ENGR6201_StressStrainBW.pdf - Theory of Stress and Rate of...

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Theory of Stress and Rate of Strain in Fluid Mechanics(Notes Based on References in Course Outline)
2Encompasses the unified study for the transfer ofFluid dynamics: Transport of momentum(= mass • velocity)Heat Transfer: Transport of energyMass Transfer: Transport of massof various chemical speciesEquations governing motion of viscous compressible fluid express concepts ofConservation of massConservation of momentumConservation of energyDue to presence of viscosityComplicated viscous forces arise due toShear stressesLeading to dissipation of energy due to work done by shear stressesRecall workWdone by a constant force of magnitude ܨԦon a pointP Point Pdisplaces in a straight line in the direction of the force These stresses and resulting rate of strain essential in developing governing equations of motionFor viscous compressible (and incompressible) flowsAlejandro Allievi – ENGR 6201Stress and Rate of Strain in F.M.2
3Forces acting on an fluid element of continuous media, two typesExternal or bodyforcesInternal or surfaceforces (also called contact forces) • Example bodyforcesElectromagnetic, gravitationalDistributed over the entire volume of the mediumUsually expressed per unit mass• Example surfaceforcesAct on the boundary of a volume (or volume element)Consider a surfaceAof a fluid in contact with a solidForces exerted on the fluidacross A(in the limit dAas in figure)Equal and opposite toForces exerted on the solidacross A (Newton’s 3rdlaw)Provided inertia and body forces are negligibly smallThen forces on a surface transmitted from surroundingsEquivalent to a force ܨԦand a couple ܯDefining stress as limit of the ratio of force to area on which force actsWhen area shrinks topoint (within limitations of continuum), couple arm must vanishTherefore, we can restrict ourselves to the study of force per unit area, i.e., STRESSAlejandro Allievi – ENGR 6201Stress and Rate of Strain in F.M.3
4Consider the parallelepiped with dimensions x, y, zWhose center is located at location (x, y, z). Note we are NOT saying it is a FLUID• Consider Pi=1,…,6are the mean stresseson each face All applied at the center of each face of the parallelepiped, so thatNote optical illusion Face where P4acts is parallel to x-zplane butdoes notlie on x-z planeAlejandro Allievi – ENGR 6201Stress and Rate of Strain in F.M.4x, y, z
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