Fluid Mechanics: 2D Laminar Flow

[jdawg]

Homework Statement

A reasonable approximation for the two-dimensional incompressible laminar boundary layer on the flat surface in Fig. P4.17 is

u ≈ U(2*(y/δ) - 6*(y³/δ⁴) + y⁴/δ⁴)

δ = Cx^1/2 where C is a constant

y ≤ δ

Homework Equations

The Attempt at a Solution

Incompressible so density is constant, 2D flow so no z component, continuity equation reduces to
∂u/∂x + ∂v/∂y = 0For the x component, I got [ (-2y/δ²) + (6*y³/δ⁴) - (4*y⁴/δ⁵) ]

But there is supposed to be a dδ/dx term attached to each term. Where does this term come from? Is it some kind of chain rule thing? Thanks!

 

[haruspex]

U(2*(y/δ) - 6*(y³/δ⁴) + y⁴/δ⁴)

I assume you mean 6(y³/δ³).

Is it some kind of chain rule thing?

Yes. ∂/∂x of f(y, z) where z is a function of x is given by ∂f/∂z dz/dx.