MPhys Physics with Astrophysics

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.

Students must have taken MATH20401 OR MATH20411

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.

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.

• 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]

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Students can also get feedback on their understanding during the lecturer's office hours.

The following are further optional reading and are not required by this course:

• 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.

The independent study hours will normally comprise the following. During each week of the taught part of the semester:
 
•         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
•         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
•         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
•         In some weeks you may be preparing coursework or revising for mid-semester tests
 
Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.
 
The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.