}}{\lambda^{3/4} \beta^{1/4}}\breve{H}. \end{gathered}\right\} \end{equation}

These expressions (2.20) show how the standard affine transformations of the triple-deck theory is related to Prandtl's transposition. The latter introduces the body-fitted coordinates $(\breve {X},\breve {Y})$ with $\breve {X}$ measured along the interface and $\breve {Y}$ in the normal direction. As a result, the governing equations for the lower tier assume the canonical form

(2.21a)$$\begin{gather} \breve{U} \frac{\partial \breve{U}}{\partial \breve{X}}+\breve{V}\frac{\partial \breve{U}}{\partial \breve{Y}} ={-} \frac{{\rm d} \breve{P}}{{\rm d} \breve{X}}+\frac{\partial^2 \breve{U}}{\partial \breve{Y}^2}, \end{gather}$$

(2.21b)$$\begin{gather}\frac{\partial \breve{U}}{\partial \breve{X}}+\frac{\partial \breve{V}}{\partial \breve{Y}}=0, \end{gather}$$

whereas the boundary conditions become

(2.21c)$$\begin{gather} \breve{U}=\breve{V} =0 \quad \text{at} \ \breve{Y}=0, \end{gather}$$

(2.21d)$$\begin{gather}\breve{U}=\breve{Y}+\cdots \quad \text{as} \ \breve{X} \rightarrow -\infty, \end{gather}$$

(2.21e)$$\begin{gather}\breve{U}= \breve{Y}+\breve{A}(\breve{X})+\cdots \quad \text{as} \ \breve{Y} \rightarrow \infty. \end{gather}$$

We introduce the following transformations for the upper tier:

(2.22a,b)\begin{equation} y_{*}=\frac{\mu_{w}^{{-}1/4} \rho_{w}^{{-}1/2}}{\lambda^{5/4} \beta^{7/4}}\breve{y},\quad p_{*}=\frac{\mu_{w}^{1/2}}{\lambda^{{-}1/2} \beta^{1/2}}\breve{p}, \end{equation}

which turn the governing equations and boundary condition for the inviscid region (2.13) into

(2.23a)$$\begin{gather} \frac{\partial^2 \breve{p}}{\partial \breve{X}^2}+\frac{\partial^2 \breve{p}}{\partial \breve{y}^2}=0, \end{gather}$$

(2.23b)$$\begin{gather}\frac{\partial \breve{p}}{\partial \breve{y}}=\frac{\partial^2 \breve{A}}{\partial \breve{X}^2}-\frac{\partial^2 \breve{H}}{\partial \breve{X}^2} \quad \text{at} \ \breve{y}=0, \end{gather}$$

(2.23c)$$\begin{gather}\breve{p} \rightarrow 0 \quad\text{as} \ \breve{X}^2+\breve{ y}^2 \rightarrow \infty. \end{gather}$$

The solution for the pressure at the bottom of the upper deck may be written as

(2.24)\begin{equation} \frac{\partial \breve{p}}{\partial \breve{X}}={-}\frac{1}{\rm \pi} \int_{-\infty}^{\infty} \frac{\breve{A}''(s)-\breve{H}''(s)}{s-\breve{X}} \,{\rm d}s, \end{equation}

see Ruban (Reference Ruban2015) for more details. Now we turn our attention to the film flow and introduce the following affine transformations:

(2.25)\begin{equation} \left.\begin{gathered} U_{*}'=\frac{\mu_{w}^{5/4} \rho_{w}^{{-}1/2}}{\lambda^{{-}1/4} \beta^{1/4}}\breve{U}', \quad V_{*}'=\frac{\mu_{w}^{7/4} \rho_{w}^{{-}1/2}}{\lambda^{{-}3/4} \beta^{{-}1/4}}\breve{V}',\\ P_{*}'=\frac{\mu_{w}^{1/2}}{\lambda^{{-}1/2} \beta^{1/2}}\breve{P}', \quad Y_{w}^{*} =\frac{\mu_{w}^{1/4} \rho_{w}^{{-}1/2}}{\lambda^{3/4} \beta^{1/4}} \breve{Y}_{w},\\ H_{0}^{*}=\frac{\mu_{w}^{{-}1/4} \rho_{w}^{1/2}}{\lambda^{{-}3/4} \beta^{{-}1/4}}\breve{H}_{0}^{*}, \quad \varrho=\frac{\mu_{w}^{{-}1/4} \rho_{w}^{{-}1/2}}{\lambda^{5/4} \beta^{7/4}}\breve{\varrho}. \end{gathered}\right\} \end{equation}

These transformations (2.25) turn governing equations and boundary conditions of the film flow (2.15) to the following canonical form:

(2.26)\begin{equation} \left.\begin{gathered} \frac{{\rm d} \breve{P}'}{{\rm d} \breve{X}}= \frac{\partial^2 \breve{U}'}{\partial \breve{Y}^2}, \\ \frac{\partial \breve{U}'}{\partial \breve{X}}+\frac{\partial \breve{V}'}{\partial \breve{Y}}=0. \end{gathered}\right\} \end{equation}

No-slip conditions (2.17) assume the form

(2.27)\begin{equation} \breve{U}'=0, \quad \breve{V}'=\frac{\partial \breve{Y}_{w}}{\partial \breve{T}} \quad \text{at} \ \breve{Y}=\breve{Y}_{w}(\breve{T},\breve{X}),\end{equation}

and the conditions (2.15) at the interface turn into

(2.28a)$$\begin{gather} \frac{\partial \breve{U}'}{\partial \breve{Y}'} = \left.\frac{\partial \breve{U}'}{\partial \breve{Y}'} \right|_{\breve{Y}=0} \quad \text{at} \ \breve{Y}'=\breve{H}(\breve{T},\breve{X}), \end{gather}$$

