NAME: ______________________________________________

CHE 530 Fall 2015   Homework #5: The Navier-Stokes Equations Due: Friday, 10-2-15 by 9:30 AM Note: Please print this coversheet and staple it to the front of your homework. 1) A rectangular duct extends in the z-direction for a length L and has a rectangular cross-section

of width 2W and height 2h.

A) Write the appropriate forms of both the continuity and Navier-Stokes equations for this

problem. Simplify them in accordance with the physical situation.

B) Assuming that the width of the duct is much larger than the height, find the approximate velocity distribution for steady, laminar flow.

C) If the viscous liquid flowing through the duct has a viscosity of 5 cP and a density of 1.1 g/cm3, what pressure drop will be required to provide a Reynolds number of 2200?

2) Consider the case of fully developed laminar flow of a Newtonian fluid in a straight duct of

length L and having a cross-section of triangular shape bounded by the plane surfaces: A) Write the appropriate forms of the momentum equation and boundary conditions for this

flow.

B) If the solution for the velocity profile is of the form

𝑣! = 𝐾 𝑦 − 𝐻 3𝑥! − 𝑦! Where K is a constant, determine K in terms of the parameters of the problem.

y=h$

1$cm$

3$cm$

x=W$

2h$=$1$cm$

y$x$

z$

NAME: ______________________________________________

3) Consider steady, continuous, incompressible, and fully developed laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D inclined at an angle α. The fluid is not open to the atmosphere and flows down the pipe due to an applied pressure gradient and from gravity. Derive an expression for the wall shear stress using the the Equation of Motion or the Navier-Stokes equation.

4) In a velocity field given by vx = 400(y/L)2, vy = 400(x/L)2, vz = 0, determine expressions for pressure gradients in x and y at the point (L,2L). The y-axis is vertical, the fluid density is ρ, and the flow may be considered inviscid.