Please Make a Note is a collection of science & technology tips and derivations that will make it easier for research scientists & engineers to perform the various tasks they are faced with. These notes cover a wide range of scientific topics, software, media, and data analysis utilities.
We start by selecting a spherical control volume dV. As shown in the figure below, this is given by
where r, θ, and φ stand for the radius, polar, and azimuthal angles, respectively. The azimuthal angle is also referred to as the zenith or colatitude angle.
The differential mass is
We will represent the velocity field via
In an Eulerian reference frame mass conservation is represented by accumulation, net flow, and source terms in a control volume.
The accumulation term is given by the time rate of change of mass. We therefore have
The net flow through the control volume can be divided into that corresponding to each direction.
Starting with the radial direction, we have
The inflow area Ain is a trapezoid whose area is given by
The key term here is the sine term. Note that the mid segment is the average of the bases (parallel sides). Upon expansion of Ain, and in the limit of vanishing dθ, we have
substitution into Ain yields
where high order terms have been dropped.
The outflow in the radial direction is
By only keeping the lowest (second & third) order terms in the resulting expression, we have
Note, that in the expression for Aout, we kept both second order and third order terms. The reason for this is that this term will be multiplied by "dr" and therefore, the overall order will be three. In principle, one must carry all those terms until the final substitution is made, and only then one can compare terms and keep those with the lowest order.
At the outset, the net flow in the radial direction is given by
Polar Flow (θ)
The inflow in the polar direction is
The outflow in the θ direction is
Upon expansion, and keeping both second and third order terms, we get
Finally, the net flow in the polar direction is
Azimuthal Flow (φ)
The inflow in the azimuthal direction is given by
while the outflow is
At the outset, the net flow in the polar direction is
Now, by collecting all mass fluxes we have
which, upon dividing by dV and combining terms, reduces to
which is the continuity equation in spherical coordinates.