By I.G. Currie

ISBN-10: 146651700X

ISBN-13: 9781466517004

Half I: Governing EquationsBasic Conservation LawsStatistical and Continuum MethodsEulerian and Lagrangian CoordinatesMaterial DerivativeControl VolumesReynolds' shipping TheoremConservation of MassConservation of MomentumConservation of EnergyDiscussion of Conservation EquationsRotation and price of ShearConstitutive EquationsViscosity CoefficientsNavier-Stokes EquationsEnergy EquationGoverning Equations forRead more...

summary: half I: Governing EquationsBasic Conservation LawsStatistical and Continuum MethodsEulerian and Lagrangian CoordinatesMaterial DerivativeControl VolumesReynolds' delivery TheoremConservation of MassConservation of MomentumConservation of EnergyDiscussion of Conservation EquationsRotation and cost of ShearConstitutive EquationsViscosity CoefficientsNavier-Stokes EquationsEnergy EquationGoverning Equations for Newtonian FluidsBoundary ConditionsFlow KinematicsFlow LinesCirculation and VorticityStream Tubes and Vortex TubesKinematics of Vortex LinesSpecial kinds of the Governing EquationsKelvin'

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**Extra info for Fundamental Mechanics of Fluids, Fourth Edition**

**Sample text**

There are no preferred directions in the fluid, so that the fluid properties are point functions. Condition 1 requires that the stress tensor σij be of the form σij= −pδij + τij where τij depends upon the motion of the fluid only and is called the shear-stress tensor. The quantity p is the thermodynamic pressure and δij is the Kronecker delta. The pressure term is negative since the sign convention being used here is that normal stresses are positive when they are tensile in nature. The remaining unknown in the constitutive equation for stress is the shearstress tensor τij.

T ∂t δt ∂x δt ∂y δt ∂z The left-hand side of this expression represents the total change in α as observed in the Lagrangian framework during the time δt, and in the limit, it represents the time derivative of α in the Lagrangian system, which will be denoted by Dα/Dt. It may be also noted that in the limit as δt → 0, the ratio δx/δt becomes the velocity component in the x direction, namely, u. Similarly, δy/δt → v and δz/δt → w as δt → 0, so that in the limit, the expression for the change in α becomes Dα ∂α ∂α ∂α ∂α = +u +v + w .

During the time δt, the change in α may be calculated from differential calculus to be ∂α ∂α ∂α ∂α δt + δx + δy + δz. ∂t ∂x ∂y ∂z Equating the preceding change in α to the observed change δα in the Lagrangian framework and dividing throughout by δt gives δα ∂α δx ∂α δy ∂α δz ∂α . = + + + δt ∂t δt ∂x δt ∂y δt ∂z The left-hand side of this expression represents the total change in α as observed in the Lagrangian framework during the time δt, and in the limit, it represents the time derivative of α in the Lagrangian system, which will be denoted by Dα/Dt.

### Fundamental Mechanics of Fluids, Fourth Edition by I.G. Currie

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