Derivation of continuity equation in spherical coordinates?

Answer 1

Derivation of the Continuity Equation in Spherical Coordinates

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.

AccumulationThe 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.

Radial FlowStarting 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

but

where

and

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

where

The outflow in the θ direction is

where

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

with

while the outflow is

and

At the outset, the net flow in the polar direction is

Continuity EquationNow, 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