By Günter Brenn
This e-book offers analytical ideas to a few classical difficulties in delivery techniques, i.e. in fluid mechanics, warmth and mass move. increasing computing energy and extra effective numerical equipment have elevated the significance of computational instruments. even though, the translation of those effects is frequently tough and the computational effects must be validated opposed to the analytical effects, making analytical options a useful commodity. in addition, analytical strategies for shipping tactics supply a far deeper knowing of the actual phenomena fascinated by a given procedure than do corresponding numerical suggestions. notwithstanding this ebook essentially addresses the wishes of researchers and practitioners, it might even be priceless for graduate scholars simply getting into the sector.
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Additional info for Analytical Solutions for Transport Processes: Fluid Mechanics, Heat and Mass Transfer
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 . This is an inhomogeneous form of a Bessel-type ordinary differential equation. The solution of the homogeneous equation is composed of the two zero-order Bessel functions of the first and second kinds, J0 (ξ ) and Y0 (ξ ).
Rev. 42, S269–S282 (1989) 12. : Exact solutions of the steady-state Navier-Stokes equations. Annu. Rev. Fluid Mech. 23, 159–177 (1991) 13. : Free Liquid Jets and Films: Hydrodynamics and Rheology. Longman Scientific & Technical, New York (1993) Chapter 2 The Equation for the Stokesian Stream Function and Its Solutions Abstract This chapter presents and discusses the equation for the Stokesian stream function. 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.
12) The solution of the equation is obtained by the same means as for the previous version. 13) where l 2 = k 2 +(−α +ikU0 )/ν. We will discuss applications of this stream function to the stability analysis of plane liquid sheets in Chap. 6. For a spatially two-dimensional linear unsteady flow, with the flow velocity varying with the coordinate y, but not with x, in contrast, the differential equation for the stream function reads ∂ 2 ∂ 2 ψr 1 ∂ + 2 =0. 15) ψr − ψr,0 = C1 y + C2 + C1 eiqy + C2 e−iqy e−αt , where we have defined q = (α/ν)1/2 .