Problem 5.4.8

Solution

The fundamental reason for the transfer of rotation from the lower disk to the upper one lies in the continuous exchange of tangential momentum. The gas filling the vessel acts as a physical mediator in this exchange; however, the mechanics of this mediation transform radically depending on the density of the medium.

The Molecular Bullet Regime (Rarefied Gas, $\lambda \gg d$)

Imagine a deep vacuum where only occasional, solitary molecules fly by. In this ultra-rarefied limit, the gas cannot even be considered a continuous medium—it is an ensemble of independent particles whose trajectories practically never intersect. A molecule striking the surface of the rotating lower disk undergoes an act of diffuse scattering. It is adsorbed by the crystal lattice for microseconds, comes into thermal equilibrium with the surface, and breaks off again, acquiring the directed tangential velocity of the disk at the point of detachment.

Since the mean free path $\lambda$ vastly exceeds the distance between the disks $d$, this molecule flies through the gap like a microscopic bullet, meeting no obstacles on its way. It strikes precisely the upper suspended disk, giving up its acquired tangential momentum entirely. The torque transferred in such a ballistic regime is determined exclusively by the intensity of the molecular "fire"—the number of molecules crossing the gap per unit time. The higher the gas pressure $P$, the proportionally greater the concentration of particles in the volume, and the more intense the bombardment of the target. Doubling the pressure means strictly doubling the number of independent momentum carriers; therefore, the transferred torque and, as a consequence, the twist angle $\varphi$ grow strictly linearly with increasing pressure.

The Viscous Continuum Regime (High Pressure, $\lambda \ll d$)

Continuing to pump gas into the vessel, we transition the system into an entirely different physical regime. The mean free path drops rapidly. Molecules begin to collide with each other long before they reach the opposite disk. Local thermodynamic equilibrium emerges, a continuous macroscopic velocity profile forms—a continuous medium is born from the chaos. Now the torque is transferred not by direct sniper shots from disk to disk, but through the slow diffusion of momentum via myriads of intermolecular collisions. The mechanism is described by the macroscopic viscous friction of a Couette shear flow.

Here, one of the most striking and counterintuitive results of kinetic theory takes the stage—the great Maxwell paradox. In 1866, James Clerk Maxwell, in his famous Bakerian lecture, mathematically predicted and experimentally proved that the dynamic viscosity of an ideal gas $\eta$ is absolutely independent of its pressure. The physical nature of this macroscopic paradox is elegant in its microscopic symmetry: as we increase the pressure, we increase the density of the gas, meaning the number of momentum carriers per unit volume grows linearly. However, strictly in proportion to this growth, the mean free path $\lambda$ shrinks—the distance over which each carrier is capable of transporting this momentum without scattering. The increase in density is perfectly and completely compensated by the drop in the long-range action of the molecules.

The gas becomes denser, but momentum transfer within it becomes exactly as much more localized. As a result, the dynamic viscosity of the medium is fixed at a rigid constant depending exclusively on the temperature and the quantum-mechanical scattering cross-section of the molecules themselves. The transferred torque ceases to feel the injected pressure, and the thread's twist angle reaches a horizontal asymptote—the immutable constant $\varphi_2$.

Angle-Pressure Dependence at $\varphi \ll \varphi_2$
The condition $\varphi \ll \varphi_2$ geometrically implies that the system is at the very beginning of its evolutionary path. Small deflection angles are realized exclusively in the regime of ultra-low pressures, where the gas operates in the ballistic regime of independent molecular bullets. Since in this asymptotic region the torque grows strictly proportional to the pressure, the angle's dependence takes the form of a direct proportionality. Knowing the reference point ($P_1$, $\varphi_1$), we obtain a strict linear dependence.

For those interested in the mathematical details of the model

Such physical poetry demands a rigorous mathematical framework. Let us build an exact analytical model that connects both asymptotic regimes into a single, mathematically flawless structure. This will require diving into the kinetic theory of gases and the equations of hydrodynamics.

The Ultra-Rarefied Limit and Momentum Accommodation

Let us turn to kinetic theory. Suppose the pressure $P$ is vanishingly small, so that intermolecular collisions in the gap can be neglected. We isolate an elementary ring of radius $r$ and width $dr$ on the lower rotating disk, whose area is $dS = 2\pi r dr$.

