Chapter 8 : Viscous Incompressible Flows
Lecture 24 :


Vector Notation & derivation in Cylindrical Coordinates - Navier-Stokes equation

  • Using, vector notation to write Navier-Stokes and continuity equations for incompressible flow we have
(24.21)

and

(24.22)
  • we have four unknown quantities, u, v, w and p ,
  • we  also have four equations, - equations of motion in three directions and the continuity equation.
  • In principle, these equations are solvable but to date generalized solution is not available due to the complex nature of the set of these equations.
  • The highest order terms, which come from the viscous forces, are linear and of second order
  •  The first order convective terms are non-linear and hence, the set is termed as quasi-linear.
  • Navier-Stokes equations in cylindrical coordinate (Fig. 24.2) are useful in solving many problems. If  , and $ V_{z}$ denote the velocity components along the radial, cross-radial and axial directions respectively, then for the case of incompressible flow, Eqs (24.21) and (24.22) lead to the following system of equations:

FIG 24.2 Cylindrical polar coordinate and the velocity components

 

 

 (24.23a)

 

 (24.23b)

 

 (24.23c)
(24.24)


               
                          

            End of Lecture 24!

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