# What is Boundary: Definition and 1000 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. ### A Boundary conditions in ##\delta I=0## to derive Einstein's equations

Dirac derives Einstein's field equations from the action principle ##\delta I=0## where $$I=\int R\sqrt{-g} \, d^4x$$ (##R## is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-g} \, d^4x$$ where ##L## involves only ##g_{\mu\nu}## and its first derivatives, unlike...
2. ### I Definition of manifolds with boundary

In differential geometry, we typically define the boundary ##\partial M## of a manifold ##M## as all ##p \in M## for which there exists a chart ##(U,\varphi), p \in U## such that ##\varphi(p) \in \partial\mathbb{H}^n := \{ x \in \mathbb{R}^n : x^n = 0 \}##. Consequently, we also demand that...
3. ### I Klein-Gordon Equation with boundary conditions

I am trying to find solutions for the Klien-Gordon equations in 1-d particle in a box. The difference here is the box itself oscillating and has boundary conditions that are time dependent, something like this L(t)=L0+ΔLsin(ωt). My initial approach is to use a homogeneous solution and use...
4. ### How do you solve for multiple constants given the wave functions at the boundary?

From my understanding, you can equate ψ1(x) and ψ2(x) at the boundary of x = a, so I plugged in the values of a into x for both equations and I got ψ1(x) = 0 and ψ2(x) = ## (a-d)^2-c ##. I am a bit stuck on where to go from here.
5. ### I Green's function for 2-D Laplacian within square/rectangular boundary

From the table of Green functions on Wikipedia we can get the generic 2-D Green's function for the Laplacian operator. But how would one apply boundary conditions like u = 0 along a rectangular boundary? Would we visualize a sort of rectangle-based, tilted pyramid, with logarithmically changing...
6. ### A Topology - Boundary of a ball without a point

Let us say we have f analytic in ##Ball_1(0)##. which means, radius 1, starting at ##z_0 = 0## point. If I want to find the boundary of ##Ball_1(0)##. Will the boundary be ##{0}## or ##{\emptyset}##? Not homework, just an intuition to understand ##f(z)=\frac 1 z## function ( for example ) better.
7. ### A Minkowski Spacetime 3+1 Dimensions: Conformal Diagram w/Boundary

The Minkowski spacetime in 3+1 dimensions does not have a boundary. Yet, its conformal diagram (see the left diagram in the attached picture) has a timelike boundary ##r=0##. A spacetime with a timelike boundary (another example is AdS) has a different causal structure than a spacetime without a...
8. ### A Boundary conditions for variable length bar

Suppose I'm looking at a bar of length L(t) in 1D and I have the conservation of mass: \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)=0 In order to make things easier, I make the change of variable x'=x/L(t) so that in this frame of reference, the length remains constant...
9. ### Engineering Fluid dynamics problem involving the Prandtl boundary layer equation for a two dimensional steady laminar flow

I have tried to approach in the following way I am stuck. How should I approach this next.please help
10. ### A Two degrees of freedom in GR after boundary conditions specified?

I read there are 2 degrees of freedom in GR after boundary conditions specified. Does that mean 2 equations are enough for EFE equivalent? Those two seem like the amplitude and a phase.
11. ### Fortran Need help with Jacobi relaxation method for Dirichlet boundary conditions

program r_jacobi implicit none !!!!Variables!!! real*8 V, V_1, V_2, Lx, Ly integer n ,i , j, k, nx, ny real*8, allocatable :: arrx(:), arry(:), phi(:,:,:) real*8 x, xi, xf, y, yi, yf, dx, dy real*8 d, q, bx, by V=1 V_1=V V_2=-V Lx = 2 Ly = 1 nx = 200 ny = nx/2...
12. ### A The boundary conditions in reference to Laplace's equation

We have inhomogenous dirichlet boundary conditions (well understood)....the laplace equation is a steady state equation and we can clearly see that in 2D..it will be defined by 4 boundary conditions and NO initial condition...having said that; kindly have a look at the continuation below... I...
13. ### Checking assumptions in boundary conditions of double well system

