• Learning Objectives:

By the end of this lecture, students will be able to be able to 1) Employ Navier-Stokes equations to solve general fluid mechanics problems, such as, general velocity profile, calculate volumetric flow rate of poiseuille flow between parallel plates or in an Annular Die (important for blow molding). 2) Describe general features of Newtonian poiseuille flow. 3) Analysis non-Newtonian effects.                                                                                                                   

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• Sample Homework:

1. Couette Flow between Parallel Plates

Figure 1: Simple Shear between Parallel Plates. The bottom plate is fixed and the top plate moves with velocity V in the x₁ direction.

The velocity profile is: 

a) Show that this velocity profile satisfies the incompressible Continuity Equation.

b) Show that this velocity profile satisfies the Navier-Stokes Equations.

c) What Boundary Conditions are appropriate for this problem?

d) If V = 10 cm / s and h = 12 cm, what is the average fluid velocity?

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2. Question from 2003’s First Exam

A Newtonian polymer melt flows in a circular pipe 48.2 meters long, with a constant radius R = 2 cm. The flow is driven by a pressure of 11 atm at the beginning of the pipe and the end of the pipe is simply at atmospheric pressure (1 atm). What is the pressure in the polymer half-way along the pipe? (24.1 meters from either end).

The relevant Navier-Stokes Equations are:

The Continuity Equation in Cylindrical Coordinates is:

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3. Poiseuille Flow through a Round Pipe

Figure 2: Pressure-Driven Flow through a Circular Pipe. Pressure P₁ at the left end of the pipe is greater than pressure P₂ at the right end.

Gravitational forces are negligible compared to viscous forces.

　

a) Show that this velocity profile satisfies the Continuity Equation

b) Determine the pressure distribution in the pipe using the Navier-Stokes Equations.

c)Where is the velocity maximum?

d) Use the Navier-Stokes Equations to show that

e) Calculate the volumetric flow rate in terms of 4P, μ, L and R. This is a very important equation known as the Hagen-Poiseuille Law.

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4. Question from 2003’s First Exam

Consider a Poiseuille flow through a circular pipe (shown in Fig. 2). The velocity profile is:

Determine the average velocity.           

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• Prerequisite Knowledge:

  Students that have not previously been exposed to Fluid Mechanics might benefit from any of a number of textbooks on the subject. A large fraction of the class has some Fluid Mechanics background, and while we will cover necessary Fluid Mechanics, we will do so quickly. I list two excellent readable examples of Fluid Mechanics textbooks below:

  • D. Pnueli and C. Gutfinger, Fluid Mechanics, Cambridge (1992).

  • J. R. Welty, C. E. Wicks and R. E. Wilson, Fundamentals of Momentum, Heat and Mass Transfer, Wiley (1984).

  • First reading assignment: Chapter 1 of Dealy and Wissbrun                                                                 

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