By Günter Brenn
This booklet presents analytical strategies to a couple of classical difficulties in shipping procedures, i.e. in fluid mechanics, warmth and mass move. increasing computing energy and extra effective numerical tools have elevated the significance of computational instruments. notwithstanding, the translation of those effects is frequently tricky and the computational effects must be established opposed to the analytical effects, making analytical strategies a precious commodity. in addition, analytical options for shipping tactics supply a far deeper realizing of the actual phenomena keen on a given technique than do corresponding numerical strategies. even though this publication basically addresses the wishes of researchers and practitioners, it could actually even be useful for graduate scholars simply coming into the sphere.
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Extra resources for Analytical Solutions for Transport Processes: Fluid Mechanics, Heat and Mass Transfer
The quantity C in Eq. 69) is then zero. 70) only. The function f (r ) is determined by the same differential equation as for the impulsively started flow, with the same solution as there. 83) where ψcθ,0 is a constant. These flows will be discussed in detail in Chap. 3. e. 84) E cθ 4 is given in Eq. 62). Separation of the stream function ψcθ into functions where E cθ of the radial coordinate r and the axial coordinate z, with the assumption that the solution is periodic in the axial direction with a wavenumber k = 2π/λ, where λ is the wavelength, leads to the following ODE for the function f (r ) representing the radial dependency of the stream function r d dr 1 d d r r dr dr f r − 2k 2 r d dr f r + k4 f = 0 .
The equation emerges as the one non-zero component of the curl of the two-dimensional momentum equation with the velocity components given as spatial derivatives of the stream function. The stream function is defined such that its derivatives yield a solenoidal velocity field. The analyses of the flows discussed in Part I of this book are based on this function. In view of our search for analytical solutions, we are restricted to laminar two-dimensional flow in simple geometries. The equations of change therefore need no turbulence modelling, the concept of the Stokesian stream function can be applied for representing the flow velocity, and the boundary conditions are easy to formulate and implement analytically in the general solutions.
The first minus sign in the equation differs from the plus in the Bessel differential equation. The solution of this equation has the form C1 r I1 (kr ) + C2 r K 1 (kr ), where I1 and K 1 are modified Bessel functions of the first and second kinds, respectively . For ψcθ,1 we therefore obtain ψcθ,1 = [C1r I1 (kr ) + C2 r K 1 (kr )] eikz−αt + const1 . 66) The quantity k = 2π/λ is the wave number of a spatially periodic variation of the stream function with the wavelength λ in the direction of the axial coordinate z.