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Geosciences LibreTexts

3: Flow Past a Sphere II - Stokes' Law, The Bernoulli Equation, Turbulence, Boundary Layers, Flow Separation

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    • 3.1: Introduction and the Navier-Stokes Equation
      An equation of motion for a viscous fluid when the forces acting in or on the fluid are those of viscosity, gravity, and pressure, is called the Navier–Stokes equation. The Navier–Stokes equation is notoriously difficult to solve in a given flow problem to obtain spatial distributions of velocities and pressures and shear stresses. Basically the reasons are that the acceleration term is nonlinear, meaning that it involves products of partial derivatives, and the viscous-force term contains secon
    • 3.2: Flow Past a Sphere at Low Reynolds Numbers
      We will make a start on the flow patterns and fluid forces associated with flow of a viscous fluid past a sphere by restricting consideration to low Reynolds numbers.
    • 3.3: Inviscid Flow
      Over the past hundred and fifty years a vast body of mathematical analysis has been devoted to a kind of fluid that exists only in the imagination: an inviscid fluid, in which no viscous forces act. This fiction (in reality there is no such thing as an inviscid fluid) allows a level of mathematical progress not possible for viscous flows, because the viscous-force term in the Navier–Stokes equation disappears, and the equation becomes more tractable.
    • 3.4: The Bernoulli Equation
      In the example of inviscid flow past a sphere described in the preceding section, the pressure is high at points where the velocity is low, and vice versa. It is not difficult to derive an equation, called the Bernoulli equation, that accounts for this relationship.
    • 3.5: Turbulence
      Most of the fluid flows of interest in science, technology, and everyday life are turbulent flows—although there are many important exceptions to that generalization.  Turbulence might be loosely defined as an irregular or random or statistical component of motion that under certain conditions becomes superimposed on the mean or overall motion of a fluid when that fluid flows past a solid surface or past an adjacent stream of the same fluid with different velocity.
    • 3.6: Boundary Layers
      A boundary layer is the zone of flow in the immediate vicinity of a solid surface or boundary in which the motion of the fluid is affected by the frictional resistance exerted by the boundary. The no-slip condition requires that the velocity of fluid in direct contact with solid boundary be exactly the same as the velocity of the boundary; the boundary layer is the region of fluid next to the boundary across which the velocity of the fluid grades to that of the unaffected part of the flow.
    • 3.7: Flow Separation
    • 3.8: Flow Past a Sphere at High Reynolds Numbers
    • 3.9: Settling of Spheres
      This section deals with some basic ideas about settling of solid spheres under their own weight through still fluids. This is an important topic in meteorology (hailstones), sedimentology (sediment grains), and technology (cannon balls and spacecraft). In this section we will look at the terminal settling velocity of spheres as an applied problem. At the end I will make some comments about the complicated matter of the time and distance it takes for a sphere to attain its terminal settling veloc