## Viscous Fluid Flow

 Unit code: MATH35001 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

Students must have taken MATH20401 OR MATH20411

#### Aims

The course will provide an introduction to the mathematical theory of viscous fluid flows. After deriving the governing equations from a general continuum mechanical approach, the theory will be applied to a variety of practically important problems.

#### Overview

This course is concerned with the mathematical theory of viscous fluid flows. Fluid mechanics is one of the major areas for the application of mathematics and has obvious practical applications in many important disciplines (aeronautics, meteorology, geophysical fluid mechanics, biofluid mechanics, and many others). Using a general continuum mechanical approach, we will first derive the governing equations (the famous Navier-Stokes equations) from first principles. We will then apply these equations to a variety of practical problems and examine appropriate simplifications and solution strategies.

Many members of staff in the School have research interests in fluid mechanics and this course will lay the foundations for possible future postgraduate work in this discipline.

#### Learning outcomes

On successful completion of this course unit students will be able to:

• Analyse and interpret the kinematics of fluid flow in terms of suitable mathematical quantities, such as the rate of strain tensor, the rate of rotation tensor, the material derivative, etc.
• Derive the Navier-Stokes equations from the underlying physical principles and be able to express them in non-dimensional form.
• Simplify the Navier-Stokes equations by making use of the parallel flow assumption and apply the resulting equations to analyse steady or unsteady flows that are driven by physical effects such as wall motion, applied pressure drops or body forces.
• Formulate flow problems using the Navier Stokes equations in appropriate coordinate systems; apply suitable boundary conditions (such as no slip, traction, free surface conditions); solve the resulting mathematical problem; and, where appropriate, interpret the results in physical terms.
• Formulate the Stokes equations in terms of a streamfunction and apply the resulting equations to physically relevant scenarios.

#### Assessment Further Information

End of semester examination: two hours weighting 100%

#### Syllabus

• Introduction; overview of the course; introduction to index notation. [2 lectures]
• The kinematics of fluid flow: The Eulerian velocity field; the rate of strain tensor and the vorticity vector; the equation of continuity. [3]
• The Navier-Stokes equations: The substantial derivative; the stress tensor; Cauchy's equation; the constitutive equations for a Newtonian fluid. [4]
• Boundary and initial conditions; surface traction and the conditions at a free surface. [1]
• One-dimensional flows: Couette/Poiseuille flow; flow down an inclined plane; the vibrating plate. [3]
• The equations in curvilinear coordinates; Hagen-Poiseuille flow; circular Couette flow. [2]
• Dimensional analysis and scaling; the dimensionless Navier-Stokes equations and the importance of the Reynolds number; limiting cases and their physical meaning; lubrication theory. [3]
• The stream function/vorticity equations. [2]
• Stokes flow (zero Reynolds number flow). [2]
• High-Reynolds number flow; boundary layers; the Blasius boundary layer. [2]

• Spiegel, M., Vector Calculus, McGraw Hill (Schaum's Outline series) 1974.
• Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge 1967.
• Sherman, F.S., Viscous Flow, McGraw Hill 1990.
• McCormack , P.S. and Crane, L.J.,Physical Fluid Dynamics, Academic Press 1973.
• Panton, R.L., Incompressible Flow, (second edition), Wiley 1996.
• White, F.M., Viscous Fluid Flow, (second edition), McGraw Hill 1991.

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Matthias Heil - Unit coordinator