The kinetic particle flux, i.e., the number of molecules striking a unit surface area per unit time, is determined by the fundamental formula:
$$\nu = \frac{1}{4} n \langle v \rangle$$
where $n$ is the volumetric concentration of molecules, and $\langle v \rangle = \sqrt{\frac{8kT}{\pi m}}$ is the mean arithmetic thermal velocity.

Upon collision of a molecule with the surface, a momentum exchange occurs, the efficiency of which is characterized by the Maxwell tangential momentum accommodation coefficient (TMAC), denoted by $\sigma$. The physical meaning of this coefficient lies in what fraction of molecules is completely thermalized by the surface (diffuse scattering) and what fraction bounces off absolutely elastically, preserving its tangential momentum (specular reflection).

To understand the scale of real physical values of TMAC obtained in precision disk spindown experiments, it is useful to refer to the table below:

To construct an idealized analytical model, we will adopt the hypothesis of complete diffuse scattering, setting $\sigma = 1$. This means that every molecule leaving the lower disk carries away a macroscopic tangential momentum $p_\tau = m v_{disk} = m \omega r$. Reaching the upper stationary disk, it transfers this momentum to it entirely.

The elementary force of viscous friction acting on the isolated annular section of the upper disk equals the momentum flux through this section:
$$dF = p_\tau \cdot \nu \cdot dS = (m \omega r) \left( \frac{1}{4} n \langle v \rangle \right) 2\pi r dr$$

The elementary torque is calculated as the product of the force and the lever arm $r$ ($dM = r dF$):
$$dM = \frac{\pi}{2} m n \langle v \rangle \omega r^3 dr$$

Integrating over the entire area of the disk from the center to the outer radius $R$ yields the exact value of the torque:
$$M_1 = \int_0^R \frac{\pi}{2} m n \langle v \rangle \omega r^3 dr = \frac{\pi}{8} m n \langle v \rangle \omega R^4$$

Using the ideal gas equation of state $P = n k T$, we can express the concentration as $n = P/kT$ and obtain the sought functional dependence of the torque on the macroscopic pressure:
$$M_1(P) = \left( \frac{\pi m \langle v \rangle \omega R^4}{8 k T} \right) P$$
Mathematics inexorably confirms physical intuition: in the ultra-rarefied limit, the torque is strictly linear with respect to pressure.

The Gas-Dynamic Limit and Couette Flow
In the opposite extreme of high pressures, the gas represents a continuous medium. The geometry of the problem reduces to a classic shear flow, known in hydrodynamics as Taylor-Couette flow in the planar limit. At low Reynolds numbers, the velocity profile between the disks is strictly linear. The velocity vector is directed tangentially to concentric circles, and its magnitude changes from $\omega r$ on the rotating lower disk to zero on the suspended upper one according to the law:
$$v(z) = \omega r \frac{z}{d}$$
where the coordinate $z$ is measured downwards from the stationary upper disk.

Newton's law of viscous friction relates the shear stress $\tau$ to the velocity gradient via the dynamic viscosity $\eta$:
$$\tau = \eta \frac{\partial v}{\partial z} = \eta \frac{\omega r}{d}$$

We calculate the total torque by integrating this shear stress over the entire area of the disk:
$$M_2 = \int_0^R r \cdot \tau \cdot 2\pi r dr = \int_0^R r \left( \eta \frac{\omega r}{d} \right) 2\pi r dr = \frac{2\pi \eta \omega}{d} \int_0^R r^3 dr = \frac{\pi \eta \omega R^4}{2 d}$$

The resulting expression represents the limiting constant. Equating this torque to the elastic torque of the thread $\gamma \varphi_2$, we obtain an analytical expression for the maximum twist angle:
$$\varphi_2 = \frac{\pi \eta \omega R^4}{2 d \gamma}$$

Maxwell's Paradox in Formulas
To rigorously prove the independence of the asymptotic torque $M_2$ from pressure, we must reveal the internal structure of the dynamic viscosity of an ideal gas $\eta$. According to kinetic theory, momentum transport in a gas is described by the formula:
$$\eta \approx \frac{1}{3} \rho \langle v \rangle \lambda$$

