Viscous Stresses in 2D (Cartesian Coordinates)
To account for viscous stresses in the momentum equations, consider an element as in the sketch, Fig. 8.4.
The viscous forces in the xdirection are
( Ax Ax
&1,1(x + —, y, t) – ah1(x – —, y, t) Ay
( Ay Ax
+ 02,1 (x, y + 2, t) – <72,1 (x, y – —, t) Ax (8.13)
Dividing by Ax. Ay and taking the limit gives
д71,1 до2,1
dx + dy
Fig. 8.4 Viscous stresses in 2 D
Similarly, the viscous forces in the ydirection is
Ax Ax
01,2 (x + —, y, t) – 01,2(x – —, y, t) Ay
( Ay Ax
+ 02,2(X, y + —, t) – 02,2(X, y – —, t) Ax
Again, dividing by Ax. Ay and taking the limit gives


Note: taking moments about the centroid (x, y)
0 = ^ M = o1,2Ax. Ay – ff2,iAx. Ay
hence 01,2 = 02,1.
The momentum equations for viscous fluid flow become:
dp. do1,1 , do2,1 dx + dx + dy d p. d01,2 , do2,2 dy + + ~dy~










Fig. 8.5 Boundary layer development on a thin airfoil 
The boundary conditions for the flow past a profile or a plate are the following (Fig.8.5):
• on the solid surface, no penetration, no slip,
• in the far field, no disturbance.
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