An Introduction to Fluid Dynamics. By G. K. Batchelor. Pp. 615. 75s. (Cambridge.)

TL;DR: In this paper, the Navier-Stokes equation is derived for an inviscid fluid, and a finite difference method is proposed to solve the Euler's equations for a fluid flow in 3D space.

Abstract: This brief paper derives Euler’s equations for an inviscid fluid, summarizes the Cauchy momentum equation, derives the Navier-Stokes equation from that, and then talks about finite difference method approaches to solutions. Typical texts for this material are apparently Acheson, Elementary Fluid Dynamics and Landau and Lifschitz, Fluid Mechanics. 1. Basic Definitions We describe a fluid flow in three-dimensional space R as a vector field representing the velocity at all locations in the fluid. Concretely, then, a fluid flow is a function ~v : R× R → R that assigns to each point (t, ~x) in spacetime a velocity ~v(t, ~x) in space. In the special situation where ~v does not depend on t we say that the flow is steady. A trajectory or particle path is a curve ~x : R→ R such that for all t ∈ R, d dt ~x(t) = ~v(t, ~x(t)). Fix a t0 ∈ R; a streamline at time t0 is a curve ~x : R→ R such that for all t ∈ R, d dt ~x(t) = ~v(t0, ~x(t)). In the special case of steady flow the streamlines are constant across times t0 and any trajectory is a streamline. In non-steady flows, particle paths need not be streamlines. Consider the 2-dimensional example ~v = [− sin t cos t]>. At t0 = 0 the velocities all point up and the streamlines are vertical straight lines. At t0 = π/2 the velocities all point left and the streamlines are horizontal straight lines. Any trajectory is of the form ~x = [cos t + C1 sin t + C2] >; this traces out a radius-1 circle centered at [C1 C2] >. Indeed, all radius-1 circles in the plane arise as trajectories. These circles cross each other at many (in fact, all) points. If you find it counterintuitive that distinct trajectories can pass through a single point, remember that they do so at different times. 2. Acceleration Let f : R × R → R be some scalar field (such as temperature). Then ∂f/∂t is the rate of change of f at some fixed point in space. If we precompose f with a 1 Fluid Dynamics Math 211, Fall 2014, Carleton College trajectory ~x, then the chain rule gives us the rate of change of f with respect to time along that curve: D Dt f := d dt f(t, x(t), y(t), z(t)) = ∂f ∂t + ∂f ∂x dx dt + ∂f ∂y dy dt + ∂f ∂z dz dt = ( ∂ ∂t + dx dt ∂ ∂x + dy dt ∂ ∂y + dz dt ∂ ∂z ) f = ( ∂ ∂t + ~v · ∇ ) f. Intuitively, if ~x describes the trajectory of a small sensor for the quantity f (such as a thermometer), then Df/Dt gives the rate of change of the output of the sensor with respect to time. The ∂f/∂t term arises because f varies with time. The ~v ·∇f term arises because f is being measured at varying points in space. If we apply this idea to each component function of ~v, then we obtain an acceleration (or force per unit mass) vector field ~a(t, x) := D~v Dt = ∂~v ∂t + (~v · ∇)~v. That is, for any spacetime point (t, ~x), the vector ~a(t, ~x) is the acceleration of the particle whose trajectory happens to pass through ~x at time t. Let’s check that it agrees with our usual notion of acceleration. Suppose that a curve ~x describes the trajectory of a particle. The acceleration should be d dt d dt~x. By the definition of trajectory, d dt d dt ~x = d dt ~v(t, ~x(t)). The right-hand side is precisely D~v/Dt. Returning to our 2-dimensional example ~v = [− sin t cos t]>, we have ~a = [− cos t − sin t]>. Notice that ~v · ~a = 0. This is the well-known fact that in constant-speed circular motion the centripetal acceleration is perpendicular to the velocity. (In fact, the acceleration of any constant-speed trajectory is perpendicular to its velocity.) 3. Ideal Fluids An ideal fluid is one of constant density ρ, such that for any surface within the fluid the only stresses on the surface are normal. That is, there exists a scalar field p : R × R → R, called the pressure, such that for any surface element ∆S with outward-pointing unit normal vector ~n, the force exerted by the fluid inside ∆S on the fluid outside ∆S is p~n ∆S. The constant density condition implies that the fluid is incompressible, meaning ∇ · ~v = 0, as follows. For any region of space R, the rate of flow of mass out of the region is ∫∫ ∂R ρ~v · ~n dS = ∫∫∫