What is Boundary: Definition and 999 Discussions

In physics and fluid mechanics, a boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. The liquid or gas in the boundary layer tends to cling to the surface.
The boundary layer around a human is heated by the human, so it is warmer than the surrounding air. A breeze disrupts the boundary layer, and hair and clothing protect it, making the human feel cooler or warmer. On an aircraft wing, the boundary layer is the part of the flow close to the wing, where viscous forces distort the surrounding non-viscous flow. In the Earth's atmosphere, the atmospheric boundary layer is the air layer (~ 1 km) near the ground. It is affected by the surface; day-night heat flows caused by the sun heating the ground, moisture, or momentum transfer to or from the surface.

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  1. G

    A Laplace eq. in cylindrical coordinates and boundary conditions

  2. Z

    Tackling Boundary Conditions in Python (Griffins Example)

    How to run a numerical simulation of Laplace equation if one of the boundary condition is like this: $$V(x,y) = 0 \text{ when } x \to \infty$$ I am trying to use Python to plot the solution of this Example 3.5. in Griffins EM
  3. HansBu

    Laplace's Equation and Boundary Condition Problem

    I really have no idea as to how to attack the problem in the first place. I am here to ask for some generous help on how to start. The figure is shown below for reference.
  4. Leonardo Machado

    A Improper boundary in non-linear ODE (pseudospectral methods)

    Hello, I am trying to compute some non-linear equations with pseudospectral/collocation methods. Basically I am expanding the solution as $$ y(x)=\sum_{n=0}^{N-1} a_n T_n(x), $$ Being the basis an Chebyshev polynomial with the mapping x in [0,inf]. Then we put this into a general...
  5. Tony Hau

    Boundary conditions of linear materials

    The question is as follows: Solutions given only contain part a) to c), which is as follows: So I now try to attemp d), e) and f). d) The magnetic field of a uniformly magnetized sphere is: $$ \vec B =\frac{2}{3}\mu_{o} \vec M = \mu_{o}\vec H$$ $$\frac{2}{3}\vec M = \vec H$$ The perpendicular...
  6. C

    I The diffusion equation with time-dependent boundary condition

    Hi everyone, I am trying to solve the 1 dimensional diffusion equation over an interval of 0 < x < L subject to the boundary conditions that C = kt at x = 0 and C = 0 at x = L. k is a constant. The diffusion equation is \frac{dC}{dt}=D\frac{d^2C}{dx^2} I am using the Laplace transform method...
  7. LCSphysicist

    I Boundary conditions for TM and TE waves

    Theta in the incident angle Phi is the refraction angle '' denotes everything that propagates to the other medium, that is, everything related to refraction ' denotes the reflection in the original medium I am rather confused, would appreciate any help. I see the second equation of TE is...
  8. jk22

    A Exploring the Boundary of the 3-Sphere: A Topological Analysis

    On wikipedia : https://en.m.wikipedia.org/wiki/3-sphere We learn that this manifold is without boundary. Is there a simple analytical method to obtain out if its parametrization the fact that the boundary is empty ?
  9. D

    A Name for a subset of real space being nowhere a manifold with boundary

    I was wondering if anyone knew of a name for such a set, namely a subset S \subseteq \mathbb{R}^n which at every point x \in S there exists no open subset U of \mathbb{R}^n containing x such that S \cap U is homeomorphic to either \mathbb{R}^m or the half-space \mathbb{H}^m = \{(y_1,...,y_m)...
  10. C

    Today's Climate Change and the Permian-Triassic Boundary

    This video shows how geologists figured out that a huge greenhouse effect nearly wiped out life on Earth 250 million years ago. The explanation is premised on an analysis of rock samples found at the Permian-Triassic Boundary in Utah. It involves a lot of Chemistry and Earth science so I...
  11. Z

    Boundary Value and Separation of Variables.

