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MA40255: Viscous fluid dynamics

[Page last updated: 23 October 2023]

Academic Year: 2023/24
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:
  • Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA30253
Learning Outcomes: Students should be able to derive the Navier-Stokes 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 Navier-Stokes 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 semi-infinite flat plate. Derivation of Prandtl's boundary-layer equations and similarity solutions for flow past a semiinfinite flat plate. Discussion of separation and application to the theory of flight. Jeffrey-Hamel flows.
Slow flow: Slow flow past a cylinder and sphere. Non-uniformity of the two-dimensional approximation; Oseen's equation. Flow around corners and eddies.
Lubrication theory: bearings, squeeze films, thin films; Hele-Shaw cell and Saffman-Taylor instability.

Course availability:

MA40255 is Optional on the following courses:

Department of Mathematical Sciences
  • RSMA-AFM16 : Integrated PhD Statistical Applied Mathematics
  • TSMA-AFM17 : MRes Statistical Applied Mathematics
  • TSMA-AFM19 : MSc Mathematics with Data Science for Industry
  • TSMA-AWM19 : MSc Mathematics with Data Science for Industry
  • TSMA-AFM16 : MSc Statistical Applied Mathematics
  • USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
  • USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
  • USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
  • USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
  • USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
  • USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
  • USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
  • USXX-AFB03 : BSc(Hons) Mathematics and Physics (Year 3)
  • USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
  • USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
  • USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 4)
  • USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
  • USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)

Notes:

  • This unit catalogue is applicable for the 2023/24 academic year only. Students continuing their studies into 2024/25 and beyond should not assume that this unit will be available in future years in the format displayed here for 2023/24.
  • Courses 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 pre-requisite rules.
  • Find out more about these and other important University terms and conditions here.