Elasticity and Viscous Fluid Dynamics

MATH35020 Elasticity and Viscous Fluid Dynamics 20 Full year Undergraduate Level 3 Ashleigh Hutchinson Unit coordinator Joel Daou Unit coordinator 1 FHEQ level (Framework for Higher Education Qualifications) ' Last part of a Bachelors ' European Credit Transfer & Accumulation System Rating : 10.0 This course unit discusses some elements of elasticity theory and viscous fluid flows, which are essential building blocks in continuum mechanics. Initially we start with analysis of stress and strain and look at constitutive relations and examine the differences between solids and fluids. We then concentrate on the linearised theory of elasticity. A typical problem of the subject is as follows: Suppose an elastic body (e.g. an underground oil pipe) is subjected to some loading on its outer surface. What is the stress distribution which is generated throughout the body? Does this stress distribution have unexpectedly large values which might lead to failure? The subject is developed, and particular problems solved, from a mathematical standpoint. In the second half of the course, we continue the discussion of stresses and strains but now focussing on how they impact viscous fluid flow. We first derive the governing nonlinear equations – the Navier-Stokes equations using conservation principles. We then go on to study a number of exact solutions such as the flow in a pipe or channel, flow between rotating cylinders. Finally we look at asymptotic approaches to solving flows at small and large Reynolds numbers.  This course unit discusses some elements of elasticity theory and viscous fluid flows, which are essential building blocks in continuum mechanics. Initially we start with analysis of stress and strain and look at constitutive relations and examine the differences between solids and fluids. We then concentrate on the linearised theory of elasticity. A typical problem of the subject is as follows: Suppose an elastic body (e.g. an underground oil pipe) is subjected to some loading on its outer surface. What is the stress distribution which is generated throughout the body? Does this stress distribution have unexpectedly large values which might lead to failure? The subject is developed, and particular problems solved, from a mathematical standpoint. In the second half of the course, we continue the discussion of stresses and strains but now focussing on how they impact viscous fluid flow. We first derive the governing nonlinear equations – the Navier-Stokes equations using conservation principles. We then go on to study a number of exact solutions such as the flow in a pipe or channel, flow between rotating cylinders. Finally we look at asymptotic approaches to solving flows at small and large Reynolds numbers.  This course aims to:(i) Introduce students to the mathematical theory of linear elasticity and viscous flows. (ii) Develop and apply the theory of linear elasticity to a number of practical problems in solid mechanics. (iii) Develop and apply the theory governing viscous fluid flow to solve a number of problems in fluid mechanics. (iv) Introduce a range of analytical techniques to solve the differential equations that arise in problems involving linear elasticity. (v) Discuss the exact solution of a number of problems in viscous flow governed by the Navier-Stokes equations.  ILO 1identify, calculate and provide physical interpretations for displacements, strains, rotation tensors, stresses and tractions for given deformations of linear elastic materials,  ILO 2use constitutive laws to relate strain and stress fields,  ILO 3write down and solve the Navier-Lamé equations for linear elasticity, and write down the Navier-Stokes equations for fluid flow in various coordinate systems.ILO 4solve two-dimensional problems in linear elasticity (plane strain or plane stress) using the Airy stress function or related approaches,ILO 5formulate, solve and provide physical interpretations for the solutions of boundary value problems in linear elasticity.ILO6  analyze 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.ILO7simplify the Navier-Stokes equations by making use of suitable assumptions such as for unidirectional flows, to analyse the steady and unsteady flows in plane channels and pipes.  ILO8Conduct asymptotic analysis to derive and analyse the Stokes equations for low Reynolds numbers, the boundary layer equations for high Reynolds numbers, and apply the analysis to physically relevant scenarios.    Syllabus:Elasticity (semester one, weeks 1-5)   - Analysis of strain: the infinitesimal strain tensor, derivation and interpretation; maximum normal strain; strain invariants; equations of compatibility of strain.     - Analysis of stress: the traction vector and the stress tensor; maximum normal stress. Stress equations of motion and their linearisation.   - Constitutive equations stress-strain relations. Elastic and linearly elastic materials; isotropic materials.   - Governing Equations: Navier's equation of motion for the displacement vector; equations of compatibility of stress for an isotropic materials in equilibrium (Beltrami-Michell equations) Viscous Flow (semester one, weeks 7-12) Introduction and fundamental principles.  - The kinematics of fluid flow: The Eulerian velocity field; the rate of strain tensor and the vorticity vector; the equation of continuity.  - The Navier-Stokes equations: The substantial derivative; the stress tensor; the constitutive equations for a Newtonian fluid. The streamfunction and vorticity equations.- Boundary and initial conditions; conditions at an interface.-  Uni-directional flows: Couette/Poiseuille flow; flow down an inclined plane; the vibrating plate.   Elasticity  (semester two Weeks 1-6).- Formulation of boundary value problems of linear elastostatics:  One-dimensional problems. A selection of solvable problems (which are effectively one-dimensional) in Cartesian, cylindrical polar or spherical polar coordinates. St. Venant's principle.  Plane strain problems. Theory of plane strain, Airy stress function. A selection of soluble two-dimensional problems using plane-strain theory. Viscous Flow (semester two, Weeks 7-11)-  The Navier-Stokes equations in curvilinear coordinates; Hagen-Poiseuille flow; circular Couette flow.  -  Dimensional analysis and scaling; the dimensionless Navier-Stokes equations and the importance of the Reynolds number; limiting cases and their physical meaning; lubrication theory.  -  Stokes flow (zero Reynolds number flow)  - High-Reynolds number flow; boundary layers; the Blasius boundary layer.   3 contact hours per week, divided between 2 lectures and one tutorial class. Most of content delivered in traditional in person mode but supplemented with some online notes and short videos. 0 Other 10% 1 Written exam 80% 2 Written assignment (inc essay) 10% 80% Summer exam  3 hours - General feedback provided after exam is marked.10% Written coursework ( week 8, semester 1) 1 week - Individual feedback provided10% Written coursework ( week 7, semester 2) 1 week - Individual feedback provided  PHYS20171 Mathematics of Waves and Fields Pre-Requisite Compulsory MATH24420 Partial Differential Equations & Vector Calculus Pre-Requisite Compulsory MATH20401 Partial Differential Equations and Vector Calculus A Pre-Requisite Compulsory MATH35020 Requisites: (A) Pre-Requisites - MATH24420 or PHYS20171 or MATH20401; (B) Anti-Requisites - Students are not permitted to take both PHYS30652 and MATH35020Maths & Physics or Physics students can replace the prerequisite MATH24420 by PHYS20171 N P.L. Gould & Y. Feng.  Introduction to Linear Elasticity, 4th Edition, Springer, 2018  Green, A.E. & Zerna, W. Theoretical Elasticity. Dover (1992) paperback reprint of theoriginal version from Oxford University press.D.J. Acheson Elementary Fluid Dynamics, OUP, 1990.R.L. Panton Incompressible Flow (4th edition), Wiley, 2013.A.I. Ruban & J.S.B. Gajjar Fluid Dynamics, Part 1, Classical fluid dynamics,  OUP,  2014 Lectures 44 Tutorials 22 0 134