By Günter Brenn
This ebook presents analytical strategies to a few classical difficulties in delivery strategies, i.e. in fluid mechanics, warmth and mass move. increasing computing strength and extra effective numerical tools have elevated the significance of computational instruments. although, the translation of those effects is usually tough and the computational effects have to be demonstrated opposed to the analytical effects, making analytical suggestions a useful commodity. additionally, analytical strategies for shipping tactics supply a far deeper knowing of the actual phenomena concerned with a given procedure than do corresponding numerical ideas. notwithstanding this ebook basically addresses the wishes of researchers and practitioners, it could actually even be priceless for graduate scholars simply getting into the field.
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Extra info for Analytical Solutions for Transport Processes: Fluid Mechanics, Heat and Mass Transfer (Mathematical Engineering)
E. 2) for constant fluid density is automatically satisfied. 2) for the stream function ψr . In this and all the following corresponding equations formulated in different coordinate systems, the Jacobian reads ∂( f, g) ∂ f /∂ x ∂ f /∂ y . 4) which is the Laplace operator in Cartesian coordinates, and ∇4 = respectively. 1 Linear, Unsteady Flow Analytical solutions of Eq. 2) are found for linear flow fields, where the Jacobian is either negligible or vanishes exactly. The linearisation leads to the equation − 1 ∂ + ∇ 2 ∇ 2 ψr = 0 .
In their arguments we have defined q = (α/ν)1/2 . 3 The Equation for the Stream Function in Spherical Coordinates 47 where ψs,m,0 is a constant. The general form of the stream function is a sum over all the modes m. When applying the general description for the Stokesian stream function to flows in special geometries, the values of the coefficients Ci,m are determined by initial and boundary conditions of the problem. Terms of the solution must be discarded in regions of the flow field where they diverge by setting the related coefficients Ci,m to zero.
1, we may start from Eq. 35) ψr = C0 + C1 y + C2 y 2 + C3 y 3 for the stream function, derived in Sect. 2. Corresponding to the flow field, this stream function does not depend on the coordinate x in the flow direction.