Continuity Equation - Differential Form
- The point at which the continuity equation has to be derived, is enclosed by an elementary control volume.
- 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
|(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
This is the Equation of Continuity for a compressible fluid in a rectangular cartesian coordinate system.