Question

A compressible fluid with negligible viscosity is initially at rest with uniform density \(\rho_{0}\) and pressure \(p_{0}\), with no body forces. A small perturbation is then introduced so that there is a velocity \(u(r, t)\) and the density becomes \(\rho_{0}+\rho_{1}(r, t)\). (a) Assuming that products of the small quantities \(\boldsymbol{u}\) and \(\rho_{1}\) can be neglected, show that the equation for conservation of mass (5.9) becomes $$ \frac{\partial \rho_{1}}{\partial t}+\rho_{0} \nabla \cdot u=0 $$ (b) Assuming that the perturbation \(p_{1}\) to the pressure is related to \(\rho_{1}\) by \(p_{1}=a \rho_{1}\) where \(a\) is constant, show that the Navier- Stokes equation reduces to $$ \rho_{0} \frac{\partial u}{\partial t}=-a \nabla \rho_{1} $$ (c) Hence show that the density perturbation \(\rho_{1}\) obeys the wave equation and interpret this result physically.

Short answer

The conservation of mass reduces to \(\frac{\partial \rho_{1}}{\partial t} + \rho_{0} abla \cdot \boldsymbol{u} = 0\), the Navier-Stokes equation simplifies to \(\rho_{0} \frac{\partial \boldsymbol{u}}{\partial t} = -a abla \rho_{1}\), and the density perturbation obeys the wave equation \(\frac{\partial^2 \rho_{1}}{\partial t^2} = a abla ^2 \rho_{1}\), representing sound waves in the fluid.

Step-by-step solution

Apply the conservation of mass

The general form for the conservation of mass (continuity equation) for a compressible fluid with no body forces is given by \[ \frac{\partial \rho}{\partial t} + abla \cdot (\rho \boldsymbol{u}) = 0 \]. Since the fluid is initially at rest with uniform density \(\rho_{0}\), and the velocity and density perturbations are small, \(\rho_{1}\) and \(\boldsymbol{u}\) can be considered as small quantities.

Simplify the equation considering small perturbations

Because \(\rho_{1}\) and \(\boldsymbol{u}\) are small, their product can be neglected. This simplifies the conservation of mass to \[ \frac{\partial (\rho_{0} + \rho_{1})}{\partial t} + \rho_{0} abla \cdot \boldsymbol{u} + \rho_{1} abla \cdot \boldsymbol{u} \approx 0 \], ignoring higher order terms like \(\rho_{1} abla \cdot \boldsymbol{u}\). Since \(\rho_{0}\) is constant, the derivative \(\frac{\partial \rho_{0}}{\partial t}\) is zero, and the equation simplifies further to \[ \frac{\partial \rho_{1}}{\partial t} + \rho_{0} abla \cdot \boldsymbol{u} = 0 \].

Apply Navier-Stokes equation

The Navier-Stokes equation for an inviscid fluid with no body forces is \[ \rho \frac{\partial \boldsymbol{u}}{\partial t} + \rho (\boldsymbol{u} \cdot abla) \boldsymbol{u} = -abla p \]. Due to the small perturbations assumption, the term \(\rho (\boldsymbol{u} \cdot abla) \boldsymbol{u}\) can be neglected.

Simplify Navier-Stokes with perturbation conditions

Assuming that the pressure perturbation \(p_{1}\) is related to the density perturbation \(\rho_{1}\) by \(p_{1} = a \rho_{1}\), where \(a\) is a constant, and knowing that \(p = p_{0} + p_{1}\) and \(\rho = \rho_{0} + \rho_{1}\), the simplified Navier-Stokes equation can be written as \[ (\rho_{0} + \rho_{1}) \frac{\partial \boldsymbol{u}}{\partial t} = -abla (p_{0} + p_{1}) \]. Ignoring the product of small quantities again leaves us with \[ \rho_{0} \frac{\partial \boldsymbol{u}}{\partial t} = -abla (a \rho_{1}) \] or equivalently \[ \rho_{0} \frac{\partial \boldsymbol{u}}{\partial t} = -a abla \rho_{1} \].