(2.28b)$$\begin{gather}\breve{V}'=\frac{\partial \breve{H}}{\partial \breve{T}}+\breve{U}'\frac{\partial \breve{H}}{\partial \breve{X}} \quad \text{at} \ \breve{Y}'=\breve{H}(\breve{T},\breve{X}). \end{gather}$$

Further, we transformed the normal stress condition (2.18) as

(2.29)\begin{equation} \breve{P}=\breve{P}'+\breve{\varrho} \frac{\partial^2 \breve{H}}{\partial \breve{X}^2}. \end{equation}

Lastly, the initial condition for the film takes the following form:

(2.30)\begin{equation} \breve{U}'|_{\breve{X}={-}\infty}=\breve{Y}' \quad \text{at} \ \breve{y}' \in [0,\breve{H_{0}}]. \end{equation}

We realised that the canonical interaction problem coincides with the interaction problem describing an incompressible flow. Hence, we can use the asymptotic solutions to a subsonic flow to describe the flow behaviour just before encountering the transonic flow regime, see Khoshsepehr & Ruban (Reference Khoshsepehr and Ruban2022) for more details.

3.1. Upper tier

The structure of the interaction region over a vibrating surface is presented in figure 2. While considering (3.1), we find that in the transonic flow regime the longitudinal extend of the interaction region yields

(3.4)\begin{equation} \Delta \hat{x}\sim L \,Re^{{-}1/3} \sigma^{{-}1/3}, \end{equation}

with the characteristic time being

(3.5)\begin{equation} \Delta \hat{t}\sim \frac{L}{V_{\infty}}L \,Re^{{-}2/9} \sigma^{{-}11/9}. \end{equation}

Figure 2.

Triple-deck structure with a liquid film over a vibrating wall where $\delta = Re^{-11/18}\sigma _{\mu }^{-1/9}$.

The lateral coordinate $\hat {y}$ may be estimated in the upper tier by considering (2.2) and its affine transformation (2.22a,b) and these yield to

(3.6)\begin{equation} \hat{y}\sim L\,Re^{{-}5/18} \sigma_{\mu}^{{-}7/9}. \end{equation}

To find the order quantity of the pressure in the transonic regime, we consider lateral and longitudinal orders of the viscous sublayer (2.7a–c) along with the asymptotic expansions of the upper tier (2.11) and the affine transformations defined in (2.22a,b) and (3.1) which yield to

(3.7)\begin{equation} \frac{\hat{p}-p_{\infty}}{\rho_{\infty}V_{\infty}^2}\sim Re^{{-}2/9} \sigma_{\mu}^{{-}2/9}. \end{equation}

Equation (3.7) suggests that the solution in the upper tier should be sought in the form of asymptotic expansions:

(3.8)\begin{equation} \left.\begin{gathered} u=1+Re^{{-}2/9}\sigma_{\mu}^{{-}2/9}u_{3}(t_{*},x_{*},y_{3})+Re^{{-}1/3}\sigma_{\mu}^{2/3}u_{33}(t_{*},x_{*},y_{3})+\cdots,\\ v=Re^{{-}5/18}\sigma_{\mu}^{2/9}v_{3}(t_{*},x_{*},y_{3})+Re^{{-}7/18}\sigma_{\mu}^{10/9}v_{33}(t_{*},x_{*},y_{3})+\cdots,\\ p=Re^{{-}2/9}\sigma_{\mu}^{{-}2/9}p_{3}(t_{*},x_{*},y_{3})+Re^{{-}1/3}\sigma_{\mu}^{2/3}p_{33}(t_{*},x_{*},y_{3})+\cdots,\\ \rho=1+Re^{{-}2/9}\sigma_{\mu}^{{-}2/9}\rho_{3}(t_{*},x_{*},y_{3})+Re^{{-}1/3}\sigma_{\mu}^{2/3}\rho_{33}(t_{*},x_{*},y_{3})+\cdots,\\ h=\frac{1}{(\gamma-1)}-Re^{{-}1/9}\sigma_{\mu}^{8/9}\frac{2M_{1}}{(\gamma-1)}+Re^{{-}2/9}\sigma_{\mu}^{{-}2/9}h_{3}(t_{*},x_{*},y_{3})\\ +\,Re^{{-}1/3}\sigma_{\mu}^{2/3}h_{33}(t_{*},x_{*},y_{3})+\cdots, \end{gathered}\right\} \end{equation}

with the independent variables

(3.9a–c)\begin{equation} t_{*}=\frac{t}{Re^{{-}2/9}\sigma_{\mu}^{{-}11/9}}, \quad x_{*}=\frac{x-1}{Re^{{-}1/3}\sigma_{\mu}^{{-}1/3}}, \quad y_{3}=\frac{y}{Re^{{-}5/18}\sigma_{\mu}^{{-}7/9}}. \end{equation}

We substitute (3.8) and (3.9a–c) into the Navier–Stokes equations (2.3) which leads to the following leading-order approximations:

(3.10)\begin{equation} \left.\begin{gathered} \frac{\partial u_{3}}{\partial x_{*}}+\frac{\partial p_{3}}{\partial x_{*}}=0, \\ \frac{\partial v_{3}}{\partial x_{*}}+\frac{\partial p_{3}}{\partial y_{3}}=0,\\ \frac{\partial h_{3}}{\partial x_{*}}-\frac{\partial p_{3}}{\partial x_{*}}=0, \\ \frac{\partial \rho_{3}}{\partial x_{*}}+\frac{\partial u_{3}}{\partial x_{*}}=0, \\ h_{3}+\frac{\rho_{3}}{\gamma-1}-\frac{\gamma}{\gamma-1} p_{3}=0 . \end{gathered}\right\} \end{equation}

