Analytical solution for an acoustic boundary layer around an oscillating rigid sphere

The solution of the order $\epsilon$ equation (5) due to the oscillatory motion of a rigid sphere along the $z$-direction with velocity amplitude, $U_{0}$ can only depend on the vector $\bm{U}_{0}$ and the position vector $\bm{x}$ with the origin at the centre of the sphere. Symmetry consideration implies that the solution to (5) has the general form

where $u_{r}(r)$ and $u_{\theta}(r)$ are only functions of the radial distance, $r$ from the centre of the sphere and $\bm{e}_{r}$ and $\bm{e}_{\theta}$ are unit vectors in the direction of increasing radial and polar coordinates relative to the $z$-direction. In general, a vector field, $\bm{u}$ can be expressed as the sum of a divergence free component, $\bm{u}_{T}$ with $\nabla\cdot\bm{u}_{T}=0$ and an irrotational component, $\bm{u}_{L}$ with $\nabla\times\bm{u}_{L}=\bm{0}$, as shown in Landau and Lifshitz [30, p. 101–106, §22]. Since $\bm{u}(\bm{x})$ is independent of the azimuthal angle $\varphi$ and has no components in the $\bm{e}_{\varphi}$ direction, the irrotational longitudinal component of $\bm{u}$ can be represented as $\bm{u}_{L}(\bm{x})=\nabla[\phi(r)\cos\theta]\,U_{0}$, where $\phi(r)$ satisfies $(\nabla^{2}+k_{L}^{2})[\phi(r)\cos\theta]=0$. Similarly, the divergence free transverse component of the velocity can be represented as $\bm{u}_{T}(\bm{x})=-\nabla\times[h(r)\sin\theta\;\bm{e}_{\varphi}]\,U_{0}$ with $(\nabla^{2}+k_{T}^{2})[h(r)\cos\theta]=0$. The representation (7) then follows.

The introduction of $\phi(r)$ and $h(r)$ simplifies the expression for the pressure and the vorticity of the primary flow. From the continuity equation and the equation of state of the compressible fluid, the pressure is:

The pressure in the incompressible limit where $\nabla\cdot\bm{u}\rightarrow 0$ and the speed of sound, $c_{0}\rightarrow\infty$, reduces to the familiar acoustic result: $p(\bm{x})=\mathrm{i}{\,}\omega\rho_{0}\phi(r)\cos\theta\,U_{0}$. From (7), we find for the vorticity:

The order $\epsilon$ equation (5) that governs the primary flow, $\bm{u}(\bm{x})$ has the same mathematical form as the equation for elastic waves in linear elasticity if the longitudinal and transverse velocity of the elastic waves are identified formally as: $c_{T}^{2}\equiv\omega^{2}/k_{T}^{2}$ and $c_{L}^{2}\equiv\omega^{2}/k_{L}^{2}$, respectively. This analogy is explored further in Sec. III.2.

In a linear dynamic elastic system, equilibrium of forces requires $\nabla\cdot\bm{\Sigma}=\rho\partial^{2}\bm{U}/\partial t^{2}$, with $\bm{\Sigma}$ the stress tensor, $\bm{U}$ the displacement, $\rho$ the density and $t$ time. Assuming harmonic motion with angular frequency $\omega$, one can write $\bm{\Sigma}=\bm{\sigma}e^{-\mathrm{i}{\,}\omega t}$ and $\bm{U}=\bm{u}e^{-\mathrm{i}{\,}\omega t}$, then (in the frequency domain) the equation of motion becomes $\nabla\cdot\bm{\sigma}=-\rho\omega^{2}\bm{u}$. Linear isotropic homogeneous materials satisfy Hooke’s law as:

with $\mu$ the shear modulus, $\bm{I}$ the identity matrix and superscript ${}^{`T\text{'}}$ indicating the transpose. The longitudinal ($k_{L}$) and transverse ($k_{T}$) wave numbers in linear elasticity are defined as $k_{L}^{2}=\omega^{2}\rho/(\lambda+2\mu)$ and $k_{T}^{2}=\omega^{2}\rho/\mu$, ($\lambda$ is one of the Lamé constants) and are real valued quantities. When Hooke’s law is substituted into the equation of motion, one gets:

