All Questions
Tagged with dimensional-analysis heat-equation
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Applying conservation of energy to dimensionless form of heat equation
This is part of a large exercise about dimensional analysis. Basically we have the 1-D heat equation in a rod with infinite length:
$$\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\...
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Modeling a Heat PDE
I am trying to answer the following question...
Consider a wall made of brick $10$ centimeters thick, which separates a room in a house from the outside. The room is kept at $20$ degrees. Initially ...
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How to achieve (approx) unit scaling of a non-linear diffusion (heat) equation with a wildly varying diffusion coefficient?
I have numerical issues with a poorly scaled one-dimensional non-linear diffusion equation in physical co-ordinates
$$ \frac{\partial{u}}{\partial{t}}(x,t) = \frac{\partial}{\partial{x}}\left(D(u) \...
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Scaling in heat transfer PDE and dimensionless group
I found the following in my Scaling in Heat Transfer notes:
Rod Conduction
A rod of length $L$ initially at $T_0$ then (at $t = 0$) one end is raised to $T_1$. Find $T(x, t)$.
$$T_t = \kappa T_{x x}$$...
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Showing a function is a function of only a variable $\eta(x,t)$
Suppose I have a differential equation $$\frac{\partial\theta}{\partial t}=\frac{\partial^2\theta}{\partial x^2}$$ satisfied by $\theta(x,t)$ and I then had that $\theta(x,t)=k F(x,t)$, and I know $F(...
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Intuition behind similarity solution for Heat equation
The heat equation is $$\frac{\partial\theta}{\partial t}=\kappa\frac{\partial^2\theta}{\partial x^2}.$$ From this, one can look at dimensions on both sides, and conclude that the quantity $$\eta=\frac{...
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Scaling Two Equations
I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated.
An incompressible thermal conducting fluid is contained between two infinite ...
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