It is easily seen that this set of (3.10) is degenerate, as the equations are not independent of each other. Indeed, by manipulating these equations we find that

(3.11a–c)\begin{equation} u_{3}={-}p_{3}, \quad \rho_{3}=p_{3}, \quad h_{3}=p_{3}, \end{equation}

with $p_{3}$ remaining arbitrary. To find $p_{3}$ we need to consider the next approximation equations which are written as

(3.12)\begin{equation} \left.\begin{gathered} \frac{\partial u_{33}}{\partial x_{*}}+\frac{\partial p_{33}}{\partial x_{*}}={-}\frac{\partial u_{3}}{\partial t_{*}}, \\ \frac{\partial v_{33}}{\partial x_{*}}+\frac{\partial p_{33}}{\partial y_{3}}={-}\frac{\partial v_{3}}{\partial t_{*}}, \\ \frac{\partial h_{33}}{\partial x_{*}}-\frac{\partial p_{33}}{\partial x_{*}}=\frac{\partial p_{3}}{\partial t_{*}}-\frac{\partial h_{3}}{\partial t_{*}}, \\ \frac{\partial \rho_{33}}{\partial x_{*}}+\frac{\partial u_{33}}{\partial x_{*}}={-}\frac{\partial \rho_{3}}{\partial t_{*}}-\frac{\partial v_{3}}{\partial y_{3}}, \\ h_{33}+\frac{\rho_{33}}{\gamma-1}-\frac{\gamma }{\gamma-1} p_{33}=\frac{2M_{1}}{\gamma -1}\rho_{3}. \end{gathered}\right\} \end{equation}

Note that the first-order approximations are $O(\sigma _{\mu }^{1/9} Re^{1/9})$ and the second order approximations are $O(\sigma _{\mu })$. By algebraic manipulations, we simplify these equations (3.12) and find the governing equation for pressure as

(3.13)\begin{equation} 2M_{1} \frac{\partial^2 p_{3}}{\partial x_{*}^2}+2\frac{\partial^2 p_{3}}{\partial t_{*} \partial x_{*}}-\frac{\partial^2 p_{3}}{\partial y_{3}^2}=0. \end{equation}

3.2. Lower tier

The fluid-dynamic functions for the viscous sublayer, region 1 in figure 2, are represented in the form of asymptotic expansions

(3.14)\begin{equation} \left.\begin{gathered} u=Re^{{-}2/18}\sigma_{\mu}^{{-}2/18}u_{1}(t_{*},x_{*},y_{1})+\cdots, \\ v=Re^{{-}7/18}\sigma_{\mu}^{2/18}v_{1}(t_{*},x_{*},y_{1})+\cdots, \\ p=Re^{{-}2/9}\sigma_{\mu}^{{-}2/9}p_{1}(t_{*},x_{*},y_{1})+\cdots,\\ h=h_{w}+O(Re^{{-}1/9}),\\ \rho=\rho_{w}+O(Re^{{-}1/9}), \\ \mu=\mu_{w}+ O(Re^{{-}1/9}), \end{gathered}\right\} \end{equation}

with the independent variables defined as

(3.15a–c)\begin{equation} t_{*}=\frac{t}{Re^{{-}2/9}\sigma_{\mu}^{{-}11/9}}, \quad x_{*}=\frac{x-1}{Re^{{-}1/3}\sigma_{\mu}^{{-}1/3}}, \quad y_{1}=\frac{y}{Re^{{-}11/18}\sigma_{\mu}^{{-}1/9}}, \end{equation}

where $h_{w}$, $\rho _{w}$ and $\mu _{w}$ are positive constants representing enthalpy, density and viscosity on the wall surface. We substitute (3.14) and (3.15a–c) into the Navier–Stokes equations (2.3) and we find the following governing equations:

(3.16a)$$\begin{gather} \rho_{w}\left(u_{1} \frac{\partial u_{1}}{\partial x_{*}}+v_{1}\frac{\partial u_{1}}{\partial y_{1}}\right) ={-} \frac{\partial p_{1}}{\partial x_{*}}+\mu_{w}\frac{\partial^2 u_{1}}{\partial y_{1}^2}, \end{gather}$$

(3.16b)$$\begin{gather}\frac{\partial p_{1}}{\partial y_{1}}=0, \end{gather}$$

(3.16c)$$\begin{gather}\frac{\partial u_{1}}{\partial x_{*}}+\frac{\partial v_{1}}{\partial y_{1}}=0. \end{gather}$$

3.3. Film flow

We define the fluid-dynamic functions for the film flow as

(3.17)\begin{equation} \left.\begin{gathered} u=Re^{{-}2/18}\sigma_{\mu}^{{-}2/18}u_{1}'(t_{*},x_{*},y_{1})+\cdots, \\ v=Re^{{-}7/18}\sigma_{\mu}^{2/18}v_{1}'(t_{*},x_{*},y_{1})+\cdots, \\ p=Re^{{-}2/9}\sigma_{\mu}^{{-}2/9}p_{1}'(t_{*},x_{*},y_{1})+\cdots, \end{gathered}\right\} \end{equation}

with the independent variables defined in (3.15a–c). By substituting (3.17) and (3.15a–c) into Navier–Stokes equations (2.3) we derive the lubrication equations that describe the motion of the film:

(3.18)\begin{equation} \left.\begin{gathered} \frac{\partial p_{1}'}{\partial x_{*}}=\frac{\partial^2 u_{1}'}{\partial y_{1}^2}, \\ \frac{\partial u_{1}'}{\partial x_{*}}+\frac{\partial v_{1}'}{\partial y_{1}}=0. \end{gathered}\right\} \end{equation}

We solve these (3.18) with conditions (2.27) and (2.28).

