On the convection of heat from small cylinders in a stream of fluid: Determination of the convection constants of small platinum wires, with applications to hot-wire anemometry

Abstract

Sections 1 and 2.- Until comparatively recently, the problem of solving the equations of heat conduction in the case of a solid cooled by a stream of fluid had received little attention, although the general problem was formulated by Fourier himself as long ago as 1820. In 1901 the problem was taken up by Boussinesq, and many cases were dealt with in his memoir of 1905. By means of an extremely elegant transformation Boussinesq was able to express the general equation for the two-dimensional problem in a linear form: by transforming the equation to the set of orthogonal curvilinear co-ordinates determined by the stream-lines and equipotentials of the hydrodynamical problem of the flow of a uniform stream of velocity V past the cylindrical obstacle, the equation for the temperature θ at any point of the fluid takes the form ∂ 2 θ /∂ α 2 + ∂ 2 θ /∂ β 2 = 2 n ∂ θ / ∂β , (1) where the curves α = constant represent the stream-lines and β = constant the equipotentials. The constant n is given by the relation 2 n = c V/ k = s σV/ k , where c is the specific heat of the fluid per unit volume, s that per unit mass, σ its density, and k its thermal conductivity. If the surface of the cylinder be the particular stream-line α = 0, and the critical equipotentials be the curves β =o and β = β 0 , the heat-flux per unit length of the cylinder is given by H = -∫ β 0 0 k ∂ θ /∂ α 0 dβ . (2) where the integral is taken to include the two branches of the stream-lines α = 0.