Continuity Equation- Cylindrical Polar Coordinate System

    The continuity equation in any coordinate system can be derived in either of the two ways:-

1. By expanding the vectorial form of general continuity equation, Eq. (9.3) with respect to the particular coordinate system.

2. By considering an elemental control volume appropriate to the reference frame of coordinates system and then by applying the fundamental principle of conservation of mass.

Fig 9.7   A Cylindrical Polar Coordinate System

     First Approach:-

     The term in a cylindrical polar coordinate system (Fig. 9.7) can be written as

(9.9)

Therefore, the equation of continuity in a cylindrical polar coordinate system can be written as

(9.10)

     Second Approach:-

     Consider the mass fluxes in the control volume shown in Fig. 9.8.

Fig 9.8   A control volume appropriate to a cylindrical polar coordinate system

• Rate of mass entering the control volume through face
  

• Rate of mass leaving the control volume through the face
  

• Therefore, the net rate of mass efflux in the r direction =    where      (the elemental volume)
  

• The net rate of mass efflux from control volume in θ direction   = (Mass leaving through face ADHE) - (Mass entering through face BCGF)
  
• It can be written as
  
• The net rate of mass efflux in z direction can be written in a similar fashion as
  
• The rate of increase of mass within the control volume becomes
  
  
• Hence, the final form of continuity equation in a cylindrical polar coordinate system becomes

or,

• In case of an incompressible flow,

(9.11)

• The equation of continuity in a spherical polar coordinate system  can be written by expanding the term of Eq. (9.3) as

(9.12)

• For an incompressible flow, Eq. 9.12 reduces to

(9.13)