Unformatted text preview: 40 Physics Formulary by ir. J.C.A. Wevers When the ﬂow velocity is v at position r holds on position r + dr : v (dr ) = v (r )
translation T can be split in a part pI representing the normal tensions and a part T representing the shear stresses: T = T + pI, where I is the unit tensor. When viscous aspects can be ignored holds: divT= −gradp. + dr · (gradv )
rotation, deformation, dilatation The quantity L:=gradv can be split in a symmetric part D and an antisymmetric part W. L = D + W with Dij := 1 2 ∂ vj ∂ vi + ∂ xj ∂ xi , Wij := 1 2 ∂ vj ∂ vi − ∂ xj ∂ xi For a Newtonian liquid holds: T = 2η D. Here, η is the dynamical viscosity. This is related to the shear stress τ by: ∂ vi τij = η ∂ xj When the rotation or vorticity ω = rotv is introduced holds: W ij = 1 εijk ωk . ω represents the local rotation 2 velocity: dr · W = 1 ω × dr. 2 For compressible media can be stated: T = (η divv )I + 2η D. From equating the thermodynamical and mechanical pressure it follows: 3η + 2η = 0. If the viscosity is constant holds: div(2D) = 2 v + grad divv . The conservation laws for mass, momentum and energy for continuous media can be written in both integral and differential form. They are: Integral notation:
1. Conservation of mass: ∂ ∂t ∂ ∂t d3 V + (v · n )d2 A = 0 v (v · n )d2 A = f0 d3 V + n · T d2 A 2. Conservation of momentum: 3. Conservation of energy: ∂ ∂t − v d3 V + ( 1 v 2 + e) d3 V + 2 ( 1 v 2 + e) (v · n )d2 A = 2 (v · n T)d2 A (q · n )d2 A + (v · f0 )d3 V + Differential notation:
1. Conservation of mass: ∂ + div · ( v ) = 0 ∂t ∂v +( v· ∂t )v = f0 + divT = f0 − gradp + divT 2. Conservation of momentum: 3. Conservation of energy: T de p d ds = − = −divq + T : D dt dt dt Here, e is the internal energy per unit of mass E/m and s is the entropy per unit of mass S/m. q = −κ T is the heat ﬂow. Further holds: ∂e ∂E ∂e ∂E =− , T= = p=− ∂V ∂ 1/ ∂S ∂s so ∂e ∂h CV = and Cp = ∂T V ∂T p with h = H/m the enthalpy per unit of mass. ...
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