Problem 1:

Consider a flow field with constant density (ρ) and viscosity (ν) and assume the below equations represent the conservation equations for this flow field.

∂u/∂t + u.∂u/∂x + v.∂u/∂y = g + ν[∂²u/∂x² + ∂²u/∂y²]

∂v/∂t + u.∂v/∂x + v.∂v/∂y = g + ν[∂²v/∂x² + ∂²v/∂y²]

∂u/∂x + ∂v/∂y = 0

(a) List the assumptions made in order to simplify the Navier-Stokes and Mass Conservation Equations to the ones show above
(b) Differentiate the x-momentum equation with respect to y
(c) Differentiate the y-momentum equation with respect to x
(d) Now subtract the equation obtained in (c) from the equation obtained in (b)
(e) Rewrite the equation in terms of ω_z. Assume that that ω_z = (∂v/∂x - ∂u/∂y)

(f) Use the definition of ψ(v = ∂ψ/∂x, u = ∂ψ/∂y), the stream function, to rewrite ω_z in terms of ψ

(g) Using the new definition of ω_z rewrite the Equation obtained in (e) to show that it is now:

∂/∂t(∇²ψ) + ∂ψ/∂y.∂/∂x(∇²ψ) - ∂ψ/∂x.∂/∂y(∇²ψ) = υ(∇⁴ψ)

(h) Expand ∇⁴? What is it?

Problem 2:

For practical applications we can assume that an airplane wing is a flat plate. A plane wing flying at altitude of 10km (ρ_air = 0.4135 kg/m³ and μ = 1.458 x 10^-5 kg/ (m-s)) can be approximated by a stationary flat plate with a uniform velocity fluid (U) passing over it. If you assume the below velocity distribution, use the momentum integral and determine the drag force per unit width on a 0.2m long wing flying at 100 m/s.

u/U = a + b(Y/δ) + c(y/δ)² + d(y/δ)³

You will need to use boundary conditions below with the momentum integral to find the values of the coefficients.

At y = 0 u = 0 At y = δ  ∂u/∂y = 0

At y =0 ∂²u/∂y² = 0 At y = δ u = U

d/dx ₀∫^∞ (U_∞ - u)u dy + dU_∞/dx ₀∫^∞ (U_∞ - u)dy  = ν(∂u/∂y)_y=0

You may estimate drag force on the wing using the below relationships (notice that dz = 1):

Τ_o = μdu/dy|_y=0

C_f,x = Τ_o/(1/2ρU²)

C‾_f,L = 1/L ₀∫^L C_f,x dx

Drag = 1/2C‾_f,LρA_sU²