MA40255: Viscous fluid dynamics
[Page last updated: 27 September 2022]
Academic Year:  2022/23 
Owning Department/School:  Department of Mathematical Sciences 
Credits:  6 [equivalent to 12 CATS credits] 
Notional Study Hours:  120 
Level:  Masters UG & PG (FHEQ level 7) 
Period: 
 Semester 2

Assessment Summary:  EX 100% 
Assessment Detail:  
Supplementary Assessment: 
 Likeforlike reassessment (where allowed by programme regulations)

Requisites: 
Before taking this module you must take MA30253

Learning Outcomes:
 Students should be able to derive the NavierStokes equations for the flow of viscous fluids and analyse these in different flow situations using asymptotic analysis and partial differential equation theory.

Aims:  To describe viscous flow phenomena and analyse these with the necessary mathematical theory.

Skills:  Numeracy T/F, A
Problem Solving T/F, A
Written Communication F (on problem sheets).

Content:  Review of Lagrangian and Eulerian descriptions: Jacobian, Euler's identity and Reynold's transport theorem. The continuity equation and incompressibility condition.
Cauchy's stress theorem and properties of the stress tensor. Cauchy's momentum equation.
Constitutive law for a Newtonian viscous fluid. The incompressible NavierStokes equations. Vorticity and Helmholtz's theorems. Energy. Exact solutions for unidirectional flows; Couette and Poiseuille flow.
Dimensional analysis, Reynolds number. Derivation of equations for high and low Reynolds number flows.
Further topics chosen from the following:
Boundary layer theory: thermal boundary layer on a semiinfinite flat plate. Derivation of Prandtl's boundarylayer equations and similarity solutions for flow past a semiinfinite flat plate. Discussion of separation and application to the theory of flight. JeffreyHamel flows.
Slow flow: Slow flow past a cylinder and sphere. Nonuniformity of the twodimensional approximation; Oseen's equation. Flow around corners and eddies.
Lubrication theory: bearings, squeeze films, thin films; HeleShaw cell and SaffmanTaylor instability.

Programme availability: 
MA40255 is Optional on the following programmes:
Department of Mathematical Sciences
 RSMAAFM16 : Integrated PhD Statistical Applied Mathematics
 TSMAAFM17 : MRes Statistical Applied Mathematics
 TSMAAFM19 : MSc Mathematics with Data Science for Industry
 TSMAAWM19 : MSc Mathematics with Data Science for Industry
 TSMAAFM16 : MSc Statistical Applied Mathematics
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 3)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
 USMAAFB13 : BSc(Hons) Mathematics (Year 3)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
 USMAAFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
 USMAAAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
 USMAAKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
 USMAAFB05 : BSc(Hons) Statistics (Year 3)
 USMAAAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
 USMAAKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
 USMAAFM14 : MMath(Hons) Mathematics (Year 3)
 USMAAFM14 : MMath(Hons) Mathematics (Year 4)
 USMAAAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
 USXXAFB03 : BSc(Hons) Mathematics and Physics (Year 3)
 USXXAAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
 USXXAKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
 USXXAFM01 : MSci(Hons) Mathematics and Physics (Year 4)
 USXXAAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
 USXXAKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)

Notes:  This unit catalogue is applicable for the 2022/23 academic year only. Students continuing their studies into 2023/24 and beyond should not assume that this unit will be available in future years in the format displayed here for 2022/23.
 Programmes and units are subject to change in accordance with normal University procedures.
 Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any prerequisite rules.
 Find out more about these and other important University terms and conditions here.
