Continuum Modeling: An Approach Through Practical Examples by Adrian Muntean

By Adrian Muntean

This e-book develops continuum modeling abilities and techniques the subject from 3 aspects: (1) derivation of world imperative legislation including the linked neighborhood differential equations, (2) layout of constitutive legislation and (3) modeling boundary procedures. the focal point of this presentation lies on many useful examples masking elements equivalent to coupled move, diffusion and response in porous media or microwave heating of a pizza, in addition to site visitors matters in bacterial colonies and effort harvesting from geothermal wells. the objective viewers includes basically graduate scholars in natural and utilized arithmetic in addition to operating practitioners in engineering who're confronted via nonstandard rheological issues like these regularly coming up within the meals industry.

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By Adrian Muntean

This e-book develops continuum modeling abilities and techniques the subject from 3 aspects: (1) derivation of world imperative legislation including the linked neighborhood differential equations, (2) layout of constitutive legislation and (3) modeling boundary procedures. the focal point of this presentation lies on many useful examples masking elements equivalent to coupled move, diffusion and response in porous media or microwave heating of a pizza, in addition to site visitors matters in bacterial colonies and effort harvesting from geothermal wells. the objective viewers includes basically graduate scholars in natural and utilized arithmetic in addition to operating practitioners in engineering who're confronted via nonstandard rheological issues like these regularly coming up within the meals industry.

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We refer the reader to [BOB14] and references cited therein for more details on this topic. We wish to build here a possible model to describe this particular geothermal scenario. Let x be a point in a macroscopic domain Ω ⊂ R3 surrounding the well (borehole), and consider a typical pore Yx ⊂ R3 positioned at x. Aquifers are by definition saturated with water. This fact allows us to model Yx as composed of a (connected and simply connected) solid part Bx1 and a fluid part Bx2 . Mathematically, Yx = Bx1 ∪ Bx2 , with the two (closed) domains intersecting along the boundary Γx = Bx1 ∩ Bx2 .

Boundary conditions and discontinuities appear as prominent issues. We give a few examples of constitutive equations for the stress tensor and for the transport fluxes. A set of worked-out practical modeling scenarios is emphasized. 1 Notation, Body and Configurations, Coordinates Throughout this chapter, we use scalars in R (like m, ρ, θ), vectors in Rd (d ∈ {1, 2, 3}) (like x, v), and tensors in Rd × Rd (d ∈ {1, 2, 3}) (like F, I, T). Sets will typically refer to either bodies or configurations.

3 Propose a complete mathematical model able in principle to describe the balance of the total energy of the geometry Ω sketched in Fig. 2. 30 2 Balance Equations in Continuum Physics Fig. 2 How to ensure the thermal comfort in my flat? 6 Comments on Conservation of Momenta Recall the definition of the linear momentum (Ω (t), t) := Ω (t) vρdx and that of the angular momentum α(Ω (t), t) := Ω (t) x × vρdx. 1 Show that for every part Ω (t) ⊂ Ω(t), we have d (Ω (t), t) = dt d α(Ω (t), t) = dt Ω (t) Ω (t) ρ Dv dx Dt x×ρ Dv dx.

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