Page 92 - Computational Fluid Dynamics for Engineers

Problems 77

<P V ~ (£1 + £ 2 V + gtU = 0 (P2.10.2)

(a) In order that the disturbances in the boundary layer decay near the edge of
the boundary layer so that (j> and <\J —> 0 as y —> oc, show that the solutions of
Eq. (P2.10.2) must be of the form given by Eq. (P2.7.5).
(b) Show that, with D = d/dy, the "edge" boundary conditions on 0 can be
written as
(
(D 2 - fi)4> + 6 + & ) P + £i)0 = 0 (P2.10.3)
2
(D + &)(D -£)<{> = 0 (P2.10.4)

2-11. Noting that the characteristic equation for Eq. (P2.10.1) is

verify Eq. (P2.7.5).

2-12. Classify the following equations into hyperbolic, elliptic or parabolic type:
(a) u xx + 2u xy + u yy - (l + xy)u = 0
2
(b) Uyy + (1 + X )U X — Uy + U = 0
(c) 5u xx — Suyy -f u x -\- u = s'mxy

2-13. Determine the regions where the Tricomi's equation

^xx ~r "E^yy == ^

is of elliptic, parabolic or hyperbolic type.

2-14. For what values of x and y are each of the following equations hyperbolic,
parabolic, elliptic?
(a) (y + l)u xx + 2xu xy + u yy = x + y
(b) XU XX - yU Xy + XUyy + U X = 0
(c) (y + 1 ) 1 ^ + 2xu xy + 2/iA^y = u
(d) (1 - y)n x x + 2(1 - x)u xy + (1 + y)u yy + yn x = sinx

2-15. Suppose that u, u x and u^ are known at every point on a curve T in the
xy-plane, where u is the solution of the equation

A(x, y)u xx + B(x, y)u xy + C(x, i/)ix yy = 0
Show that if the slope ?/ of T satisfies the auxiliary equation

A(x, y){y') 2 - B(x, y)y' + C(x, y) = 0