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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 6.055 / Art of approximation 72 8.3 Drag Pendulum motion is not a horrible enough problem to show the full benefit of dimensional analysis. Instead try ﬂuid mechanics – a subject notorious for its mathematical and physical complexity; Chandrasekhar’s books [ 10 , 11 ] or the classic textbook of Lamb [ 12 ] show that the mathematics is not for the faint of heart. The next examples illustrate two extremes of ﬂuid ﬂow: oozing and turbu Density ρ fl Viscosity ν ρ obj R v lent. An example of oozing ﬂow is ions transporting charge in seawater ( Section 8.3.6 ). An example of turbulent ﬂow is a raindrop falling from the sky after condens ing out of a cloud ( Section 8.3.7 ). To find the terminal velocity, solve the partialdifferential equations of ﬂuid mechanics for the incompressible ﬂow of a Newtonian ﬂuid: ∂ v 1 + ( v · ∇ ) v = − ρ ∇ p + ν ∇ 2 v , (3 eqns) ∂ t ∇ · v = . (1 eqn) Here v is the ﬂuid velocity, ρ is the ﬂuid density, ν is the kinematic viscosity, and p is the pressure. The first equation is a vector shorthand for three equa tions, so the full system is four equations. All the equations are partialdifferential equations and three are nonlinear. Worse, they are coupled: Quantities appear in more than one equation. So we have to solve a system of coupled, nonlinear, partialdifferential equations. This solution must satisfy boundary con ditions imposed by the marble or raindrop. As the object moves, the boundary conditions change. So until you know how the object moves, you do not know the boundary condi tions. Until you know the boundary conditions, you cannot find the motion of the ﬂuid or of the object. This coupling between the boundary conditions and solution compounds the difficulty of the problem. It requires that you solve the equations and the boundary conditions together. If you ever get there, then you take the limit t → ∞ to find the terminal velocity. Sleep easy! I wrote out the Navier–Stokes equations only to scare you into using dimen sional analysis and specialcases reasoning. The approximate approach is easier than solv ing nonlinear partialdifferential equations. 8.3.1 Naive dimensional analysis To use dimensional analysis, follow the usual steps: Choose relevant variables, form di mensionless groups from them, and solve for the terminal velocity. In choosing quantities, do not forget to include the variable for which you are solving, which here is v . To decide on the other quantities, split them into three categories (divide and conquer): 1. characteristics of the ﬂuid, 2. characteristics of the object, and 3. characteristics of whatever makes the object fall. 73 8 Special cases The last category is the easiest to think about, so deal with it first. Gravity makes the object The last category is the easiest to think about, so deal with it first....
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 Spring '08
 SanjoyMahajan
 Fluid Dynamics, Drag force, terminal velocity

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