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. andyrk

    Boundary Condition for tension

    A rope is tied at one end then rotated in a vertical circle. Why do we take the tension at the free end of the rope as 0(Boundary Condition)?
  2. mishima

    D and E at boundary (dielectrics)

    Homework Statement Electromagnetics, Kraus, 4th edition problem 4.7.3 The y-z plane is the boundary between 2 dielectrics of relative permittivities εr = 2 and εr = 5. For negative values of x, E = (3,0,2) V/m. Find D (magnitude and direction) for positive values of x. Homework Equations...
  3. W

    The best method to solve Helmholtz equation for a irregular boundary

    i have an almost square region. By 'almost' i mean the edges are curvy, not completely straight. i now need to solve the Helmholtz equation with Dirichlet boundary condition what is the best numerical method? how is Finite element, though i do not know what Finite element is
  4. G

    Comsol 3.5x: Obtaining boundary coordinates from irregular geom obj

    I don't have access to Comsol 4.x. I imported a 3D mesh generated from point cloud data and generated a geometry. (A hollow almost-ellipsoid.) I solve my system on the surface/boundary alone; there is no volumetric data. I need to extract 1D data from the surface/boundary at points other...
  5. A

    Dielectric Boundary Condition Question

    Hi, I have a question regarding the boundary condition present for a dielectric immersed in a static field. I hope one of you physics guru's can shed some light on this. Suppose we have a dielectric in space subjected to some external static electric field. I have read (without explanation)...
  6. Y

    Question on phasors at boundary of oblique incidence.

    A plane wave travels in ##\hat k_I=\hat x \sin\theta_I+\hat z \cos \theta_I## direction hitting a boundary formed by xy plane ( z=0). The incidence wave is in the plane of incident formed by xz plane where y=0. We let ##\tilde E_I(\vec k_I)= \hat x E_{I_x}+\hat y E_{I_y}+\hat z E_{I_z}...
  7. Z

    Wavefunction boundary condition solve for k

    Homework Statement A wave function is given by: \Psi (x) = a cos(2\pi x) + b sin (2\pi x) for\: x<0 \\ and\\ \Psi (x) = Ce^{-kx} for\: x>0 \\ Determine the constant k in terms of a, b and c using the boundary conditions and discuss the case a >> b. Homework Equations...
  8. T

    Evaluate the integral with C be the boundary of the domain

    Homework Statement Let C be the boundary of the domain enclosed between y = x^2 and y = x. Assuming C is oriented counterclockwise. Evaluate the integral ∫c (6xy+e^(-x^2))dx Homework Equations I was thinking of using Green's Theorem. Would be the approach be correct? The Attempt at a...
  9. S

    Boundary Conditions for a beam with two supports

    Homework Statement I'm trying to find the boundary conditions for the beam shown in the figure. Homework Equations Notation: V= Shear force M= Bending momentThe Attempt at a Solution at x=0 V=R1, M=0 at x=9 V=R3, M=0 In the solution provided at x=9 V=-R2. I don't understand why it's...
  10. M

    Incompressibility in boundary layer (Fluid Dynamics)

    I have started studying fluid mechanics recently and seems to be a very basic conceptual question that is bugging me and unfortunately I am unable to find a reasonable explanation for it. Your help would be more than appreciated. The mathematical definition for incompressiblility in fluid...
  11. R

    Additional boundary conditions for inclined flow?

    Homework Statement I am solving an inclined flow problem, and am stuck. The problem is to find the volumetric flow rate of inclined flow in a square channel. Once I have the velocity profile, I can just integrate over that to get the flow rate. 2. The attempt at a solution Letting the...
  12. G

    Boundary Layer Equations: Neglecting the term for x-axis momentum

    I'm having a little difficulty in the topic 'Boundary layer Equations' in Fluid Mechanics due to my weak math skills. With reference to the figure in attachment, if we say that "we neglect \frac{∂^{2}u}{∂x^{2}}", does it mean that we will only consider the portion where \delta(x) is almost...
  13. E

    Surface impedance - Boundary condition

    Hello! Let a plane wave propagate towards the -y direction. It is normally incident upon the plane (x,z) (whose normal unit vector is the y-direction unit vector, \mathbf{\hat{u}}_y): the plane represents the interface between the free space (in y > 0) and a general lossy medium (in y < 0). We...
  14. K