    If the boundary condition is not provided in the form of electric potential, how do we solve such problem? In this case, I want to use ##V = - \int \vec{E} \cdot{d\vec{l}}##, but I don't know how to choose an appropriate reference point.
  12. JD_PM

    I Assuming boundary conditions when integrating by parts

    Let's present two examples $$-\frac 1 2 \int d^3x'\big (-i \phi(x', t)\nabla^2\delta^3(x-x') \big )$$ Explicit evaluation of this integral yields $$-\frac 1 2 \int d^3x'\big (-i \phi( \vec x', t)\nabla'^2\delta^3(\vec x-\vec x') \big ) =\frac{i}{2}\phi(\vec x', t) \nabla' \delta^3(\vec...
  13. Rzbs

    I Electron scattering in the Brillouin zone boundary

    I what to know what is electron scattering in Brillouin zone boundary? What exactly happen for electron in Brillouin zone boundary; what happen for it in real space and what happen for it in reciprocal space? And is electron scattering from a Brillouin zone boundary could be a source for...
  14. K

    Automotive Modal analysis - Input and boundary conditions given?

    My understanding in modal analysis is very limited. All I know is it helps to find a specific mode of vibration and the natural frequency corresponding to it. While I was discussing about this with my NVH team colleague, he told me that there is no force input or excitation input given to a...
  15. M

    I Vibrations of waves with pinned vs free boundary conditions

    Hi PF! Can someone explain to me why in math/physics the frequencies associated with waves (or say drum heads) tend to be larger when the boundaries are pinned as opposed to free? If possible, do you know any published literature on this? Thanks!
  16. Riccardo Marinelli

    A Boundary conditions of eigenfunctions with Yukawa potential

    Hello, I was going to solve numerically the eigenfunctions and eigenvalues problem of the schrödinger equation with Yukawa Potential. I thought that the Boundary condition of the eigenfunctions could be the same as in the case of Coulomb potential. Am I wrong? In that case, do you know some...
  17. E

    B Gradient of scalar field is zero everywhere given boundary conditions

    I'm struggling with a few steps of this argument. It's given that we have a surface ##S## bounding a volume ##V##, and a scalar field ##\phi## such that ##\nabla^2 \phi = 0## everywhere inside ##S##, and that ##\nabla \phi## is orthogonal to ##S## at all points on the surface. They say this is...
  18. R

    Engineering Relationship between the solution convergence and boundary conditions

    I create an algorithm that can solve [K]{u}={F} for atomic structure, but the results are not converge Do the boundary conditions affect the convergence of the resolution of a system of nonlinear partial equations? And how to know if the solution is diverged because of the boundary conditions...
  19. Leonardo Machado

    A Boundary conditions in the time evolution of Spectral Method in PDE

    Hi everyone! I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example, $$ u_t=k u_{xx}, $$ $$ u(t,-1)=\alpha, $$ $$ u(t,1)=\beta, $$ $$ u(0,x)=f(x), $$ $$...
  20. person123

    I Boundary Conditions For Modelling of a Fluid Using Euler's Equations

    Hi! I want to use Euler's equations to model a 2 dimensional, incompressible, non-viscous fluid under steady flow (essentially the simplest case I can think of). I'm trying to use the finite difference method and convert the differential equations into matrices to be solved using MATLAB. I set...
  21. jk22

    I Do there exist surfaces whose boundary is a closed knot?

    I ask this for the condition of application of Stoke's theorem.
  22. S

    Heat Equation with Periodic Boundary Conditions

    I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be: $$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...
  23. B

    Topological insulators and their optical properties

    I have tried to write down the boundary conditions in this case and looked into them. As conditions i) and ii) were trivial, i looked into iii) and iv) for information that I could use. But all I got was that for the transmitted wave to have an angle, the reflective wave should also have an...
  24. L

    A Proving that the real projective plane is not a boundary

    While all orientable 2 dimensional compact smooth manifolds are boundaries e.g. the sphere is the boundary of a solid sphere, the torus is the boundary of a cream filled doughnut - not all unorientable surfaces are boundaries. For instance, The Projective Plane is not the boundary of any 3...
  25. M_Abubakr

    Wave dispersion in 2D Unit cell subjected to a periodic boundary

    How do I get the wave dispersion for a 2D continuum unit cell subjected to a periodic boundary which is excited longitudinally? I'll be applying forces in ABAQUS with varying frequencies. I have come across Blochs theorem but I can't find any application of it in continuous systems. Every...
  26. dRic2