This equation is identical to (5). Since the boundary conditions are also identical being $\bm{u}=\bm{U}_{0}$, then these two systems should have identical mathematical solutions. In Klaseboer et al. [19], an analytical solution was given for the elastic waves emitted by a rigid sphere oscillating in an infinite elastic medium. The solution of the elastic wave problem can be represented as a combination of the Green’s function and a dipole field and explicit solutions for the velocity components were given in Klaseboer et al [19] as:

with the functions $F(y)\equiv-1-3\mathrm{i}{\,}/y+3/y^{2}$, $G(y)\equiv\mathrm{i}{\,}/y-1/y^{2}$ and $\bm{x}$ the position vector with respect to the center of the sphere. This solution will also be valid in the acoustic boundary layer context. Note however that in linear elasticity $k_{L}$ and $k_{T}$ are real parameters, while in the current case of acoustic streaming they are complex parameters as given in Eq. (6).

When an axial symmetric coordinate system is used, using $\bm{U}_{0}=U_{0}\cos\theta\bm{e}_{r}-U_{0}\sin\theta\bm{e}_{\theta}$, $\bm{x}=r\bm{e}_{r}$ and $(\bm{x}\cdot\bm{U}_{0})=rU_{0}\cos\theta$, Eq. (12) becomes:

It is more convenient, instead of the constants $c_{1}$ and $c_{2}$ of Eq. (12), to introduce new constants $C_{1}=-2c_{1}$ and $C_{2}=k_{L}^{2}a^{2}[c_{1}2/(k_{T}^{2}a^{2})+c_{2}]$ in (13). The functions $h(r)$ and $\phi(r)$ defined in (7) can now seen to be:

From the no-slip boundary conditions at the surface $r=a$: $u_{r}(a)=U_{0}$ and $u_{\theta}(a)=-U_{0}$ we find:

The complex valued $k_{T}$ and $k_{L}$ are taken to have positive imaginary parts to ensure a finite solution at infinity. We can now also identify terms in $\exp(\mathrm{i}{\,}k_{L}r)$ and $\exp(\mathrm{i}{\,}k_{T}r)$ to be the longitudinal, $\bm{u}_{L}$ and transverse, $\bm{u}_{T}$ components of $\bm{u}$ respectively.

The solutions (13) and (14) reduce to potential flow solutions when $|k_{T}a|\gg 1$, as shown in Section V.1. When the radius, $a$ is large, at $\theta=\pi/2$ the Stokes solution for an oscillating plate is recovered and at $\theta=0$ a plane sound wave is recovered, as illustrated in Section V.2. In the incompressible limit we recover the result given by Landau and Lifshitz [31, p. 89, §24 Problem 5] and Stokes flow if the frequency $\omega$ tends to zero, as demonstrated in Section V.3. The recovery of these classical solutions gives us further confidence that the constructed solution is indeed correct.

If the sphere has a zero tangential stress boundary condition, the vanishing of the tangential stress due to the primary flow will determine different constants $C_{1}$ and $C_{2}$.

The secondary ‘streaming’ flow will now be obtained, assuming the primary flow is as given in Sec. III.3. The streaming velocity is determined by the order $\epsilon^{2}$ terms of the momentum equation (Eq. 1b):

For harmonic time dependence of all quantities: $\overline{\phi}(\bm{x},t)\sim e^{-\mathrm{i}{\,}\omega t}$, we define the corresponding steady, time-independent streaming quantities by taking the time average: $\phi(\bm{x})\equiv<\overline{\phi}(\bm{x},t)>=(1/T)\int_{0}^{T}\overline{\phi}(\bm{x},t)\,\mathrm{d}t$ over one period of oscillation: $T=1/f=2\pi/\omega$.

By time-averaging the order $\epsilon^{2}$ momentum equation (16), we obtain the desired Stokes equation relating the time-averaged body force, $\bm{\mathcal{F}}(\bm{x})$ to the time-average pressure, $P(\bm{x})\equiv<\overline{p}_{2}(\bm{x},t)>$, and the streaming velocity, $\bm{U}(\bm{x})\equiv<\overline{\bm{v}}_{2}(\bm{x},t)>$:

