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Diffusion equation in polar coordinates

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. I've never used option 3, so I cannot speak anything about it. Option 1 works in certain cases, but not for all.

Option 2 I have worked with and have seen in other hydrodynamics codes that I've used, so it might be your best bet. The timestep, however, is limited by the Courant-Friedrichs-Levy condition. For time-stepping purposes, this is going to be the slow point of your code. It might be worth looking into implicit methods instead of the explicit method you are currently working with.

Heat Equation in Cylindrical and Spherical Coordinates

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Discretizing the Wave Equation in polar coordinates Ask Question. Asked 6 years, 2 months ago. Active 6 years, 2 months ago. Viewed 1k times. Andy Andy 1 1 silver badge 10 10 bronze badges. Active Oldest Votes. Kyle Kanos Kyle Kanos You want the max to ensure the diffusion is appropriately captured.

Using the minimum of the two ensures that the physics is contained in the cell. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Feedback post: New moderator reinstatement and appeal process revisions.

The new moderator agreement is now live for moderators to accept across the….Hot Threads. Featured Threads. Log in Register. Search titles only. Search Advanced search…. Log in. Contact us. Close Menu. Support PF! Buy your school textbooks, materials and every day products Here! JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding. Diffusion equation in polar coordinates. Thread starter robinegberts Start date Mar 14, Tags diffusion equation fourier transform polar coordinates.

Homework Statement I am trying to solve the axisymmetric diffusion equation for vorticity by Fourier transformation. Any ideas? Orodruin Staff Emeritus.

Science Advisor. Homework Helper. Insights Author. Gold Member. If you really want to use transform methods, the transform you would be looking for is the Hankel transform. Thanks, that makes sense. Ray Vickson Science Advisor.Danladi Eli 1Gyuk P. All Rights Reserved. A general solution for transverse magnetization, the nuclear magnetic resonance NMR signals for diffusion-advection equation with spatially varying velocity and diffusion coefficients, which is based on the fundamental Bloch NMR flow equations, was obtained using the method of separation of variable.

It assumed that the velocity component is proportional to the coordinate and that the diffusion coefficient is proportional to the square of the corresponding component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient. After some assumptions, the 3-D equation degenerates to a 2-D problem.

The solution to this equation is useful in describing physical phenomenon such as transport of materials in a fluid. The result obtained in this study can have applications in functional magnetic resonance imaging fMRI with more accurate information. Cite this paper: Danladi Eli, Gyuk P. Article Outline 1. Introduction 2. The equation which describes such a process is known as the advection equation Awojoyegbe et. The diffusion—advection equation a differential equation describing the process of diffusion and advection is obtained by adding the advection operator to the main diffusion equation.

In the spherical coordinates, the advection operator is Where the velocity vector v has components, and in the, and directions, respectively.

diffusion equation in polar coordinates

Generally speaking, the advection term for the transverse magnetization is. After expansion we obtained When the fluid velocity is constant, and then this is similar to a case of incompressible fluid in fluid dynamics.

Since perfusing substances obey the advection equation, the appropriate equation to accurately describe a flow process in a spherical geometry based on equation r.

Awojoyobe el. In this paper, we are interested in getting the general solution of the equation using the method of separation of variables. If we define and dada et. And if we assume E. Danladi et. The method of the separation of variables relies upon the assumption that a function of the form Where is a function of r, and t, and F is a function of r, and G is a function of t alone, will be the solution to the partial differential equation above. Equation 7 can now be written as: Divide through by Equate both sides with a constant to have 7 8 From 79 Let 10 Putting 10 into 9 Therefore 11 Also from 8 12 Equation 12 can also be separated by equating both sides with a constant and by re-arrangement we obtain 13 14 Equation 14 can also be written as 15 From 13 16 Substitute 16 into 13 Therefore, 17 From 17 let And making we have: From 15let 18 Put 18 into 15 Therefore, 19 From 19let If we let we have: Then the transverse magnetization becomes 20 The solution above is the NMR transverse magnetization and signal in spherical geometry.

This NMR signal is a function of diffusion coefficient. The solution can be a tool to accurately understand the combined effect of diffusion and perfusion processes in human physiological and pathological flow systems. There seems to be evidence to suggest that in some transport processes the velocity and diffusion coefficients are not constants but functions of time and space Zoppou and, Knight.

