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 = 2D. Here, is the dynamical viscosity. This is related to the shear stress by: vi ij = xj When the rotation or vorticity = rotv is introduced holds: W ij = 1 ijk k . represents the local rotation 2 velocity: dr W = 1 dr. 2 For compressible media can be stated: T = ( divv )I + 2D. From equating the thermodynamical and mechanical pressure it follows: 3 + 2 = 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 - vd3 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 = - 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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