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. S

    How do I get this interface condition?

    Homework Statement I have an infinitely long cylinder of a dielectric material, surrounded by another dielectric material and coated with graphene which has surface conductivity \sigma, implying it has a superficial current. The sheet of graphene is very thin, and the dielectrics are asumed to...
  2. S

    Interface conditions on a graphene interface

    Homework Statement I'm trying to do all the calculations for the attached paper, and I'm having trouble with the boundary conditions for P polarization. My question is, how can I arrive to those conditions? The problem is 2D, an infinite dielectric cylinder coated with a layer of graphene...
  3. Pushoam

    Force exerted by a particle in a box on the boundary

    Homework Statement Homework EquationsThe Attempt at a Solution The energy eigen - value of a particle in a box is given by ## E_n = \frac { n^2 h^2}{8mL^2} ## . Now, applying classical mechanics , ## \frac { p^2}{2m} = \frac { n^2 h^2}{8mL^2} ## . ## p \propto \frac { 1} L ## , So...
  4. T

    Boundary condition for PDE heat eqaution

    Homework Statement I am having an issue, not with the maths but with the boundary conditions for this question. A bar 10 cm long with insulated sides, is initially at ##100 ^\circ##. Starting at ##t=0## Find the temperature distribution in the bar at time t. The heat flow equation is...
  5. K

    Boundary conditions of a bending plate

    Homework Statement I'm trying to find the boundary conditions for the following problem: A plate with length 2L is placed on supports at x = L/2 and x = - L/2. The plate is deforming elastically under its own weight (maximum displacement bowing up at x = 0). Both ends of the plate are free...
  6. PsychonautQQ

    A Boundary operator on p-chains

    Can someone help me to understand what the boundary operator on a p-chain is doing exactly? Or boundary operators in general? I really need to develop a better intuition on the matter.
  7. A

    A Fourier Transform for 3rd kind of boundary conditions?

    I am studying online course notes from University of Waterloo on 'Analytical mathematics in geology' in which the author describes a 'modified Fourier transform' which can be used to incorporate 3rd kind of boundary conditions. The formula is ## \Gamma \small[ f(x) \small] = \bar{f}(a) =...
  8. G

    Boundary layer thickness in partial air vacuum

    How does low ambient pressure effect on the thickness of the boundary layer for given flow conditions of gas? In absolute vacuum the thickness of the boundary layer is 0 of course, but it seems that all the boundary layer thickness correlations use the Reynolds number, which doesn’t change much...
  9. T

    A Free Simulator Software for Vacuum Field BVPs | Tom

    Hi, I am interested in simulating the vacuum field equations, but solving a full boundary value problem rather than the initial value problem. i.e. I might have boundary conditions in all spatial and temporal extents/extremes, rather than just an initial 3D surface. Does anyone know any free...
  10. M

    Kinematic Boundary Condition Fluids

    Hi PF! Given a fluid/fluid interface ##F(t,x,y,z) = 0## the kinematic boundary condition states ##DF/Dt = 0##. Given ##y=F## the text states ##v = \partial_tF+u\partial_xF+w\partial_zF##. How is this possible? I thought ##DF/Dt = 0 = \partial_tF+u\partial_xF+v\partial_yF+w\partial_zF## and...
  11. M

    Spherical Boundary condition problem

    Homework Statement I'm trying to keep the post brief and will post more info if needed. But I am trying to understand how the value of two "A" constants were found. This is from Griffiths Electrodynamics. In this part of the problem, I am given a boundary condition that is a function of theta...
  12. M

    Electrostatics Boundary condition and coordinate choice

    Homework Statement So I have an equation V = Ae(kx)+Be(-kx) And boundary conditons V= V0 when x=0 and V= 0 when x=b 2. Homework Equations I have solved ones where v=0 at x=0 where it nicely simplifies as the exponentials =1 and the Coeffecients A=-B which leads to a sinh function and I...
  13. F

    A Please verify integral and approximation, boundary theory

    I used Newtons method and taylor approximations to solve this equation $$f'''+\frac{m+1}{2}ff''+m(1-f^{'2})=0$$ It solves for velocity of air over a flat plate. The velocity is a constant ##u_e## everywhere except in a boundary layer over the plate, where the velocity is a function of distance...
  14. F

