Vector Laplacian in spherical coordinates: on one uncorrected misprint in “Fluid mechanics” by L.D. Landau and E.M. Lifshitz

The problems of acoustic physics describing the propagation of acoustic effects or associated flows rely on the equations of hydroaeromechanics, in particular, on the Navier–Stokes equations containing the vector Laplacian of the velocity of a liquid or gaseous medium. One of the most commonly used coordinate systems is spherical one. The paper discusses a fundamental misprint in “Fluid mechanics” by L.D. Landau and E.M. Lifshitz (Course of Theoretical Physics, Volume VI), which has been replicated up to the 6th revised edition of 2021, and which also made it into the English version of the Course published by Pergamon. It concerns the Navier-Stokes equation in spherical coordinates (15.21), which describes the r-component of the vector Laplacian. A typo migrates to secondary publications and can complicate theoretical studies related to the application of the Navier–Stokes equations, which the author of this note has personally encountered. The r-component of the vector Laplacian is derived in detail from general principles to show what the corresponding Navier–Stokes equation should look like and that the typo is indeed present. In addition, more compact equivalent forms of the Navier–Stokes equations in spherical coordinates are proposed.