Undergrad Diffusion and convection in a moving fluid

[fahraynk]

In a fluid moving at constant velocity V, a concentration will undergo diffusion and convection

$$U_t=\alpha^2 U_{xx}+VU_x$$

I understand that in this situation you can transform to a moving coordinate system and the convection term will disappear and you can solve for diffusion alone and transform back.

My question is for a situation where the fluid streamlines are moving at different velocities. In this situation doesn't the diffusion depend on the speed of the fluid... So will alpha^2 be a function of velocity?
Alpha^2 is The diffusivity, if heat it would be thermal conductivity/(heat capacity * density)

If it is a function of velocity... how do I determine it? If not... what is the other term other than diffusion and convection that I put into the PDE?

An example of this situation would be couette flow, fluid flow between 2 moving plates infinite in x direction finite in Y.

 

[Chestermiller]

The diffusivity does not change with velocity. But even in uniform flow, you have omitted terms in the transient balance: $$U_t+VU_x=\kappa(U_{xx}+U_{yy}+U_{zz})$$You can have heat or mass diffusing normal to the streamlines. In the example of couette flow, the main heat flow is normal to the steamlines. And, for couette flow, you need to properly express the heat balance in cylindrical coordinates:
$$U_t+\frac{v_{\theta}(r)}{r}U_{\theta}=\kappa\left(\frac{1}{r}(rU_r)_r+\frac{1}{r^2}U_{\theta \theta}\right)$$

 

[fahraynk]

The diffusivity does not change with velocity. But even in uniform flow, you have omitted terms in the transient balance: $$U_t+VU_x=\kappa(U_{xx}+U_{yy}+U_{zz})$$You can have heat or mass diffusing normal to the streamlines. In the example of couette flow, the main heat flow is normal to the steamlines. And, for couette flow, you need to properly express the heat balance in cylindrical coordinates:
$$U_t+\frac{v_{\theta}(r)}{r}U_{\theta}=\kappa\left(\frac{1}{r}(rU_r)_r+\frac{1}{r^2}U_{\theta \theta}\right)$$

Thank you for your help.

So, absent of viscous dissipation when talking about heat... so say a mass is diffusing... Velocity has nothing to do with it? (say flat plate 2d couette flow with only velocity in x direction but plates infinite in x direction so convection should not matter since no velocity in y direction)

This would mean that stirring a glass of chocolate milk would have nothing to do with stirring the chocolate milk, so you might as well just stab the chocolate powder with a knife to have same affect? (Bc all the mixing would come from moving the particles randomly and the speed of the rotating milk would have no added bonus)

 

[Chestermiller]

Thank you for your help.

So, absent of viscous dissipation when talking about heat... so say a mass is diffusing... Velocity has nothing to do with it? (say flat plate 2d couette flow with only velocity in x direction but plates infinite in x direction so convection should not matter since no velocity in y direction)

This would mean that stirring a glass of chocolate milk would have nothing to do with stirring the chocolate milk, so you might as well just stab the chocolate powder with a knife to have same affect? (Bc all the mixing would come from moving the particles randomly and the speed of the rotating milk would have no added bonus)

What mixing does is change (increases) the temperature gradients and the concentration gradients. This enhances the diffusion rate. If you have axial flow in a tube with an axial velocity that varies radially, the flow affects the temperature or concentration gradient in the radial direction, which enhances the radial diffusion rate.

 

[fahraynk]

What mixing does is change (increases) the temperature gradients and the concentration gradients. This enhances the diffusion rate. If you have axial flow in a tube with an axial velocity that varies radially, the flow affects the temperature or concentration gradient in the radial direction, which enhances the radial diffusion rate.

Okay... so where I am lost is... how is the change in concentration gradient caused by velocity represented in the PDE? Or is it represented by the k∇^2U term? (Also I know its wrong coordinate system but my intuition is better in Cartesian).

I was thinking it can't be represented by k∇^2U BC it represents the average between 2 instantaneous points? But Maybe this is where I am mistakern

 

[Chestermiller]

Okay... so where I am lost is... how is the change in concentration gradient caused by velocity represented in the PDE? Or is it represented by the k∇^2U term? (Also I know its wrong coordinate system but my intuition is better in Cartesian).

I was thinking it can't be represented by k∇^2U BC it represents the average between 2 instantaneous points? But Maybe this is where I am mistakern

Suppose the diffusion were suddenly shut off, but there were temperature or concentration variations already in the fluid and the flow velocity where one of shearing (like flow in a pipe). Would the advection flow not cause the temperature gradients to change?

 

[fahraynk]

Suppose the diffusion were suddenly shut off, but there were temperature or concentration variations already in the fluid and the flow velocity where one of shearing (like flow in a pipe). Would the advection flow not cause the temperature gradients to change?

AH! Thanks for your patience. I was being stupid. Convection is the velocity times the gradient and its in every direction where the gradient is strongest it will send mass in that direction.

so, if the velocity has an x and y component, than
$$U_t=-V*[U_x+U_y+U_z]$$
I thought convection just carried the mass down in 1 direction at a speed V because I was studying method of characteristics for uniform flow. I did not realize. thanks