    Partial differential equation boundary

    Homework Statement I have to calculate the stationary field inside a room. Homework EquationsThe Attempt at a Solution I used the diffusion equation to calculate the temperature, which is T(x,y)=(Eeknx+Fe-knx)cos(kny), k=(n*pi/a), a is the length of the room. Now i have to satisfy boundary...
  15. Marcin H

    Boundary Conditions & finding E

    Homework Statement Find E1, E3, and ps2 Homework Equations boundary conditions The Attempt at a Solution (these are class notes)[/B] I understand how to find E1, but I am a bit confused about the reasoning behind finding E3... Why do we leave the 2x(hat) for E3...? I though that only...
  16. A

    I How to simulate a lattice with boundary conditions?

    Hello, My question is very simple but I do not have a lot of experience with simulation. I want to write some code to simulate a lattice with boundary conditions and then I will perform calculations with the Hubbard model to find different kinds of properties of interest. I would like to know...
  17. D

    Boundary conditions for a step potential

    Homework Statement A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##. What is the boundary conditions for this eigenfunctions? Find the degeneracy level for the energy, when it is ##E<V_0## Homework Equations Radial equation \begin{equation}...
  18. D

    Boundary conditions with l=0

    The question is basically find the boundary conditions when ##l=0##, for energies minor than 0. Homework Equations $$V(r)=\begin{cases} & 0\text{ $r<a_0$}\\ &V_0\text{ $a_0<r<a_1$}\\ & 0\text{ $r>a_1$}\\ \end{cases} $$ $$...
  19. D

    Boundary conditions for eigenfunctions in a potential step

    1. Homework Statement A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##. How is the general form for the eigenfunctions? What is the boundary conditions for this eigenfunctions? Find the degeneracy level for the energy, when it is ##E<V_0## 2...
  20. L

    B Understanding Extremum and Derivatives at Boundary Points

    It's understandable that finding absolute extremum is impossible for a function with restricted boundary conditions. But why does the derivative of similar functions is not zero when the extremum is on the end points? To be precisely short with my question, why does the derivative gives only...
  21. M

    Newtonian energy integral and suitable boundary conditions

    I have a (somewhat) strange energy equation which has the following form: KE = A + B W + C \exp(-D W), where A,B,D are known constant, C is an unknown constant to be determined and kinetic and potential energy are given by KE and W respectively with W\equiv W(r) i.e. is a function of...
  22. W

    Solving Boundary Value ODE: y''+λy=0

    Homework Statement ## y''+\lambda y = 0 ; y(0) = 0, y(\pi)-y'(\pi) = 0## Homework EquationsThe Attempt at a Solution So, we have to test when lambda is equal to, less than and greater than 0. Let ## \lambda = 0## thus, the ODE becomes ## y'' = 0 ## which implies solutions of the form ## y(t) =...
  23. W

    Boundary Value ODE: Eigenvalues & Functions

    Homework Statement Find the eigenvalues and eigenfunctions of the following boundary-value problem. ## y''+\lambda y = 0 ; y(0) = 0, y'(L) = 0 ## Homework EquationsThe Attempt at a Solution So, we have to test when lambda is equal to, less than and greater than 0. Let ## \lambda = 0## thus...
  24. A

    Why Set the Inner Boundary Condition to -1 in Radial Flow Equations?

    Hello gents, Q:/ what is the reason for letting the inner boundary condition = (-1) when solving the radial flow of infinite form of diffusivity equation, and i would like to know what will happened if i didn't equate it with (-1). as in the attached pic: https://i.imgur.com/AVesmHM.png
  25. A

    Implementing symmetry boundary condition for the diffusion equation

    The following lines of codes implements 1D diffusion equation on 10 m long rod with fixed temperature at right boundary and right boundary temperature varying with time. xsize = 10; % Model size, m xnum = 10; % Number of nodes xstp =...
  26. C

