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XLIX. The Equations of Motion of a Viscous Fluid. By 

 Gr. B. Jeitery, M.A., B.Sc, Assistant in the Department 

 of Applied Mathematics, University College, London *. 



THE transformation of the equations of elasticity to curvi- 

 linear coordinates has been discussed by Lame and 

 others, but the analogous equations for the motion of a 

 viscous fluid do not appear to have attracted the same 

 attention. The first section of this paper deals with the 

 transformation of the equations of motion. The terms in 

 the equations which express the resultant force on an element 

 of fluid are mutatis mutandis identical with the corresponding 

 terms in the equations for an elastic solid, but they are given 

 in a form which I believe to be new, and which lends itself 

 more readily to applications to particular systems of co- 

 ordinates. The terms which express the accelerations of the 

 fluid are different owing to the different assumptions which 

 underlie the two theories. The second section is devoted 

 to a discussion of the components of stress in curvilinear 

 coordinates. The theory is illustrated by applications to 

 cylindrical and spherical polar coordinates. In the remaining 

 two sections we discuss the special cases of axial and plane 

 motion respectively. 



§ 1, Transformation of the Equation of Motion. 



If ?i, w, w, X, Y, Z, are the components of velocity and 

 body force respectively, p the mean pressure, p the density, 

 and fju the coefficient of viscosity, the equations of motion of 

 a viscous fluid are f 



and two similar equations. 



These three equations may be replaced by a single vector 

 equation. If v be the velocity and F the externally applied 

 force, then using the relation 



curl. curl v = grad. div. v— V 2 v J, 



* Communicated by Prof. Karl Pearson, F.P.S. 

 f Lamb, ' Hydrodynamics/ p. 688. 

 X In the usual vector notation 



curl, curl v = [v[v . ▼]] 



= v(vv)-v' : v 



= arrad. div. v - v-v. 



