region does not grow with x because the approaching flow confines the vorticity generated associated with the no-slip condition next to the wall. Similarly, we can solve the axial symmetric problem numerically, and find
109 75 . 2 / = ν δ B for the axial symmetric stagnation flow. (3-86b) The shear stress at the wall is calculated as ν µ ν µ µ τ B F n Bx B F n Bx x v y u y w ) 0 ( ' ' 1 0 ) 0 ( ' ' 1 0 − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = = , (3-87) which increases linearly with x. There is no contribution from the term 0 | / = ∂ ∂ y y v µ . For the planar case, we have 2 = n , and 23 . 1 ) 0 ( ' ' = F according to the numerical result. The shear stress at wall is usually written in dimensionless form as R 1 ) 0 ( ' ' 2 2 1 2 − = ≡ n F u C I w f ρ τ , (3-87a) where f C is called the skin friction coefficient, and ν / x u I = R is the Reynolds number. Finally, with the velocity field obtained, we can evaluate the pressure field by integrating (3-81a). (iii) Other nonlinear exact solutions There are also some other nonlinear problems having exact solutions. For examples, the flow induced by a rotating disk (the “von Karman’s viscous pump”, Problem 4 of the Homework), the steady jet from a point source of momentum (called the Squire or Landau’s jet, see Batchelor’s book), and some other interesting problem in Yih’s book (pp.332-334). (IV) Concluding Remarks For steady incompressible flow with constant viscosity, the Reynolds number is the only dimensionless parameter governing the flow. Mathematically, exact solution is valid for all Reynolds number. Unfortunately, this is not true! As claimed by L.D. Landau and E. M. Lifshitz (1959), “Yet not every solution of the equation of motion, even if it is exact, can actually occur in nature. The flows that occur in nature must not only obey the equations of fluid dynamics, but also be stable.” For example, we may observe parabolic velocity profile in a circular pipe under the steady, fully developed condition only when the Reynolds number (based on the mean velocity and pipe diameter) is less than a
110 critical value, say, less than 2100. When the Reynolds number exceeds the critical value, the flow becomes unstable, and we observe turbulent flow when the Reynolds number is sufficiently large. The shape of the mean turbulent profile is more flat (may be approximated as a 1/7-power profile) in comparing with the parabolic profile. The book by Drazin & Reid is a good introduction to the study of the stability of fluid motion (P. G. Drazin and W. H. Reid, “Hydrodynamic stability,” Cambridge University Press, 1981).
111 Figure 3-15: The Fortran program for solving the planar stagnation flow using shooting method (with fourth order Runge-Kutta method).
112 Figure 3-16: Numerical results for the planar stagnation flow.