    [calculus] question about identify boundary curve between two surface

    Homework Statement I have two questions. 1) generally speaking, when we are given two equations both describing surface in R3: f1(x,y,z)=k and f2(x,y,z)=C, The intersection of the two will be a curve that's by solving both equations. My question is, by solving f1 and f2 to get anther...
  15. G

    Applying boundary condition on heat equation

    Homework Statement hey, i have a heat equation question which asks to solve for u(x,t) given that u(0,t)=Q_0 + ΔQsin(ωt).Homework Equations d_xx u = k d_t u u(0,t)=Q_0 + ΔQsin(ωt) The Attempt at a Solution so you can solve the equation pretty easily with separation of variables, i.e...
  16. D

    Mass Transfer - Boundary Conditions

    Homework Statement An airborne spherical cellular organism, 0.015 cm in diameter, utilizes 4.5 gmol O2/(hour kg of cell mass). Assume Sh = 4 for external convective resistance to O2 transfer to the cell. (Sh = kd/D is based on diffusivity in the gas phase). Assume zero-order kinetics for...
  17. A

    Integration/ boundary conditions

    Hi guys, I regard a particle in an Potential. I have callculated the partition function and the probability density function F_{1}. $$ H= \frac{p^{2}_{x}}{2m} + \frac{p^{2}_{z}}{2m}+ \frac{p^{2}_{\phi}}{2I}+ mgz $$ For callculating an average value I do: $$ <mgz>=\int...
  18. L

    Solid State - Phonons at Brillouin Zone Boundary

    Homework Statement Homework Equations {3.9b} A[2\mu -m\omega ^2 ]=2\mu Bcos(\frac{ka}{2}) B[2\mu -M\omega ^2 ]=2\mu Acos(\frac{ka}{2}) The Attempt at a Solution All I can think of is setting k =\frac{\pi}{a} so that B[2\mu -M\omega ^2 ]=A[2\mu -m\omega ^2 ] solve for omega...
  19. J

    D.E. Boundary Value Problem: Finished the work, seem to get wrong ans

  20. S

    Radiative/Convective Boundary Conditions for Heat Equation

    Hi everyone, I'm attempting to create a computer program to solve the transient 3d heat equation using the Crank Nicolson method. I would like to model the boundaries of my domain as losing heat via convection and radiation due to the temperature difference between the boundary and the air in...
  21. B

    Does the boundary of the causal universe function as an event horizon?

    The universe is expanding as described by Hubble's law, which means that at a certain distance from an observer, expansion exceeds the speed of light, so all waves become infinitely red-shifted. In other words, if an goes beyond this point, no information about it can ever come back to the...
  22. Y

    Find the lowest order solution for a boundary value problem

    Hello, I need help in solving the problem: " find the lowest order uniform approximation to the boundary value problem εy''+y'sinx+ysin(2x)=0. y(0)=(pi), y(pi)=0. " what I did: y(out)=Ʃ(ε^n)y(n) εy''(out)+y'(out)*sinx+y(out)*sin(2x)=0 for order 0: y'(out)*sinx+y(out)*sin(2x)=0...
  23. D

    Boundary of a Mobius band - I think S1 V S1, everyone else says S1?

    Boundary of a Mobius band - I think S1 V S1, everyone else says S1?? Hey I am having a huge problem! There are a few problems where I'm using Van Kampen's theorem and for one part of the problem I need to compute the fundamental group of the boundary of the Mobius band. Everyone keeps telling...
  24. B

    Compatibilty of the Dirichlet boundary condition

    Hi, I am trying to solve a Poisson equation \nabla^2 \phi = f in \Omega, with Dirichlet boundary condition \phi = 0 on \partial \Omega. My problem is that I am trying to understand the condition under which a solution exists. All the text I consulted says that the problem is solvable. However...
  25. J

    Correcting Mistakes in Representing Constants for a Differential Equation?

    I'm not sure if my answer is correct. Did I make a mistake somewhere? I'm not sure the ± needs to be there.
  26. J

    D.E. Boundary Value Problem: Finished the work but it might be wrong

    I get a different answer from my classmates. Where did I go wrong?
  27. D

    Boundary of an open set in R2 is a limit point?

