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. bearcharge

    Vorticity boundary condition

    I'm solving a 2D forced convection problem by using finite difference method. It involves solving the vorticity-stream function equations. I met some problem when I'm trying to solve the vorticity equation, which can be stated as follows: How to deal with the vorticity boundary condition? after...
  2. M

    Wave equation with initial and boundary conditions.

    Hallo Every one, Homework Statement y(x,t)=sin(x)cos(ct)+(1/c)cos(x)sin(ct) Boundary Condition: y(0,t)=y(2pi,t)=(1/c)sin(ct) fot t>0 Initial Condition : y(x,0)=sin(x),( partial y / Partial t ) (x,0) = cos(x) for 0<x<2pi show that y(x,t)=sin(x)cos(ct)+(1/c)cos(x)sin(ct)...
  3. R

    Two-Point Boundary Value Problem

    Homework Statement y''+\lambday=0 y'(0)=0 y'(pi)=0 Homework Equations The Attempt at a Solution What's puzzling me is the case when we check if the eigenvalue is zero. y''=0 y'=C1 y=C1x+C2 Now when I check the first boundary value I get C1=0 now How do I check the second one ? with the...
  4. J

    Two-Point Boundary Value Problem

    Homework Statement y'' + ßy = 0, y'(0)=0, y'(L)=0 Homework Equations Meh The Attempt at a Solution I so already did the ß>1 and ß<1; I'm stuck on the ß=0. It seems easy enough. y'' = 0 -----> y' = A -----> 0=A, 0=A (from the two initial conditions) ------> No...
  5. P

    Magnetostatics - boundary condition

    Let's consider two media with magnetic permeability \mu_1, \mu_2 . What's the condition for magnetostatic vector potential \vec{A} on the boundary. Is it true that its tangent component should be continuous. Thanks for replay.
  6. P

    Green's function and Dirichlet boundary problem

    Is it true that there always exists Green's function for Dirichlet boundary problem. I mean a function G(r,r') which fullfils the following conditions: div (\epsilon grad G(r,r')) =- \delta(r,r') inside volume V and G(r,r') is 0 on boundary of V. If V is whole space there exists obvious...
  7. B

    Cauchy Boundary Conditions on a Wave

    Homework Statement So using the D'Alembert solution, I know the solution of the wave equation is of the form: y(x,t) = f(x-ct) + g(x+ct) I'm told that at t=0 the displacement of an infinitely long string is defined as y(x,t) = sin (pi x/a) in the range -a<= x <= a and y =0...
  8. R

    Limit of x^n+y^n as n -> ∞: max(x, y)

    show for two positive numbers x,y>0 that limit for n->infinity : \sqrt[n]{x^n+y^n} = max {x,y} i don't know how to make a upper boundary(lower boundary is >0 i suppose) something like assume for instance x bigger than y and than make a boundary with it, but how?
  9. A

    Neumann boundary conditions on S^1/Z_2

    Hello everybody, I've been puzzling over something (quite simple I assume). Take S^1. Now consider the action of a Z_2 which takes x to -x, where x is a natural coordinate on the cylinder ( -1< x <1). Now we mod out by this action. The new space is an orbifold: smooth except at x=0. It...
  10. P

    What would be my boundary conditions? Heat Equation

    1. I have a rod of length 4,cross section 1 and thermal conductivity 1.Nothing is mentioned about the end at the origin x=0, but at the opposite end x=4, the rod is radiating heat energy at twice the difference between the temperature of that end and the air temperature of 23 celcius. Find the...
  11. T

    Observable Boundary of Expanding Universe: Is There a Limit?

    If space is expanding then at at some point we must reach the point where it's expansion is faster than C. Do we know where that is? Is the expansion uniform or is it dependent on something like Dark Matter clusters? Is the expansion accelerating or decelerating at two fixed points? In...
  12. Topher925

    Overflow boundary condition?