3.4. Main part of boundary layer

The middle tier must have the same longitudinal orders as the rest of the interaction regions and its thickness coincides with the thickness of the Blasius boundary layer, region 2 in figure 2. By analysing the solution in the overlap region that lies between regions 1 and 2 and using the fact that $y_2= (Re~\sigma _{\mu })^{-1/9} y_1$, we seek the solutions in region 2 in the form of the asymptotic expansions

(3.19)\begin{equation} \left.\begin{gathered} u=U_{00}+Re^{{-}2/18} \sigma_{\mu}^{{-}2/18} u_{2}(t_{*},x_{*},y_2)+\cdots, \\ v=Re^{{-}5/18}\sigma_{\mu}^{4/18}v_{2}(t_{*},x_{*},y_2)+\cdots, \\ p=Re^{{-}2/9}\sigma_{\mu}^{{-}2/9}p_{2}(t_{*},x_{*},y_2)+\cdots,\\ \rho=\rho_{00}+Re^{{-}2/18} \sigma_{\mu}^{{-}2/18} \rho_2(t_{*},x_{*},y_2) +\cdots, \\ \mu=\rho_{00}+Re^{{-}2/18} \sigma_{\mu}^{{-}2/18} \mu_2(t_{*},x_{*},y_2) +\cdots, \\ h=h_{00}+Re^{{-}2/18} \sigma_{\mu}^{{-}2/18} h_2(t_{*},x_{*},y_2) +\cdots, \end{gathered}\right\} \end{equation}

with the independent variables

(3.20a–c)\begin{equation} t_{*}=\frac{t}{Re^{{-}2/9}\sigma_{\mu}^{{-}11/9}} t, \quad x_{*}=\frac{x-1}{Re^{{-}1/3}\sigma_{\mu}^{{-}1/3}}, \quad y_{2}=\frac{y}{Re^{{-}1/2}}. \end{equation}

In (3.19), the leading-order terms are defined as

(3.21)\begin{equation} \left.\begin{gathered} U_{00}(y_2)=\lambda y_2+\cdots \\ \rho_{00}(y_2)=\rho_{w}+\cdots \\ \mu_{00}(y_2)=\mu_{w}+\cdots \\ h_{00}(y_2)=h_{w} +\cdots \end{gathered}\right\} \quad \text{as} \ y_2 \rightarrow 0. \end{equation}

These functions (3.21) satisfy the classical boundary-layer equations and admit a self-similar solution for a thermally isolated wall in case of a flow past a flat wall.

We found that the perturbation terms exhibit the following behaviour at the bottom of region 2:

(3.22)\begin{equation} \left.\begin{gathered} u_{2}=A(t_{*},x_{*})+\cdots \\ v_{2}={-}\frac{\partial A}{\partial x_{*}} y_2+\cdots \end{gathered}\right\} \quad \text{as} \ y_2 \rightarrow 0. \end{equation}

The substitution of (3.20a–c) and (3.19) into the Navier–Stokes equations (2.3) results in inviscid leading-order terms to be written as

(3.23)\begin{equation} \left.\begin{gathered} U_{00}(y_2)\frac{\partial u_2}{\partial x_{*}}+ v_2\frac{{\rm d} U_{00}}{\partial \,{\rm d} y_2}=0, \\ \frac{\partial p_{2}}{\partial y_2}=0,\\ U_{00}(y_2) \frac{\partial h_2}{\partial x_{*}}+v_2\frac{{\rm d}h_{00}}{{\rm d}y_2},\\ \rho_{00}(y_2)\frac{\partial u_2}{\partial x_{*}}+U_{00}(y_2)\frac{\partial \rho_2}{\partial x_{*}}+\rho_{00}(y_2)\frac{\partial v_2}{\partial y_2}+v_2\frac{{\rm d} \rho_{00}}{{\rm d}y_2}=0 ,\\ h_{00}=\frac{1}{(\gamma-1)M_{\infty}^2} \frac{1}{\rho_{00}}, \\ h_2={-}\frac{1}{(\gamma-1)M_{\infty}^2} \frac{\rho_2}{\rho_{00}^2}. \end{gathered}\right\} \end{equation}

Note that the middle tier displays its usual behaviour which is inability to make any contributions to the displacement effect of the boundary layer. The middle tier simply transmits the streamline deformations produced by the viscous sublayer in region 1 to the upper tier in region 3. Equations (3.23) are easily solved by eliminating $\rho _{00}$ and $h_{00}$ which lead to

(3.24)\begin{equation} \frac{\partial}{\partial y_2}\left( \frac{v_2}{U_{00}}\right)=0. \end{equation}

By integrating (3.24) with the limits defined in (3.21) and (3.22) we find

(3.25)\begin{equation} \frac{v_2}{U_{00}}={-}\frac{1}{\lambda}\frac{\partial A}{\partial x_{*}}. \end{equation}

Note that the longitudinal velocity component $u_2$ can simply be found by using the continuity equation. The viscous–inviscid interaction problem is composed of (i) governing equations of the viscous lower tier, (ii) the equation describing pressure in the upper tier and (iii) governing equations of the film flow. Now, we need to introduce a new set of affine transformations for all the tiers in the transonic regime. The transformations for the airflow are

