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The continuum hypothesis and fields
Why a fluid made of molecules can be described by smooth density, velocity and pressure fields, and when that description fails.
Water and air are made of molecules, yet almost every useful statement about them treats the fluid as a continuous medium. The step that makes this legitimate is an average. Around a point \(\mathbf{x}\) take a small volume \(\delta V\) that is huge compared with the spacing between molecules but tiny compared with the length \(L\) over which the flow changes. The density is the mass in that volume divided by \(\delta V\), \[ \rho(\mathbf{x},t) = \frac{\delta m}{\delta V}, \] and the velocity \(\mathbf{u}(\mathbf{x},t)\) is the average momentum of the molecules inside divided by their mass. If such an intermediate scale exists, the averages do not depend on the exact size of \(\delta V\), and we obtain fields: functions of position and time.
The ratio that decides whether an intermediate scale exists is the Knudsen number \(\mathrm{Kn} = \lambda/L\), where \(\lambda\) is the mean free path, the average distance a molecule travels between collisions. For air at sea level \(\lambda\) is about \(68\) nanometres, so a flow around a 1 millimetre wire has \(\mathrm{Kn} \approx 7\times10^{-5}\). The continuum description is reliable when \(\mathrm{Kn}\) is well below about \(0.01\). The upper atmosphere, micro-channels and vacuum pumps can violate this, and then the kinetic theory of gases (the Boltzmann equation) replaces fluid mechanics.
A fluid is defined by what it does under shear: it keeps deforming for as long as any tangential stress, however small, is applied. A solid resists with a finite deformation. This single property is why a fluid at rest can only push normally on a surface, a fact that fixes the form of the stress in the lessons ahead.
Once the fields exist, global quantities are integrals. The mass inside a region \(V\) is \(M = \int_V \rho\,dV\); for a ball of radius 1 whose density rises from the centre as \(\rho = 1000(1+r)\), spherical shells give \(M = \int_0^1 1000(1+r)\,4\pi r^2\,dr = 7000\pi/3\). Momentum is \(\int_V \rho\mathbf{u}\,dV\) and kinetic energy \(\int_V \tfrac12\rho|\mathbf{u}|^2\,dV\). The laws of fluid dynamics are balance laws for these integrals, turned into differential equations.
Picture it: a photograph of a gas at molecular resolution is a spray of dots; blur it at the right scale and you see a smooth shade of grey, and the shade is the density field. Think it: from here on the fields are treated as functions, and the equations they satisfy are partial differential equations. Whether solutions of those equations stay smooth is a mathematical question, not a physical assumption, and this course ends by studying exactly what is known about it.
ምሳሌ · integrate 1000*(1 + r)*4*pi*r^2 dr from 0 to 1
Integrate pi·r^2·(4000r + 4000) from 0 to 1
እርምጃ በደረጃ
- \int_{0}^{1} \pi r^{2} \left(4000 r + 4000\right)\, dr
First find an antiderivative F, then evaluate F(b) − F(a).
- \int \pi r^{2} \left(4000 r + 4000\right)\, dr = \pi \int r^{2} \left(4000 r + 4000\right)\, dr
Pull the constant \pi out of the integral.
- r^{2} \left(4000 r + 4000\right) = 4000 r^{3} + 4000 r^{2}
Rewrite the integrand into a friendlier form.
- \int 4000 r^{3} + 4000 r^{2}\, dr = \int 4000 r^{3}\, dr + \int 4000 r^{2}\, dr
The integral of a sum is the sum of the integrals.
- \int 4000 r^{3}\, dr = 4000 \int r^{3}\, dr
Pull the constant 4000 out of the integral.
- \int r^{3}\, dr = \frac{r^{4}}{4}
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
- \int 4000 r^{2}\, dr = 4000 \int r^{2}\, dr
Pull the constant 4000 out of the integral.
- \int r^{2}\, dr = \frac{r^{3}}{3}
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
- F(1) - F(0) = \left(\frac{7000 \pi}{3}\right) - \left(0\right)
Fundamental theorem of calculus: plug in the limits.
- = \frac{7000 \pi}{3} \approx 7330.4
Simplify.
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እንዴት: The continuum hypothesis and fields
- Compare the mean free path with the smallest length of the flow: compute Kn and check it is well below 0.01.
- Name the fields the problem needs: density, velocity, pressure, temperature.
- Write every global quantity (mass, momentum, energy) as an integral of a field over a region.
- Choose coordinates that match the symmetry (shells for a ball, rings for a pipe) before integrating.
ጥያቄዎች
Is the continuum hypothesis ever wrong for water?
Only at nanometre scales, for example in flows through carbon nanotubes or at a moving contact line. For anything you can see, water is a continuum to excellent accuracy.
Why is velocity an average and not the speed of a molecule?
Molecules move at hundreds of metres per second in random directions even in still air. The fluid velocity is the mean of those motions; the random part shows up as pressure and temperature.
What do I need before starting fluid dynamics?
Multivariable calculus (divergence, curl, the divergence and Stokes theorems), linear algebra (symmetric matrices and eigenvalues) and partial differential equations (the heat and Laplace equations). The last lessons also use Sobolev spaces, which are introduced where they are needed.
What are the Navier-Stokes equations in one sentence?
Newton's second law for each particle of a viscous incompressible fluid: acceleration equals the pressure force plus viscous diffusion of momentum, with the constraint that the velocity field is divergence-free.
Why is two-dimensional flow easier than three-dimensional flow?
In 2D the vorticity is a scalar that is only carried and diffused, so its maximum never grows; in 3D vortex lines can be stretched, which amplifies vorticity, and no known bound rules out unlimited growth.
Is this course physics or mathematics?
Both, in order. The first half derives the equations from physical principles and solves classical flows; the second half treats the equations as mathematical objects and studies which of their properties can be proved.
በ Fluid Dynamics
Eulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklinesConservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowVorticity, circulation and Kelvin's circulation theoremThe stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equationViscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetryStokes flow at low Reynolds numberBoundary layers