    Apparently Fluent has this thing called an "overflow" boundary condition. I tried searching the help files and found absolutely nothing about it. Does anyone know what this is and why someone would want to use it?
  13. P

    Laplace Eq with Dirichlet boundary conditions in 2D (solution check)

    Homework Statement The steady state temperature distribution, T(x,y), in a flat metal sheet obeys the partial differential equation: \frac{\partial^2{T}}{\partial{x}^2}+{\frac{\partial^2{T}}{\partial{y}^2}}=0 Separate the variables and find T everywhere on a square flat plate of sides S with...
  14. M

    Waves on Tight Strings: Boundary Condition Problem

    Homework Statement A string is attached to a ring of mass m which is free to move up and down a frictionless pole. The string is subject to tension T and its mass per unit length is \rho. The displacement of the string from its equilibrium position along the x -axis is y(x,t). The boundary...
  15. B

    2nd-Order (Linear?) Non-Homogeneous ODE, Two Point Boundary Value

    Homework Statement Find the solution to the two-point boundary value problem u'' + 4u' + exu = sin(8x) with u(-1) = u(1) = 0. Homework Equations The Attempt at a Solution I haven't taken an ODE course in years but I need to verify that my numerical solution to the ODE is accurate to the...
  16. B

    Boundary Value Problem for the 1-D Wave

    So here's the problem: I'm asked to find the solutions to the 1-D Wave equation u_{tt} = u_{xx} subject to u(x,0) = g(x), u_t(x,0) = h(x) but also u_t(0,t) = A*u_x(0,t) and discuss why A = -1 does not allow valid solutions. I can't figure it out at all. The solutions to...
  17. Y

    2nd order Boundary Value Problem.

    I want to solve: y(x)''-(\frac{m\pi}{a})^2y(x)=0 With boundary condition y(0)=y(a)=0. First part is very easy using constant coef. which give: y(x)=c_1 cosh(\frac{m\pi}{a}x) + c_2 sinh(\frac{m\pi}{a}x) y(0)=0 \;\Rightarrow\; c_1=0 \;\Rightarrow\; y(x) = c_2 \; sinh(\frac{m\pi}{a}...
  18. T

    PDE-Heat Equation with weird boundary conditions help

    Homework Statement Consider the Heat Equation： du/dt=k(d2u/dx2), where d is a partial and d2 is the second partial. The B.C.'s are u_x(0,t)=u(0,t) and u_x(L,t)=u(L,t), where u_x is the partial of u with respect to x. The I.C is u(x,0)=f(x) Now, consider the Boundary Value Problem...
  19. U

    Boundary layer thickness over square cylinder

    Hello all, I am looking for a reference (text or peer-reviewed journal article) for the derivation of the boundary layer thickness over a short square cylinder as a function of distance from the leading edge for the simplest case of steady, uniform and laminar axial flow of an inviscid...
  20. J

    Simplifying a Boundary Layer Theory Equation

    Homework Statement The problem is write this d\Psi/dy(d^2\Psi/dxdy)-d\Psi/dx(d^2\Psi/dy^2=-\nu(d^3\Psi/dy^3) in the form of -ff''=f''' where \Psi(x,y)=-sqrt(V*\nu*x)f(\eta) f(\eta)=integral(from 0 to \eta)(\Pi')*(\overline{\eta})*d(\overline{\eta}) where \overline{\eta} is a dummy...
  21. A

    Approximation of boundary value problem using finite differences

    Homework Statement A hot fluid is flowing through a thick-walled cylindrical metal tube at a constant temperature of 450C. The cylinder wall has an inner radius of 1 cm and an outer radius of 2 cm and the surrounding temperature is 20C. The temperature distribution u(r) in the metal is defined...
  22. U

    Fluid mechanics boundary layer solution

    Hello, I am looking for a reference which has solutions for the laminar flow boundary layer for the following scenario: circular cylinder, L>>d, length in direction of flow, with flat circular cap uniform laminar flow inviscid, incompressible fluid In other words, I would like the...
  23. F

    Why is the boundary of the rationals (Q) equal to R?

    I was reading a website that said the boundary of a set's boundary is equal to the first boundary. Visually, this makes sense for subsets of R1 and R2 because the first boundary will not have an interior (no ball about the points will fall into the boundary). However, the reading went on to...
  24. G

    Waves under Boundary Conditions

    For a string with one endpoint attached to a wall and the other to an oscillator (so that it is under boundary conditions), what is the character of waves that are not at a harmonic frequency?
  25. L

    Heat Transfer Boundary Conditions

    Homework Statement A high temperature, gas cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element (Ka=57 W/m*K) is encased in graphite (Kb= 3 W/m*K) and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium...
  26. W

    Boundary conditions on D-Branes

    Hi there, I recently read that the equations of motion for an classical open string naturally give rise to two boundary conditions, namely Dirichlet and Neumann boundary conditions. (i) Could someone explain to me what do these boundary conditions physically mean, in particular for open...
  27. Z