(3.26)\begin{equation} \left.\begin{gathered} x_{*}=2^{{-}1/3}\mu_{w}^{{-}1/3}\lambda^{{-}4/3} \rho^{{-}1/3}X, \quad y_{1}=2^{{-}1/9}\mu_{w}^{2/9}\lambda^{{-}7/9} \rho^{{-}4/9}(Y+H), \\ u_{1}=2^{{-}1/9}\mu_{w}^{2/9}\lambda^{2/9} \rho^{{-}4/9}U, \quad v_{1}=2^{1/9}\mu_{w}^{7/9}\lambda^{7/9} \rho^{{-}5/9}\left[V+u_{1}\frac{\partial H}{\partial x_{*}}\right], \\ p_{1}=2^{{-}5/9}\mu_{w}^{1/9}\lambda^{{-}8/9} \rho^{{-}2/9}P, \quad A=2^{{-}1/9}\mu_{w}^{2/9}\lambda^{{-}7/9} \rho^{{-}4/9}A_{*}-\lambda H(x_{*}),\\ H=2^{{-}1/9}\mu_{w}^{2/9}\lambda^{{-}7/9} \rho^{{-}4/9}H_{*}, \quad M_{1}=2^{1/9}\mu_{w}^{{-}2/9}\lambda^{{-}2/9} \rho^{4/9} \tilde{M}, \\ y_{3}=2^{{-}7/9}\mu_{w}^{{-}4/9}\lambda^{{-}13/9} \rho^{{-}1/9}Y_{*},\quad t_{*}=2^{{-}2/9}\mu_{w}^{{-}5/9}\lambda^{{-}14/9} \rho^{{-}1/9} T, \end{gathered}\right\} \end{equation}

and for the film

(3.27)\begin{equation} \left.\begin{gathered} x=2^{{-}1/3}\mu_{w}^{{-}1/3}\lambda^{{-}4/3} \rho^{{-}1/3}X, \quad y'=2^{{-}1/9}\mu_{w}^{2/9}\lambda^{{-}7/9} \rho^{{-}4/9}Y', \\ u'=2^{{-}1/9}\mu_{w}^{11/9}\lambda^{2/9} \rho^{{-}4/9}U', \quad v'=2^{1/9}\mu_{w}^{16/9}\lambda^{{-}1/9} \rho^{{-}5/9}V' ,\\ p'=2^{{-}5/9}\mu_{w}^{1/9}\lambda^{{-}8/9} \rho^{{-}2/9}P',\quad \varrho=2^{{-}7/9}\mu_{w}^{{-}4/9}\lambda^{{-}13/9} \rho^{{-}1/9}\tilde{\varrho}. \end{gathered}\right\} \end{equation}

The transformation of the interaction problem into a canonical form is presented in Appendix B.

4.1. Receptivity analysis

The full derivation of solutions to the IVP problem is presented in Appendix C.3. We found that the general solution to the longitudinal velocity in the viscous sublayer is described by the Airy equations which is written as

(4.5)\begin{equation} \frac{{\rm d} \bar{u}}{{\rm d} z}=3\bar{A}\,{\rm Ai}(z), \quad z=({\rm i}k)^{1/3}Y; \end{equation}

see Abramowitz & Stegun (Reference Abramowitz and Stegun1964) for more details on Airy equations. We found the general solution of the pressure takes the form

(4.6)\begin{equation} \bar{p}=\frac{{\rm i}k^2}{\varkappa } (\bar{A}-\bar{H}). \end{equation}

The solution (4.6) depends on the sign of $\tilde {M}$ and we define $\varkappa$ as

(4.7a)$$\begin{gather} \varkappa =\begin{cases} |\omega k+ k^2 \tilde{M} |^{1/2} & \text{if} \ \ k \in (-\infty, -\frac{\omega}{|\tilde{M}|} ) \cup (0, \infty) , \\ {\rm i}|\omega k+ k^2 \tilde{M} |^{1/2} & \text{if} \ \ k \in [-\frac{\omega}{|\tilde{M}|}, 0], \end{cases} \quad \text{for} \ \tilde{M}>0, \end{gather}$$

(4.7b)$$\begin{gather}\varkappa =\begin{cases} |\omega k+ k^2 \tilde{M} |^{1/2} & \text{if} \ \ k \in ( 0, \frac{\omega}{|\tilde{M}|} ), \\ {\rm i}|\omega k+ k^2 \tilde{M}|^{1/2} & \text{if} \ \ k \in ( -\infty, 0] \cup [\frac{\omega}{| \tilde{M}|}, \infty), \end{cases} \quad \text{for} \ \tilde{M}<0. \end{gather}$$

We are interested in the relations between the function describing the initial film thickness and air pressure, thus we eliminate $\bar {A}$ in our solutions (4.5) and (4.6). As a result, we found

(4.8)\begin{equation} \bar{H}= \bar{p}\left[\frac{ ({\rm i}k)^{1/3}}{3{\rm Ai}'(0)} - \frac{\varkappa}{{\rm i} k^2} \right] . \end{equation}

Now we focus on the solution of film equations, see Appendix C.3 for more details. The velocity components of the film are written as

(4.9a)$$\begin{gather} \bar{u}'=\frac{{\rm i}k}{2}\bar{p}'Y^2+\varXi Y-\bar{y}_{w}(k), \end{gather}$$

(4.9b)$$\begin{gather}\bar{v}'=\frac{k^2}{6}\bar{p}'Y^3-\varXi \frac{Y^2}{2}+{\rm i}k\bar{y}_{w}(k)Y+{\rm i}\omega \bar{y}_{w}(k), \end{gather}$$

where $\varXi = [(\textrm {i}k)^{2/3}({\textrm {Ai}(0)}/{\textrm {Ai}'(0)})\bar {p}-\textrm {i}kH_{0} \bar {p}' ]$.

