Derivation of continuity equation in spherical coordinates? - Answers

Derivation of the Continuity Equation in Spherical CoordinatesWe start by selecting a spherical control volume dV. As shown in the figure below, this is given bywhere 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 isWe will represent the velocity field viaIn 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 byThe 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 havesubstitution into Ain yieldswhere high order terms have been dropped.The outflow in the radial direction isbutwhereandBy only keeping the lowest (second & third) order terms in the resulting expression, we haveNote, 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 byPolar Flow (θ)The inflow in the polar direction is whereThe outflow in the θ direction iswhereUpon expansion, and keeping both second and third order terms, we getFinally, the net flow in the polar direction isAzimuthal Flow (φ)The inflow in the azimuthal direction is given by withwhile the outflow isandAt the outset, the net flow in the polar direction isContinuity EquationNow, by collecting all mass fluxes we have which, upon dividing by dV and combining terms, reduces towhich is the continuity equation in spherical coordinates