The idea here (as I'm told) is to use the boundary conditions to get a transcendental equation, and then that transcendental equation can be solved numerically. So I'm making a few assumptions in this problem: 1. The potential ##V(x)## is even, so the wavefunction ##\psi(x)## is either even or...
14. ### A Solve the heat equation having Dirichlet boundary conditions

I am going through these notes...they are pretty easy to follow. I would like more insight on the initial condition. In this problem, (attachment below), i guess the choice of initial condition is convenient as its easier to plug in the values of ##n=2## and ##b=3## (highlighted on the...
15. ### Solution to Differential Equation with Limit Boundary Condition

The original differential equation is: My solution is below, where C and D are constants. I have verified that it satisfies the original DE. When I apply the first boundary condition, I obtain that , but I'm unsure where to go from there to apply the second boundary condition. I know that I...
16. ### I Query on Electromagnetic Theory (Dielectric Boundary Conditions)

The given question from Electromagnetic Theory (which is based on Dielectric Boundary Conditions) is as follows: Interface b/w two dielectric medium has a surface charge density (suppose xyz C / (m ^ 2) ). Using boundary condition find field in 1 (relative permittivity =xyz) if field in 2...
17. ### MATLAB No End" Boundary Conditions in Wave on a String

How did they impose boundary conditions here if "no end" is selected? Here: https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html I would like to do the same thing without changing the wave equation of the string.
18. ### MATLAB Boundary conditions in the resolution of a PDE with the FFT method

How to impose boundary conditions when solving a PDE with fft? For example here: If I copy this code I get periodic boundary conditions. Thank you
19. ### A Connectedness of Boundary Points in Schwarzschild Penrose Chart

In the Penrose chart for Schwarzschild spacetime, the boundary "at infinity" appears to be connected all the way around. I want to explore what that means physically and whether particular boundary points that appear to be connected on the chart actually are. I am using the following notes as a...
20. ### Engineering Hand tool boundary conditions - Forces determination

Hello, I' m trying to make a linear static analysis (Finite Element Analysis) on the following hand tool. I want to determine the boundary conditions. In order to do that I have decided to use a force couple to represent the forces that a bolt exerts on the jaws of this spanner. Despite using...
21. ### Exploring Electric Field Boundaries at a Charge Density Boundary

Hey, I have a really short question about electrostatics. The boundary conditions are : \mathbf{E}^{\perp }_{above} - \mathbf{E}^{\perp}_{below} = -\frac{\sigma}{\varepsilon_{0}}\mathbf{\hat{n}} , \mathbf{E}^{\parallel }_{above} = \mathbf{E}^{\parallel}_{below}. My question is what is...
22. ### Boundary Conditions for an infinite rectangular pipe

Does setting up the problem symmetrically on this axis and the boundary conditions applied make sense? I don't believe I will have a problem solving for the potential inside, but i just want to make sure I have my B.C and axis correct before proceeding. EDIT: Or should this be a 2-D lapace...
23. ### Evaluating the boundary conditions for a rectangular pipe

I have attached an image of the pipe in the attachmnts. The pipe is parallel to z-axis form (-∞,∞) and sides of length a. So my boundary conditions for this problem are as follows 1.) V=0 at y=0 2.)V=0 at y=a 4.)∂v/∂x=0 @ x=0 3.)V0 @ x=a I am a little confused on the fourth boundary...

25. ### I Is a Manifold with a Boundary Considered a True Manifold?

