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How to constrain two PDE variables to be the same value on boundary?

I have a 4 pdes for 4 variables: v, v2, ua, and p. I would like to specify that v==v2 on the left and right boundaries at all times. Here are the equations: I discretise the system using PDEtoODE. ...
Ariana Fenris's user avatar
1 vote
0 answers
59 views

Why is NDSolve ignoring two PDEs out of three ones I am solving?

I am solving the following coupled system of 3 PDEs modelling a 1D membrane coupled to a 1D fluid flow field underneath. However, on putting them into NDSolveValue (and trying with FEM), it says the &...
Ariana Fenris's user avatar
6 votes
1 answer
154 views

How do I couple an 1D PDE to a 2D Laplace equation?

Recently I've come across the following system governing the spreading of an evaporating droplet. The height of the droplet $h$ is defined by the following equation $$\frac{\partial h}{\partial t}=-\...
FLP's user avatar
  • 547
4 votes
2 answers
171 views

Solve Richards equation for unsaturated soils with FEM (`NDSolveValue`)

I am attempting to solve the nonlinear Richards equation for the dependent variable $\Psi(t,x,y)$. The equation is as follows: $\frac{\partial\theta(\Psi)}{\partial t}=\nabla. (k(\Psi)\nabla(\Psi + y) ...
Stratus's user avatar
  • 2,954
6 votes
1 answer
133 views

Avoiding artificial diffusion and minimize changes to code

I am currently working through Solving Partial Differential Equations with Finite Elements specifically the fluid flow problems. I took the Stokes flow problem and replaced it with Euler's equations ...
Kendall's user avatar
  • 363
4 votes
1 answer
167 views

Solving Integro-Differential Equation with Numerous Dependencies Using DSolveValue

I am attempting to solve equation (3.9a) from the paper "On the equation for spherical-particle motion: effect of Reynolds and acceleration numbers" to characterize the response of a ...
Benjamin Tarver's user avatar
5 votes
1 answer
338 views

Solving third order DE from fluid dynamics

I am trying to use DSolve to solve a differential equation from G. Batchelor: An Introduction to Fluid Dynamics, eq. 5.12.4: [...] equation now reduces to $$\boxed{...
simon's user avatar
  • 47
7 votes
2 answers
271 views

How to improve FDM solver for unsteady viscous flow?

To solve the problem that is discussed in the paper Finite Difference Analysis of Time-Dependent Viscous Nanofluid Flow Between Parallel Plates we developed FDM solver based on the code from the blog ...
Alex Trounev's user avatar
3 votes
4 answers
188 views

NDSolve over derivatives of Heaviside function

I am trying to numerically solve for the strain of a Maxwell material in response to a step stress. The governing equations are $$\dot{\sigma} + \sigma = \dot{\varepsilon}$$ and I want to find $\...
JamesVR's user avatar
  • 407
2 votes
0 answers
69 views

Can Mathematica's FEM solve coupled Dirichlet Boundary Conditions?

I am solving a coupled system of PDEs using Mathematica's FEM capabilities. Specifically, the Navier-Stokes equations with a no-flux stress-free boundary. To do this, I need to specify a Dirichlet ...
Eric Hester's user avatar
6 votes
0 answers
703 views

Code for quasi 1D nozzle flows

The quasi-one-dimensional model describing the flow of compressible gas in rocket nozzles is very common. The corresponding equations have a divergent non dimensional form $ \frac{\partial \mathbf{U}}...
Alex Trounev's user avatar
2 votes
1 answer
140 views

Infinite expresssion 1/0 encountered when use NDSolve for 3D axisymmetric Navier-Stokes (Euler) equations

The PDEs we are interested in solving using NDSolve is the vorticity-stream formulation of the 3D axisymmetric Navier-Stokes (Euler) equations (Ref.1 :T. Y. Hou, Potential singularity of the 3D Euler ...
mike's user avatar
  • 303
3 votes
1 answer
163 views

Invalid PeriodicBoundaryCondition for Navier Stokes problem using NDSolveValue

Background: I'm looking to have a 2D re-entrant channel of a geophysical flow that is forced by a wind stress at the surface ("taux"; below) and experiences planetary rotation ("cor&...
Chris's user avatar
  • 33
2 votes
1 answer
90 views

How to extract gauss coordinates in AceGen/ AceFEM?

I am trying to implement a 2D turbulence problem with the model of Spalart-Allmaras. Therefore, I need to extract the global gauss coordinates to define parameter 'd' which stands for the distance to ...
user1234321's user avatar
1 vote
0 answers
83 views

NDSolve stops solving past a certain time

This is a continuation of the post I've made Unable to solve Delay PDEs Error in Boussinesq Approximation. I apologise if I shouldn't have posted a seperate question for this but I think that the ...
Lucas's user avatar
  • 103
3 votes
1 answer
207 views

Steady state fluid flow for downward flow past an obstacle

I was attempting to simulate fluid flow past a circular obstacle. The following is the code which I used ...
XU KANGYOU's user avatar
1 vote
0 answers
82 views

