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Passive transport in the interstitium and circulation: basics
Published in Benjamin Loret, Fernando M. F. Simões, Biomechanical Aspects of Soft Tissues, 2017
Benjamin Loret, Fernando M. F. Simões
To build the solution, the important property that 1/r is harmonic is used. A function is harmonic in a certain domain if its Laplacian vanishes in this domain. The velocity is sought in the form () w=∇ϕ+w1,w1≡(−c/r,0,0),cconstant,
Engineering Problems and Partial Differential Equations
Published in Guigen Zhang, Introduction to Integrative Engineering, 2017
The Laplacian operator plays a very important role in differential equations describing many engineering problems. The Laplacian of a field allows us to quantitatively compare the field at a selected point with those at neighboring points. An intuitive way to understand this is to recall the way we find extremes (maximum and minimum) of a 1D function: first, we set the first-order derivative of the function to zero to find the locations of the extremes, and then to know whether an extreme point is a maximum or minimum, we evaluate the corresponding second-order derivative. If the second-order derivative is greater than zero, we have a minimum; if it is zero, we have a local constant field; and if it is less than zero, we have a maximum.
Numerical differentiation and integration
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
The Laplacian is a differential operator given by the divergence of the gradient of a function, often denoted by ∇2 or Δ. In Cartesian coordinates, the Laplacian is given by the sum of second partial derivatives of the function with respect to x, y, and z.
Finite element solution of vibrations and buckling of laminated thin plates in hygro-thermal environment based on strain gradient theory
Published in Mechanics of Advanced Materials and Structures, 2023
Michele Bacciocchi, Nicholas Fantuzzi, Raimondo Luciano, Angelo Marcello Tarantino
The constitutive laws based on the strain gradient theory defined in [29] allow to relate the stresses in the kth layer to the corresponding strain components as shown below in which the nonlocal parameter is able to control the effect of the interactions at the micro/macro-scale. The dependency of the stresses on the strain distribution within the medium is emphasized by the presence of the Laplacian in Cartesian coordinate system: It should be specified that the theoretical approach presented by [73] is followed here. Thus, the nonlocal effect is involved only in this constitutive relation. Different dynamic approaches that could be used are reviewed, for completeness purposes, in the paper by Askes and Aifantis [29]. On the other hand, represents the plane stress-reduced stiffness coefficients matrix of the kth layer. The terms of this matrix depend on the orthotropic properties of the layer (Young’s moduli Poisson’s ratio and shear modulus ), as well as by an arbitrary orientation Their well-known explicit definition can be found in the book by Reddy [45].
Analytical modeling of sound transmission through FGM sandwich cylindrical shell immersed in convected fluids
Published in Mechanics of Advanced Materials and Structures, 2023
The acoustic velocity potentials of Eqs. (15), (17), and (18) for an inviscid, irrotational, and incompressible fluid moving should satisfy the convected wave equation, given by where is Laplacian operator in the cylindrical coordinate system. Substitution of Eqs. (15), (17), and (18) into Eqs. (19) and (20), one obtains the wave number in the incident side and transmitted side fluid mediums as where and are the wave number in the flow fluid, while and are the Mach number of mean flow in the incident field and that in the transmitted field, respectively. The acoustic wave propagating in the fluid medium close to the shell and the bending wave propagating in the shell should be consistent with each other in their wavelengths [36, 37], that is, so that the refraction angle is expressed as follows: which also describes the refraction laws for sound transmission from one medium to another.
Annular crack in a thermoelastic half-space
Published in Journal of Thermal Stresses, 2020
In the absence of thermal sources, the heat conduction problem under consideration is governed by the following Laplace’s equation where the symbol denotes the Laplacian operator expressed in the cylindrical coordinate system. This partial differential equation can be readily reduced to an ordinary one by employing the Hankel transform technique. We define the direct and inverse zero-order Hankel transforms of a function f by respectively, whereas J0 is the Bessel function of the first kind of order zero. Therefore, using the Hankel transform method, an appropriate integral representations of the temperature field for the two regions k = 1, 2, satisfying the Eq. (6) can be taken in the following forms where are unknown functions in ξ to be determined through thermal boundary conditions.