<Moderator note: thread split from https://www.physicsforums.com/threads/speed-of-light.1012508/#post-6601734 > Is a manifold with a boundary still a manifold?
26. ### Error when creating boundary conditions (OpenFOAM)

Let's give some context. I created a Mesh via snappyHexMesh and next I am setting the boundary conditions (BCs). First I copied the patches folder "createPatchDict" to my system's folder via the command "cp \$FOAM_ETC/caseDicts/mesh/manipulation/patches/createPatchDict ./system" Taking into...
27. ### Boundary conditions ##\vec{B}## and ##\vec{H}##

When asking for boundary conditions I'm wondering if this is enough in this situation to give ##\vec{\nabla} \cdot \vec{B} = 0 , B_{2\perp} - B_{1 \perp} = 0## ##\vec{\nabla} \cdot \vec{H} = - \vec{\nabla} \cdot \vec{M}, H_{2\perp} - H_{1 \perp} = - (M_{2\perp} - M_{1 \perp})## ##\vec{\nabla}...
28. ### Boundary conditions (E and D) for a dielectric sphere

Since there is no free charge ##\int_S \vec{D} \cdot d\vec{a} = 0## and ##\rho_f = 0## ##\sigma_f = 0## ##\vec{nabla} \cdot \vec{P} = 0## since P is a constant ##\rho_b = - \vec{nabla} \cdot \vec{P} = 0## For a simple surface we can find the boundary conditions for ##\vec{E}## using a Gauss'...
29. ### Perturbation Methods: Guarantee a boundary layer?

Hi, I was working on the following problem: Question: A small parameter multiplying the highest derivative does not guarantee that the solution will have a boundary layer for small values of ##\epsilon##. This may be due to the form of the differential equation, or the particular boundary...
30. ### I Non-Homogeneous Robin Boundary conditions and Interpretations of Signs

I have been solving the constant coefficient 1D advection-diffusion equation ##\frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} = D\frac{\partial^2 c}{\partial x^2}## on ##0<x<L,t>0## with a variety of robin BC's. Namely $$vc + D\frac{\partial c}{\partial x} = J^f ~~at~~ x=L$$...
31. ### A Solving Laplace's equation in polar coordinates for specific boundary conditions

Hello everybody, Currently I am doing my master's thesis and I've encountered a physics problem which is very difficult for me to solve. The problem I have is finding equations for the magnetic scalar potential inside and outside a ferromagnetic wire for specific boundary conditions...
32. ### Confused About Boundary Conditions for ##y## and ##x##

I am going through this notes, i can follow quite well...my only issue is on the highlighted part...i thought that we had two boundary conditions for ##y## ( of which one of them is non homogenous) and two boundary conditions for ##x##( of which both are homogenous)...kindly clarify on this part...
33. ### Boundary conditions in Ansys

Hi, I have one question related to the boundary conditions I should apply in a Static Structural simulation for the following support. The support is subjected to the following loading conditions shown below...
34. ### I Unraveling Laplace's Equation: Exploring Valid Domains and Boundary Conditions

Hi! This thread might well be similar to: https://www.physicsforums.com/threads/thread-about-jacksons-classical-electrodynamics-3rd-edition.910410/ I'm self-studying Vanderlinde and having a great time. However, I think that I am conflating and confusing many different things. Let me just ask...
35. ### Boundary conditions for 2 ropes fixed to a massless ring with a damper

Hi, I'm not quite sure if I'm correct. I need to find the boundary conditions for 2 ropes ##T_1 \mu_1, T_2 \mu_2## fixed at ##x=0## to a massless ring with a massless damper of force ##F_d - -bv_y## Here what I think, since the ring and the damper is massless ##\sum F_y = 0##. Thus, ##-T_1...
36. ### I Help Regarding Application of Bernoulli in a Boundary Layer

Hey all, I recently took an aerodynamics exam that included the question "Please Explain how the Bernoulli Equation can be Applied Inside a Boundary Layer". Now, it is my belief that the Bernoulli equation, defined by my textbook as P+0.5ρV2=ℂ, requires inviscid flow to be properly applied...
37. ### Periodic boundary conditions -> Shouldn't supports hinder all motion?

