By Günter Brenn
This booklet presents analytical ideas to a couple of classical difficulties in delivery strategies, i.e. in fluid mechanics, warmth and mass move. increasing computing strength and extra effective numerical equipment have elevated the significance of computational instruments. despite the fact that, the translation of those effects is frequently tough and the computational effects must be confirmed opposed to the analytical effects, making analytical ideas a helpful commodity. moreover, analytical options for shipping approaches offer a miles deeper figuring out of the actual phenomena eager about a given strategy than do corresponding numerical recommendations. notwithstanding this e-book essentially addresses the wishes of researchers and practitioners, it will possibly even be worthy for graduate scholars simply coming into the sector.
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Extra info for Analytical Solutions for Transport Processes: Fluid Mechanics, Heat and Mass Transfer
Therefore, after introduction of the stream function, there is no need to take the curl of the momentum equation. In these cases, we obtain from the momentum equation in the main flow direction x the PDE for the stream function in Cartesian coordinates ∂ψr ∂ 2 ψr ∂ 3 ψr ∂ψr ∂ 2 ψr − = ν . 38) The stream function is composed of a mapping function of the spatial coordinate x only and a function of a self-similar coordinate. The self-similar coordinate is a y coordinate transverse to the main flow direction, normalised by the boundary-layer thickness, which itself depends on the x position in the field.
1 Linear, Unsteady Flow Analytical solutions of Eq. 95) are found for linear flow fields, where the Jacobian is either negligible or vanishes exactly. The linearisation for spherical flows in the r, θ plane leads to the equation − 1 ∂ + E s2 ν ∂t E s2 ψs = 0 . 97) In solving this equation, we use the approach of Tomotika developed for the cylindrical case for the spherical problem as well . The details are given in Sect. 4. 99) respectively. The two Eqs. 99) are solved by separation of variables, which reveals the functions ψs,1 and ψs,2 as products of eigenfunctions of the two operators in the radial and angular coordinates and in time.
44) to itself. The operator reads ∂ 1 ∂ ∂ 1 ∂ 1 ∂ ∂ r r + r r ∂r ∂r r ∂r ∂r r ∂r ∂r 1 ∂ 1 ∂4 ∂3 + 3 . 51) is the biharmonic equation. 53) where f = d f /dr . 54) which, together with the periodic solution in the angular coordinate θ , yields the stream function ψczs,m − ψczs,m,0 = C1,m r m + C2,m r −m + C1,m r m+2 + C2,m r −m+2 eimθ . 2 The Equation for the Stream Function in Cylindrical Coordinates 37 In case of axial symmetry of the flow field, where the dependency of the field on the polar angle θ vanishes and there is no component of a pressure gradient in that direction, the value of the mode number m is zero and the highest order derivative is the third.