Since $\partial(\overline{\rho}_{1}\;\overline{\bm{v}}_{1})/{\partial t}$ and $\partial(\overline{\bm{v}}_{2})/{\partial t}$ are periodic, their time average is zero. The time averaging also removes the periodic part of $(\overline{\bm{v}}_{1}\;\overline{\bm{v}}_{1})$, $\overline{p}_{2}(\bm{x},t)$ and $\overline{\bm{v}}_{2}(\bm{x},t)$. Furthermore, the streaming velocity, $\bm{U}(\bm{x})$ is divergence free, $\nabla\cdot\bm{U}=0$ [21, 18], based on a physical argument that there are no steady sources or sinks. However, a formal proof of this result is not that straightforward. The secondary velocity satisfies $\nabla\cdot\bm{U}(\bm{x})=-\nabla\cdot<\overline{\bm{v}}_{1}\overline{\rho}_{1}>/\rho_{0}$. Thus it appears that both sides of this equation are of order $\epsilon^{2}$. While Nyborg [21] assumed a solenoidal secondary flow from the onset, Lee and Wang [18] proved more rigorously that $\nabla\cdot\bm{U}(\bm{x})=0$ using a low viscosity argument.

Accepting that the streaming velocity, $\bm{U}$ is divergence free, its governing equation (17) is identical to a Stokes flow in the presence of a non-conservative body force, $\bm{\mathcal{F}}$ due to the time-averaged Reynolds stress tensor, $<-\rho_{0}(\overline{\bm{v}}_{1}\;\overline{\bm{v}}_{1})>$ that is given in terms of the first order velocity field, $\overline{\bm{v}}_{1}$.

This is the starting point for the general small amplitude theory of the streaming velocity by calculating $\bm{\mathcal{F}}$ from the solution of the primary flow. The desired physical solution is $\overline{\bm{v}}_{1}(\bm{x},t)=\text{Real}\{\bm{u}(\bm{x})e^{-\mathrm{i}{\,}\omega t}\}$ where $\bm{u}(\bm{x})=\bm{u}^{\prime}(\bm{x})+\mathrm{i}{\,}\bm{u}^{\prime\prime}(\bm{x})$ is in general complex with real and imaginary parts $\bm{u}^{\prime}(\bm{x})$ and $\bm{u}^{\prime\prime}(\bm{x})$ and the time-averaged body force in (17) becomes, $\bm{\mathcal{F}}={\textstyle\frac{1}{2}}\rho_{0}\nabla\cdot[\bm{u}^{\prime}\bm{u}^{\prime}+\bm{u}^{\prime\prime}\bm{u}^{\prime\prime}]$.

Drawing on the mathematical concepts used in electrophoresis of a spherical particle, explicit solutions for the streaming vorticity and velocity fields can be given in terms of the body force. The derivation is rather tedious but otherwise relatively straightforward and is provided in Appendix A.

If viscous effects are small, we expect a very thin boundary layer. From (6), a vanishing viscosity corresponds to the limit $|k_{T}a|\gg|k_{L}a|$ and hence $e^{\mathrm{i}{\,}k_{T}r}\ll 1$ for $r>a$ due to the imaginary part of $k_{T}$. From (15) we find

and (13) becomes:

This solution is consistent with Landau and Lifshitz [31, p. 286, §74 Problem 1].

If in addition, $|k_{L}a|\rightarrow 0$, we recover the potential flow solution for the fluid velocity around a sphere moving at a constant velocity:

This solution is consistent with Landau and Lifshitz [31, p. 21–22, §10 Problem 2].

Fig. 4 shows the flow patterns with a carefully selected set of $k_{T}a$ and $k_{L}a$ values, $k_{T}a=14+\mathrm{i}{\,}\,14$ and $k_{L}a=0.05+\mathrm{i}{\,}\,2.7\times 10^{-4}$ since it is close to the thin boundary layer limit as $|k_{T}a|\gg 1$ and $|k_{T}a|\gg|k_{L}a|$. As demonstrated in Fig. 4, when $|k_{T}a|\gg|k_{L}a|$, outside the thin boundary layer, the results obtained with (13) for the primary velocity field are in good agreement with the results using (18) for the thin boundary layer potential flow limit. Also, for this particular case, as the imaginary part of $k_{L}a$ is close to zero, the imaginary parts of the primary flow velocity components calculated by (18) almost vanish, as displayed by the green circles in Fig. 4. Note that $\mathrm{Re}[u_{\theta}(r)]$ closely follows the potential flow results for $r/a>1.5$, but, in order to satisfy the no-slip condition, bends over sharply in the region $r/a=1$ to $1.5$ to satisfy the no slip condition $u_{\theta}(a)=-1$. In contrast, the potential flow solution is $u_{\theta}(a)=0.5$.

