# Deriving the 1-D Linear Convection Equation

• Mr_Acceleration
In summary, the conversation discusses the derivation of the 1-D nonlinear convection equation and the process of linearizing it by assuming a constant velocity of wave propagation. This results in the 1-D linear convection equation, where one of the u values is replaced with c. The conversation also mentions the need for a full derivation and the reasoning behind linearization with perturbation.
Mr_Acceleration
With the assumptions of Inviscid flow, no pressure gradient and no body force terms in 1-D Navier Stokes becomes 1-D nonlinear convection equation;

And if we assume velocity of wave propagation is constant value c, equation becomes 1-D linear convection equation;

This is online derivation and my question is why only the u value outside of partial derivatives is replaced with c? we have 3 u term there. I was thinking they were all same term. So why do we only replace one of them?

I don't think this is correct. Can you please provide the full derivation ?

dRic2 said:
I don't think this is correct. Can you please provide the full derivation ?

Here is the derivation.

This is correct so far. The problem is this:

Mr_Acceleration said:
And if we assume velocity of wave propagation is constant value c, equation becomes 1-D linear convection equation;

This is not. That's why I'd like a full derivation.

Anyway the reasoning is the following:
$$\frac {\partial u} {\partial t} + u \frac {\partial u} {\partial x} = 0$$
is NON-linear. What you usually do is linearize it, by writing ##u = u_0 + u'## where ##u_0## is an unperturbed constant velocity and ##u'## is a small perturbation. If you substitute into the diff equation ##\frac {\partial u_0} {\partial t} = \frac {\partial u_0} {\partial x} = 0## because ##u_0## is a constant. You finally end up with
$$\frac {\partial u'} {\partial t} + u_0 \frac {\partial u'} {\partial x} = 0$$

SCP and Mr_Acceleration

## 1. What is the 1-D Linear Convection Equation?

The 1-D Linear Convection Equation is a mathematical model used to describe the movement of a fluid in one dimension, such as along a pipe or channel. It takes into account the velocity and direction of the fluid flow, as well as any external forces acting on the fluid.

## 2. How is the 1-D Linear Convection Equation derived?

The 1-D Linear Convection Equation is derived from the Navier-Stokes equations, which describe the motion of a fluid in three dimensions. By simplifying these equations for one-dimensional flow and assuming a constant velocity, the 1-D Linear Convection Equation can be derived.

## 3. What are the applications of the 1-D Linear Convection Equation?

The 1-D Linear Convection Equation is commonly used in various fields such as fluid mechanics, heat transfer, and meteorology. It can be used to model the flow of air or water in pipes, the transfer of heat in a fluid, and the movement of atmospheric fronts, among other applications.

## 4. What are the limitations of the 1-D Linear Convection Equation?

The 1-D Linear Convection Equation assumes a constant velocity and does not account for turbulence or other complex flow phenomena. It also does not consider the effects of viscosity or compressibility, which may be significant in certain situations.

## 5. How is the 1-D Linear Convection Equation solved?

The 1-D Linear Convection Equation can be solved using numerical methods such as finite difference or finite element methods. These methods involve discretizing the equation into smaller parts and solving for the solution at each point. Analytical solutions may also be possible for simplified cases.

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