    I have kind of a simple point set topology question. If I am in ℝ2 and I have a connected open set, call it O, then is it true that all points on the boundary ∂O are limit points of O? I guess I'm stuck envisioning as O as, at least homeomorphic, to an open disk of radius epsilon. So it seems...
  28. U

    The Hydrodynamic and Thermal boundary layers

    Hi, I'm doing 'Heat and Mass transfer' at college and we're covering the topic on the hydrodynamic and thermal boundary layers. I have a couple of questions, the answers to which are not given explicitly in any of my textbooks. 1. During open flow, why does laminar flow eventually have...
  29. W

    Isothermal Expansion of Water/Moving Boundary

    Hello all, I've been wondering how water reacts in a closed, rigid system with one moving boundary. Assuming the system is perfectly filled with water, and one side of the boundary moves (increasing the volume), how does this affect the pressure in the system? Since water is...
  30. M

    Boundary conditions on a conductor?

    I've been trying to get my head arround this problem for several days now, and while I deemed it relatively simple at first it turns out that I can't figure out the BCs on a conductor, to which we apply a potential U. In the simplified version of the problem, there is a rectangular conductor...
  31. D

    Deriving d'Alemberts solution - Boundary conditions

    Hi, I shall show (using Fourier transform) that the solution to \frac{\partial^2 u(x,t)}{\partial t^2} = \frac{\partial^2 u(x,t)}{\partial x^2}\\ u(x,0) = f(x) \\ u_t(x,0) = 0 is u(x,t) = (f(x+t) + f(x-t))/2 I got it almost: Taking the Fourier transform in the variable x...
  32. J

    Is the Speed of Light Boundary Relative to a Fixed Point in the Universe?

    Speed of light "boundary" ? I have a simple question : When everyone is talking about the "speed of light boundary" what is it relative to ? Speed is ALWAYS relative to "something" else, otherwise it doesn't even make any sense. Which brings a second point : if there is indeed a speed...
  33. A

    PDE: Initial Conditions Contradicting Boundary Conditions

    Suppose we have the following IBVP: PDE: u_{t}=α^{2}u_{xx} 0<x<1 0<t<∞ BCs: u(0,t)=0, u_{x}(1,t)=1 0<t<∞ IC: u(x,0)=sin(πx) 0≤x≤1 It appears as though the BCs and the IC do not match. The derivative of temperature with respect to x at position x=1 is a constant 1...
  34. S

    Comsol moving boundary with loss of mass

    I need some help!My problem is a problem of moving boundary with loss of mass... I started to use COMSOL and I need to simulate one plate with a hole on the center. And this hole is increasing with the time according to one equation (like a velocity, in m/s) which depend of the stress. Someone...
  35. T

    Definition of the boundary map for chain complexes

    I've been poking around, learning a little about homology theory. I had a question about the boundary operator. Namely, how it's defined. There's two definitions I've seen floating around. The first is at: http://en.wikipedia.org/wiki/Simplicial_homology The second, at...
  36. M

    Criteria of periodic boundary condition

    We used to apply periodic boundary condition to simulate an infinite system. What will happen if the interactions between atoms do not drop to zero even when they are infinitely far away? Is the periodic boundary still valid? How can I prove the periodic boundary condition is valid or not? thanks.
  37. G

    Condition of continuity of E field at a boundary

    I am trying to understand the derivation of Snell's law using Maxwell's equation and got stuck. My textbook says that "the E field that is tangent to the interface must be continuous" in order to consider refraction of light. If it were static E field I understand this is true because in...
  38. J

    PDE with variable boundary condition

    Homework Statement I am trying to solve this PDE with variable boundary condition, and I want to use combination method. But I have problem with the second boundary condition, which is not transformed to the new variable. Can you please give me some advise? Homework Equations (∂^2 T)/(∂x^2...
  39. S

    Laplace equation in a square with mixed boundary conditions

    The length of the side of the square is a. The boundary conditions are the following: (1) the left edge is kept at temperature T=C2 (2) the bottom edge is kept at temperature T=C1 (3) the top and right edges are perfectly insulated, that is \dfrac{\partial T}{\partial x}=0,\dfrac{\partial...
  40. nomadreid

    In holographic principle, the boundary is wrt which light cone?