    A How to calculate the extrinsic curvature of boundary of AdS_2

    I have a simple but technical problem: How to calculate the extrinsic curvature of boundary of AdS_2? I am not very familiar with this kind of calculation. The boundary of AdS2metric $$ds^2=\frac{dt^2+dz^2}{z^2}$$ is given by (t(u),z(u)). The induced metric on the boundary is...
  27. Ibix

    I Deriving Schwarzschild Interior Boundary Conditions

    I've been playing around with Maxima and it's ctensor library for tensor manipulation. I decided to have a crack at deriving Schwarzschild's solution for the interior of a constant-density sphere. I've managed to derive a static, spherically symmetric solution, but am struggling a bit with the...
  28. R

    Boundary conditions in dielectric problems

    Q) A conducting sphere of radius R floats half submerged in a liquid dielectric medium of permittivity e1. The region above the liquid is a gas of permittivity e2. The total free charge on the sphere is Q. Find a radial inverse-square electric field satisfying all boundary conditions and...
  29. A

    A Boundary conditions on the Euclidean Schwarzschild black hole

    This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf). The Euclidean Schwarzschild black hole $$ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{1-\frac{2M}{r}} +...
  30. C

    I Laplace equation boundary conditions

    Hi, I need to solve Laplace equation ##\nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ##r_1<r<r_2##, ##0<z<L##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ &\Phi(L,r)=V_P \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial...
  31. T

    How electric potential boundary condition works

    Homework Statement [/B] Inside a sperical dielectric mass there is a electric dipole on the center of the sphere. The sphere has radius a. This dieletric sphere is inside and on the center of a conductive spherical shell of radius b. The problem asks to find the potentials and then the...
  32. syamsul

    Static structural ansys, rigid body boundary condition

    hi guys, i am new user on ansys i have task from my lecturer. it is about static test for seat bus. i try simulate the seat bus on ansys statical structural. you can see my model on attachment (untitle1.png). at that picture, there are 2 bars will push the seat back. the stiffness beahvior of...
  33. RamosPinto

    I Two point boundary problem - Shooting method

    I'm currently trying to solve the following two-point boundary problem by means of the shooting method: To clarify, I'm investigating the optimal route of aerial/marine vehicles from one point to another point, considering a flow field. * The starting and ending locations are set as x0, y0...
  34. davidge

    I Boundary and homeomorphism

    The 2-sphere ##\mathbb{S}^2## can be expressed as the product ##\mathbb{S}^1 \times \mathbb{S}^1## Now can we express ##\mathbb{S}^1## as ##\mathbb{S}^1 \subset (-a,a)##, where ##(-a,a)## is some open interval of ##\mathbb{R}##? If so, then (I think) ##\mathbb{S}^1## is homeomorphic to...
  35. T

    Don't know how to solve for Boundary values

    Homework Statement http://imgur.com/7FKgeYs[/B] Here is the link to the problem I am stuck on. I don't know what question b) means by solve for the boundary values? Are these the same as boundary conditions or are boundary values explicit values? Homework Equations NA The Attempt at a...
  36. davidge

    I Do these manifolds have a boundary?

    Any open subset of ##\mathbb{R}^{n}##; The n-Sphere, ##\mathbb{S}^n##; The Klein Bottle. I guess they don't have a boundary, as a neighborhood of any point of them is homeomorphic to ##\mathbb{R}^n##. I'd like to know whether my guess is correct and whether the reason I'm giving for them not to...
  37. davidge

    I Ignoring boundary term in the action

    We can ignore the boundary contribution, namely the term that arises when we solve $$\int d^{4}x \sqrt{-g}g^{ab} \delta R_{ab}$$ when varying the action. According to Wikipedia it's appropriate to do so only when the underlying manifold is both without boundary and compact. That condition seems...
  38. 1

    Boundary conditions of a plane wave on a conductor

    Homework Statement Consider a plane monochromatic wave incident on a flat conducting surface. The incidence angle is ##θ##. The wave is polarized perpendicular to the plane of incidence. Find the radiation pressure (time-averaged force per unit area) exerted on the surface. Homework Equations...
  39. M

