By Günter Brenn
This ebook offers analytical strategies to a few classical difficulties in shipping approaches, i.e. in fluid mechanics, warmth and mass move. increasing computing strength and extra effective numerical equipment have elevated the significance of computational instruments. notwithstanding, the translation of those effects is usually tough and the computational effects have to be demonstrated opposed to the analytical effects, making analytical ideas a priceless commodity. moreover, analytical suggestions for delivery strategies supply a far deeper knowing of the actual phenomena serious about a given technique than do corresponding numerical options. notwithstanding this booklet basically addresses the wishes of researchers and practitioners, it might even be useful for graduate scholars simply coming into the sphere.
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Additional resources for Analytical Solutions for Transport Processes: Fluid Mechanics, Heat and Mass Transfer
This behaviour is seen already when transforming the equations of motion in cylindrical and spherical coordinates into the self-similar forms. 32) ∇ 4 ψr = 0 . 32) is the biharmonic equation. It is solved with the aim to find the solution in terms of eigenfunctions of the differential operators involved. Separating the stream function into one function of the coordinate x and one of the coordinate y, and assuming the stream function to be wave-like in the x direction of the main flow, we obtain the solution ψr,sx = C1 eky + C2 e−ky C3 eikx + C4 e−ikx .
46) may be used to derive a dispersion relation for a disturbed jet. The most unstable disturbance wavelength predicted by this relation differs from Rayleigh’s result by less than 2 % . We will discuss the stability of two-phase flows with interfaces in Chap. 6. 5 Quasi One-Dimensional Flow in a Boundary Layer Another example for a quasi one-dimensional description of a flow is the integral formulation of the equations of motion in the boundary-layer form. The idea behind this formulation is to integrate the momentum equation in the coordinate direction across the thickness of a boundary layer as in Fig.
We first solve Eq. 69), for the case that C = C1 + Ct e−αt . 70) ψcθ = ψcθs (r ) + f (r )e−αt , which is composed of a steady and an unsteady part. 71) where the prime denotes the derivative with respect to the coordinate r . 72) ψcθs = C1r 4 + C2 r 2 ln r − + C3r 2 + C4 . 70) of the stream function into Eq. 71) for the steady part, we obtain the following ODE for the unknown function f (r ): f d α f 1 d r = Ct . 74) which we rewrite to obtain r 2h + r h + α 2 r h = Ct r 2 . 76) ξ 2 k + ξ k + ξ 2 k = Ct r 2 .