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## MA40255: Viscous fluid dynamics

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

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.

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.
Programme availability:

#### MA40255 is Optional on the following programmes:

Department of Mathematical Sciences
• RSMA-AFM16 : Integrated PhD Statistical Applied Mathematics
• TSMA-AFM17 : MRes Statistical Applied Mathematics
• 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-AFB05 : BSc(Hons) Statistics (Year 3)
• 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)

 Notes: This unit catalogue is applicable for the 2018/19 academic year only. Students continuing their studies into 2019/20 and beyond should not assume that this unit will be available in future years in the format displayed here for 2018/19. 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 pre-requisite rules. Undergraduates: Find out more about these and other important University terms and conditions here. Postgraduates: Find out more about these and other important University terms and conditions here.