    It is stated that in the holographic principle (e.g., in http://en.wikipedia.org/wiki/Holographic_principle) that the the description of a volume of space is encoded on a light-like boundary to the region. But with respect to which position in the volume? In a black hole, it is clear, because...
  41. C

    2-D Poisson Equation Boundary Value Prob

    Homework Statement Solve the equation: ∂2F/∂x2 + ∂2F/∂y2 = f(x,y) Boundary Conditions: F=Fo for x=0 F=0 for x=a ∂F/∂y=0 for y=0 and y=b Homework Equations How can I find Eigengunctions of F(x,y) for expansion along Y in terms of X? The Attempt at a Solution I can't imagine...
  42. K

    Defenition of prior austenite grain boundary

    What is prior austenite grain boundary?Grain boundary and prior austenite grain boundary are same terminology or different?I am confused:confused: Thanks
  43. S

    Two-point boundary value problem

    Homework Statement Solve the given BVP or show that it has no solution. (It does have a solution) y"+2y = x, y(0)=y(\pi)=0 Homework Equations Characteristic polynomial is r^2 + 2 = 0. μ = √2 The Attempt at a Solution The solution to the complementary homogeneous equation is y_h...
  44. M

    Green's Function fo a Boundary Value Problem

    Homework Statement L[y] = \frac{d^2y}{dx^2} Show that the Green's function for the boundary value problem with y(-1) = 0 and y(1) = 0 is given by G(x,y) = \frac{1}{2}(1-x)(1+y) for -1\leq y \leq x \leq 1\ G(x,y) = \frac{1}{2}(1+x)(1-y) for -1\leq x \leq y \leq...
  45. D

    Boundary value problem - constrained paramter

    Let's say I have a set of nonlinear differential equations of the form. x' = f(x,y) \\ y' = g(x,y) Where f and g contain some parameter 'a' that is constrained to within certain values. Let's say I know x(0), y(0) and x(T), y(T) where T isn't a set value. What methods can I use to...
  46. L

    Divergence Theorem on a surface without boundary

    Reading through Spivak's Calculus on Manifolds and some basic books in Analysis I notice that the divergence theorem is derived for surfaces or manifolds with boundary. I am trying to understand the case where I can apply the divergence theorem on a surface without boundary.
  47. J

    PDE Separation of Variables with Nonzero Boundary Conditions

    Homework Statement Solve the diffusion equation: u_{xx}-\alpha^2 u_{t}=0 With the boundary and initial conditions: u(0,t)=u_{0} u(L,t)=u_{L} u(x,0=\phi(x) The Attempt at a Solution I want to solve using separation of variables... I start by assuming a solution of the form...
  48. B

    What Equation Models Boundary Layer Thickness in Early Stage Pipe Flow?

    Hi I cannot find an equation for a boundary layer in a pipe flow (laminar). I am looking for an equivalent of the equation δ(x)=4.91x/(√Re) that works for a flow between plates (x is the distance downstream). The thing is- I am looking for BL thickness for still undeveloped flow. I would be...
  49. D

    MHB Solution of the Damped Wave Equation under Certain Boundary Conditions

    $$ u_{tt} + 3u_t = u_{xx}\Rightarrow \varphi\psi'' + 3\varphi\psi' = \varphi''\psi. $$ $$ u(0,t) = u(\pi,t) = 0 $$ $$ u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10 $$ \[\varphi(x) = A\cos kx + B\sin kx\\\] \begin{alignat*}{3} \psi(t) & = & C\exp\left(-\frac{3t}{2}\right)\exp\left[t\frac{\sqrt{9...
  50. E

    How to set up Neumann boundary condition for a PDE in a coordinate-invariant form?

    I'm having trouble finding out how to set up Neumann (or, rather, "Robin") boundary conditions for a diffusion-type PDE. More specifically, I have a scalar function f(\boldsymbol{x}, t) where \boldsymbol{x} is n-dimensional vector space with some boundary region defined by A(\boldsymbol{x})=0...
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