An analytical solution to the Navier–Stokes equation for incompressible flow around a solid sphere

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Ahmad Talaei, Timothy J. Garrett


This paper is concerned with obtaining a formulation for the flow past a sphere in a viscous and incompressible fluid, building upon previously obtained well-known solutions that were limited to small Reynolds numbers. Using a method based on a summation of separation of variables, we develop a general analytical solution to the Navier--Stokes equation for the special case of axially symmetric two-dimensional flow around a sphere. For a particular set of mathematical conditions, the solution can be expressed generally as a hypergeometric function. It reproduces streamlines and flow velocities close to a moving sphere, and provides the angular location immediately behind the sphere where there is a separation between laminar flow and a stagnant region. To produce eddies around a fast-moving sphere, we present a solution obtained using a variable substitution that does not require the separation of variables and is a function of Bessel functions of the first and second kind. For particular boundary conditions, it exhibits eddies behind a fast-moving sphere.



Applied Mathematics, Earth Sciences, Engineering, Fluid Dynamics, Mechanical Engineering, Other Mechanical Engineering, Partial Differential Equations, Physical Sciences and Mathematics, Physics, Special Functions


angle of separation, associated Legendre function of the first kind, Bessel functions of the first kind, Bessel functions of the second kind, hypergeometric function, Legendre functions of the first kind, Legendre functions of the second kind, modified Bessel functions of the first kind, modified Bessel functions of the second kind, Navier–Stokes equation, partial differential equation, stream function


Published: 2020-08-25 21:54

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GNU Lesser General Public License (LGPL) 2.1

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In this paper, we provide an analytical solution for the flow surrounding a moving sphere in a viscous and incompressible fluid. Previous analytical solutions have been limited to particle Reynolds numbers Re<1, or to approximate solutions for Re<100. We provide two solutions to the Navier–Stokes equation. The first analytical solution extends previously well-known work by applying a summation of the separation of variables technique. The solution is expressed as a function of a hypergeometric function and reproduces streamlines and velocities around a sphere moving with high Reynolds numbers that are qualitatively consistent with the experiment. It also provides known angular locations on the surface of the sphere where a stagnant boundary layer separates from the main streamlines. The solution, however, is unable to reproduce vortices and eddies around a fast-moving sphere due to the simplification employed for the radial functionality of the stream function. To reproduce these phenomena, we adopt a change of variable to yield a solution that is a function of Bessel functions of the first and second kind. Graphical representation of the solution for particular boundary conditions simulates a train of vortices behind a sphere moving with high Reynolds numbers.