MTH 375 MTH 575 - Fluid Mechanics
Fluid Mechanics is an application area of the subject of Partial Differential Equations. In general terms, the subject of fluid mechanics deals with any physical problem that involves fluid flows. Some examples are flow of air past an aircraft, flow of blood in our vasculature, flow of air as we breathe, flows past submarines and ships, etc. This course will build from first principles, and after briefly considering the relatively simple fluid statics scenarios, will derive the integral as well as differential forms of euations that govern fluid dynamics. Different approximations : incompressible, irrotational, inviscid, and boundary layers : will be considered and the contexts in which each of these approcimations are useful will be clearly elaborated upon. At the end of the course, students will be able to solve problems analytically wherever possible and possess an intial framework to employ numerical methods when the problem complexity is too high to allow an analytical approach. The course will prepare the student to undertake a semester long or year lomg project in various application areas of fluid mechanics / computational fluid dynamics.
CO1: Students will be able to solve solve fluid statics and simple kinematics and dynamics problems involving pressure distributions and buoyancy.
CO2: Students will be able to apply physical principles of mass consrvation, Newton's second law, and energy conservation to derive the intergral as well as differential forms of governing equations (dimensional as well as nondimensional forms) of fluid mechanics and apply them to solve a range of physical problems.
CO3: Students will be able to apply scaling principles to derive the boundary layer equations and apply boundary layer concepts to calculate drag in simple situations.
CO4: Students will be able to solve flows in the incompressible-inviscid-irrotational regime as well as compressible-inviscid regimes.
CO5: Students will be able to set up the discretized version of governing equations along with a mesh and appropriate boundary conditions to computationally approach problems that are too complicated to handle analytically.
Week 1: The concept of a fluid, Fluid as a continuum, Properties of the velocity field, Thermodynamic properties of a fluid, Viscosity and other secondary properties, Flow patterns: Streamlines, streaklines and pathlines.
Week 2: Pressure distribution in a fluid: Pressure and pressure gradient, Equilibrium of a fluid element, Hydrostatic pressure distributions, Application to manometry, Hydrostatic forces on plane and curved surfaces and in layered fluids, Buoyancy and stability, Pressure distribution in rigid body motion, Pressure measurement.
Weeks 3 & 4: Integral relations for a control volume: Basic physical laws of fluid mechanics, The Reynolds Transport Theorem, Conservation of mass, Linear momentum equation, Angular momentum theorem, Energy equation, Frictionless flow: Bernoulli equation.
Weeks 5 & 6: Differential relations for a fluid particle: The acceleration field of a fluid, Differential equations of mass conservation, linear momentum, angular momentum and energy, Boundary conditions, Stream function, Some illustrative incompressible viscous flows.
Week 7: Dimensional analysis and similarity: The principle of dimensional homogeneity, The Pi theorem, Nondimensionalization of governing equations, Modeling and its pitfalls.
Week 8: Flow past immersed bodies: Reynolds number and geometry effects, Momentum integral estimates, Boundary layer equations, Flat plate boundary layer, Boundary layers with pressure gradient.
Weeks 9 & 10: Potential flows: Vorticity and irrotationality, Frictionless irrotational flows, Elementary plane flow solutions, Superposition of plane flow solutions, Plane flow past closed body shapes, Airfoil theory, Axisymmetric potential flow.
Weeks 11 & 12: Compressible flows: Basic elements of thermodynamics, Speed of sound, Adiabatic and isentropic steady flow, Normal shock wave, Two dimensional supersonic flow, Prandtl Meyer expansion waves.
Weeks 13 & 14: A brief introduction to Computational Fluid Dynamics (CFD): Mesh generation, Finite difference and finite volume approaches to spatial discretization of governing equations, temporal integration, Order of accuracy, Numerical stability.
Project / Term Paper: 20
Mid Semester Exam: 30
End Semester Exam: 50
MTH204 Mathematics IV [Differential Equations]
When: Tuesdays and Fridays: 9:00 - 10:30 AM.
References (A few copies of each are being procured for the library):
Fluid Mechanics by Frank M. White.
Fluid Mechanics by Robert W. Fox, Alan T. McDonald, Philip J. Pritchard and John W. Mitchell.
Fluid Mechanics by Yunus Cengel and John Cimbala.
Fluid Mechanics by Victor Streeter and E. Benjamin Wylie.
Fundamentals of Aerodynamics by John. D. Anderson.
Elements of Gas Dynamics by Liepmann and Roshko.
Computational Fluid Dynamics by John. D. Anderson.