June 26, 2011
June 26, 2011
June 29, 2011
Energy Conversion and Conservation
22.257.1 - 22.257.18
Assessment of Navier-Stokes’ Theorem in a Fluid Mechanics CourseAbstractThe Navier-Stokes equations are indeed the foundation of fluid mechanics. Stokes’theorem is used in nearly every branch of mechanics as well as electromagnetics.Stokes' Theorem also plays a vital role in many secondary theorems such as thosepertaining to vorticity and circulation. However, the divergence theorem is amathematical statement of the physical fact that, in the absence of the creation ordestruction of matter, the density within a region of space can change only by having itflow into, or away from the region through its boundary. This is also known as Gauss’sTheorem. At present, our students who take a fluid mechanics course in their junior orsenior year are not learning about Stokes’ Theorem or Green’s Theorem or Gauss’Theorem. They also do not have a background of vector differential calculus or vectorintegral calculus. Many scholars believe that Navier Stokes Equations and theContinuum Equations form the two giant cornerstones of modern day fluid dynamics.In the year 2000 the Navier-Stokes Equation was designated as a Millennium Problem.The solution for each Millennium Problem is worth a Million Dollars. Navier-StokesEquation is one of seven mathematical problems selected by the Clay MathematicsInstitute of Cambridge, Massachusetts for this special million dollar award. It isessential to generate new techniques and modern mathematical methods for addressingand analyzing flow in complex fluids such as gels, suspensions, liquid crystals and foams.We all should however be recognize the fact that classical fluid mechanics has been verysuccessful in providing us with a quantitative understanding of turbulent flow such asshock waves. The students are unable to calculate the Gradient of a scalar field. Theyare unaware of of the need for determining the Divergence of a vector field or theimportance of calculating the Curl of vector field. The author introduced the importanceof these concepts to the students when he taught a course in hydraulics and fluidmechanics about 12 years ago. He also collected some assessment data and he presentshis findings in this paper.
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