By Roustem N. Miftahof, Hong Gil Nam
Mathematical modelling of physiological structures delivers to boost our figuring out of complicated organic phenomena and pathophysiology of ailments. during this ebook, the authors undertake a mathematical method of symbolize and clarify the functioning of the gastrointestinal method. utilizing the mathematical foundations of skinny shell conception, the authors patiently and comprehensively advisor the reader during the primary theoretical thoughts, through step by step derivations and mathematical routines, from simple thought to complicated physiological versions. purposes to nonlinear difficulties concerning the biomechanics of stomach viscera and the theoretical barriers are mentioned. specific cognizance is given to questions of advanced geometry of organs, results of boundary stipulations on pellet propulsion, in addition to to medical stipulations, e.g. sensible dyspepsia, intestinal dysrhythmias and the influence of gear to regard motility problems. With finish of bankruptcy difficulties, this e-book is perfect for bioengineers and utilized mathematicians.
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Mathematical modelling of physiological structures can provide to increase our figuring out of complicated organic phenomena and pathophysiology of ailments. during this publication, the authors undertake a mathematical method of symbolize and clarify the functioning of the gastrointestinal approach. utilizing the mathematical foundations of skinny shell thought, the authors patiently and comprehensively consultant the reader in the course of the basic theoretical techniques, through step by step derivations and mathematical routines, from easy thought to advanced physiological versions.
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M Kirchhoff for thin plates and later applied by Love for thin shells. They are known as the Kirchhoff–Love hypotheses and are fundamental in the theory of thin shells. Additional hypotheses will be introduced in the text as needed. 6 Equations of compatibility For the surface to retain continuity during deformation, the parameters εik and æik (i, k = 1, 2) must satisfy the three differential equations called the equations of continuity of deformations. They can be obtained by subtracting the Gauss– Codazzi equations formulated for the undeformed state from the corresponding equations for the deformed conﬁguration.
11)) is called the ﬁrst fundamental form of the surface. It allows us to calculate the length of line elements, the angle between coordinate curves and the surface area pﬃﬃﬃ (1:12) dsΔ ¼ jr1 Â r2 jdα1 dα2 ¼ a dα1 dα2 ; and therefore it fully describes the intrinsic geometry of S. 2 Extrinsic geometry Let Γ be a non-singular curve on S parameterized by arc length s (Fig. 2 Extrinsic geometry m n 9 α2 τ Γ S n nb α1 Fig. 3 The extrinsic geometry of the surface and a local base fn; nb ; τ g associated with a curve Γ.
2 Parameterization of the equidistant surface 31 Δ where i ð. Þ is the covariant derivative in metric ajk ; δik is the Kronecker delta and of S. By substituting bik and bik are the components of the second fundamental form Ã Ã Eq. 4) the components of the metric tensor A on S are found to be Ã aik ¼ aik þ 2εik ; (2:13) where εik are the components of the tensor of tangent ﬁctitious deformations given by (compare this with Eqs. 67) in Chapter 1) Ã Ã 2εik ¼ ri rk À ri rk ¼ eik þ eki þ ajs eij eks þ $i $k : (2:14) In just the same way as we introduced bending deformations æik, we introduce bending ﬁctitious deformations of the surface S: Ã æik ¼ bik À bik : (2:15) The components εik and æik are interdependent and satisfy conditions of continuity similar to those given by Eqs.
Mathematical Foundations and Biomechanics of the Digestive System by Roustem N. Miftahof, Hong Gil Nam