Dimensional Analysis for Fluids

Introducing fluid dynamics using dimensional analysis Jens Højgaard Jensen Citation: Am. J. Phys. 81, 688 (2013); doi: 10.1119/1.4813064 View online: http://dx.doi.org/10.1119/1.4813064 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v81/i9 Published by the American Association of Physics Teachers Related Articles Towards the Kelvin wake and beyond Am. J. Phys. 81, 245 (2013) Why do bubbles in Guinness sink? Am. J. Phys. 81, 88 (2013) The Electric Whirl in the 19th and 21st Centuries Phys. Teach. 50, 536 (2012) Beyond the Van Der Waals loop: What can be learned from simulating Lennard-Jones fluids inside the region of phase coexistence Am. J. Phys. 80, 1099 (2012) The Enigma of the Aerofoil: Rival Theories in Aerodynamics, 1909–1930. Am. J. Phys. 80, 649 (2012) Additional information on Am. J. Phys. Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html Downloaded 10 Sep 2013 to 129.25.7.39. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission Introducing fluid dynamics using dimensional analysis Jens Højgaard Jensena) IMFUFA and “Glass and Time,” Department of Sciences, Roskilde University, DK-4000 Roskilde, Denmark (Received 25 October 2012; accepted 21 June 2013) Many aspects of fluid dynamics can be introduced using dimensional analysis, combined with some basic physical principles. This approach is concise and allows exploration of both the laminar and turbulent limits—including important phenomena that are not normally dealt with when fluid dynamics is first introduced as a precise mathematical discipline. In this paper, we use dimensional analysis to understand drag forces on bodies, flow speeds through tubes, and lift forces. VC 2013 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4813064] I. INTRODUCTION arguments. In Sec. VI, we discuss how the usual mathematical way of presenting fluid dynamics has overshadowed the In this article, we describe a method, using dimensional more intuitive approach of Secs. III–V. Finally, in Sec. VII, analysis, for introducing fluid dynamics to undergraduate the pros and cons of the approach are discussed. physics students. Dimensional analysis, which emphasizes thinking more than memorizing, is not commonly used in in- 1–6 II. DIMENSIONAL ANALYSIS troductory textbooks when discussing fluids, even though such as approach facilitates a clearer understanding of the In recognition of the centenary of its introduction, we will effects of turbulence when considering drag and lift forces, use the Bohr model of the hydrogen atom as an example to and flow resistance in tubes. introduce the method of dimensional analysis. Fluid dynamics is a huge and difficult subject, with a long When presenting his model in 1913, Niels Bohr gave a history, not only in the field of physics but also in the fields dimensional analysis argument in the introduction of his arti- of mathematics and engineering. In spite of this, a complete cle.10 The classical mechanical problem of calculating the understanding of turbulence remains an unsolved problem, orbits of the electron around the proton involves only two 7 as several famous physicists have noted. input variables: the mass of the electron and the constant 2 Turbulence is common in daily life. The transition from kC ¼ e =4p0 that determines the strength of the force laminar to turbulent flow is seen, for example, in smoke ris- between the two particles. But these two input variables can- ing from a cigarette. Turbulence affects the air resistance act- not be combined into a physical quantity with dimensions of 8 ing on a bicyclist and on falling objects such as the coffee length, needed to explain the size of the atom. On the other 9 filters or muffin forms often used in physics teaching. In hand, when Planck’s constant is introduced as a third vari- these cases, the air resistance is proportional to the square of able, dimensional analysis gives a characteristic length—the the object’s velocity, and this quadratic dependence on ve- Bohr radius—that agrees with the known order of magnitude locity, in incompressible fluids, is connected to turbulence. of the size of the atom. Let us see how. Nevertheless, in physics instruction drag forces are often We stress that the symbols used in physics formulas refer assumed to be linear in the velocity (as is the case for lami- to physical quantities—not merely to the numerical values of nar flow) rather than quadratic, even though the latter is these quantities in a particular system of units. For example, more commonly encountered. if we write d1 ¼ 2d2, to indicate that one distance is twice as Within the paradigm of fluid dynamics as an