Continuity Equation - Differential Form

    Derivation
  1. The point at which the continuity equation has to be derived, is enclosed by an elementary control volume.
  2. The influx, efflux and the rate of accumulation of mass is calculated across each surface within the control volume.


Fig 9.6   A Control Volume Appropriate to a Rectangular Cartesian Coordinate System

Consider a rectangular parallelopiped in the above figure as the control volume in a rectangular cartesian frame of coordinate axes.
  • Net efflux of mass along x -axis must be the excess outflow over inflow across faces normal to x -axis.
  • Let the fluid enter across one of such faces ABCD with a velocity u and a density ρ.The velocity and density with which the fluid will leave the face EFGH will be   and  respectively (neglecting the higher order terms in δx).
  • Therefore, the rate of mass entering the control volume through face ABCD = ρu dy dz. 
  • The rate of mass leaving the control volume through face EFGH will be
 

 (neglecting the higher order terms in dx) 

  • Similarly influx and efflux take place in all y and z directions also.
  • Rate of accumulation for a point in a flow field
 
  • Using, Rate of influx = Rate of Accumulation + Rate of Efflux
 

  • Transferring everything to right side
 
(9.2)

     This is the Equation of Continuity for a compressible fluid in a rectangular cartesian coordinate system.

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