Hello everyone, I am currently trying to understand periodic boundary conditions for the mechanical investigation of mechanical properties of a RVE. I found a good video explaining the theory behind it: But something is unclear to me: At the above linked time step, the individual conical...
38. ### B Boundary between a particle in two energy states

Hi I'm new to quantum mechanics, Looking for some help regarding a concept i am struggling to solve. I am curious if I had a cube of particles in a ground state and another cube with the same particle in a higher energy state. If I placed one upon another, is there anything in quantum mechanics...
39. ### Unusual boundary conditions in FEA software

For a 2D problem with unknown displacements u(x,y) and v(x,y), is it allowed to give such a set of BCs u(0,y)=1 and vy(0,y)=0, the former being a displacement BC, the latter being a force BC (vy is the y strain)? How is this implemented in FEA software?
40. ### Treatment of source term on the boundary: FEA and Comsol

I would like to solve a coupled system of two PDEs using Comsol for the following geometry: Equation 1 (valid for 0⩽Z⩽bm): The initial and boundary conditions are: Tm(r,t→0)=20 Tm(r→rw,t)=70 Tm(r→∞,t)=20 However, for bm⩽Z⩽bm+b2, the equation to solve is: With the following initial and...
41. ### B Equation of Circle: Boundary or Surface?

Equation of circle: ##x ^2+y ^2=1##. Is this equation describing boundary or a surface? Thanks.
42. ### Set up boundary conditions for a simple elasticity problem

[Mentor Note -- Thread moved to the ME forum to get better views] Let's consider an incompressible block of Neo-Hookean material. Let the initial reference geometry be described by ##B=[0,b] \times [0,b] \times [0,h]##. The professor gave me the following task: Of course there can be many...
43. ### Boundary condition: null traction on the boundary of an elastic block

Hi everyone, I'm trying to understand the rationale behind the boundary condition for the problem "Finite bending of an incompressible elastic block". (See here from page 180).Here we have as Cauchy Stress tensor (see eq. (5.82)): ##T = - \pi I + \mu (\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r...
44. ### Maple Boundary finite element method in Maple

I would like any tips about a Maple ''home made'' program that I received for a project but this program seems to stop before the very end of the code. I want to find de lift of an airfoil with Boundary finites elements method. I have this error at the very end : Error, (in fprintf) number...
45. ### Frequency of boundary layer instability

I've been searching the net but can not find any freely available literature. Can someone give me a quick lesson on boundary layer instability and its frequencies? I have an equation that claims to scale the instability frequency by: F = U/2*delta U is leading edge velocity delta is layer...
46. ### I Recession Velocity near Cosmological Boundary: Hubble's Law?

I'm confused whether Hubble's Law applies to objects near the cosmological horizon (CH). I'm told that objects asymptotically approach the CH and freeze there (v -> 0) in the same way that occurs during in-fall towards a black hole. But Hubble's Law says that velocity is proportional to...
47. ### A Ising model open chain and periodic boundary conditions

One dimensional Ising model is often treated as open chain system with free ends. Then when external field is added it is treated with cyclic boundary condition. Can someone explain me are those methods equivalent, or not?
48. ### Is d'Alembert's Formula Correct for Neumann Boundary Conditions in PDEs?

Hi all, I was hoping someone could check whether I computed part (4) correctly, where i find the solution u(t,x) using dAlembert's formula: $$\boxed{\tilde{u}(t,x)=\frac{1}{2}\Big[\tilde{g}(x+t)+\tilde{g}(x-t)\Big]+\frac{1}{2}\int^{x+t}_{x-t}\tilde{h}(y)dy}$$ Does the graph of the solution look...
49. ### A Implementing FDM Boundary Conditions at a Red Point

in Finite Difference Method (FDM), the boundary conditions can be implemented by applying the continuity of parallel component of magnetic field intensity. when it comes to the interface of two areas, it is done at ease, but consider this case at the red point: in FDM we exactly require on...
50. ### Question about boundary layer growth in convergent and divergent ducts

Hi, I was recently reading about convergent-divergent nozzles and was wondering about how boundary layers grow in them. Question: How does a boundary layer grow in a convergent duct in subsonic flow? How does this compare to the growth of a boundary layer in a divergent duct in subsonic flow...