exact mathe- large as another, we are making a statement that is independ- matical discipline, drag forces proportional to velocity ent of the particular units and numbers used to specify d1 and (Stokes’ law) and other phenomena (e.g., Poiseuille’s law for d2. the flow speed through tubes) in the laminar flow regime can We also note that it is not possible to add quantities of dif- be derived. But due to the nonlinearity of the fundamental ferent dimension—for example, a length and a mass. On the equations, this approach does not give results in the turbulent other hand, quantities with different dimensions can be mul- limit. In such situations dimensional analysis is especially tiplied and divided to obtain quantities with new dimensions useful, in that much of the behavior can be revealed without (for example, when we calculate a density from a mass di- actually solving the nonlinear equations. vided by a length cubed). In general, if a physical formula In Sec. II, we give a brief introduction to dimensional expresses some output quantity Q1 in terms of several input analysis, meant for readers unfamiliar with the method. In quantities Q2, Q3,…, then this formula must take the form Secs. III and IV, we estimate drag forces on bodies and flow speeds through tubes in both the turbulent and laminar limits a b c Q1 ¼ g Á Q2 Á Q3 Á Q4…; (1) using dimensional analysis combined with work/energy-flow arguments. Instead of comparing the relative strengths of the where the powers a, b, c,… yield the correct dimensions for terms (with dimensions of force) in the Navier-Stokes’ equa- Q1 and the dimensionless factor g can be either a pure num- tion, as is usually done in introductory textbooks on fluid dy- ber or a function of dimensionless power-law combinations 11 namics, we ask whether the work done by driving forces is of Q2, Q3, …, if such dimensionless combinations exist. converted primarily into thermal energy or primarily into Returning to the Bohr atom, we denote the Bohr radius a0, macroscopic kinetic energy. In Sec. V, we calculate lift the mass of the electron me, and write the Coulomb force 2 forces in both the turbulent and laminar limits using dimen- between the electron and the proton as kC=r . We will use sional analysis combined with force/momentum-flow square brackets to denote the dimensions of a quantity, and 688 Am. J. Phys. 81 (9), September 2013 http://aapt.org/ajp VC 2013 American Association of Physics Teachers 688 Downloaded 10 Sep 2013 to 129.25.7.39. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission the symbols L, T, and M to denote the fundamental mechani- It is not common for physicists to follow Niels Bohr and cal dimensions length, time, and mass, respectively. We can present dimensional analysis in their formal writings. But 3 À2 then write ½a0¼L; ½me¼M, and ½kC¼M Á L Á T (note dimensional analysis is a powerful tool that physicists often that the final expression is equivalent to force times L2). use before developing a more formal line of reasoning. Now suppose that a0 were a function of only me and kC. Solving problems using dimensional analysis is also a good Then, following the form of Eq. (1), we could write approach for teaching students how to think like a physicist.11 a b a0 ¼ g ðmeÞ ðkCÞ (2) III. DRAG for some a and b. Taking the dimensions of both sides of this equation gives We now turn to fluid dynamics, and use dimensional analysis to calculate the drag force on a body moving through a L ¼ Ma ðM Á L3 Á TÀ2Þb ¼ Maþb Á L3b Á TÀ2b: (3) fluid. For motion through air, drag is often called air resistance. But the formulas we derive below are valid for the Since none of the fundamental dimensions L, T, and M can motion of bodies in both gases and liquids. We consider the be written in terms of the others, the powers of each dimen- steady motion of a body pulled though a fluid, as sketched in sion must match up separately between the two sides. But Fig. 1. The drag D is the force exerted by the fluid in the this is impossible—matching powers of L implies b ¼ 1=3, direction opposite to the motion. If the motion is steady, the while matching powers of T implies b ¼ 0. drag has the same magnitude as the pulling force. Bohr’s dimensional analysis thus shows that classical The first thing to do is to ask which physical quantities the mechanics cannot produce a characteristic atomic length. drag can depend on. The pulling force and the drag depend But including Planck’s constant (divided by 2p), which has on the velocity v of the body; they also depend on the shape dimension ½h¼M Á L2 Á TÀ1, changes the situation.

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