    Moving Boundary (Piston/Cylinder) -- Work Done

    Homework Statement a) [/B]There is a piston-cylinder arrangement; initial pressure is 200 kPa, initial volume is 0.04m^3. Volume of the gas increases to 0.1 m3 while the pressure remains constant, ie a constant pressure process. Calculate the work. b) There is a piston-cylinder arrangement...
  40. maistral

    A Implementing a weird-looking boundary condition (PDE/FDM)

    So I have this problem, taken from Kraus's heat transfer book. So deriving the computational molecule, the conditions for (3.251a), (3.251b) is a bit of a no brainer. The issue I am having is about the boundaries for (3.251c) and (3.251d). This is actually the first time I have seen this...
  41. maistral

    A PDE discretization for semi-infinite boundary?

    Hi. Been a while since I logged in here, I missed this place. Anyway, I have a question (title). Is that even possible? Say for example I have the standard heat equation (PDE) subject to the boundary conditions: T(0,t) = To T(∞,t) = Ti And the initial condition: T(0,t) = Ti I am aware of how...
  42. henry wang

    Why can we use periodic boundary conditions?

    (Mentor note: moved here from noon homework thread hence no template) I was studying vibration of a one-dimensional monatomic chain and the textbook used periodic boundary condition (PBC). I wanted to justify the use of PBC, so I came up with this: atoms deep inside the crystal sees an...
  43. M

    I Boundary Regions in LAMMPS MD Simulation

    Hi, I'm working on a project using LAMMPS. I am very new to the software and find the documentation doesn't always answer the fundamental questions I have. I noted previous LAMMPS queries were posted on this forum, although I am from a mechanical/materials engineering background. I am trying...
  44. ramzerimar

    Is BEM a More Efficient Alternative to FEA in Computational Science?

    Hello! I'm starting a project with one of my professors at college whose subject is the Boundary Element Method. I've studied a little bit of Finite Element Method, but BEM is new to me. This project will envolve a lot of programming in Python and Matlab. I would like to know how does BEM...
  45. S

    I Understanding the Definition of Boundary in Set Theory for Topological Spaces

    Hello! This is more of a set theory question I guess, but I have that the definition of the boundary of a subset A of a topological space X is ##\partial A = \bar A \cap \bar B##, with ##B = X - A## (I didn't manage to put the bar over X-A, this is why I used B). I think I have a wrong...
  46. Toby_phys

    Mechanical wave reflection at a boundary

    Homework Statement Homework Equations The right hand section (A) has an incident and reflected wave $$y_1=Ae^{i(kx+\omega t)} +A'e^{i(-kx+\omega t)} $$ The middle section (B) has a transmission reflected wave $$y_2=Be^{i(k_2x+\omega t)} +B'e^{i(-k_2x+\omega t)}$$ Section (C) just has the...
  47. N

    Reflection and Transmission of acoustic waves at a boundary

    Homework Statement An interface is formed between a block of aluminium (density = ##2.70 \times 10^3 kg/m^3##, speed of sound =##6.40 \times 10^3m/s##) and a block of copper (density = ##8.96 \times 10^3 kg/m^3##, speed of sound =##4.76 \times 10^3m/s##). Longitudinal waves traveling through...
  48. FranciscoSili

    I Help Solving an Equation with a Boundary Condition

    Hello everybody. I'm about to take a final exam and I've just encountered with this exercise. I know it's simple, but i tried solving it by Separation of variables, but i couldn't reach the result Mathematica gave me. This is the equation: ∂u/∂x = ∂u/∂t Plus i have a condition...
  49. F

    Shear stress at the boundary of beam

    Homework Statement We all know that $$\tau = VQ/ It $$ how to determine the shear stress at **G** ? I'm having problem of finding Ay centroid if the solid that i found earlier is y = 98.5mm There's a dashed line from G to the bottom . It's making me confused . for Q , should i consider the...
  50. abilolado

    A Free boundary conditions on vibrating rectangular membranes

    I've been trying to come up with wave equations to describe the motion on vibrating rectangular (more specifically, square) membranes. However, most paper I find assume fixed edges. What are the boundary conditions I need to apply to the 2D wave equations in order to have an free boundary in a...
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