    I The density of states independent of Boundary Conditions

    Most undergrad textbook simply say that it is intuitive that boundary conditions should not play a role if the box is very large. Other textbooks suggest that this should be taken for granted since the number of particles at the surface are orders of magnitude smaller that the number of bulk...
  27. M_Abubakr

    RVE Periodic Boundary Conditions in ANSYS Workbench Modal Analysis

    Hi I have a project regarding micromechanics of composites. I'm starting my analysis on the Fiber Matrix RVE. Right now I'm trying to find the natural frequency of the unit cell. The Unit cell has some unique geometry which I will keep on changing to see how natural frequency changes. I have...
  28. patricio ramos

    Question about initial and boundary conditions with the heat equation

    I am seeing the heat conduction differential equation, and I was wondering about a boundary condition when the equation is of transient (unsteady) nature. When analyzing boundary conditions at the surface of say, a sphere, the temperature does not depend on time. For example, if you have...
  29. LordGfcd

    What is the continuity condition for the heat flux through a boundary?

    Assume there is a boundary separates two medium with different heat conductivity [κ][/1] and [κ][/2]. In one medium, the temperature distribution is [T][/1](r,θ,φ) and on the other medium is [T][/2](r,θ,φ). What is the relationship between [T][/1] and [T][/2] ? Is it - [κ][/1]grad [T][/1]=-...
  30. Miles123K

    Wave behavior across two semi-infinite membranes with a special boundary

    Since the membrane doesn't break, the wave is continuous at ##x=0## such that ##\psi_{-}(0,y,t) = \psi_{+}(0,y,t)## ##A e^{i(k \cos(\theta)x + k \sin(\theta)y - \omega t)} = A e^{i(k' \sin(\theta ') y- \omega t)}## Which is only true when ## k' \sin(\theta ') = k \sin(\theta) ##. From the...
  31. F

    Boundary conditions for a purely inductive load in an AC circuit

    Hi all, Kirchhoff's equation for this simple circuit is equivalent to \dot I=\frac{V}{L} Where V=V_0 \sin(\omega t). Integrating both sides should give I(t) = -\frac{V_0}{L\omega} \cos(\omega t)+c where c is an arbitrary constant (current). Here, most of the derivations I've found simply drop...
  32. Kaushik

    Reflection of a wave by a rigid boundary

    I found this on the internet. Source How does the crest reach the end of the medium? As the other end is fixed there is no way the crest can reach the interface. Isn't it? My book gave an alternative explanation. It stated that as there is no net displacement at the interface, we can use the...
  33. D

    Compute the flux of a vector field through the boundary of a solid

    is it correct if i use Gauss divergence theorem, computing the divergence of the vector filed, that is : div F =2z then parametrising with cylindrical coordinates ##x=rcos\alpha## ##y=rsin\alpha## z=t 1≤r≤2 0≤##\theta##≤2π 0≤t≤4 ##\int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{4} 2tr \, dt \, dr...
  34. Leonardo Machado

    A Boundary conditions for the Heat Equation

    Hello guys. I am studying the heat equation in polar coordinates $$ u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}) $$ via separation of variables. $$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$ which gives the ODEs $$T''+k \lambda^2 T=0$$ $$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$...
  35. currently

    Deriving ground state electron energy using Boundary Value

    This is the equation given. I attempted to use Radial Equation, obtained from separating variables, to solve for ##E_1##.
  36. W

    A Conditions to extend functions Continuously into the Boundary (D^1/S^1)

    Other than for null-homotopic maps, which continuous maps defined on ##D^1 \rightarrow D^1## (Open disk)extend continuously to maps ##B^1 \rightarrow B^1 ## ,(##B^1## the closed disk) which maps can be extended in opposite direction, i.e., continuous maps ## f: S^1 \rightarrow S^1 ## that...
  37. tworitdash