Unable to solve Delay PDEs Error in Boussinesq Approximation

I'm trying to solve the set of equations below describing the flow of a pot of water being heated slightly. The equations are 2D axisymmetric in nature. ...
Lucas's user avatar
  • 103
5 votes
1 answer
135 views

Spalart-Allmaras turbulence model

In this post Alex gives an implementation of the Spalart-Allmaras turbulence model [1, 2]. The example produces reasonable results, as far as I can tell. However, the implementation Alex uses deviates ...
user21's user avatar
  • 39.9k
4 votes
1 answer
433 views

Implementation of FEM to steady-state coupled fluid flow and heat transfer

This is a steady-state conjugate heat transfer problem (the time-independent version of this problem). The problem is conjugate as the energy equation is being solved in thermally connected solid and ...
Avrana's user avatar
  • 297
4 votes
1 answer
196 views

NDSolveValue gives up without warning or solution on PDEtoODE system

I have the following pair of PDEs that I want to solve in the half-space x>=0:  I followed this post to decompose the 4th-order height equation into two 2nd-order ones to respect the hyperbolic ...
Ariana Fenris's user avatar
7 votes
1 answer
361 views

Refining mesh size leads to absurd results for a coupled heat transfer FEM model

I have been recently solving a conjugate heat transfer problem, which involves fully-reversing or reciprocating flow of fluid over a heated block of solid. The problem is 2D and the temperature field ...
Avrana's user avatar
  • 297
2 votes
1 answer
258 views

Speeding up NDSolve to reasonable speeds to solve a coupled PDE system

Problem Statement I am planning to solve a PDE system which consists of a fluid droplet spreading on a non-Newtonian substrate. The system consists of the following equations: $$\frac{\partial p_1}{\...
FLP's user avatar
  • 547
1 vote
1 answer
223 views

How to use VectorPlot3D to visualize a solution of Navier-Stokes equations in cylindrical coordinates?

I try to solve axially symmetric Navier-Stokes equations in cylindrical coordinates. I have successfuly solved the equations, but I have troubles when I try to plot the solution, as VectorPlot3D does ...
lemurman's user avatar
  • 412
5 votes
0 answers
198 views

Resolving singularity in convection-diffusion equation using pdetoode

Building on the system of equations in this post, I attempted to solve an additional convection-diffusion equation describing the concentration of solute in the lens, which affects its spreading. ...
FLP's user avatar
  • 547
2 votes
1 answer
160 views

Directly calculating the cyclic steady state of a time-periodic conjugate heat transfer problem

Context The following transient problem is the reciprocating (i.e., fully reversing) flow of a fluid $0<x<L, 0<y<d$ over a thick heated block $0<x<L, -e<y<0$ until the system ...
Avrana's user avatar
  • 297
3 votes
1 answer
149 views

Conjugate heat transfer modelling of reciprocating flow crashes for long flow times

The following transient problem is essentially the reciprocating (i.e., fully reversing) flow of a fluid over a thick heated block until the system reaches a cyclic steady-state (i.e., the system ...
Avrana's user avatar
  • 297
21 votes
4 answers
1k views

What's wrong with this FFT-based Von Kármán vortex street simulation?

About 9 years ago, I came across this interesting website, and found the following paragraph with a broken Mathematica code sample: When fluid passes an object, it can leave a trail of vortices ...
xzczd's user avatar
  • 66.7k
2 votes
1 answer
120 views

Preventing Mathematica from applying product rule

I am programming viscous terms of compressible Navier-Stokes equations ...
dylewiczk's user avatar
4 votes
1 answer
164 views

NDSolve for Fluid Flow - Monitor Residuals

I'm using NDSolve for fluid flow and would like to monitor the convergence of the solution more closely. I have a simple stationary case set up. NDSolve should return the solutions for u,v and p. ...
kienappel's user avatar
  • 421
9 votes
1 answer
566 views

Fourth order, nonlinear PDE depicting dynamics of liquid lenses

PROBLEM STATEMENT Recently, I was trying to verify the solution of a set of thin film spreading equations given by this paper. Where $$p1=-(h_{1xx}+\sigma h_{2xx})$$ $$p2=-\sigma h_{2xx}-\Pi[h2-h1]$$ ...
FLP's user avatar
  • 547
1 vote
1 answer
120 views

Transient darcy flow equation

I can solve the following stationary Darcy problem: $div(\nabla u)=1$ with $ u = 0 \in \Gamma_d$ ...
Stratus's user avatar
  • 2,954
0 votes
0 answers
89 views

Plotting contours of a two-variable function containing a sum

I'm trying to use Mathematica to plot contours of a rather intricate two-variable function. The equation describes the velocity profile for laminar flow in a tube of rectangular section, namely: $$ {w^...
Lucas's user avatar
  • 21
0 votes
0 answers
135 views

How to find non-trivial solution to homogeneous linear equations by Mathematica using code (Input form)?

