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VISCOUS FLUID FLOW VISCOUS FLUID FLOW Tasos C. Papanastasiou Georgios C. Georgiou Department of Mathematics and Statistics University of Cyprus Nicosia, Cyprus Andreas N. Alexandrou Department of Mechanical Engineering Worcester Polytechnic Institute Worcester, MA by Boca Raton London New York Washington, D.C. CRC Press [...]... stagnation-point and rotating boundary-layer flows are also covered It takes three to four hourly lectures to cover Chapter 8 Chapter 9, on “Nearly Unidirectional Flows,” addresses lubrication and thinfilm flows Typical lubrication-flow applications considered are piston-cylinder and piston-ring lubrication of engines, journal-bearing system lubrication, and flows in nearly rectilinear channel or pipe Flows... Bidirectional Flows,” examines slow flows dominated by viscous forces, or, equivalently, small Reynolds number flows In the limit of zero Reynolds number, the equations of flow are simplified to the so-called Stokes equations Stokes flow is conveniently studied with the introduction of the stream function, by means of which the system of the governing conservation equations is reduced to a much-easier-to-handle... Time Unit tangent vector Absolute temperature Reference temperature Total stress tensor ij-component of the total stress tensor Vector; velocity vector Mean velocity Radial velocity component Slip velocity (at a wall) x-velocity component y-velocity component z-velocity component azimuthal velocity component φ-velocity component Velocity (magnitude of); internal energy per unit mass, dU ≡ Cv dT Terminal... vector in the x-direction First invariant of a tensor Unit tensor Second invariant of a tensor Third invariant of a tensor Cartesian unit vector in the y-direction nth-order Bessel function of the first kind Linear momentum, J ≡ mu Rate of momentum convection Angular momentum, Jθ ≡ r × J Thermal conductivity; diffusion coefficient; Boltzman constant; index Cartesian unit vector in the z-direction Length;... lectures to cover Chapter 7 In Chapter 8, on “Laminar Boundary Layer Flows,” we examine laminar, highReynolds-number flows in irregular geometries and over submerged bodies Flows are characterized as potential flows, away from solid boundaries, and as boundarylayer flows, in the vicinity of solid boundaries Following the development of the boundary-layer equations by means of the stream function, exact solutions... Double dot product × Cross product Superscripts T Transpose (of a matrix or a tensor) −1 Inverse (of a matrix or a tensor) ∗ Dimensionless variable Abbreviations 1D One-dimensional 2D Two-dimensional 3D Three-dimensional CFD Computational Fluid Dynamics ODE(s) Ordinary differential equation(s) PDE(s) Partial differential equation(s) © 2000 by CRC Press LLC Chapter 1 VECTOR AND TENSOR CALCULUS The physical... volume Diameter; distance Differential arc length Differential surface Directed differential surface, dS ≡ ndS Differential length Differential volume Diameter Rate-of-strain tensor, D ≡ 1 [∇u + (∇u)T ] 2 Substantial derivative operator Unit vector in the xi -direction energy ˙ E E2 E4 Eu f F FD Fr g g G h H ˙ H i i I I II III j Jn J ˙ J Jθ k k L m m ˙ M M n N O p p0 © 2000 by CRC Press LLC Rate of energy conversion... spherical coordinate; angle Second viscosity coefficient Kinematic viscosity, ν ≡ η ρ Dimensionless number Stretching coordinate; similarity variable Density Surface tension Tensor ij-component of σ Viscous stress tensor; tensor ij viscous stress component Wall shear stress Spherical coordinate; angle; scalar function Stream function Vorticity; angular frequency Vorticity vector Angular velocity Angular velocity... conservation equations is reduced to a much-easier-to-handle single fourth-order PDE Representative creep- © 2000 by CRC Press LLC ing flow examples, such as the flow near a corner and the flow past a sphere, are discussed in detail It takes two to three hourly lectures to cover Chapter 10 All chapters are accompanied by problems, which are often open-ended The student is expected to spend time understanding the... i.e., addition and multiplication, follow the rules of elementary algebra A scalar field is a real-valued function that associates a scalar (i.e., a real number) with each point of a given region in space Let us consider, for example, the right-handed Cartesian coordinate system of Fig 1.1 and a closed three-dimensional region V occupied by a certain amount of a moving fluid at a given time instance t . VISCOUS FLUID FLOW VISCOUS FLUID FLOW Tasos C. Papanastasiou Georgios C. Georgiou Department. LLC 3CONSERVATIONLAWS 3.1ControlVolumeandSurroundings 3.2TheGeneralEquationsofConservation 3.3TheDifferentialFormsoftheConservationEquations 3.4Problems 3.5References 4STATICEQUILIBRIUMOFFLUIDSANDINTERFACES 4.1MechanicsofStaticEquilibrium 4.2MechanicsofFluidInterfaces 4.2.1InterfacesinStaticEquilibrium 4.3Problems 4.4References 5THENAVIER-STOKESEQUATIONS 5.1TheNewtonianLiquid 5.2AlternativeFormsoftheNavier-StokesEquations 5.3BoundaryConditions 5.4Problems 5.5References 6UNIDIRECTIONALFLOWS 6.1Steady,One-DimensionalRectilinearFlows 6.2Steady,AxisymmetricRectilinearFlows 6.3Steady,AxisymmetricTorsionalFlows 6.4Steady,AxisymmetricRadialFlows 6.5Steady,SphericallySymmetricRadialFlows 6.6TransientOne-DimensionalUnidirectionalFlows 6.7SteadyTwo-DimensionalRectilinearFlows 6.8Problems 6.9References 7APPROXIMATEMETHODS 7.1DimensionalAnalysis 7.1.1Non-dimensionalizationoftheGoverningEquations 7.2PerturbationMethods ©