Considering the kinematic condition and pressure condition with the transverse velocity solution of the film at the interface yields to another relationship between the pressure and film surface function which is written as

(4.10)\begin{equation} \bar{H}=\frac{{\rm i}(\omega+kH_0)\bar{y}_{w}-\bar{p} \left(\dfrac{k^2 H_{0}^3}{3}+({\rm i}k)^{5/3} \dfrac{H_{0}^2}{2}\dfrac{{\rm Ai}(0)}{{\rm Ai}'(0)}\right) }{ \dfrac{k^4 H_{0}^3 \tilde{\varrho}}{3}+{\rm i}(\omega +kH_0)}. \end{equation}

We found the final solution for the pressure by eliminating $\bar {H}$ in (4.8) and (4.10) which results in

(4.11)\begin{equation} \bar{p}=\frac{{\rm i}(\omega+kH_{0})\bar{y}_{w}(k)}{D(k)}, \end{equation}

where the denominator $D(k)$ is defined as

(4.12)\begin{equation} D(k)=\frac{\left(\dfrac{({\rm i}k)^{1/3}}{3{\rm Ai}'(0)}-\dfrac{\varkappa}{{\rm i} k^2} \right)}{\left( {\rm i}\omega+{\rm i}kH_{0}+\dfrac{k^4H_{0}^3\tilde{\varrho}}{3} \right)^{{-}1}} + \frac{k^2 H_{0}^3}{3}+({\rm i}k)^{5/3} \frac{H_{0}^2}{2}\frac{ {\rm Ai}(0)}{{\rm Ai}'(0)}. \end{equation}

We can derive the dispersion equation, describing the instability modes of the airflow over a vibrating section of the wall that is coated with a thin film, by equating the denominator of the first term in (4.12) to zero and let

(4.13)\begin{equation} (\omega+\tilde{M}k)={-}1. \end{equation}

Note that this condition implies that the airflow is incompressible and because the airflow in the viscous sublayer has a relatively slow speed we assume the flow to be incompressible in this layer. Thus, the dispersion equation for the incompressible flow is valid in the lower tier of the transonic flow. We assume that $\omega \in \mathbb {R}$ and $k \in \mathbb {C}$ in our receptivity analysis. By obtaining the roots of the dispersion equation $k_0$ we can determine at which frequency values the airflow is unstable. We seek the root of the dispersion equation in the $k$-complex plane, where $k=\textrm {Re}\{k\}+\textrm {i} \textrm {Im}\{k\}$, using Newton's iteration method. Based on the perturbations in (4.3) we can see that if $\textrm {Im}\{k\}>0$, then the airflow is stable, and if $\textrm {Im}\{k\}<0$, the flow is unstable. Figure 3 shows that the airflow is unstable for the case where surface tension $\tilde {\varrho }$ is zero however the airflow becomes stable when $\tilde {\varrho }>0$. The results presented in figure 3(b) show the airflow is unstable for small values of $\textrm {Re}\{k_0\}$ however the airflow becomes stable as $\textrm {Re}\{k_0\}$ increases. Frequency values at which $\textrm {Im}\{k\}= 0$ is referred to as neutral frequency $\omega _0$. The value of neutral frequency varies as Kàrmàn–Guderley parameter changes as shown in figure 4. As the value of thickness or surface tension increase, $\omega _0$ becomes sensitive to the presence of the film flow compared with zero-valued surface tension.

Figure 3.

Solution to $D(k)=0$ is denoted by $k_{0}$. These graphs display $k_{0}$ in the $k$-complex plane for different values of $\tilde {M}$: (a) $\tilde {\varrho }= 0$, $H_0=1.5$ and (b) $\tilde {\varrho }= 0.5$, $H_0=1.5$.

Figure 4.

Neutral frequency as a function of $\tilde {M}$: (a) $H_0=0.01$, (b) $H_0=1.5$ and (c) $H_0=1.5$.

We apply inverse Fourier transform to the pressure presented in (4.11) which leads to the following expression:

(4.14)\begin{equation} \tilde{p}= \frac{1}{2 {\rm \pi}} \int_{-\infty}^{\infty} \frac{{\rm i}(\omega+kH_{0})}{D(k)} \bar{y}_{w}(k) \,{\rm e}^{{\rm i}kX} \,{\rm d}k. \end{equation}

To choose the contour integration for the integral (4.14) we turn our attention to the frequency values that are larger than the neutral frequencies for both positive and negative $\tilde {M}$. The behaviour of the root when $\tilde {\varrho } = 0$ is different from that of the root when $\tilde {\varrho } \neq 0$. In the first case, the root stays in the third quadrant as the frequency increases (figure 3a). In the latter case, increasing the frequency further the trajectory of the root crosses the real axis to the second quadrant and remains there in the complex $k$-plane (figure 3b). We turn our attention to root in the second quadrant where $\tilde {\varrho } \neq 0$ which represents the Tollmien–Schlichting wave.

Considering the analytical branch of the function $(\textrm {i}k)^{1/3}$ we set a branch cut along the positive axis in the $k$-plane. We split integration interval in (4.14) into two parts, negative and positive real semi-axes. The chosen contour of integration for integral (4.14) is displayed in figure 5(a). For $\tilde {M}<0$, the contour along the negative real semi-axis is closed by the arc $C^{-}_{R}$ with large radius $R$, ray $C_{-}$ along the negative real semi-axis and ray $C'_{-}$ along the positive imaginary semi-axis. We denote by $\tilde {p}^{-}$ the analytic extension of the integrand (4.11) in the region enclosed inside this contour along the real negative semi-axis. On the positive semi-axis, the contour is composed of two closed contours $(C'_{+}, C^{+}_1, C^{+}_{L_1}, C^{+}_{R_1})$ and $(C^{+}_{L_2}, C^{+}_3, C^{+}_{R_2})$. Note that the position of $( C^{+}_{L_1}, C^{+}_{L_2})$ depends on value of $\omega /|\tilde {M}|$. We denote by $\tilde {p}^{+}_{1}$ and $\tilde {p}^{+}_2$ the analytic extensions of the integrand (4.11) in the regions enclosed inside the contours along $C^{+}_1$ and $C^{+}_2$, respectively. The analytic extension of the denominator (4.11) for contour $C_{-}$ is given by the expression

