MT4261: Viscous Fluid Flow

Credit rating 10.

Two lectures each week and weekly example classes.

General description 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 department have research interests in fluid mechanics and this course will lay the foundations for possible future postgraduate work in this discipline.

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.

Objectives On successful completion of the course unit students will be able to

• understand the continuum mechanical derivation of the Navier-Stokes equations and the appropriate boundary conditions.
• understand the kinematics of fluid flow.
• apply the equations to various fluid problems giving a mathematical description of the flow, and to solve some of these problems.

Prerequisites Students need to be familiar with the vector calculus part of MT2121. Background knowledge from MT2272 is useful.

Syllabus

• 1. Introduction; overview of the course; introduction to index notation. [2]
• 2. The kinematics of fluid flow: The Eulerian velocity field; the rate of strain tensor and the vorticity vector; the equation of continuity. [3]
• 3. The Navier-Stokes equations: The substantial derivative; the stress tensor; Cauchy's equation; the constitutive equations for a Newtonian fluid. [4]
• 4. Boundary and initial conditions; surface traction and the conditions at a free surface. [1].
• 5. One-dimensional flows: Couette/Poiseuille flow; flow down an inclined plane; the vibrating plate. [3]
• 6. The equations in curvilinear coordinates; Hagen-Poiseuille flow; circular Couette flow. [2]
• 7. 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]
• 8. The streamfunction/vorticity equations [2]
• 9. Stokes flow (zero Reynolds number flow); flow past a sphere. [2]
• 10. High-Reynolds number flow; boundary layers; the Blasius boundary layer. [2]

Textbooks

Acheson, D.J. Elementary Fluid Dynamics. Clarendon Press, Oxford, 1990.

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. & 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.

Assessment

Continual assessment (15%) plus end of semester examination (85%).

Lecturer

Dr Matthias Heil (Room 18.07, Telephone 5808)