The gas density is trivially expressed through the concentration and the mass of a single molecule: $\rho = m n$. The mean free path $\lambda$, in turn, is inversely proportional to the concentration of particles and their effective scattering cross-section $\sigma_{scat} = \pi d_{mol}^2$ (where $d_{mol}$ is the gas-kinetic diameter of the molecule):
$$\lambda = \frac{1}{\sqrt{2} \sigma_{scat} n}$$

Let us substitute these fundamental microscopic quantities into the macroscopic expression for viscosity:
$$\eta = \frac{1}{3} (m n) \langle v \rangle \left( \frac{1}{\sqrt{2} \sigma_{scat} n} \right) = \frac{m \langle v \rangle}{3 \sqrt{2} \sigma_{scat}}$$

The concentration $n$, and along with it the thermodynamic pressure $P$, mathematically annihilate, canceling out in the numerator and denominator. The dynamic viscosity $\eta$, and hence the limiting twist angle $\varphi_2$, turn out to depend exclusively on the mass of the molecules, their cross-section, and the temperature of the system.

Bridging Regimes via the Knudsen Number (Kn)
The true beauty of theoretical physics manifests not in the analysis of extremes, but in the ability to describe the continuous transition between asymptotics. To construct a unified theory, we will need a dimensionless similarity criterion—the Knudsen number, defined as the ratio of the mean free path to the characteristic macroscopic dimension of the system: $Kn = \lambda / d$.

In the transition regime, when the mean free path $\lambda$ becomes comparable to the gap $d$, the phenomenon of viscous boundary slip comes into play. This effect was first discovered experimentally by Kundt and Warburg in 1875 when observing the damping of disk oscillations in a rarefied gas, and in 1879 it received a theoretical foundation from Maxwell himself.

The essence of the phenomenon is that the gas located directly at the surface of a solid body does not rest relative to it (the classical no-slip condition is violated). A so-called Knudsen layer forms, in which the macroscopic velocity of the gas at the wall $v_{slip}$ differs from the velocity of the wall itself. The magnitude of this slip velocity is directly proportional to the velocity gradient in the bulk and the mean free path:
$$v_{slip} = \zeta \lambda \left\vert \frac{\partial v}{\partial z} \right\vert$$
where $\zeta$ is a dimensionless coefficient of order unity (often called the slip coefficient), which depends on that very momentum accommodation coefficient $\sigma$. For absolutely diffuse scattering $\zeta \approx 1$, but for real surfaces it can be larger.

The presence of slip on both disks leads to a flatter velocity profile. The velocity gradient decreases, as if the physical gap between the disks had expanded beyond its geometric limits. A rigorous solution of the differential equation with slip boundary conditions gives a redefined gradient:
$$\frac{\partial v}{\partial z} = \frac{\omega r}{d + 2\zeta \lambda}$$

Substituting this gradient into the torque integral, we obtain a generalized formula describing momentum transfer at any pressure:
$$M(P) = \frac{\pi \eta \omega R^4}{2(d + 2\zeta \lambda)}$$

This formula is a genuine triumph of analytical thought. Let us test its resilience in our two limits.

At high pressures $P \to \infty$, the mean free path tends to zero ($\lambda \to 0$). The term $2\zeta\lambda$ in the denominator vanishes, and the formula degenerates into the purely hydrodynamic solution $M \to M_2$, independent of pressure.

At ultra-low pressures $P \to 0$, conversely, $\lambda \to \infty$. The geometric gap $d$ in the denominator can be neglected compared to the giant mean free path. The equation then takes the form:
$$M(P) \approx \frac{\pi \eta \omega R^4}{4\zeta \lambda}$$

Here it is necessary to recall that the macroscopic viscosity $\eta$ contains the mean free path implicitly ($\eta \propto \rho \lambda \propto n \lambda$). The ratio $\eta / \lambda$ turns out to be a quantity strictly dependent only on the particle concentration $n$, and thus linearly dependent on the pressure $P$. Substitution of the kinetic expressions for $\eta$ and $\lambda$ instantly and losslessly transforms this fraction into the molecular bullet formula $M \propto P$.

The mathematical structure of the model brilliantly reflects the deep unity of nature: from isolated molecular bombardments to monolithic macroscopic flows, the thread's twist angle smoothly, elegantly, and inexorably evolves from linear growth to an unshakable constant.