(4.15)\begin{align} D^{-}(k)&=\frac{\left(\dfrac{({\rm i}k)^{1/3}}{3{\rm Ai}'(0)}-\dfrac{(-\omega k- \tilde{M}k^2)^{1/2}}{k^2} \right)}{\left( {\rm i}\omega+{\rm i}kH_{0}+\dfrac{k^4H_{0}^3\tilde{\varrho}}{3} \right)^{{-}1}}+ \frac{k^2 H_{0}^3}{3}\nonumber\\ &\quad +({\rm i}k)^{5/3} \frac{H_{0}^2}{2}\frac{ {\rm Ai}(0)}{{\rm Ai}'(0)} \quad k \in (-\infty,0). \end{align}

Here $k_0$ is the solution to $D^{-}(k)=0$ and it is found numerically by employing Newton's iteration. The value of $k_0$ depends on values of the surface tension $\tilde {\varrho }$ and initial film thickness $H_0$.

Figure 5.

Deformation of the contour of integration in the complex $k$-plane for (a) negative $\tilde {M}<0$ and (b) positive $\tilde {M}>0$ values of Kàrmàn–Guderley parameter.

The integrand (4.11) along the negative real semi-axis can by divided into the sum of three integrals, written as

(4.16)\begin{equation} \tilde{p}^{-}= \frac{1}{2 {\rm \pi}} \left( \int_{C_{-}}+ \int_{C'_{-}} +\int_{C^{-}_{R}} \right) \frac{{\rm i}(\omega+kH_{0})}{D^{-}(k)} \bar{y}_{w}(k) \,{\rm e}^{{\rm i}kX} \,{\rm d}k. \end{equation}

The function $\bar {p}$ has a residue which is a simple pole at $k_{0}$. By the residue theorem we find the solution to integral (4.16) to be written as

(4.17)\begin{equation} \left( \int_{C_{-}}+ \int_{C'_{-}} +\int_{C^{-}_{R}} \right) \frac{{\rm i}(\omega+kH_{0})}{D^{-}(k)} \bar{y}_{w}(k) \,{\rm e}^{{\rm i}kX} \,{\rm d}k= 2{\rm \pi} {\rm i}\,\text{Res}(\tilde{p}^{-}, k_0 ). \end{equation}

The residue is evaluated as

(4.18)\begin{equation} 2{\rm \pi} {\rm i}\,\text{Res}(\tilde{p}^{-} \,{\rm e}^{{\rm i}kX}, k_{0} )={-} 2{\rm \pi} \frac{(k_{0}H_{0}+\omega)}{{\rm d}D^{-}(k_{0})/{\rm d}k} \bar{y}_{w}(k_0) \,{\rm e}^{{\rm i}k_{0}X}. \end{equation}

According to the Jordan lemma the integral along arc $C_{R}^{-}$ is zero as the arc radius $R \rightarrow \infty$. The integral along $C'_{-}$ recovers the original integral. The integral along $C'_{-}$ is a Laplace-type integral and it can be evaluated with the help of Watson's lemma at large values of $X$. First, we need to determine the behaviour of the integrand $\bar {p}^{-}$ in (4.16) at small values of $k$. We found the dominant term is written as

(4.19)\begin{equation} \bar{p}^{-}=\frac{{\rm i}(\omega+kH_{0})\bar{y}_{w}(k)}{D^{-}(k)} \approx \frac{{\rm i}k^{3/2}}{\sqrt{\omega}} \bar{y}_{w}(k) \quad \text{as} \ k\rightarrow 0, \ \omega={O}(1). \end{equation}

The integral along $C'_{-}$ can be rewritten as

(4.20) \begin{equation} \int_{C^{-}} \bar{p}^{-}_1 \,{\rm e}^{{\rm i}kX} \,{\rm d}k \approx {\rm i} \frac{\bar{y}_{w}(0)}{\sqrt{\omega}} \int_{C^{-}} k^{3/2} \,{\rm e}^{{\rm i}kX} \,{\rm d}k . \end{equation}

We perform the calculation of (4.20) along the left-hand side of the branch cut, in this case we found that $k^{3/2}=|k|^{3/2} \frac {\sqrt {2}}{2} (1-i)$. The solution to the integral (4.20) yields

(4.21)\begin{equation} \int_{C^{-}} \bar{p}^{-} \,{\rm e}^{{\rm i}kX} \,{\rm d}k = \frac{\bar{y}_{w}(0)}{X^{5/2}\sqrt{\omega}}\frac{\sqrt{2}}{2}(1+i) \varGamma\left(\frac{5}{2}\right) \quad \text{as} \ X \rightarrow \infty. \end{equation}

Now, we turn our attention to the integral (4.11) along the positive real semi-axis which consists of two contours. The analytic extensions of the integrand's denominator (4.12) for the contours along $C^{+}_1$ and $C^{+}_2$ are expressed as

(4.22)$$\begin{gather} D^{+}_1(k)=\frac{\left(\dfrac{({\rm i}k)^{1/3}}{3{\rm Ai}'(0)}-\dfrac{(\omega k +\tilde{M}k^2)^{1/2}}{{\rm i}k^2} \right)}{\left( {\rm i}\omega+{\rm i}kH_{0}+\dfrac{k^4H_{0}^3\tilde{\varrho}}{3} \right)^{{-}1}} + \frac{k^2 H_{0}^3}{3}\nonumber\\ \qquad\qquad+({\rm i}k)^{5/3} \frac{H_{0}^2}{2}\frac{ {\rm Ai}(0)}{{\rm Ai}'(0)} \quad k \in \left(0,\frac{\omega}{|\tilde{M}|}\right), \end{gather}$$

