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Lecture-5

# In section i i we performed such a verification for a

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Unformatted text preview: using any vector identity) show that the acceleration of a fluid particle is given by auV (- I )+ w x u, a=-+ q2 at 2 where q is the magnitude of velocity u and w is the vorticity. 14. The definition of the streamfunction in vector notation is u = -k x V1/, where k is a unit vector perpendicular to the plane of flow. Verify that the vector definition is equivalent to equations (3.35). Supplemental Reading Ars, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ: 29 Prentice-HalL. (The distinctions among streamlines, path lines, and streak lines in unsteady flows are Problem 3.13 30 the acceleration of a fluid particle is given by auV (- I )+ w x u, a=-+ q2 at 2 Problem 3.14 where q is the magnitude of velocity u and w is the vorticity. 14. The definition of the streamfunction in vector notation is u = -k x V1/, where k is a unit vector perpendicular to the plane of flow. Verify that the vector definition is equivalent to equations (3.35). Supplemental Reading Ars, R. (1962). Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Englewood Cliffs, NJ: Prentice-HalL. (The distinctions among streamlines, path lines, and streak lines in unsteady flows are explained; with examples.) Prandtl, L. and O. C. Tietjens (1934). Fundamentals of Hydro- and Aeromechanics, New York: Dover Publications. (Chapter V contains a simple but useful treatmnt of kinematics.) Prandtl, L. and O. G. Tietjens (1934). Applied Hydro- and Aeromechanics, New York: Dover Publications. (Ths volume contains classic photographs from Prandtls laboratory.) 31 Problem 3.14 32 Equations of Fluid Dynamics (Conservation Laws) • • • Continuity (Mass) Navier-Stokes (Force, Momentum) Energy 33 Continuity Mass 34 Newton’s Law 35 Integral Relations (Section 4.2) 36 Integral Relations (Section 4.2) 37 Momentum 38 Models for Stress 39 Navier & Stokes Claude-Lewis Henri Navier (1785-1836) George Stokes (1819-1903) 40 Stokesian Fluid 41 Navier-Stokes Equation 42 Navier-Stokes & Euler 43 Energy 44 The Importance of Viscosity 45 Creation of Vorticity (Note: Flow at thin layer at surface of cylinder vanishes.) 46 Summary • The equations of ﬂuid dynamics are dynamical conservation equations: • • Mass conservation • Energy conservation Momentum changes via total forces (body and surface forces) 47...
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