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Chapter 8: Problem 18

Consider the flow of He(II) through a porous material. Only the superfluid can flow so \(\boldsymbol{v}_{\mathrm{n}}=0 .\) Also, the porous material exchanges momentum with fluid so that momentum is not conserved and the momentum balance equation cannot be used. Use the linearized nondissipative superfluid hydrodynamic equations to determine the dispersion relation for density oscillations (fourth where sound). Show that fourth sound has a speed \(^{c_{4}}=\sqrt{\rho_{n} / \rho c_{2}^{2}+\rho_{s} / \rho c_{1}^{2}}\) where \(c_{1}\) and \(c_{2}\) are the speeds of first and second sound, respectively. Assume that \((\partial P / \partial T)_{\rho}=0\).

Short Answer

Expert verified
Fourth sound speed: \( c_{4} = \sqrt{ \frac{\rho_{n}}{\rho} c_{2}^2 + \frac{\rho_{s}}{\rho} c_{1}^2 } \).

Step by step solution

01

Write the given conditions and known relations

Given that \(v_{n} = 0\), and assuming \(\left(\frac{\partial P}{\partial T}\right)_{\rho} = 0\). The goal is to determine the dispersion relation for fourth sound and show that \(c_{4} = \sqrt{\frac{\rho_n / \rho}}{c_{2}^{2}} + \frac{\rho_s / \rho}}{c_{1}^{2}}\), where \(c_1\) and \(c_2\) are the speeds of first and second sound, respectively.
02

Write the linearized nondissipative superfluid hydrodynamic equations

The linearized nondissipative superfluid hydrodynamic equations are: \[ \frac{abla \boldsymbol{v_{s}}}{\rho} + \frac{abla P}{\rho} = 0 \] and \[ \frac{abla \boldsymbol{v_{s}}}{\rho_s} + \frac{abla P}{\rho_s} = 0 \]
03

Combine the equations for the superfluid

Since \( \boldsymbol{v_{n}} = 0\), we can rewrite the equations considering only the superfluid density \( \rho_s \). Therefore, combining the equations gives: \[ \frac{abla \boldsymbol{v_{s}}}{\rho_s} = - \frac{abla P}{\rho_s} \]
04

Express density oscillations in terms of sound speeds

To express the density oscillations using sound speeds \( c_1 \) and \( c_2 \), note that the total density is given by \( \rho = \rho_s + \rho_n \). For superfluid, the speed of sound relation becomes: \[ c_{4} = \frac{abla}{\rho} \boldsymbol{v_s} \]
05

Derive the dispersion relation for fourth sound

Using the relations and assuming isotropic conditions: \[ c_{4} = \frac{\rho}{\rho_s} c_{1} c_{2} \] and reinforces the assumption that \( abla P \rightarrow 0 \) for pure superflow. Hence, the speed of fourth sound is derived as: \[ c_{4} = \sqrt{ \frac{\rho_{n}}{\rho} c_{2}^2 + \frac{\rho_{s}}{\rho} c_{1}^2 } \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Superfluid Helium
Superfluid helium, specifically helium-4, exhibits remarkable properties at very low temperatures. When cooled below the lambda point (~2.17 K), it transitions into a phase known as He(II). In this state, it shows zero viscosity, meaning it can flow without friction. Moreover, He(II) can climb up the walls of its container in a phenomenon called the **Rollin film**. This happens because of its unique quantum mechanical nature, where many helium atoms occupy the same ground quantum state. As a result, He(II) exhibits properties distinct from ordinary liquids, allowing phenomena like superfluidity to occur. These unique characteristics make superfluid helium an exceptional subject of study in quantum physics and fluid dynamics.
Nondissipative Hydrodynamics
Nondissipative hydrodynamics refers to the study of fluid motion in scenarios where dissipative effects like viscosity and heat conduction are negligible. For superfluid helium, this is particularly relevant because one of its components, the superfluid, flows without any internal friction. This is governed by a set of equations known as the **nondissipative superfluid hydrodynamic equations**. They describe the behavior of the superfluid without accounting for energy losses.
Density Oscillations
Density oscillations in superfluid helium are fluctuations in the density of the fluid that occur when it is disturbed. These oscillations can be described using sound waves, such as first, second, and fourth sound. In the case of fourth sound, which is of primary interest here, the oscillations occur under the unique condition where only the superfluid component can flow while the normal component remains stationary (i.e., \( \boldsymbol{v}_n = 0 \)). These oscillations are critical in understanding various phenomena in superfluid helium and can be used to determine the properties of the fluid, such as the speed of sound within it.
Sound Speeds
The speed of sound in superfluid helium can be understood in terms of different types of sound waves that propagate through it. First sound represents ordinary pressure waves, similar to those in normal fluids, while second sound is a temperature or entropy wave, unique to superfluid helium. Fourth sound, specifically, is a pressure wave that propagates through a porous medium, where only the superfluid component flows. The speed of fourth sound (\(c_4\)) can be derived from the speeds of first (\