(4.23)$$\begin{gather}D^{+}_2(k)=\frac{\left(\dfrac{({\rm i}k)^{1/3}}{3{\rm Ai}'(0)}-\dfrac{(-\omega k- \tilde{M}k^2)^{1/2}}{k^2} \right)}{\left( {\rm i}\omega+{\rm i}kH_{0}+\dfrac{k^4H_{0}^3\tilde{\varrho}}{3} \right)^{{-}1}}+ \frac{k^2 H_{0}^3}{3}\nonumber\\ \qquad\qquad+({\rm i}k)^{5/3} \frac{H_{0}^2}{2}\frac{ {\rm Ai}(0)}{{\rm Ai}'(0)} \quad k \in \left(\frac{\omega}{|\tilde{M}|},\infty \right), \end{gather}$$

respectively. According to Cauchy's theorem, the closed contours ($C^{+}_1$, $C^{+}_{L_1}$, $C^{+}_{R_1}$, $C'_{+}$) and ($C^{+}_2$, $C^{+}_{R_2}$, $C^{+}_{L_2}$) are equal to zero because these integrals do not enclose any singularities; as a result, we found that

(4.24)$$\begin{gather} \int_{C^{+}_1} \bar{p}^{+}_1 \,{\rm e}^{{\rm i}kX} \,{\rm d}k ={-}\left(\int_{C^{+}_{L_1}} +\int_{C^{+}_{R_1}} +\int_{C'_{+}} \right) \bar{p}^{+}_1 \,{\rm e}^{{\rm i}kX} \,{\rm d}k , \end{gather}$$

(4.25)$$\begin{gather}\int_{ C^{+}_2} \bar{p}^{+}_2 \,{\rm e}^{{\rm i}kX} \,{\rm d}k={-}\left(\int_{C^{+}_{R_2}} +\int_{C^{+}_{L_2}} \right)\bar{p}^{+}_2 \,{\rm e}^{{\rm i}kX}\, {\rm d}k. \end{gather}$$

The integrals along $C^{+}_{L_1}$ and $C^{+}_{L_2}$ are along the same line with opposite directions, therefore these two integrals cancel each other as the integration is performed. According to Jordan's lemma the integrals along arcs $C^{+}_{R_1}$ and $C^{+}_{R_2}$ are also zero as the radii tend to infinity. The remaining integral is along ray $C'_{+}$ which is of Laplace type and may be evaluated by Watson's lemma. We found the behaviour of the integrand $\bar {p}^{+}_{1}$ at small values of $k$ to be expressed as

(4.26)\begin{equation} \bar{p}^{+}_{1}=\frac{{\rm i}(\omega+kH_{0})\bar{y}_{w}(k)}{D^{-}(k)} \approx{-} \frac{{\rm i}k^{3/2}}{\sqrt{\omega}} \bar{y}_{w}(k)\quad \text{as} \ k\rightarrow 0. \end{equation}

The integral along $C'_{+}$ can be rewritten as

(4.27)\begin{equation} \int_{C^{+}_1} \bar{p}^{+}_1 \,{\rm e}^{{\rm i}kX} \,{\rm d}k \approx{-}{\rm i} \frac{\bar{y}_{w}(0)}{\sqrt{\omega}} \int_{C^{+}_1} k^{3/2} \,{\rm e}^{{\rm i}kX} \,{\rm d}k . \end{equation}

We perform the calculation of (4.27) along the right-hand side of the branch cut, in this case we found that $k^{3/2}=|k|^{3/2} \frac {\sqrt {2}}{2} (-1+i)$. Consequently, we find the solution to the integral (4.20) is expressed as

(4.28)\begin{equation} \int_{C^{+}_1} \bar{p}^{+}_1 \,{\rm e}^{{\rm i}kX} \,{\rm d}k\approx \frac{\bar{y}_{w}(0)}{X^{5/2}\sqrt{\omega}}\frac{\sqrt{2}}{2}({-}1+i) \varGamma\left(\frac{5}{2}\right) \quad \text{as} \ X \rightarrow \infty. \end{equation}

By considering the solutions (4.18), (4.21) and (4.28) we find the pressure solution $\tilde {p}$ is

(4.29)\begin{equation} \tilde{p}(T,X)=\mathcal{K}(\omega)\bar{y}_{w}(k_0) \,{\rm e}^{{\rm i}k_{0}X} - \sqrt{\frac{1}{2\omega}} \frac{\bar{y}_{w}(0)}{{\rm \pi} X^{5/2}} \varGamma\left(\frac{5}{2}\right)+\cdots, \end{equation}

where the receptivity coefficient is written as

(4.30)\begin{equation} \mathcal{K}(\omega)={-}\frac{(k_{0}H_{0}+\omega)}{{\rm d}D^{-}(k_{0})/{\rm d}k}, \end{equation}

which represents the amplitude of the Tollmien–Schlichting wave in a transonic boundary layer over the vibrating section of a flat surface, coated by a thin liquid film. We obtain the receptivity coefficient of the transonic flow where $\tilde {M}>0$ (figure 6) with a similar procedure except that the integration contour takes a different form which is presented in the $k$-plane shown in figure 5(b). This led us to find the same expression for the receptivity coefficient as in (4.30) in the transonic flow regime with positive $\tilde {M}$.

Figure 6.

Modulus of the receptivity coefficient for various values of $\tilde {M}$: (a) $\tilde {\varrho }= 0$ and $H_0=0.5$ and (b) $\tilde {\varrho }= 1.5$ and $H_0=1.5$.