Friday, September 10, 2010

Joseph Fourier

The August 2010 issue of Physics Today, published by the American Institute of Physics, contains an article by T. N. Narasimhan about Thermal Conductivity Through the 19th Century. A large part of the article deals with Joseph Fourier (1768-1830), the French physicist and mathematician. Russ Hobbie and I discuss Fourier’s mathematical technique of representing a periodic function as a sum of sines and cosines of different frequencies in Chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology. Interestingly, this far-reaching mathematical idea grew out of Fourier’s study of heat conduction and thermal conductivity. Russ and I introduce thermal conductivity in homework problem 15 of Chapter 4 about diffusion. This is not as odd as it sounds because, as shown in the problem, heat conduction and diffusion are both governed by the same partial differential equation, typically called the diffusion equation (Eq. 4.24). The concept of heat conduction is crucial when developing the bioheat equation (Chapter 14), which has important medical applications in tissue heating and ablation.

Narasimhan’s article provides some interesting insights into Fourier and his times.
“In 1802, upon his return to France from Napoleon’s Egyptian campaign, Fourier was appointed perfect of the department of Isere. Despite heavy administrative responsibilities, Fourier found time to study heat diffusion. He was inspired by deep curiosity about Earth and such phenomena as the attenuation of seasonal temperature variations in Earth’s subsurface, oceanic and atmospheric variations in Earth’s subsurface, oceanic and atmospheric circulation driven by solar heat, and the background temperature of deep space. […]

Thermal conductivity, appropriate for characterizing the internal conduction, was defined by Fourier as the quantity of heat per unit time passing through a unit cross-section divided by the temperature difference of two constant-temperature surfaces separated by unit distance […] Fourier presented his ideas in an unpublished 1807 paper submitted to the Institut de France.

Fourier was not satisfied with the 1807 work. It took him an additional three years to go beyond the discrete finite-difference description of flow between constant-temperature surfaces and to express heat flow across an infinitesimally thin surface segment in terms of the temperature gradient.

When Fourier presented his mathematical theory, the nature of heat was unknown […] Fourier considered mathematical laws governing the effects of heat to be independent of all hypotheses about the nature of heat. […] No method was available to measure flowing heat. Consequently, in order to demonstrate that his mathematical theory was physically credible, Fourier had to devise suitable experiments and methods to measure thermal conductivity.

It is not widely recognized that in his unpublished 1807 manuscript and in the prize essay he submitted to the Institut de France in 1811, Fourier provided results from transient and steady-state experiments and outlined methods to invert exponential data to estimate thermal conductivity. For some reason, he decided to restrict his 1822 masterpiece, The Analytical Theory of Heat, to mathematics and omit experimental results.”
For more insight on Fourier’s life and times, see Keston’s article Jospeh Fourier: Policitian and Scientist. It begins
“The life of Baron Jean Baptiste Joseph Fourier (1768 - 1830) the mathematical physicist has to be seen in the context of the French Revolution and its reverberations. One might say his career followed the peaks and troughs of the political wave. He was in turns: a teacher; a secret policeman; a political prisoner; governor of Egypt; prefect of Isère and Rhône; friend of Napoleon; and secretary of the Académie des Sciences. His major work, The Analytic Theory of Heat, (Théorie analytique de la chaleur) changed the way scientists think about functions and successfully stated the equations governing heat transfer in solids. His life spanned the eruption and aftermath of the Revolution; Napoleon's rise to power, defeat and brief return (the so-called Hundred Days); and the Restoration of the Bourbon Kings.”

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