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. V

    Neumann Boundary Conditions question

    So I'm reading through Jackson's Electrodynamics book (page 39, 3rd edition), and they're covering the part about Green's theorem, where you have both \Phi and \frac{\delta \Phi}{\delta n} in the surface integral, so we often use either Dirichlet or Neumann BC's to eliminate one of them. So for...
  2. T

    Find the upper and lower boundary curve to find the area between two curves.

    How do I know which function is the upper boundary curve and which is the lower boundary curve. For example find the area between the curves e^x and x bounded on the sides x=0 and x=1. We can draw it and we know that e^x is the upper curve and x is the lower curve. Thus the area is ∫e^x-∫x...
  3. R

    Hyperbolic Boundary Valued Problem

    Hi, I am trying to understand solving boundary valued partial differential equations and it's relation to hyperbolic functions. In one of my problems, there is a PDE and the solution contains the hyperbolic function "cosh". I was just curious if anyone has any information for me to read up on...
  4. R

    Third-order nonlinear ODE with boundary condition

    I'm trying to solve a third-order nonlinear ordinary differential equation. I couldn't get the answer even using Mathematica. The equation is: u'''(t) + u/2 u''(t) = 0 with conditions u(0)=0, u'(0)=0, u(10)=1. I need to get both analytic solution and numerical solution. For the...
  5. K

    Axisymmetric vs cyclic symmetry boundary conditions

    Hi, Can anyone explain the difference between axisymmetric and cyclic symmetry boundary conditions? Isn't it the same i.e. bith cyclic symmetry and axisymmetric?
  6. C

    [Holography] Global symmetry in boundary corresponds to gauge symmetry in bulk?

    I hear the statement that global symmetries in the boundary field theory corresponds to gauge symmetries in the bulk. 1) Is this a generic statement that is expected to hold for all holography pairs? (Maldacena states this towards the end of his first lecture at PiTP2010, which was supposed to...
  7. A

    Comp Sci Java Maze Boundary Check Method for Pathfinding

    I'm writing a method that checks a given coordinates neighbors to see if they are part of the path. the problem I am having with is the conditions that ihave to make sure i don't go outside of the grid. it doesn't seem that the two conditions for col are being checked. i put print statements...
  8. Z

    FEM software? Poisson equation: Boundary conditions for a charged boundary

    Hi all, I want to calculate the electrostatic potential for an two-dimensional area with given Dirichlet boundary conditions (say, a square) with a charged ring in it (like a wedding ring, but inifinitely thin) with a given line charge density. I figured out that the problem should be...
  9. M

    Boundary Behaviour: Reflection, Refraction & Diffraction Explained

    Hellllllllllo World 1- http://www.physicsclassroom.com/Class/refrn/u14l3a1.gif When a light ray falls on the boundary between two media , some of the energy is reflected back to the first medium,some energy refracts and some energy is absorbed. how is this exlplained?does it has something...
  10. stripes

    Accumulation points must be interior or boundary points.

    Homework Statement Prove the following: an accumulation point of a set S is either an interior point of S or a boundary point of S. Homework Equations None The Attempt at a Solution Suppose x is not an interior point. Then you cannot find a neighborhood around x such that N is a...
  11. J

    What does this symbol mean (Countour integration over a boundary)?

    I keep seeing this symbol, something like \oint_{{\partial}\omega}\omega, I know it is a contour integral and read that {\partial}\omega is called a boundary, but I don't know what it means or why there isn't a differential at the end of it. Can someone please answer these questions and explain...
  12. D

    Boundary points of subsets when viewed with the subset topology

    Hi! I have this two related questions: (1) I was thinking that \mathbb{Q} as a subset of \mathbb{R} is a closed set (all its points are boundary points). But when I think of \mathbb{Q} not like a subset, but like a topological space (with the inherited subspace topology), are all it's...
  13. Y

    Solving a PDE with Boundary Conditions: A Minimization Problem

    Homework Statement Let Ω\subsetR2 be a region with boundary \Gamma=\Gamma1\bigcup\Gamma2. On Ω we must solve the PDE -{div}(\frac{h^{3}}{12\mu}{grad} p+\frac{h}{2}{u})+kp=f with h and f functions of the spatial coordinates, \mu and k given constants, u a given constant velocity...
  14. S

