G,5 • TRANSPIRATION-COOLED PIPE FLOW 



In the present treatment, exact solutions of the Navier-Stokes equations, 

 the continuity equation, and the energy equation in cyhndrical coordi- 

 nates are obtained. The assumptions made here are: (1) the fluid is in- 

 compressible, i.e. the mass density and the viscosity of the fluid are 

 assumed to be constant, (2) the fluid flowing in the axial direction and 

 the fluid flowing through the porous wall are assumed homogeneous, 

 (3) the maximum axial velocity at the entrance of the porous-wall pipe 

 is equal to the maximum axial velocity in the Poiseuille flow, (4) the 

 fluid flowing through the porous wall is uniform throughout, (5) the free 

 convection, radiation, and dissipation are neglected, and (6) the wall tem- 

 perature is constant and changes discontinuously at a; = (see Fig. G5,a). 

 If a curvilinear coordinate system is introduced (see Fig. G,5a) with 

 the origin at the center of the cross section of a circular pipe, where x is 

 taken in the direction of the flow, r in the radial direction, and ■& the 

 azimuthal angle, then with axial symmetry of flow the Navier-Stokes 

 equations become 



du du _ 1 dp /d^u I du d'^u\ , ^. 



dx dr p dx \dr'^ r dr dx^J 



dv , dv I dp , /d^V , d^v , 1 dv v\ ,_ ^. 



dx dr p dr \dx^ dr^ r dr r^/ 



The continuity equation: 



^ + ^ = (5-3) 



dx dr 



The energy equation: 



( dT . dT\ dp , . 



The boundary conditions are 



ld_{ dT\ d^T 

 r dr\ dr I dx'^ 



+ 4> (5-4) 



r = 0: y = ^ = (5-5) 



dr 



r < R and x = 0: T = Ti (5-6) 



r = R: u = 0, v = —Vy, = const (5-7) 



r = R and x > 0: T ^ T^ (5-8) 



Velocity distribution and skin friction. For two-dimensional incom- 

 pressible flow a stream function exists such that 



ru = -^, —rv = -^ (5-9) 



dr dx 



and the continuity equation (Eq. 5-3) is satisfied. For a constant fluid 

 injection or suction at the porous wall and the given boundary conditions, 



< 461 ) 



