Need a boundary condition

hunt_mat · May 25, 2023, 08:40

I am studying sintering. To start a simple approach, I am looking at a 1D unsteady system to begin with.

I have a compressible elastic rod. I have three governing partial differential equations and one ODE governing the length.

I have a conservation of mass equation, a conservation of momentum, a temperature equation, and an ODE governing the length of the rod.

I have boundary conditions for the conservation of momentum, and temperature equation which helps me fill in the velocity and temperature arrays, but I don't know what to do for the conservation of mass. The mass is constant overall. Any suggestions?

LuckyTran · May 25, 2023, 10:13

You should have BCs for continuity (i.e. no mass across the boundaries) but they will be shared with the momentum equation.

Sry I don't know off the top of my head what are the governing equations for a sintering rod. Would be helpful if you provide them.

But in Navier-Stokes for example we have also a situation where you have mass, momentum, and energy conservation equations but no separate BCs for mass and momentum since a velocity specification in the momentum equation is shared with the continuity equation. The approach for fluid flow then is to solve a constrained system of equations. That is, you solve the momentum equation subject to the continuity equation constraint. See for example, lagrange multipliers.

May 25, 2023, 08:40	Need a boundary condition	#1
hunt_mat Senior Member Matthew Join Date: Mar 2022 Location: United Kingdom Posts: 184 Rep Power: 5	I am studying sintering. To start a simple approach, I am looking at a 1D unsteady system to begin with. I have a compressible elastic rod. I have three governing partial differential equations and one ODE governing the length. I have a conservation of mass equation, a conservation of momentum, a temperature equation, and an ODE governing the length of the rod. I have boundary conditions for the conservation of momentum, and temperature equation which helps me fill in the velocity and temperature arrays, but I don't know what to do for the conservation of mass. The mass is constant overall. Any suggestions?

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May 25, 2023, 10:13		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,843 Rep Power: 68	You should have BCs for continuity (i.e. no mass across the boundaries) but they will be shared with the momentum equation. Sry I don't know off the top of my head what are the governing equations for a sintering rod. Would be helpful if you provide them. But in Navier-Stokes for example we have also a situation where you have mass, momentum, and energy conservation equations but no separate BCs for mass and momentum since a velocity specification in the momentum equation is shared with the continuity equation. The approach for fluid flow then is to solve a constrained system of equations. That is, you solve the momentum equation subject to the continuity equation constraint. See for example, lagrange multipliers.