    Hi,I want to show that the set of boundary points on a manifold

    Hi, I want to show that the set of boundary points on a manifold with boundary is well defined, i.e the image of a point on a manifold with boundary can not be both the interior point and boundary point on upper half space. To do this, it is enough to show that R^n can be homeomorphic to...
  15. E

    What are strings with endpoint boundary conditions NN, DD, ND and DN?

    http://en.wikipedia.org/wiki/D-brane says, "The equations of motion of string theory require that the endpoints of an open string (a string with endpoints) satisfy one of two types of boundary conditions: The Neumann boundary condition, corresponding to free endpoints moving through spacetime at...
  16. T

    Numerical solution to hyperbolic PDE - grid leapfrog - what to do at boundary

    Hi! I'm implementing a scheme to solve the following equation \frac{\partial \psi}{\partial t}=-c_{s} \cdot \frac{\partial \phi}{\partial x} \frac{\partial \phi}{\partial t}=-c_{s} \cdot \frac{\partial \psi}{\partial x} c_{s} is just the isothermal velocity of sound. The equations are for a...
  17. T

    A and A complement have the same boundary

    How would you show that if X is a non-empty set and A\subseteq X then A and A^c have the same boundary? The definition is x\in \partial A \iff there exists r>0 such that the open ball B(x,r) intersects both A and A^c but this is precisely the statement that x\in \partial A^c!
  18. Z

    Benefits of using a metal surface with glass/air boundary?

    I was looking at this webpage: http://www.ap.smu.ca/demos/index.php?option=com_content&view=article&id=120&Itemid=85 I was wondering, when n2(imag)=0 what would be the merits of using a metal surface and a glass/air boundary (ie internal reflection in a prism) as a mirror surface? Also...
  19. S

    Spherical Boundary Displace Current

    Homework Statement A current I is flowing along the y-axis and a spherical surface with radius 1 m has its center at origin, as in the figure left. A closed contour C is chosen as in the figure, which is a boundary between two semi-sphere surfaces S1 and S2. Based on the...
  20. M

    Solving Heat Equation w/ Neumann BCs Different Domain

    Hi guys! I'm to find the solution to \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} Subject to an initial condition u(x,0) = u_0(x) = a \exp(- \frac{x^2}{2c^2}) And Neumann boundary conditions \frac{\partial u}{\partial x} (-1,t) = \frac{\partial...
  21. R

    Boundary Layer Thickness of Blunt Body

    I am working on a hypersonic vehicle project and would like to calculate the boundary layer. The vehicle that I am studying is the Apollo re-entry capsule. I'm assuming a 2-D flow. Here's what I have done so far. Calculate the local surface inclination angle at any point on the blunt body...
  22. M

    Boundary conditions of ODE

    Homework Statement Given w'' - w = f(x) w'(0) = 1 w'(1) = 0 Homework Equations Find the Green's Function The Attempt at a Solution The solution to the homogeneous equation is known as: w(x) = A*exp(-x) + B*exp(x) For G's function we have: u(x) = A1*exp(-x) +...
  23. L

    A Harmonic Function is Zero on an open portion of the boundary, help

    A harmonic function in a region is zero on an open portion of the boundary, and its normal derivative is also zero on the same part, and it is continuously differentiable on the boundary. I have to show that the function is zero everywhere, but I have no idea how. I have tried this for hours...
  24. B

    How many boundary conditions should I have for a 2-D transient pde?

    Hello all: I'm a newbie, trying to write/use code for solving a 2D advection-diffusion problem. I'm not sure how many boundary conditions I should have for the property that is being transported. In my problem, I have diffusion switched off (advection only). The property being...
  25. L

    Electromagnetism - Boundary conditions for Polarization field at interface

    Not actually a homework question, this is a question from a past exam paper (second year EM and optics): Homework Statement Use a Gaussian surface and Amperian loop to derive the electrostatic boundary conditions for a polarization field P at an interface between media 1 and 2 with...
  26. Telemachus

    Heat equation with boundary conditions

    Hi. I'm trying to solve the heat equation with the initial boundary conditions: u(0,t)=f_1(t) u(x_1,t)=f_2(t) u(x,0)=f(x) 0<x<x_1 t>0 And the heat equation: \frac{\partial u}{\partial t}-k\frac{\partial^2 u}{\partial x^2}=0 So when I make separation of variables I get: \nu=X(x)T(t)...
  27. B

    Should I Scale My Boundary Condition Values for Problem Where I Scaled Interior?

