Convection Diffusion Equation

In summary, the conversation discusses using the Convection Diffusion Equation to model temperature behavior near a heat source in a water bath. The speaker initially solved the system for arbitrary boundary conditions, but encountered a problem when trying to solve for a specific boundary condition of u'(1)=0. They explain that this issue is typically addressed by allowing C2 to be a polynomial and solving for the boundary conditions. The conversation then briefly touches on the idea of C2 being a linear function.
  • #1
liquidFuzz
97
3
I'm tinkering with the Convection Diffusion Equation (a second order differential equation) to model a temperature behavior in proximity to a heat source in a water bath. Just to get going I solved the system for some arbitrarily chosen boundary conditions. The result is that the temperature at the ambient boundary has a derivative that isn't zero. This is sort of okay in my solution as it is only meant to show a likely temperature behavior. But how do I tackle this problem if I like to solve it with a boundary condition such as u'(1)=0, i.e. x=1 is at ambient temperature. The problem I have with this is that if I chose an exponential solution such as u(x) = C1 + C2 exp(x) I get C2 = 0 when I try to find the constants C.
 
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  • #2
I believe this is usually handled by letting your C2 by a polynomial and solving for the boundary conditions.
 
  • #3
Okay, so C2 equals some function f(x)? A linear function..?
 

1. What is the Convection Diffusion Equation?

The Convection Diffusion Equation is a mathematical model used to describe the transport of a substance, such as heat or a chemical, through a medium by both convection (flow) and diffusion (random motion). It takes into account the effects of both convection and diffusion on the distribution of the substance over time and space.

2. How is the Convection Diffusion Equation derived?

The Convection Diffusion Equation is derived from the conservation of mass and the first law of thermodynamics. It is a partial differential equation that describes the change in the concentration or temperature of the substance with respect to time and space.

3. What are the applications of the Convection Diffusion Equation?

The Convection Diffusion Equation has various applications in fields such as fluid dynamics, heat transfer, chemical engineering, and atmospheric sciences. It is used to model and understand processes like heat and mass transfer, dispersion of pollutants in the environment, and drug delivery in the human body.

4. What are the boundary conditions for the Convection Diffusion Equation?

The boundary conditions for the Convection Diffusion Equation include the initial concentration or temperature distribution, as well as the boundary conditions for the medium, such as the temperature or concentration at the boundaries. These conditions are necessary to solve the equation and obtain a solution that accurately represents the system.

5. How is the Convection Diffusion Equation solved?

The Convection Diffusion Equation can be solved using various numerical methods, such as finite difference, finite volume, and finite element methods. These methods involve discretizing the equation into smaller parts and solving them iteratively until a solution is obtained. Advanced techniques, such as computational fluid dynamics, can also be used to solve the equation for more complex systems.

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