    Boundary problem

    given a boundary Hermitian eigenvalue problem L= -\frac{d}{dx}\left[p(x)\frac{dy}{ dx}\right]+q(x)y=\lambda w(x)y with y=y(x) , in one dimension, can we always find two operators D_{1} = \frac{d}{dx}+f(x) and D_{2} = -\frac{d}{dx}+U(x) so L= D_{1} D_{2} , with Adj( D_{1}) =...
  28. C

    Solving du/dt = (d^2)u/d(x^2) with Boundary Conditions

    Homework Statement du/dt = (d^2)u/d(x^2), t>0, 0<x<1 u(0,t) = 0 = u(1,t) , t>0 u(x,0) = P(x), 0<x<1 P(x) = {0 , if abs(x-1/2) >epsilon/2 {u/epsilon, if abs(x-1/2) <= epsilon/2 i need to find u(1/2,1/pi^2) Homework Equations i have u(x,t) = SUM{ 2/(n*pi)...
  29. B

    Dielectric-Dielectric Boundary Conditions Problem

    Homework Statement The problem can be found http://whites.sdsmt.edu/classes/ee382/homework/382Homework4.pdf" . It is the first one. Note: The subscript x = 0 is supposed to be y = 0 (the teacher typed it in wrong). Homework Equations \vec{\boldsymbol{D}}_{2t} =...
  30. P

    Solving 1d Helmholtz with boundary conditions

    Hello all, This is to do with forced longitudinal vibration of a rod (bar). It's basically a problem to do with the linearised plane wave equation (1d). The rod is fixed firmly at one end, and excited at the other by a harmonic force. The wave equation (with constant rho/E instead of...
  31. J

    Weird Boundary Conditions?

    Homework Statement d20/de2+1=0 and the boundry condition is -d0/de(evaluated at e=+/- 1)=+/-H0(evaluated at +/-1). The final result yields 0(e)=(1/2)(1-e2)+1/H. What i don't understand is how to use this boundary condition and where the 1/H comes from. The Attempt at a Solution...
  32. M

    Boundary Conditions for infinite grounded cylinder (Laplace Equation)

    Homework Statement Find the potential outside of a long grounded conducting cylindrical rod of radius R placed perpendicular to a uniform electric field E0. Homework Equations V(s,\phi) = a_{0}+b_0{}ln(s) + \sum(A_n{}cos(n\phi)+B_n{}sin(n\phi))*(C_n{}s^n{}+D_n{}s^{-n}) The sum being...
  33. P

    COMSOL two phase flow with slip boundary condition

    Hey all, I am try to model a problem in which one fluid (of known properties) sits on top of another fluid (with different properties) and there is an imposed slip boundary condition at the interface. I'm wondering if COMSOL has a Slip boundary condition that can be readily used or if...
  34. T

    Electric field inside hollow conductor boundary value problem

    Hi, I am in Purcell's E&M book at the section explaining why the field is zero inside a hollow conductor of any shape. The proof given is that the potential function inside the conductor must obey Laplace's equation, and that the boundary of the region (in this case a rectangular metal box) is...
  35. A

    Understanding Smooth Extension to Boundary of D in C (or R^2)

    What does it mean to say something "extends smoothly" to a boundary in C (or R^2)? I'm studying Cauchy's integral formula, and one of the assumptions of the theorem is that a function be analytic on a domain D and extend smoothly to the boundary of D. What does that mean, exactly?
  36. M

    Real Analysis: Interior, Closure and Boundary

    Homework Statement Let W\subset S \subset \mathbb{R}^n. Show that the following are equivalent: (i) W is relatively closed in S, (ii) W = \bar{W}\cap S and (iii) (\partial W)\cap S \subset W. Homework Equations The only thing we have to work with is the definitions of open and closed sets...
  37. E

    Pressure at the boundary between two gas phases which are originally sepearted

    Hi. I have got two questions... 1. How can we determine the pressure at the boundary of two gas phases (say, compressed carbon dioxide released from a can and the atmosphere) which are originally separated? If we apply the ideal gas equation on each of these two phases, different pressures...
  38. A

    What Caused 33.7 Million Year Ago Eocene-Oligocene(E-O) Boundary Event?

