Studies on the Motion of Viscous Flows— III 



. -.;-::■ PART 1 



SUBSTRUCTURE FORMULATION 



In this section we shall formulate substructure equations for the flow of a 

 Newtonian viscous incompressible fluid of density p and viscosity m around a 

 circular cylinder of radius 'a' such that the velocity of the fluid at the cylinder 

 wall is zero for all time t , whereas the velocity at distances from the cylinder 

 approaching infinity is uniform in direction and with a magnitude u^, which may 

 be a constant or a function of time alone. The starting point is the set of Navier- 

 Stokes equations and the continuity equation in two dimensions. 



FUNDAMENTAL EQUATIONS IN TWO DIMENSIONS 



The fundamental equations for the flow are the two-dimensional Navier - 

 Stokes equations and the continuity equation. As the boundary of the cylinder is 

 circular, it is natiural to use a two-dimensional polar -coordinate system. The 

 reference frame is fixed to the cylinder so that the orientation of the polar axis 

 is parallel to the direction of the velocity vector in the flow field as the radius 

 vector r approaches infinity. Figure A illustrates the reference coordinate 

 system. 



Fig. A - Reference coordinate system of a cylinder 



Conservation of Momentum Equations (the Navier -Stokes Equations): 



Bu ^ du V du v^ _ i-^p fJ- jd'^ u 1 3u u 1 d^u 2 3v \ /■, -ix 



Bt dr r Bfi' r ^Br ^ \dr^ r or P P dU^ P dc^ 



Bv » Bv V Bv uv 1 ^P fi fci^v 1 Bv v 1 B^v 2 Bu 



Bt B? ? b5 ? pr d0 ^ \3f2 r dr P p 3^2 p2 9^ 



(1.2) 



501 



