This simulation generates a vector field around a sphere with diameter \( d = 1 \; \mathrm{ m } \). One of three fluids can be selected from the dropdown menu. The slider changes the fluid velocity. The vector field automaticaly updates if a parameter is changed. The fluid velocity may change when a new fluid is selected in order to keep the Reynold's number within a valid range.

The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluids. They correspond to conservation of momentum and mass¹. For an incompressible fluid and steady-state flow with radial symmetry about the x-axis, the Navier-Stokes equation in spherical coordinates is^2,3,4

$$ \frac{ Re }{ 2 } \left[ \frac{ \partial \Psi }{ \partial r } \frac{ \partial }{ \partial \theta } \left( \frac{ E^{2} \Psi }{ r^{2} \mathrm{ sin }^{ 2 } \theta } \right) - \frac{ \partial \Psi }{ \partial \theta } \frac{ \partial }{ \partial r } \left( \frac{ E^{2} \Psi }{ r^{2} \mathrm{ sin }^{ 2 } \theta} \right) \right] \mathrm{ sin } \theta = E^{ 4 } \Psi $$

where \( r \) is radius, \( \theta \) is the polar angle, \( \Psi \) is the stream function, \( Re \) is the Reynolds number and \( E \) is

$$ E^{ 2 } = \frac{ \partial^{ 2 } }{ \partial r^{ 2 } } + \frac{ \mathrm{ sin } \theta }{ r^{ 2 } } \frac{ \partial }{ \partial \theta } \left( \frac{ 1 }{ \mathrm{ sin } \theta } \frac{ \partial }{ \partial \theta } \right) $$

The Reynolds number is the ratio of inertial forces to viscous forces:

$$ Re = \frac{ \rho V D }{ \mu } $$

where \( \rho \) is the density of the fluid, \( V \) is the fluid velocity at \( r = \infty \), \( D \) is the sphere diameter, and \( \mu \) is the dynamic vicosity of the fluid. The radial velocity \( V_{ r } \) and tangential velocity \( V_{ \theta } \) at coordinate \( ( r, \theta ) \) are

$$ V_{ r } = - \frac{ 1 }{ r^{ 2 } \; \mathrm{ sin } \theta } \frac{ \partial \Psi }{ \partial \theta } $$ $$ V_{ \theta } = \frac{ 1 }{ r \; \mathrm{ sin } \theta } \frac{ \partial \Psi }{ \partial r } $$

The boundary conditions for flow around a sphere are

$$ \Psi = \frac{ 1 }{ 2 } r^{ 2 } \mathrm{ sin }^{ 2 } \theta, \; \; r \rightarrow \infty $$ $$ \Psi = \frac{ \partial \Psi }{ \partial r } = 0, \; \; r = R $$

where \( R \) is the radius of the sphere (0.5 meters in this simulation). The numerical Galerkin method used in this simulation approximates the stream function \( \Psi \) as a polynomial in which coefficients are determined experimentally. This solution is only valid for the laminar flow regime \( 0 \leq Re \leq 500 \), with error increasing as Reynold's number increases^3,4.

References:

1. n.a. (2021) Navier–Stokes Equations. Wikipedia.com. https://en.wikipedia.org/wiki/Navier–Stokes_equations
2. Kawaguti, M. (1955). The critical Reynolds number for the flow past a sphere. Journal of the Physical Society of Japan, 10(8), 694-699.
3. Hamielec, A. E., Hoffman, T. W., & Ross, L. L. (1967). Numerical solution of the Navier‐Stokes equation for flow past spheres: Part I. Viscous flow around spheres with and without radial mass efflux. AIChE Journal, 13(2), 212-219.
4. Hamielec, A. E., Johnson, A. I., & Houghton, W. T. (1967). Numerical solution of the Navier‐Stokes equation for flow past spheres: Part II. Viscous flow around circulating spheres of low viscosity. AIChE Journal, 13(2), 220-224.

This simulation was created in the Department of Chemical and Biological Engineering, at University of Colorado Boulder for LearnChemE.com by Mingyuan Lu and Neil Hendren under the direction of Professor John L. Falconer. It is a JavaScript/HTML5 implementation of a simulation by M.D. Mikhailov¹. Address any questions or comments to learncheme@gmail.com. All of our simulations are open source, and are available on our LearnChemE Github repository.

If this simulation is too big for your screen, zoom out using  +  on Mac or  +  on Windows. To zoom in, use  +  on Mac or  +  on Windows.

References:

1. Mikhailov, M.D. (2013). Flow around a Sphere at Finite Reynolds Number by Galerkin Method. A Wolfram Demonstrations Project.
2. Viscosity and density of honey: Nayik, Gulzar & Dar, B. & Nanda, Vikas. (2015). "Physico-chemical, rheological and sugar profile of different honeys from Kashmir Valley of India". Arabian Journal of Chemistry. 12. 10.1016/j.arabjc.2015.08.017.
3. Viscosity and density of water: Gerhart, P. M., Gerhart, A. L., & Hochstein, J. I. (2016). Munson, Young and Okiishi's Fundamentals of Fluid Mechanics. John Wiley & Sons.