PHY5210 W15

Chapter 16: Continuum Mechanics

Reading:

In Taylor, read sections 16.12 for today.

Hydrodynamics

The topic of hydrodynamics is also called fluid mechanics, fluid dynamics, and sometimes, hydraulics.

Convective Derivative

We want to small a follow, infinitesimal piece of fluid, and evaluate its change in density over time. The density in general can be a function of both position and time, ρ = ρ(r, t). Imagine marking a small drop of fluid with a dye and tracking the dye. The density of the dye can change because of a change in time, and because the dye moves to a new location where the velocity is different. This is captured with the relation

dρ/dt = ∂ρ/∂t + ∑ ∂ρ/∂r_i dr_i/dt = ∂ρ/∂t + (v ⋅ ∇) ρ

The operator

d/dt = ∂/∂t + (v ⋅ ∇)

is known as the convective derivative, also called the material derivative. It shows up in other areas where the rate of change of a field (a quantity defined in space and time) is evaluated.

We can do the same procedure for the velocity of the element of fluid

dv/dt = ∂v/∂t + ∑ ∂v/∂r_i dr_i/dt = ∂v/∂t + (v ⋅ ∇) v

where v ⋅ ∇ is understood to have components v_i ∂/∂r_i. This gives us the acceleration of a volume element of fluid.

Equation of Motion for an Inviscid Fluid

Now we can get an equation of motion using F = ma where we allow the force to come from gravity and gradients of the pressure:

ρ (dv/dt) = ρ g - ∇ p

It is easy to see that ρ g is the gravitational force per unit volume. To see that -∇ p is another force arising from pressure gradients, relies on a result obtained in a section that we skipped over. It is certainly reasonable that pressure gradients will produce a force on a volume element.

Bernoulli's Theorem

Recall that Bernoulli's theorem relates the pressure in a fluid to its speed. It is used, for instance, to determine how the speed of a fluid changes when moving through a pipe that changes diameter.

½ ρ (dv²/dt) + ρ d(gz)/dt + dp/dt = 0

In the steady state dρ/dt = 0, so the derivative can be moved outside of each term yielding

d/dt( ½ ρ v² + ρ gz + p ) = 0

½ ρ v² + ρ gz + p = constant

This result is used in introductory physics. It can be used to find the pressure in hydraulic systems (v = 0), or determine how high a water stream can rise.

The Continuity Equation

If we follow a particular volume element, the mass in that element will not change. (Note that this can be a confusing concept, since at the atomic scale atoms will mix. This argument lies somewhat above the atomic scale, and we imagine that it holds to a very good approximation.) That the mass does not change is expressed by

d/dt(ρdV) = dV (dρ/dt) + ρ (d(dV)/dt) = 0

The time derivative of a moving volume element was worked out in part of Chapter 13 that we did not cover (Liouville's theorem) and gives d(dV)/dt = ∇ ⋅ v dV. Using this, we get the continuity equation

dρ/dt + ρ ∇ ⋅ v = 0

There is an equivalent form that you are to find in problem 6 of this week's homework

∂ρ/∂t + ∇ ⋅ (ρ v) = 0