The equations of motion we saw in Perfect Fluids were extremely difficult systems of coupled partial differential equations. In order to make them more tractable and extract physical intuition we make approximations using something called scale analysis. Scale analysis makes use of order of magnitude estimates of velocities, lengths, time scales, pressure differences, densities, and physical constants of a given system we’re analyzing. From this analysis we can determine the dominant terms in the evolution of a system and remove terms that are small or negligible in governing the system.

In this section we will learn when we can approximate a fluid as incompressible and when we must include viscosity or acceleration terms.

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Fundamentally, the laws of physics cannot depend on an arbitrary choice of measurement units

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Rayleigh’s Method

Falling Bodies

We will start with a simple power law form for the time it takes for an object to fall from a height $h$ to the surface of the Earth and consider other effects that may influence this time to make a more accurate model that can indicate when the simple model is valid.

If we simply assume $g$ is constant and the only force on the body than Newton’s second law indicates that $\ddot z=-g$ and the time is then found as

$$ \tau_0=\sqrt\frac{2h}{g} $$

From a dimensional analysis perspective we have

$$ [\tau_0]=T,[h]=L,[g]=L/T^2 $$

Apply Rayleigh analysis we find that

$$ \tau_0\propto\sqrt{\frac{h}{g}} $$

without solving any differential equation. Also using our physical intuition, it makes sense that at a height of $0$, that the time is $0$, so we don’t need to add some offset to our equation.

However there are other forces at play in the Earth’s atmosphere that may also be influencing this time:

• $g$ changes with height, related to the radius $R$ of Earth

  • This introduces a dimensionless values $h/R$ and relationship could have some arbitrary functional relationship with $\Pi_1=h/R$

• The Earth spins at a rate of $\Omega$ and causes a centrifugal force on the Earth’s reference frame

  • This introduces another dimensionless value $\Pi_2=\Omega^2 R/g$

• Drag force on the object based on the characteristic length of the object $l$ (more specifically it’s area $l^2$), the viscosity of the fluid $\mu$, the speed of the particle $U$, and the density of to the fluid $\rho$.

  • dimensions:

  $$ [U]=LT^{-1},[\rho]=ML^{-3},[\mu]=ML^{-1}T^{-1},[l]=L^{-1} $$

  • This introduces another dimensionless value called the Reynold’s number:

  $$ \text{Re}=\frac{\rho U l}{\mu} $$

Combining these dimensionless values we could write a new model

$$ \tau_0\approx\sqrt\frac{h}{g}f\bigg(\frac hR,\frac{\Omega^2R}{g},\frac{\rho Ul}{\mu}\bigg)\\

$$

If each of these values are extremely small then we expect $f(0,0,0)=\sqrt2$ to get our simple model back from Newton’s second law. If the dimensionless values are small we can also make a first order Taylor expansion to give an approximation