    Hello all: I would very much appreciate advice on setting up a problem. Apologies in advance... This is probably a silly question--I'm more of a chemist than an engineer/math person! I have written a code for calculating changes in concentration/mass within a domain over time, as new...
  28. V

    How can I calculate the displacement thickness of a turbulent boundary layer?

    Lets go straight to the point. I need to find a way to calculate the displacement thickness of a turbulent boyndary layer. The laminar part has been simulated with Thwaite's method but I need to go from there. I'v heard of "Head's-model" but can't find the solution for it. Anyone that can...
  29. M

    Maxwell's equations on the boundary between non-conductor and conductor

    Homework Statement Hi, this is the first time I post a thread in this forum. I am not sure if I could post this question here since it is not a homework problem. I have trouble understanding two boundary condition between nonconductor and conductor from Maxwell's equations in dynamic case...
  30. L

    Is There a Solution to the Boundary Condition Problem for y-2y'+2y=0?

    I'm asked to determine if for the solution y=c_{1}e^{x}cos(x)+c_{2}e^{x}sin(x) for: y"-2y'+2y=0 whether a member of the family can be found that satisfies the boundary conditions: y(0)=1, y'(\pi)=0 Not quite sure what to do here. The examples in my book give boundary conditions for the same...
  31. jegues

    Dieletric Boundary Conditions (Parallel Plate Capacitor)

    Homework Statement See figure attached for problem statement, as well the solution. Homework Equations The Attempt at a Solution I'm confused as to how he is writing these equations from the boundary conditions. What I understand as the boundary condition for D is, \hat{n}...
  32. jegues

    Dielectric-Dielectric Boundary Conditions

    Homework Statement No problem, I just have a confusion about a certain concept. Homework Equations The Attempt at a Solution I'm confused as to how they draw the result, \oint_{C} \vec{E} \cdot \vec{dl} = E_{1t}\Delta l - E_{2t}\Delta l = 0 You don't really need to do the...
  33. stripes

    Isolated points are equal to boundary points

    Homework Statement Prove: If x is an isolated point of a set S, then x is an element of bd S. I believe this could be reworded as, if x is an isolated point of S, then x is also a boundary point of S. Homework Equations None The Attempt at a Solution My book says that an...
  34. U

    What do you call a more general boundary condition?

    Is there a term to describe something like a boundary condition but which can be applied within the domain, not just on the boundaries? For example in a heat transfer problem you might specify a constant rate of heat generation in some region. Is that still called a boundary condition...
  35. T

    Diffusion equation, boundary conditions

    EDIT: The subscripts in this question should all be superscripts! Homework Statement I'm trying to solve a temperature problem involving the diffusion equation, which has led me to the expression: X(x) = Cekx+De-kx Where U(x,y) = X(x)Y(y) and I am ignoring any expressions where...
  36. A

    How do I prove the closure and the boundary of a concrete example?

    Homework Statement Let X = R2 with the Euclidean metric and let S = {(x1, x2) : x1^2+x2^2 <1}.Prove that Closure of S ={(x1,x2):x1^2+x2^2<= 1} and that the Boundary of S= { (x1, x2) : x1^2 +x2 ^2=1 } . Homework Equations The Attempt at a Solution I was able to prove all my...
  37. M

    Proving closure and boundary points

    Homework Statement Let S = {(x,y): x^{2}+y^{2}<1}. Prove that \overline{S} is (that formula for the unit circle) \leq 1 and the boundary to be x^{2}+y^{2}=1. Homework Equations Boundary of S is denoted as the intersection of the closure of S and the closure of S complement. p \epsilon...
  38. G