Derive the wave equation for density perturbation

To derive the wave equation for \(\rho_{1}\), we take the gradient of the equation obtained from step 4, which gives us \[ abla \left( \rho_{0} \frac{\partial \boldsymbol{u}}{\partial t} \right) = -a abla (abla \rho_{1}) \]. We also differentiate the equation from step 2 with respect to time, \[ \frac{\partial}{\partial t} \left( \frac{\partial \rho_{1}}{\partial t} \right) + \rho_{0} \frac{\partial}{\partial t} abla \cdot \boldsymbol{u} = 0 \]. Substituting \( \frac{\partial}{\partial t} abla \cdot \boldsymbol{u} \) from the previous step, we obtain \[ \frac{\partial^2 \rho_{1}}{\partial t^2} = a abla ^2 \rho_{1} \], which is the wave equation for \(\rho_{1}\).

Physical Interpretation

The wave equation derived in the previous step indicates that the density perturbations in the fluid propagate as waves. The constant \(a\) relates to the speed of sound in the medium, as this equation is a typical wave equation with speed \(\sqrt{a}\) where \(a > 0\). In this context, the density perturbations can be thought of as sound waves traveling through the fluid.

Key concepts

Conservation of Mass

In the context of our exercise, where the fluid is initially at rest with a uniform density \(\rho_0\), this principle takes a simplified form when involving small velocities and density changes. The linear approximation allows us to consider only first-order changes, leading to the equation \[ \frac{\partial \rho_1}{\partial t} + \rho_0 abla \cdot u = 0 \], which encodes the essence of mass conservation in the presence of small disturbances in the fluid.

So what does this mean in simpler terms? Think of it like a traffic flow—cars (fluid particles) can speed up or slow down (velocity perturbations), and the spacing between them might change a bit (density perturbations), but the number of cars passing a point per unit time doesn't change if there are no entrances or exits (mass is conserved).

Navier-Stokes Equation

The Navier-Stokes equation is a fundamental expression in fluid mechanics, describing how fluid momentum changes due to pressure and viscous forces. For an ideal inviscid fluid—a hypothetical fluid with no viscosity—the equation simplifies to represent only changes due to pressure.

In our exercise, given the assumption of negligible viscosity and small perturbation, the Navier-Stokes equation takes on a reduced form: \[ \rho_0 \frac{\partial u}{\partial t} = -a abla \rho_1 \]. This focuses solely on how the small changes in density (\(\rho_1\)) directly influence the fluid's velocity field (\(u\)). The constant \(a\) connects the pressure change to the density change in a linear relationship, mirroring how sound waves relate pressure and density in a medium.

What we see, in essence, is that for small disturbances in a fluid at rest, the fluid's acceleration (change in velocity) can be directly linked to the spatial changes in density, ignoring the more complex behavior that arises with larger disturbances.

Wave Equation

The wave equation governs how disturbances move through a medium, and in fluids, this typically refers to how sound waves propagate. By combining the conservation of mass with the simplified Navier-Stokes equation, we arrive at a wave equation for density perturbation: \[ \frac{\partial^2 \rho_1}{\partial t^2} = a abla^2 \rho_1 \].

This equation bears a striking resemblance to the classic wave equation, with \(a\) playing the role of the wave speed squared. In this scenario, \(a\) being squared suggests that it's related to the speed of sound, which, incidentally, is how rapidly pressure variations travel through the fluid.

So, this part of the exercise confirms something important: tiny density fluctuations in our initially calm fluid behave like sound waves. They ripple outwards, unaffected by the more pedestrian movement of the fluid, giving a physical interpretation to the math we've been through.

Density Perturbation

Density perturbation refers to the variations in density within a fluid. In a sense, it's the 'cause' behind the 'effect' we witness in fluid behavior—a bit like throwing a pebble in a pond and watching the ripples.

These perturbations lead to changes in pressure (represented by \(p_1\)), which in turn drive the fluid to move (thus altering velocity \(u\)). Our exercise solidifies this connection by assuming a simple proportional relationship between pressure perturbation and density perturbation, \(p_1 = a \rho_1\), where \(a\) remains constant. This straightforward relationship helps us to see how such minor fluctuations might lead to wave-like behavior when a fluid is disturbed.

Real-world applications where understanding density perturbation is key include aeronautics, weather forecasting, and acoustics. Each of these fields deals with scenarios where seemingly small changes can have wide-ranging implications, such as the creation of sound waves, the development of storm systems, or the flight stability of aircraft.