University of Warsaw Advanced Hydrodynamics
Faculty of Physics Selected Topics in Fluid Mechanics
Summer Semester 2019/20
Homework 7 Due May 11, 2020
Solutions should be sent togustavo.abade@fuw.edu.pl
1. Consider the following boundary value problem for a function f (y),
f00+ f0 = 2, f (0) = 0, f (1) = 3, (1)
where is a small positive constant.
(a) Find the exact solution to the problem (1);
(b) Find the solution of the first order differential equation that results after setting
= 0 in (1). This is the mainstream part fM that approximates the full exact solution of (1) for y far from the boundary y = 0.
(c) Change the independent variable in (1) to ξ ≡ y/ and take the limit → 0 with ξ fixed. Solve the resulting second-order differential equation to obtain the boundary layer part fBL that satisfies the boundary condition f (0) = 0 and the matching condition
lim
ξ→∞fBL = lim
y→0fM. (2)
(d) Check if the obtained fM and fBL are particular limits of the full exact solution obtained in (a).
2. Rewrite the 2-D incompressible Navier-Stokes equations in terms of the dimensionless variables
x∗ = x
L, y∗ = y
R−1/2L, u∗ = u
U, v∗ = v
R−1/2U, p∗ = p
ρU2, (3)
where R = U L/ν. By taking the limit R → ∞, derive the boundary layer equations in their dimensionless form
u∗∂u∗
∂x∗ + v∗∂u∗
∂y∗ = −∂p∗
∂x∗ +∂2u∗
∂y∗2,
0 = −∂p∗
∂y∗, ∂u∗
∂x∗ + ∂v∗
∂y∗ = 0.
3. Consider the boundary layer near the forward stagnation point on the circular cylinder.
The mainstream flow at the edge of the boundary layer is U (x) = αx, where x is the distance along the boundary measured from the stagnation point and α a positive constant.
(a) Assuming a similarity solution of the boundary layer equations of the form u(x, y) = αxdf
dξ, ξ = y
δ(x), (4)
deduce that δ(x) must be a constant.
(b) Choose δ = (ν/α)1/2and show that the problem can be reduced to d3f
dξ3 + fd2f
dξ2 + 1 − df dξ
2
= 0. (5)
with boundary conditions f (0) = df
dξ(0) = 0, df
dξ(∞) = 1. (6)
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