    Definition of boundary operator

    Hi! I am currently studying homology theory and am using Vicks book "Homology Theory, An introduction to algebraic topology". When I was reading I found a definition that troubles me, I simply cannot get my head around it. Vick defines that: if PHI is a singular p-simplex we define di(PHI), a...
  39. K

    Open Bounded subset with non-zero measure boundary

    Homework Statement Let m be the Lebesgue measure on \mathbb R^d , and define the open sets O_n = \{ x \in \mathbb R^d : d(x,E) < \frac1n \} where d(A,B) = \inf\{ |x-y| : x \in A, y \in B \} 1) Find a closed and unbounded set E such that \lim_{n\to\infty} m(O_n) \neq m(E) . 2) Find an...
  40. C

    Boundary value problems (Ordinary differential equations) with a beam

    Homework Statement The deflection y of a non-uniform beam of length equal to 1, simply supported at both ends and with uniformly distributed load q, is governed by the equation (E*I(x)*y'')'' + k*y=q y(0)=0, y''(0)=0, y(1)=0, y''(0)=0 I(x)=A[1-0.5(1-x)2]2, 0<=x<=1 where E =Young’s...
  41. K

    Boundary conditions for buckling of column

    For fixed-fixed BC's, i have to arrest x and y displ. For simply supported case, i have to arrest y displ. Then my doubt is, while applying force at the end of the column, how the displacement will happen for fixed-fixed column.
  42. C

    Periodic Boundary Conditions on non sq lattice

    Is it possible to impose boundary conditions on the other 2d lattices like a rhombic lattice? a hexagonal lattice? an oblique lattice? How does one typically index such lattices?
  43. JK423

    Counting the states of a free particle (Periodic boundary conditions)

    Say you have a free particle, non relativistic, and you want to calculate the density of states (number of states with energy E-E+dE). In doing that, textbooks apply periodic boundary conditions (PBC) in a box of length L, and they get L to infinity, and in this way the states become countable...
  44. I

    Volume Between Two Surfaces Defined by Boundary Values

    Homework Statement Find the volume between z = x2 + y2 and z = 2 - (x2 + y2). Homework Equations The Attempt at a Solution if r2 = x2 + y2 then the lower part of the volume is defined by: r2 \leq z \leq 2 - r2 and: 0 \leq r \leq 1 the upper part by: 2 - r2 \leq z \leq r2...
  45. A

    What is the boundary of the rational numbers?

    I've read in several places that the boundary of the rational numbers is the empty set. I feel I must be misinterpreting the definition of a boundary, because this doesn't seem right to me. My understanding of the boundary of a set S is that it is the set of all elements which can be...
  46. aleazk

    Boundary and homeomorphism

    Hi, I need to know if the following statement is false or true. Given two topological spaces, X and Y, and an homeomorphism, F, between them, if bA is the boundary of the subset A of X, this implies that F(bA) is the boundary of the subset F(A) of Y?
  47. S

    Flux boundary condition on a face of a cube

    Lets us say I have a cube and I apply to a face of the cube a heat flux of 100 watt/m^2. Lets us say i divide the face of the cube into say 10 elements (area of each face of the element is 1 m^2). What will be the flux on each element , will it also be 100 watt/m^2? Sorry for a...
  48. P

    Solving a Boundary Value Problem: y + y = 0 ; 0<x<2π, y(0)=0 , y(2π)=1

    Homework Statement Determine all the solutions, if any, to the given boundary value problem by first finding a general solution to the differential equation: y" + y = 0 ; 0<x<2π y(0)=0 , y(2π)=1 The attempt at a solution So the general solution is given by: y = c1sin(x) +...
  49. V

    What is generating functional and vacuum-to-vacuum boundary conditions in QFT?

    Hello everyone :) I'm reading the book QFT - L. H. Ryder, and I don't understand clearly what are the generating functional Z[J] and vacuum-to-vacuum boundary conditions? Help me, please >"<
  50. F

    How do you I calculate the Laminar Boundary Layer Y in meters

    Homework Statement How do you I calculate the Laminar Boundary Layer Y in meters with Blasius Equation? I have an expression for U (m/s) and u/U and u(m/s) and also eta=0.1, 0.2...5.2. I am wondering how one could calculate the y laminar boundary layer (m), the y buffer boundary layer (m)...
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