Oscillatory Motion in a Viscous Fluid
- Fluid Mechanics 2.9
Oscillating Plane
Consider a plane solid surface $x=0$ bounding a viscous fluid in $x>0$. Suppose that the surface oscillates parallel to itself, in the $y$ direction, with velocity
\[ u(t)=\Re\left( {u_0e^{-i\omega t}} \right). \]
If the pressure is uniform and the motion depends only on $x$ and $t$, then the Navier-Stokes equation is reduced to
\[ \pdv{v_y}{t}=\nu\pdvn{2}{v_y}{x}. \]
Writing
\[ v_y=\Re \left( {u_0e^{i(kx-\omega t)}} \right), \]
we have
\[ i\omega=\nu k^2. \]
The solution which decays as $x\to\infty$ is obtained from
\[ k=\frac{1+i}{\delta},\qquad \delta=\sqrt{\frac{2\nu}{\omega}}. \]
Thus, for real $u_0$,
\[ v_y(x,t)=u_0e^{-x/\delta}\cos\left(\omega t-\frac{x}{\delta}\right). \]
The quantity $\delta$ is the penetration depth of the oscillatory motion. The velocity amplitude decreases exponentially, and the phase is delayed by $x/\delta$.
The shear stress is
\[ \sigma_{xy}=\mu\pdv{v_y}{x}. \]
In complex notation, at the wall,
\[ \sigma_{xy}(0,t)=\Re \left\{ \sqrt{\frac{\omega\mu\rho}{2}}(i-1)u_0e^{-i\omega t} \right\}. \]
Equivalently, the force exerted by the fluid on unit area of the wall is
\[ f_y=-u_0\sqrt{\omega\mu\rho}\cos\left(\omega t+\frac{\pi}{4}\right). \]
Hence the mean rate of dissipation per unit area is
\[ \overline{\dot E} =\frac{1}{2}u_0^2\sqrt{\frac{\omega\mu\rho}{2}}. \]
More generally, if the wall velocity $u(t)$ is not sinusoidal, the tangential force per unit area is
\[ f_y(t) =-\sqrt{\frac{\mu\rho}{\pi}} \int_{-\infty}^t \frac{\dot u(t^\prime)}{\sqrt{t-t^\prime}}\,dt^\prime. \]
This formula shows explicitly that the force is not determined by the instantaneous velocity alone. It depends on the previous motion of the wall.
Oscillating Body
For small oscillations of a body, the convective term in the Navier-Stokes equation may be omitted. Then
\[ \pdv{\b{v}}{t} =-\frac{1}{\rho}\grad p+\nu\laplacian\b{v}, \qquad \div\b{v}=0. \]
Taking the curl gives
\[ \pdv{}{t}\curl\b{v} =\nu\laplacian\curl\b{v}. \]
Thus the vorticity satisfies a diffusion equation. During one period it diffuses through a distance of order
\[ \delta=\sqrt{\frac{2\nu}{\omega}}. \]
If the characteristic length $l$ of the body satisfies
\[ l^2\omega\ll\nu, \]
then $\delta\gg l$, and the viscous diffusion is effectively instantaneous on the scale of the body. The motion is then quasi-steady. For example, a slowly oscillating sphere of radius $R$ experiences, to leading order,
\[ \b{F}=6\pi\mu R\b{u}(t), \]
provided the oscillatory $\mathrm{Re}$ is small.
The opposite case is
\[ l^2\omega\gg\nu. \]
Then vorticity is confined to a thin layer near the surface. Outside this layer the fluid is practically ideal and irrotational:
\[ \curl\b{v}=0,\qquad \div\b{v}=0. \]
The outer flow is therefore a potential flow, determined by the impermeability condition on the moving surface. The tangential no-slip condition is restored inside the thin viscous layer.
Let $x$ be the distance measured normally from the surface into the fluid, and let $v_0e^{-i\omega t}$ be the tangential velocity of the inviscid outer flow at the surface, in the reference frame of the body. Then the boundary-layer correction has the same form as for the oscillating plane:
\[ v_y(x,t) =v_0e^{-i\omega t} \left(1-\exp\left[-(1-i)\frac{x}{\delta}\right]\right), \]
The mean loss of mechanical energy is then
\[ \overline{\dot E}_{\mathrm{mech}} =-\frac{1}{2}\sqrt{\frac{\omega\mu\rho}{2}} \int v_0^2\,df, \]
where the integral is taken over the surface of the body and $v_0$ denotes the amplitude of the tangential velocity of the potential flow at the surface.
Complex Drag
For a body oscillating with velocity
\[ \b{u}=\b{u}_0e^{-i\omega t}, \]
the hydrodynamic force is usually written as
\[ \b{F}=\beta\b{u},\qquad \beta=\beta_1+i\beta_2. \]
In real notation this means
\[ \b{F} =\beta_1\b{u}-\frac{\beta_2}{\omega}\dot{\b{u}}. \]
The term with $\beta_1$ is dissipative, while the term with $\beta_2$ is an inertial correction. The mean rate of loss of mechanical energy is
\[ \overline{\dot E}_{\mathrm{mech}} =-\frac{1}{2}\beta_1u_0^2. \]
For a sphere of radius $R$, the exact linear solution gives
\[ \b{F} =6\pi\mu R\left(1+\frac{R}{\delta}\right)\b{u} +3\pi R^2\sqrt{\frac{2\mu\rho}{\omega}} \left(1+\frac{2R}{9\delta}\right)\dot{\b{u}}, \qquad \delta=\sqrt{\frac{2\nu}{\omega}}. \]
The first part is the viscous drag, and the second part is the force required to accelerate the surrounding fluid. In the limit $\omega\to0$ this reduces to Stokes’ law. In the opposite limit $R\gg\delta$,
\[ \b{F} =3\pi R^2\sqrt{2\mu\rho\omega}\,\b{u} +\frac{2}{3}\pi\rho R^3\dot{\b{u}}, \]
which is the sum of the boundary-layer friction and the added-mass force.
