By J. Tinsley Oden
A smooth method of mathematical modeling, that includes specified purposes from the sphere of mechanics
An advent to Mathematical Modeling: A direction in Mechanics is designed to survey the mathematical versions that shape the principles of contemporary technology and comprises examples that illustrate how the main winning types come up from simple ideas in glossy and classical mathematical physics. Written via an international authority on mathematical concept and computational mechanics, the publication provides an account of continuum mechanics, electromagnetic box concept, quantum mechanics, and statistical mechanics for readers with diverse backgrounds in engineering, computing device technological know-how, arithmetic, and physics.
The writer streamlines a entire knowing of the subject in 3 truly equipped sections:
Nonlinear Continuum Mechanics introduces kinematics in addition to strength and tension in deformable our bodies; mass and momentum; stability of linear and angular momentum; conservation of power; and constitutive equations
Electromagnetic box concept and Quantum Mechanics includes a short account of electromagnetic wave concept and Maxwell's equations in addition to an introductory account of quantum mechanics with comparable themes together with ab initio equipment and Spin and Pauli's principles
Statistical Mechanics offers an advent to statistical mechanics of platforms in thermodynamic equilibrium in addition to continuum mechanics, quantum mechanics, and molecular dynamics
Each a part of the ebook concludes with workout units that let readers to check their knowing of the offered fabric. Key theorems and basic equations are highlighted all through, and an intensive bibliography outlines assets for extra study.
Extensively class-tested to make sure an available presentation, An creation to Mathematical Modeling is a superb publication for classes on introductory mathematical modeling and statistical mechanics on the upper-undergraduate and graduate degrees. The e-book additionally serves as a useful reference for execs operating within the parts of modeling and simulation, physics, and computational engineering.
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Additional info for An Introduction to Mathematical Modeling: A Course in Mechanics
On the liquid/solid interface, solidiﬁcation heat is emitted. The stream of heat q released from the solidiﬁcation front is conducted through the solidiﬁed layer to the cold plate. The cold plate accumulates the emitted heat which causes an increase of its temperature. The solidiﬁcation front is sharp and rectangular. In addition, it is assumed that the heat accumulation by the solidiﬁed layer is very small in comparison with a cold plate. All thermodynamic parameters are considered constant. The energy balance describing the flow of heat from the flowing and solidifying liquid to the cold plate is deﬁned by the following equations (Lipnicki 2003) qs L dd TF À T TF À T t þ hð T 1 À T F Þ ¼ k s ; ks ¼ hCON ðT À TW Þ: d d d ð4:33Þ The temperature of a solidiﬁed layer is obtained from the above equation L dd h T ¼ TW þ qs Á t þ ðT1 À TF Þ : hCON d hCON To facilitate the analysis of the problem, a set of dimensionless was introduced: time, thickness of the solidiﬁed layer, coordinate axis, temperature and the wall temperature, appropriately deﬁned by d x T À T0 TW À T0 s ¼ Fo Á Ste; ~d ¼ ; ~x ¼ ; h ¼ ; hW ¼ ; H H T F À T0 TF À T0 Dimensionless Fourier, Stefan and Biot numbers Fo ¼ t Á js c s Á ð TF À T 0 Þ H Á hCON ; BiCON ¼ ; Ste ¼ ; L H2 ks and two dimensionless parameters were introduced ~¼ j jC ~ kC ;k¼ : jS kS Since the flowing liquid is not overheated, there is not free convection heat flow.
By substituting the above temperature function to Eq. 34), the following is obtained expðÀb Á sÞ ¼ ~d Á d~d 1 d~d þ : ds BiCON ds ð4:38Þ Integration of the above equation from s ¼ 0 to s ¼ 1 allows to deﬁne parameter b Zs expðÀbsÞ Á ds ¼ ~d2 ~d þ BiCON 2 0 Z1 expðÀbsÞ Á ds ¼ ) 0 ) b ¼ ~2 dmax 2 1 þ ~dmax BiCON : ~d2 ~dmax max þ 2 BiCON ð4:39Þ 42 4 Solidiﬁcation on a Rectangular Geometrics In this case, the thickness of the solidiﬁed layer varies from ~d ¼ 0 to ~d ¼ ~dmax , and constant b is deﬁned by ~dmax .
8 1 the solidiﬁcation process affects distribution of temperature in both the cooling plate and the solidiﬁed layer (see Figs. 16) and is very important in the solidiﬁcation process. 3 Free Convection Effect Natural convection of a liquid in a closed space occurs when it is affected by mass forces, and the temperature of walls of the reservoir differs from that of the liquid. These conditions are satisﬁed if the liquid is either heated or cooled. An omnipresent mass force is that of gravitation.