Spherical Particle of Radius - Mathematics Assignment Help

Assignment Task:

Task:

This assignment primarily involves plotting, so you can choose to submit your plots, your codes etc. online.

Links will be available in moodle website for online submission.
Only, appropriately named single *.pdf files will be accepted for online submission.
The file should also have adequate description of what the plot is - for example, titles, legends etc.

(TAsj would be happy to give a tutorial in doing this problem sheet, for
e.g.: how to plot and so on. So, if you would like to have one, let us know.)

(Dated: January 29, 2021)

1. For a spherical particle of radius a, translating with a velocity U in an otherwise quiescent fluid at Re=0 we have calculated, in the class, the velocity field in the fluid around the particle. This velocity field can be expressed as

ui = u point i + u f inite i

(1)

where

u point i = 3 4 Uj aδij r + axixj r 3

(2) u f inite i = 1 4 Uj a 3 δij r 3 − 3a 3xixj r 5

(3)
We would like to see how these velocity fields look like and the contributions by each of them (by Eq. 2 and Eq. 3). Therefore plot (i) u point and (ii) u point + uf inite. You can make a single plot and see how much is the contribution from uf inite. You can either plot velocity vectors or stream lines or both.

For convenience you may restrict the plotting to a cross sectional plane that contains the velocity vector of the particle and the centre of the particle. It would be easier if you assume the particle velocity to be along one of the principal coordinate directions (say along ˆx direction) and plot the velocity field in x − y plane. Since the velocity field is axisymmetric you will get the same solution on any other such planes.

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