Download e-book for iPad: Mathematical Foundations and Biomechanics of the Digestive by Roustem N. Miftahof, Hong Gil Nam

By Roustem N. Miftahof, Hong Gil Nam

ISBN-10: 0521116627

ISBN-13: 9780521116626

Mathematical modelling of physiological platforms offers to develop our realizing of complicated organic phenomena and pathophysiology of illnesses. during this e-book, the authors undertake a mathematical method of symbolize and clarify the functioning of the gastrointestinal method. utilizing the mathematical foundations of skinny shell concept, the authors patiently and comprehensively consultant the reader during the primary theoretical options, through step by step derivations and mathematical routines, from uncomplicated concept to advanced physiological types. functions to nonlinear difficulties with regards to the biomechanics of belly viscera and the theoretical barriers are mentioned. specified recognition is given to questions of advanced geometry of organs, results of boundary stipulations on pellet propulsion, in addition to to scientific stipulations, e.g. useful dyspepsia, intestinal dysrhythmias and the impact of gear to regard motility issues. With finish of bankruptcy difficulties, this ebook is perfect for bioengineers and utilized mathematicians.

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Setting Fi ðα1 ; α2 Þ ¼ 0 from Eq. 9) for the displacement vector, we have ∗ m m = mz ∗ S ρ2′ r2 Mz ∗ M Sz ∗ α2 ρ1′ α∗ 2 α∗ 1 r1 Hz m r2 α2 M(αi) r1 α1 S Fig. 2 Fictitious deformation of the surface. 34 Shells of complex geometry νðα1 ; α2 Þ ¼ Hðα1 ; α2 Þm:  (2:29) à  from the surface S to S . Evidently, the measured along m Here Hðαi Þ is the distance à vector equation of S can be written as à r ¼ rðα1 ; α2 Þ þ Hðα1 ; α2 Þm:  Let (2:30) à Hðα1 ; α2 Þ ¼ Hz þ Hðα1 ; α2 Þ: (2:31) Then, on substituting Eq.

By substituting Eqs. 28) are used to calculate the Christoffel symbols Ãk à ðzÞk Gij on S . Gij are calculated from Eqs. 27) by replacing Azi and their derivatives for Ai and @Ai =@α1;2 , respectively. à in Fig. 3. Let a For example, consider a shell of complex geometry S as shown à cylinder ofà constant radius R0 be the reference surface for S. 30). Introduce polar coordinates α1 and α2 on S, such that α1 is the axial and α2 is the polar angular coordinate. They are related to the global Cartesian coordinates by 40 Shells of complex geometry À Á rðαi Þ ¼ xi þ yj þ z k ¼ R0 i sin α2 þ k cos α2 þ α1 j: (2:59) The Lamé parameters Ai and curvatures kij are given by A1 ¼ 1; k11 ¼ 1=R1 ¼ 0; A2 ¼ R0 ; k12 ¼ 0; k22 ¼ 1=R2 ¼ 1=R0 : (2:60) For the coefficients θi ¼ 1 þ Hðαi Þ=Ri we have θ1 ¼ 1; θ2 ¼ 1 þ Hðαi Þ=R0 : (2:61) Hence, from Eq.

3 The extrinsic geometry of the surface and a local base fn; nb ; τ g associated with a curve Γ. τ ¼ dr dα1 dα2 þ r2 : ¼ r1 ds ds ds (1:13) By applying the Frenet–Serret formula for the derivative of τ with respect to s we get  n dτ ¼ ; ds Rc (1:14) n is the vector normal to Γ. By substituting Eq. 8) we obtain  n¼ 2 X 2 X i¼1 k¼1 rik dαi dαk dα2 dα2 þ r1 1 þ r2 2 ; ds ds ds ds (1:15) where rik ¼ @2r @2r ¼ ; @αi @αk @αk @αi rik ¼ rki :  and n such that m  n ¼ cos j. Then the Let φ be the angle between the vectors m  yields scalar product of Eq.

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Mathematical Foundations and Biomechanics of the Digestive System by Roustem N. Miftahof, Hong Gil Nam

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