1. Limit of the Validity of the Exact Solution for Creeping Flow Around a Sphere As discussed in class, the the exact solution of creeping flow around a sphere of radius R is valid only wheen the Reynold number Re 2Rpv/µ < 1. As long as R is small, Re can be held sufficiently small near the surface in order for the assumption to be valid. We wish to understand whether at large distances r from the sphere the solution is still valid. In order to do so we estimate the orders of magnitude of the convective (inertial) and viscous contributions to the fluid flow. (a) Given the exact solution, estimate the order of magnitude of the convective term in the momentum equation, |v · Vv|. (b) Repeat (a), but for the viscous term, µV?v|. (c) Form the ratio of (a) and (b). For the assumption of creeping flow to be valid, the ratio must remain small. Show that for a certain value of the radial distance r and larger the assumption breaks down. Identify that value.
1. Limit of the Validity of the Exact Solution for Creeping Flow Around a Sphere As discussed in class, the the exact solution of creeping flow around a sphere of radius R is valid only wheen the Reynold number Re 2Rpv/µ < 1. As long as R is small, Re can be held sufficiently small near the surface in order for the assumption to be valid. We wish to understand whether at large distances r from the sphere the solution is still valid. In order to do so we estimate the orders of magnitude of the convective (inertial) and viscous contributions to the fluid flow. (a) Given the exact solution, estimate the order of magnitude of the convective term in the momentum equation, |v · Vv|. (b) Repeat (a), but for the viscous term, µV?v|. (c) Form the ratio of (a) and (b). For the assumption of creeping flow to be valid, the ratio must remain small. Show that for a certain value of the radial distance r and larger the assumption breaks down. Identify that value.
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Transcribed Image Text:1. Limit of the Validity of the Exact Solution for Creeping Flow Around a Sphere
As discussed in class, the the exact solution of creeping flow around a sphere of radius
R is valid only wheen the Reynold number Re 2Rpv/µ < 1. As long as R is small,
Re can be held sufficiently small near the surface in order for the assumption to be valid.
We wish to understand whether at large distances r from the sphere the solution is still
valid. In order to do so we estimate the orders of magnitude of the convective (inertial)
and viscous contributions to the fluid flow.
(a) Given the exact solution, estimate the order of magnitude of the convective term in
the momentum equation, |v · Vv|.
(b) Repeat (a), but for the viscous term, µV?v|.
(c) Form the ratio of (a) and (b). For the assumption of creeping flow to be valid, the
ratio must remain small. Show that for a certain value of the radial distance r and larger
the assumption breaks down. Identify that value.
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