HW#5 (35 pts)

due 5pm Friday Nov 5, 2010

Q1: Application of vector calculus (10 pts)

The Navier-Stokes Equations are used to model the weather, ocean currents, water flow in a pipe, air flow around a wing, the motion of stars inside a galaxy, even the chaotic behavior of turbulence.

For laminar, incompressible flow in two-dimensions (translational invariance in the z-direction) the Navier-Stokes equations simplify to

(Eq. 1)

(Eq. 2)

where u(x,y) is the fluid velocity in the x direction and v(x,y) is the fluid velocity in the y-direction, p(x,y) is pressure, μ is viscosity which we will assume is constant, and ρg_y is the force of gravity per unit volume.

Consider steady-state pressure driven flow between two plates (diagram below): dp/dx is constant and dp/dy = 0. Fluid velocity is zero at surfaces. The force of gravity is negligible.

a) Write a vector calculus equation that relates grad(u) to laplacian(u).

b) Use boundary conditions and symmetry arguments to solve the second order ODE from part a) and find an expression for u(y).

Q2 Cylindrical coordinates (10 pts)

Starting from the scale factors for cylindrical coordinates, and the generalized expressions for grad and div in curvilinear coordinates, write the Laplacian operator using derivatives with respect to cylindrical coordinates.

Q3 Line Integrals (5 pts)

Problem 11.2

Divergence theorem (10 pts)

Problem 11.23