    Zhonghui Liu of the University of Hong Kong has made the most comprehensive deep-sea core research to date http://www.sciencemag.org/cgi/content/abstract/323/5918/1187?ck=nck. Global SST's fell by an average of 4.5 to 6 degrees F at the E-O boundary, with temps near the South Pole and North Pole...
  39. L

    Solving COMSOL Particle Tracing Problem with Boundary Coordinates

    Hello all, I have a small problem with COMSOL. I am trying to use the particle tracing feature, more precisely I want to use the boundary coordinates to specify where are the points from which to trace the electrons. From what I understand in the User's Giude, the Boundary Coordinates feature...
  40. A

    Oblique shock reflection on solid boundary

    Hi guys, i need some help on this topic. Basically, i need to calculate the angle of reflection of a oblique shock on a solid boundary. Here is the description of the question i have: an incident supersonic flow reaches a compression corner, with a deflection angle of 15 degrees...
  41. S

    Boundary value problem: local stifness matrix

    Homework Statement Given a BVP: \Delta(u)+u=1 in \Omega u=0 on \partial\Omega using linear piecewise functions, calculate the corresponding local stiffness matrix on the reference triangle : {(x,y); 0<=x<=1, 0<=y<=1-x}. The domain is a square with one point in the middle (at...
  42. MathematicalPhysicist

    Does the expansion rate of our universe depend on its mean density?

    What methods are there to figure out if our universe has a boundary or not? And if it does are we living on the boundary (like in the case of our Earth that we live on a sphere and not inside the earth)?
  43. R

    Generic question on boundary conditions

    A partial differential equation requires boundary conditions. Consider a 2-dimensional problem, where the variables are 'x' and 'y'. The boundary is the line x=0 and you are given all sorts of information about the function on that line. If you are given just the values of the function on the...
  44. P

    Helium balloon floating at air/helium boundary

    While reviewing the Archimedes Principle with my AP students using ConcepTest questions from the Wilson, Buffa, Lou text I came across an answer that I believe is incorrect. The question was the second question in a series of questions about a helium filled balloon. The first question asked...
  45. K

    1-D Wave equation with mixed boundary conditions

    Homework Statement Solve, u_{t} = u_{xx}c^{2} given the following boundary and initial conditions u_{x}(0,t) = 0, u(L,t) = 0 u(x,0) = f(x) , u_{t}(x,0) = g(x)Homework Equations u(x,t) = F(x)G(t) The Attempt at a Solution I solved it, I am just not sure if it is right. u(x,t) =...
  46. X

    Boundary conditions for a 4th order beam deflection equation

    What would the boundary conditions be for a fourth order differential equation describing the deflection (elastic curve) of a propped cantilever beam with a uniform distributed load applied? i.e. a beam with a built in support on the left and a simple support on the right. I need 4 obviously but...
  47. E

    Boundary Value Problem for y'' + y = A + sin(2x) with y(0) = y'(\pi/2) = 2

    Homework Statement a) Solve for the BVP. Where A is a real number. b) For what values of A does there exist a unique solutions? What is the solution? c) For what values of A do there exist infinitely many solutions? d) For what values of A do there exist no solutions? Homework...
  48. T

    System of ODE Boundary Value Problem with 2nd Order Backward Difference

    {\frac {{\it du}}{{\it dx}}}=998\,u+1998\,v {\frac {{\it dv}}{{\it dx}}}=-999\,u-1999\,v u \left( 0 \right) =1 v \left( 0 \right) =0 0<x<10 Second Order Backward Difference formula {\frac {f_{{k-2}}-4\,f_{{k-1}}+3f_{{k}}}{h}} I'm trying solve this numerically in matlab, but can't seem to...
  49. A

    How can I solve a system of BVPs with only boundary conditions given?

    Dear all, I have system(4 equations) of BVPs. Could anybody recommend me, how to solve this system(whatever numericaly or analytical): x'=-y/sqrt(x^2+y^2) + u y'=x/sqrt(x^2+y^2) + v u' = -xy/(x^2+y^2)^3/2 u - [1/sqrt(x^2+y^2) - x^2 /(x^2+y^2)^3/2] v v' = xy/(x^2+y^2)^3/2 v -...
  50. F

    Boundary of {1/n : n is a natural number}

    Homework Statement I'm trying to figure out the the boundary of the set of all 1/n, where n is a natural number. Consider this as a subset of R with its usual metric, nothing fancy. Homework Equations There are at least two "equivalent" definitions of the boundary of a set: 1. the boundary of...
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