Conclusions We have obtained basic expression for the transverse magnetization the NMR signals in spherical geometry based on the bloch NMR flow equations. This general solution is quite interesting and promising in the context of some recent research works on dynamical flow. The application of this fundamental solution to solve real life flow problems in which NMR-sensitive materials are transported will be presented separately.

Dada O.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.

Now I'm trying to work out how this solution has been found. Here is my attempt:. Crucially here, the dimensions of the first two terms do not involve the dimensions of the independent variable, so rescaling the independent variable does not change those terms. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Solving a diffusion equation in polar coordinates Ask Question. Asked 3 years, 9 months ago. Active 3 years, 9 months ago. Viewed 1k times. To me this looks like a modification of the standard isotropic 2D heat equation, whose fundamental solution is a 2D Gaussian with a growing variance.

Note that nothing really changed in the first two terms. Ok, how did you think of these substitutions? Active Oldest Votes. Ian Ian Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.

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The new moderator agreement is now live for moderators to accept across the…. The unofficial elections nomination thread. Linked 3. Hot Network Questions. Question feed. Mathematics Stack Exchange works best with JavaScript enabled.In engineering, there are plenty of problems, that cannot be solved in cartesian coordinates. Cylindrical and spherical systems are very common in thermal and especially in power engineering.

The heat equation may also be expressed in cylindrical and spherical coordinates. Obtaining analytical solutions to these differential equations requires a knowledge of the solution techniques of partial differential equations, which is beyond the scope of this text. On the other hand, there are many simplifications and assumptions, that can be applied to these equations and that lead to very important results.

In the next section we limit our consideration to one-dimensional steady-state cases with constant thermal conductivity, since they result in ordinary differential equations. Thermal Conduction.

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Theodore L. Bergman, Adrienne S. Lavine, Frank P. ISBN: Heat and Mass Transfer.

diffusion equation in polar coordinates

Yunus A. McGraw-Hill Education, Fundamentals of Heat and Mass Transfer. Nuclear and Reactor Physics: J. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed. Lamarsh, A. Baratta, Introduction to Nuclear Engineering, 3d ed. Glasstone, Sesonske. Nuclear and Particle Physics. Physics of Nuclear Kinetics. Addison-Wesley Pub. Paul Reuss, Neutron Physics. EDP Sciences, Advanced Reactor Physics:. See above: Thermal Conduction.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. It only takes a minute to sign up. Also, following my point above - does anyone know how to specify a delta function at the origin as an initial condition? Sorry for the poor formatting of the Mathematica code above - I don't know how to make it look neat please let me know if you do.

With this change and a few simplifications. The message you saw was caused by the verification of the PDE coefficients that where parsed. This verification happened at the coordinate 0which in this case caused the message and the rejection of the coefficient.

In newer versions this verification happens with a coordinate from some place inside the domain.

Diffusion equation in polar coordinates

The actual integration did not need to evaluate the coefficient at the singularity. It is possible to change the coordinate that is used for the verification of the coefficient.

This can be done with specifying VerificationData :. Will give the message again. Sign up to join this community.

The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Solve 2D diffusion equation in polar coordinates Ask Question.

Asked 4 years, 7 months ago. Active 3 months ago. Viewed 2k times. Best, Ben. Active Oldest Votes. I suppose that this method means that it'll be difficult to specify a Dirac Delta function as an initial condition? Any ideas on this part of the question? If you really need a delta-function solution as opposed to a high amplitude, low volume approximationlook up Green's function in, for instance, Morse and Feshbach.

diffusion equation in polar coordinates

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Derive the Laplacian for a Spherical Coordinate System in 4 Steps

Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. The Overflow Blog. Podcast a conversation on diversity and representation. Podcast is Scrum making you a worse engineer? Featured on Meta. Feedback post: New moderator reinstatement and appeal process revisions.

The new moderator agreement is now live for moderators to accept across the…. Hot Network Questions. Question feed.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I've got a question considering the polar coordinates representation of the diffusion equation.

Here I can't follow why this is true. How does this equation follows with the local circular symmetry? Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

Diffusion equation in polar coordinates Ask Question. Asked 6 months ago. Active 6 months ago. Viewed 40 times. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Feedback post: New moderator reinstatement and appeal process revisions. The new moderator agreement is now live for moderators to accept across the….

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