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## MAE 5130: VISCOUS FLOWS

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**MAE 5130: VISCOUS FLOWS**Lecture 2: Introductory Concepts August 19, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk**IMPORTANT RELATIONSHIPS**• If the curl of the velocity field is zero • Flow is irrotational • Velocity can be written as the gradient of a scalar function, f • If the divergence of the velocity field is zero • Flow is incompressible • If both are true • Laplace equation • The curl of the gradient of a scalar function is zero • The divergence of the curl of a vector is zero**SUMMARY OF VECTOR INTEGRALS**• Gradient Theorem • Vector equation involving a scalar function, a • Limits of integration such that surface encloses the volume • n points normal outward • Divergence (Gauss’) Theorem • Scalar equation • Stokes Theorem • Direction of n is given by right hand rule**V=Wa**W a EXAMPLE: VORTICITY AND STOKES’ THEOREM • Vorticity has to do with the local rate of rotation • Consider a plane flow with a small cylinder of fluid rotating with angular velocity, W • Apply Stokes’ theorem • Magnitude of vorticity is twice the local rate of fluid rotation • Physical interpretation: If a small sphere of fluid where to be instantly solidified with no change in angular momentum, local vorticity would be twice the angular velocity of the sphere • Purely a kinematic statement • Apply divergence theorem • Applies for any fluid, compressible or incompressible, viscous or inviscid • Implications: • Net flux of vorticity over any closed surface = 0 • Vortex lines can not end in fluid**KINEMATIC PROPERTIES: TWO ‘VIEWS’ OF MOTION**• Lagrangian Description • Follow individual particle trajectories • Choice in solid mechanics • Control mass analyses • Mass, momentum, and energy usually formulated for particles or systems of fixed identity (ex., F=d/dt(mV) is Lagrangian in nature) • Eulerian Description • Study field as a function of position and time; not follow any specific particle paths • Usually choice in fluid mechanics • Control volume analyses • Eulerian velocity vector field: • Knowing scalars u, v, w as f(x,y,z,t) is a solution