If the radius of the sphere, $a$, is very large, there are two locations of special interest: the front of the sphere at $\theta=0$ and the side of the sphere at $\theta=\pi/2$.

Consider first the side at which $\cos\theta=0$ and $\sin\theta=1$. From (7) we then find $\bm{u}=-u_{\theta}(r)\bm{e}_{z}$ since $\bm{U}_{0}=-U_{0}\bm{e}_{z}$. And setting $r/a\rightarrow 1$ in (13), we find

For a large sphere, $|k_{L}a|,|k_{T}a|\gg 1$, so $G(k_{L}r),G(k_{T}r)\rightarrow 0$ and $C_{1}\rightarrow-e^{\mathrm{i}{\,}k_{T}a}$ and the velocity becomes:

In the time domain, this is equivalent to the well-known Stokes oscillatory boundary thickness equation [32, 28, 33]: $\bm{u}=-\bm{U}_{0}e^{-ky}\cos(\omega t-ky)$, with $k=\sqrt{\omega\rho_{0}/(2\mu)}$, the real part of $k_{T}=(1+\mathrm{i}{\,})\sqrt{\omega\rho_{0}/(2\mu)}$, and $y=r-a$ being the distance from the flat surface. Thus, the solution at the side of the sphere tends towards the Stokes vibrating boundary layer theory for a flat plate when the radius of the sphere is large enough.

For the solution in front of the sphere at $\theta=0$, in the large $a$ or flat plate limit with $\bm{U}_{0}=U_{0}\bm{e}_{z}$, the velocity in (7) becomes $\bm{u}=u_{r}(r)\bm{e}_{z}$ and with $a/r\rightarrow 1$, we have:

Again with $G(k_{L}r),G(k_{T}r)\rightarrow 0$ for a large sphere, $|k_{L}a|,|k_{T}a|\gg 1$, we have the limiting solution

which represents a plane sound wave propagating out of an oscillating plate.

The incompressibility of the fluid implies that $c_{0}\rightarrow\infty$. In this case, we can take the limit of $|k_{L}a|\rightarrow 0$ while keeping $|k_{T}a|$ finite. As such, with the help of the Maclaurin series expansion of $e^{\mathrm{i}\,k_{L}a}$, we have

Introducing (24) into (13) we have

This solution is identical to the solution given by Landau and Lifshitz [31, p. 89, §24 Problem 5], where the velocity was written as $\bm{u}\equiv\nabla\times\nabla\times[f_{L}(r)\bm{U}_{0}]$, then $u_{r}(r)=-(2/r)\;\mathrm{d}f_{L}(r)/\mathrm{d}r$ and $u_{\theta}(r)=(1/r)\;\mathrm{d}f_{L}(r)/\mathrm{d}r+\mathrm{d}^{2}f_{L}(r)/\mathrm{d}r^{2}$. They showed that $\frac{\mathrm{d}f_{L}(r)}{\mathrm{d}r}=a_{L}\left(\frac{1}{r}-\frac{1}{\mathrm{i}{\,}k_{T}r^{2}}\right)e^{\mathrm{i}{\,}k_{T}r}+\frac{b_{L}}{r^{2}}$. This corresponds to our solution with $a_{L}=3\mathrm{i}{\,}ae^{-\mathrm{i}{\,}k_{T}a}/(2k_{T})$ and $b_{L}=-[1+G(k_{T}a)]a^{3}/2$. It is also consistent with the solution of Eq. (9) from [14]. As shown in Fig. 5, the results obtained by (13) for the primary velocity fields at $k_{L}a=0.01+\mathrm{i}{\,}\,0.001$ when (i) $k_{T}a=1+\mathrm{i}{\,}\,1$ and (ii) $k_{T}a=14+\mathrm{i}{\,}\,14$ are in good agreement with the results using (25) for the incompressible Stokes flow limit.

In order to get back the Stokes limit, we have to take the limit $|k_{T}a|\rightarrow 0$ as well. By using the Maclaurin series expansions of $e^{\mathrm{i}\,k_{T}r}$ and $e^{\mathrm{i}\,k_{T}a}$, we have

Then the velocity components become:

which is the velocity field for a sphere moving in Stokes flow Landau and Lifshitz [31, p. 50–60, §20]. Note that in the above results, terms with $1/k_{T}^{2}$ cancel each other exactly out.