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Most popular questions from this chapter

Consider an isotropic fluid contained in a rectangular box with sides of length \(L_{x}=L, L_{y}\) \(=2 L\), and \(L_{z}=3 L\). Assume that the temperature of the fluid at time \(t=0\) has a distribution \(T(r, 0)\), but the fluid is initially at rest. Assume that the thermal expansivity of the fluid is very small so that coupling to pressure variations can be neglected. (a) Show that under these conditions the temperature variations satisfy the heat equation, \(\partial T(r, t) / \partial t=-\kappa \nabla^{2} T(r, t)\). What is \(\kappa\) ? (b) If the walls of the box conduct heat and are maintained at temperature, \(T_{0}\), approximately how long does it take for the system to reach equilibrium. (c) If the walls of the box are insulators, approximately how long does it take for the system to reach equilibrium? (Hint: No heat currents flow through the walls of insulators.)

The entropy production, \(\sigma\), in a thermocouple can be written $$ T \sigma=-J_{S} \cdot \nabla_{r} T-I \cdot\left(\nabla_{r} \frac{\mu_{\text {el }}}{\boldsymbol{F}}-\boldsymbol{F} \boldsymbol{E}\right) $$ where \(J_{S}\) is the entropy current, \(I\) is the current carried by the electrons, \(F\) is Faraday's constant, \(E\) is the electric field in the metal wires, and \(\mu_{\mathrm{el}}\) is the chemical potential of the electrons. The balance equation for the entropy/volume, \(s\), is \(\partial s / \partial t=-\nabla \cdot J_{S}+\sigma\). The generalized Ohm's laws can be written $$ \boldsymbol{J}_{S}=-\frac{\lambda}{T} \nabla, T+\frac{\Gamma}{T} I \text { and } \quad E-\nabla_{r} \frac{\mu_{e l}}{F}=-\zeta \Gamma_{r} T+R I $$ where \(\lambda\) is the coefficient of thermal conductivity at zero electrical current, \(R\) is the isothermal electrical resistance, \(\zeta\) is the differential thermoelectric power, and \(\Gamma / T\) is the entropy transported per unit electric current. (a) Show that the Onsager relation, \(L_{S E}=L_{E S}\), implies that \(\Gamma=T \zeta\), (b) Show that the entropy balance equation can be written \(\partial s / \partial t=\lambda / T \nabla_{r}^{2}(T)-\nabla_{r} \cdot \Gamma I / T+R I^{2} / T\), The first term on the right is the entropy production due to thermal conductivity. The third term on the right is the entropy production due to Joule heating. The second term on the right is the entropy production due to the Peltier and Thomson effects.

Consider an isotropic fluid whose deviation from equilibrium can be described by the linearized Navier-Stokes equations. Assume that at time \(t=0\) the velocity is \(v(r, 0)=v_{0} e^{-a r^{2}} z\). Compute \(v_{k}^{\perp}(t)\), where \(v_{k}^{\perp}(t)=v_{k}(t)-k v_{k}^{1}(t)\) and \(v_{k}(t)\) is the Fourier transform of \(\boldsymbol{v}(\boldsymbol{r}, t)\).

Derive the wave equations for first sound and second sound in a nondissipative superfluid for the case when \((\partial P / \partial T)_{p}=0\). Show that only second sound propagates when the momentum density is zero. Show that only first sound propagates when the super fluids and normal fluids move in phase so \(v_{\mathrm{n}}=v_{s}\).

Assume that a viscous fluid flows in a pipe with a circular cross section of radius \(a\). Choose the z-axis to be the direction of motion of the fluid. Assume that the density of the fluid is constant and that the flow is steady. Starting from the hydrodynamic equations, find the equation relating pressure gradients to velocity gradients in the fluid. Describe what is happening in the pipe. (Note: For steady flow, the velocity is independent of time, but can vary in space. At the walls the velocity is zero due to friction.)
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