Lubrication flow

Use continuity equation to find V_y from the known V_x:

Vx = U*(h[x] - y)*(h[x] - 3*y)/h[x]^2 + Q*6*y*(h[x] - y)/h[x]^3

D[Vx,x] // Simplify

Vy = Integrate[-((2 y (3 y-2 h[x]) (-3 Q+U h[x]) h'[x])/((h[x])^4)), y] // Simplify

Case: Flat wing:

h[x_] := h0 - alpha*x

delta = Normal[Integrate[1/(h[x])^2, {x,-L/2, L/2}] /  Integrate[1/(h[x])^3, {x,-L/2, L/2}]]  // Simplify

newVx = Vx /. Q -> U*delta/2 //Simplify

newVy = Vy /. Q -> U*delta/2

Drag force:

sigmaXY = eta *(D[newVx,y] + D[newVy,x]) /. y-> 0 // Simplify

Drag = Normal[Integrate[sigmaXY, {x, -L/2, L/2}]]

Pressure:

A = Normal[Integrate[1/(h[u]^2), {u, -L/2, x}]]

B = delta*Normal[Integrate[1/(h[u]^3), {u, -L/2, x}]]

p[x_] = 6*eta*U*( A - B)  // Simplify

Plot[p[x]/(eta*U) /. {alpha -> 0.01, h0 -> 1.0, L -> 20.0}, {x, -10, 10}]

Lift:

sigmaYY = eta *D[newVy,y] /. y-> 0

Lift = Normal[Integrate[p[x], {x, -L/2, L/2}]]