    A Dirichlet and Neumann boundary conditions in cylindrical waveguides

    The book of Balanis solves the field patterns from the potential functions. Let say for TE modes, it is: F_z(\rho, \phi, z) = A_{mn} J_m(\beta_{\rho}\rho) [C_2 \cos(m\phi) + D_2 \sin(m\phi)] e^{-j\beta_z z} There is no mention of how to solve for the constant A_{mn} . Then, from a paper...
  38. C

    Please help me understand how transverse waves reflect at a boundary

    Hello, I am a student who is trying to learn some physics independently so I apologize in advance if I am not making sense. I have studied physics a bit in school but nothing very rigorous and it is a subject that I have trouble with, especially waves. This is what I have been reading...
  39. T

    I Equal or larger/smaller versus larger/smaller in boundary conditions

    Hi everyone! This is the first time I'm posting on any forum and I'm still rather unsure of how to format so I'm sorry if it seems wonky. I'll try my best to keep the important stuff consistent! I am working on infinite square well problems, and in the example problem: V(x) = 0 if: 0 ≤ x ≤ a...
  40. maistral

    Boundary conditions for convective heat and mass transfer + wall Temperature

    I am operating via finite differences. Say for example, I have this pipe that contains a fluid. I have the boundary condition at x = x1: k is the effective thermal conductivity of the fluid, T is the temperature of the fluid at any point x, hw is the wall heat transfer coefficient, and Tw is...
  41. S

    I The Multiverse and 'No boundary' conditions approach in cosmology

    Summary: Questions about the Multiverse hypothesis and the 'No boundary' conditions approach in cosmology I have some questions about James Hartle and Stephen Hawking's 'No-boundary' proposal: - In their approach multiple histories would exist. These histories could yield universes with...
  42. E

    Boundary Potential Problem- parallel alternating potential

  43. H

    A Crank-Nicholson method and Robin boundary conditions

    I have the following PDE I wish to solve: \frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{\partial x^{2}} With the following boundary conditions: \frac{\partial u}{\partial x}(t,1)+u(t,1)=f(t),\quad u(t,0)=0 Now, I wish to do this via the Crank-Nicholson method and I would naively...
  44. Quantum Alchemy

    Is the Quantum/Classical Boundary the most important question in Physics?

    I have been reading Sean Carroll's recent book called Something Deeply Hidden where he advocates a Many Worlds Interpretation of Quantum Mechanics. I was thinking that the most important question in all of Physics might be is everything quantum or is there a quantum/classical boundary. If we...
  45. sdefresco

    3-D Planar Method of Images Boundary Problem

    I understand the idea of the method of images, and its clever use of uniqueness to determine V(x,y,z) for non-trivial systems. My question now is simply about guidance for obtaining the effective "image" of this system, as it is clearly more complicated than the 2-plane analogue (in which there...
  46. Terrycho

    Partial Differential Equation: a question about boundary conditions

    Consider the following linear first-order PDE, Find the solution φ(x,y) by choosing a suitable boundary condition for the case f(x,y)=y and g(x,y)=x. --------------------------------------------------------------------------- The equation above is the PDE I have to solve and I denoted the...
  47. F

    B Why Jupiter, Saturn and the Sun have a distinctive and sharp boundary?

    Pardon the very naive question, but why does the atmosphere in these gas giants seem to have, from a distance, a very clear, sharp and distinctive boundary? When one looks at Earth's atmosphere from space, it seems to have a fuzzy bluish boundary, gracefully vanishing into the black. I read...
  48. maistral

    A Deriving the derivative boundary conditions from natural formulation

    PS: This is not an assignment, this is more of a brain exercise. I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition...
  49. Math Amateur

    MHB Understanding Topology: Closure, Boundary & Open/Closed Sets

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ... I need some help in order to fully understand a statement by Browder in Section 6.1 ... ... The...
  50. K

    I Boundary terms for field operators

    Hello! In several of the derivations I read so far in my QFT books (M. Schawarz, Peskin and Schroeder) they use the fact that "we can safely assume that the fields die off at ##x=\pm \infty##" in order to drop boundary terms. I am not sure I understand this statement in terms of QFT. A field in...
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