I have many homogeneous equations to which I am interested to find the non-trivial solution using Mathematica, But I am totally new to Mathematica, so I do no know, how to find the non-trivial ...
Noor Aslam's user avatar
10 votes
2 answers
849 views

Reciprocating flow in a channel over a heated surface

The following is a coupled heat transfer and fluid flow problem. A thick plane channel is being heated with a constant flux from the bottom (at $y=-e$) with a constant heat flux $q$ as shown in the ...
Avrana's user avatar
  • 297
21 votes
1 answer
520 views

3D stable fluids algorithm to simulate transition from laminar to turbulent flow

This algorithm is 3D extension of our 2D algorithm published on this page and here. We suppose that with this code we can simulate transition from laminar to turbulent flow. In this example we compute ...
Alex Trounev's user avatar
4 votes
1 answer
170 views

How to display warning message inside manipulate?

Using the code shown, Reynolds number is calculated for given parameters. Is it possible to display a warning message to the user if a parameter is outside applicable range? e.g., if the fluid ...
ASHISH's user avatar
  • 141
0 votes
0 answers
91 views

Can mathematica numerically solve this system of coupled integro differential equations?

Here's my system: R(x,y,t) is the concentration of a chemical that's diffusing across the box. The disk is a mobile sink for the chemical, which absorbs it in proportion to its concentration. The ...
dr_strangeloop's user avatar
1 vote
0 answers
168 views

Interaction between fluid flow and elastic obstacle (fluid-structure interaction) [closed]

I would like to model the velocity and pressure distribution of an elastic obstacle in a channel where the fluid flows. There are Navier-Stokes examples in Mathematica webpage with rigid obstacles but ...
metudust's user avatar
1 vote
1 answer
95 views

How to make a calculator to display the Non-Dimensional number

I am a beginner here and want to learn this tool. At first, I want to make one calculator that will display the value of the Non-Dimensional number of Fluid dynamics based on user inputs. In this code,...
ASHISH's user avatar
  • 141
5 votes
2 answers
326 views

Modified Heat Transfer in Fluid Flow

I am trying to simulate Modified Heat Transfer in Fluid Flow (based on Buoyancy-Driven Flow in a Square Cavity ). The modified heat transfer takes the form: with the solid volume fraction: The ...
ABCDEMMM's user avatar
  • 1,816
18 votes
1 answer
639 views

Stable fluids code for electromagnetic mixture application

This code has been translated from the original Jos Stam code and improved with some Mathematica functions. It solves problem of viscous incompressible flow with electromagnetic force in a rectangle ...
Alex Trounev's user avatar
16 votes
2 answers
1k views

Stokes equations in 2D with traction boundary conditions

This is a follow-up to a previous question (see here). We would like to solve the two-dimensional Stokes equations using the FEM package in Mathematica, when we prescribe traction boundary conditions. ...
Alex R's user avatar
  • 163
4 votes
1 answer
577 views

Stokes equations in 2D with Neumann conditions

I would like to solve the 2D Stokes equations within a unit disk, say $\Omega$, by using the finite element method (FEM) as it is implemented in NDSolve (by loading ...
Alex R's user avatar
  • 41
1 vote
1 answer
71 views

How to find invariant sets of nonlinear operators below

How to find invariant sets of nonlinear operators below. $$x[n+1]=0.9∗y[n]^2/(y[n]+2)+x[n]∗(1−0.8/(1+x[n]))$$ $$y[n+1]=0.8∗x[n]/(1+x[n])+0.6∗y[n]$$
Zafar's user avatar
  • 61
4 votes
2 answers
330 views

2D incompressible flow solver with known non-zero initial velocity distribution

I know that there are lots of questions on this forum regarding mathematica FEM, but I haven't quite found what I am looking for. I will be working with data obtained through PIV (particle image ...
Adrian del Ser's user avatar
0 votes
0 answers
114 views

How to solve this linear complementarity problem with NDSolve or other functions? The arguments should be ordered consistently

I want to find a numerical solution of $\omega(t,q)$, where $t\in[0,T]$ and $q\in[0,N]$. $q_t$ is a determinstic w.r.t $t$, other unmentioned parameters can be fullfilled or given by the user. I met ...
Hot Pizza's user avatar
0 votes
0 answers
80 views

Nonlinear Differential Equation with Paramaters in Sqrt Function

For a university project, I am trying to see if my system will have choked flow and also plot the resulting pressure spike. I set up the system below to try to model the transient response. I am able ...
OK1991's user avatar
  • 37
0 votes
1 answer
137 views

Using Solution to Differential Equation in New Function

I am trying to feed the solutions from my differential equations into another function. I am using NDSolve to solve two differential equations simultaneously. It ...
OK1991's user avatar
  • 37
6 votes
0 answers
242 views

System of coupled nonlinear pdes with FEM (magneto-hydrodynamic mass transfer)

EDIT 01 See end of post for an update. Original Question I'm trying to reproduce the magneto-hydrodynamic flow studied here using the nonlinear FEM functionality, and have been having trouble ...
George Varnavides's user avatar
0 votes
1 answer
213 views

FEM analyis of flow in a hydrocyclone [closed]

I am interested in solving the flow in a hydrocyclone (steady Navier Stokes equation). I have an STL file of the 3D geometry. My question is how to extract the internal geometry of the hydrocyclone ...
user2088899's user avatar