| Author | Message | ||
     Ivan Ognjanovic (ognjanovic) New member Username: ognjanovic Post Number: 1 Registered: 06-2006 |
Hello! I am new to the software, so I read most of the documentation and examples. However, I have two issues that I couldn't find the answer for, in provided documentation/help/examples. My equation is of the form: C1*x*dxx(P)+C2*dx(P)+C3*x*dxy(P)+C4*dy(P)+C5*x*dyy(P)+C6*x*sin(y)+C7*cos(y)=0 This equation is defined on a domain of cylindrical form, where x is z coordinate, and y is theta (angle) coordinate (cylinder surface). In order to present this domain in 2D i unfolded the cilinder, which increased the number of boundary conditions from two (for z=0 and z=L) to four (theta=0 and theta=2*pi). These last two say that the value of P at the cutting line (to enable cylinder to unfold) is unique. The problem is that I don't know how to set that boundary condition (P at theta=0 is equal to P at theta=2*pi). The second question is concerning the definition of natural boundary condition. My boundary condition at z=L says that the axial mass flux is equal to zero. The equation (boundary condition) that describes this is defined as: A*dx(P)+B*dy(P)+C=0, bur again I don't know how to write that in FlexPDE. Can You please help me? Thany You for your help and sorry if the questions are too trivial. Ivan | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 636 Registered: 06-2003 |
1. You want to specify a PERIODIC boundary condition linking theta=0 to theta=2*PI. See "Periodic Boundary Conditions" in the Help Index, and the sample problems listed there. 2. The NATURAL boundary condition for your equation specifies the value of the outward normal component of the generalized flux. If you rewrite your equation in terms of the divergence of a flux, then the mapping to a NATURAL boundary condition will be more clear. It is not clear to me why your equation includes "x" factors multiplying most of the terms. This messes up the interpretation of the flux. If this is a transport equation on the surface of a cylinder, I don't know why the x's are there. If x represents the radial coordinate, then you should use cylindrical geometry. In any case, writing in terms of a divergence would clarify it. | ||
| Ivan Ognjanovic (ognjanovic) New member Username: ognjanovic Post Number: 2 Registered: 06-2006 |
Than You for the tip about PERIODIC boundary conditions. It works fine. However, I still haven't fugured out the thing about the definition of other boundary condition. In fact my equation has become more complicated. Now it has the following form: ((L^2)/Pa)*((C1b*x)+C1a)*DXX(P1)+(L*C2/Pa)*DX(P1)+(L/Pa)*((C3b*x)+C3a)*DXY(P1)+( C4/Pa)*DY(P1)+(1/Pa)*((C5b*x)+C5a)*DYY(P1)+((C6b*x)+C6a)*sin(y)+C7*cos(y)=0 The boundary condition at one of the borders is defined as: A*dx(P)+B*dy(P)+C=0 The main equation is an equation for a pressure field in an grooved aerodynamic air bearing. I can not write the equation in terms of divergence because I have different constants that are multiplying derivatives, and on top of that, I have x coordinate that is multiplying almost every derivative. My question is - since I have my boundary condition in the form of differential equation, why can't I just write it as a boundary condition in the code? Why can't I just write that equation in place of a boundary condition? Do I always have to present the boundary condition in terms of divergence. Thank You. | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 649 Registered: 06-2003 |
As described in the documentation, the Natural boundary condition defines the values to be ascribed to the surface integrals generated by integrating the second-order terms by parts. This process is most clearly understood if you write your equation as a divergence, but this is not absolutely necessary. If you don't write your equation as a divergence, FlexPDE will construct one for you, using the identity dx(f*dx(g)) = f*dxx(g)+dx(g)*dx(f) That is, f*dxx(g) will be turned into dx(f*dx(g))-dx(g)*dx(f). The latter term is first-order, so it will merely be evaluated, not integrated. In your case, the "x" terms in the coefficients will be put inside the outer derivative and a correction term will be added to the equation. The natural bc will then be the surface-normal component of the resulting flux vector ([]*dx(P1)+[]*dy(P1), []*dy(p1)) You can control which surface the dxy term by writing it explicitly as two nested derivatives. The outer one will control which coordinate gets integrated. But I would be careful with this. In most cases, the physics is an expression of conservation of flux of some quantity, and so is legitimately a divergence of a flux. Writing coefficients outside the divergence is probably the result of a "simplification" somewhere in the derivation, and thus invalid in the general case. In most cases, including heatflow and all of electro- and magneto-statics, the natural boundary condition corresponds to a legitimate physical constraint, and is therefore the most "natural". | ||
| karunamurthy (karunamurthy_k) New member Username: karunamurthy_k Post Number: 1 Registered: 04-2007 |
what may the boundary conditions for the co-ordinate fi in cylindrical coordinates for the heat conduction equation. |