| Zimbra Collaboration Suite | fmorriso@mtu.edu |
| [cm3110-l] Re: Navier-Stokes Equation equation | Sunday, October 02, 2011 8:05:34 AM |
| From: | fmorriso@mtu.edu |
| To: | bjricchi@mtu.edu |
| Cc: | cm3110-l@mtu.edu |
Good Morning Brian,
The Navier-Stokes equation plus the continuity equation is four equations in four unknowns: v_x, v_y, v_z, and p, the pressure. When we solve the Navier-Stokes equations plus the continuity equation, we obtain the complete velocity vector and the pressure.
Pressure is not the only force in a fluid. In a flowing fluid in a shear cell, for example (Newton's experiment), the force that it takes to slide the top plate is a real force on the fluid, but it has nothing to do with pressure. Likewise, if a jet of fluid from a fire hose hits a wall, there is a force on the wall, but the fluid jet is at atmospheric pressure, and the wall not being hit by the jet is also at atmospheric pressure too. So why is there a force on the wall?
There is a force on the wall due to the jet because of the velocity change in the fluid. The velocity of the fluid goes from being high (fast flow in a fire hose) to being low (it slows down when it hits the wall) and it changes direction (remember that acceleration/deceleration can be due to change of direction as well as change of magnitude of speed). Thus, to figure out the force on the wall, we need to account for how the velocity field changes at the wall.
In my description here I will refer to equations that are in my equation handout, which is at this link:
http://www.chem.mtu.edu/~fmorriso/cm310/MorrisonCoverMatter%28c%292011.pdf
The stress generated in a moving fluid is given by the constitutive equation. If you look at the Newtonian constitutive equation on the equation handout you will see that if you know the velocity v_x, v_y, v_z (which you get from the NS equation+continuity) you can calculate Tau. If you have Tau and pressure p (which you get from the NS equation+continuity) you can calculate Pi. If you have Pi, the total stress tensor, you can calculate the force on any wall from the equation labeled "Total Molecular Fluid Force on a Finite Surface S." We should give it a name so we don't have to say that whole thing: let's call it the "Fluid Surface Force Equation."
Note that the Fluid Surface Force Equation (FSFE) has pressure in it. When pressure is the only thing that is causing force on a surface (the case you propose), this equation will be right. When velocity gradients (the Tau) also cause a contribution to the force on a surface (the more common and general case), this equation will be right. Since the equation is always right, I propose that we use it in all cases until we are expert enough to jump right to the answer of whether you can just use pressure or whether you need a Tau contribution.
One exception of the need to use the Fluid Surface Force Equation is when the fluid is stationary. If the velocity vector is zero, we can see from the Newtonian Constitutive Equation (on the formula handout) that Tau is zero. So we easily see that the total stress tensor Pi becomes a diagonal tensor with -P along the diagonal. Such a matrix will always give a force on a surface that is perpendicular to the surface and compressive. This is just the pressure you are used to.
In terms of flow rate, the formal equation for flow rate calculation in any case is labeled "Total Flow Rate out Through a Finite Surface S" in the handout. If you have velocity v_x v_y v_z you can carry out the dot product with the outwardly pointing unit normal of the surface n and then do the integral to get the flow rate. Once you have the flow rate, the average velocity is this flow rate divided by the area of the surface, which is the equation you asked about below.
I hope that helps to clarify the usefulness of the Navier-Stokes equations in calculating forces on walls and flow rates.
Sincerely,
Dr. Morrison
----- "Brian Ricchi" <bjricchi@mtu.edu> wrote:
> From: "Brian Ricchi" <bjricchi@mtu.edu>
> To: "Faith Morrison" <fmorriso@mtu.edu>
> Cc: "bjricchi" <bjricchi@mtu.edu>
> Sent: Saturday, October 1, 2011 9:11:29 PM GMT -05:00 US/Canada Eastern
> Subject: Navier-Stokes Equation
>
> Good evening Dr. Morrison,
>
> Please excuse my late email but I have a couple questions regarding
> the Navier-Stokes equation. I have been working on Homework #2 and
> doing other problems as well and don't really understand what the
> Navier-Stokes equation tells us. In the problems I've done, I have
> gotten one of the three components to equal zero, one that relates to
> the pressure of the system and one relating to the velocity profile.
> With these known, how can I determine the force on the surface?
> Wouldn't the force just be the pressure on the surface at any given
> point? Also, with the velocity profile, I can determine the
> volumetric flow rate through the equation <v> = Q/Area? If you could
> please help me with this confusion it would be much appreciated.
>
> Thank you,
>
> Brian Ricchi
> Chemical Engineering
> Michigan Technological University
> American Institute of Chemical Engineers
The Navier-Stokes equation plus the continuity equation is four equations in four unknowns: v_x, v_y, v_z, and p, the pressure. When we solve the Navier-Stokes equations plus the continuity equation, we obtain the complete velocity vector and the pressure.
Pressure is not the only force in a fluid. In a flowing fluid in a shear cell, for example (Newton's experiment), the force that it takes to slide the top plate is a real force on the fluid, but it has nothing to do with pressure. Likewise, if a jet of fluid from a fire hose hits a wall, there is a force on the wall, but the fluid jet is at atmospheric pressure, and the wall not being hit by the jet is also at atmospheric pressure too. So why is there a force on the wall?
There is a force on the wall due to the jet because of the velocity change in the fluid. The velocity of the fluid goes from being high (fast flow in a fire hose) to being low (it slows down when it hits the wall) and it changes direction (remember that acceleration/deceleration can be due to change of direction as well as change of magnitude of speed). Thus, to figure out the force on the wall, we need to account for how the velocity field changes at the wall.
In my description here I will refer to equations that are in my equation handout, which is at this link:
http://www.chem.mtu.edu/~fmorriso/cm310/MorrisonCoverMatter%28c%292011.pdf
The stress generated in a moving fluid is given by the constitutive equation. If you look at the Newtonian constitutive equation on the equation handout you will see that if you know the velocity v_x, v_y, v_z (which you get from the NS equation+continuity) you can calculate Tau. If you have Tau and pressure p (which you get from the NS equation+continuity) you can calculate Pi. If you have Pi, the total stress tensor, you can calculate the force on any wall from the equation labeled "Total Molecular Fluid Force on a Finite Surface S." We should give it a name so we don't have to say that whole thing: let's call it the "Fluid Surface Force Equation."
Note that the Fluid Surface Force Equation (FSFE) has pressure in it. When pressure is the only thing that is causing force on a surface (the case you propose), this equation will be right. When velocity gradients (the Tau) also cause a contribution to the force on a surface (the more common and general case), this equation will be right. Since the equation is always right, I propose that we use it in all cases until we are expert enough to jump right to the answer of whether you can just use pressure or whether you need a Tau contribution.
One exception of the need to use the Fluid Surface Force Equation is when the fluid is stationary. If the velocity vector is zero, we can see from the Newtonian Constitutive Equation (on the formula handout) that Tau is zero. So we easily see that the total stress tensor Pi becomes a diagonal tensor with -P along the diagonal. Such a matrix will always give a force on a surface that is perpendicular to the surface and compressive. This is just the pressure you are used to.
In terms of flow rate, the formal equation for flow rate calculation in any case is labeled "Total Flow Rate out Through a Finite Surface S" in the handout. If you have velocity v_x v_y v_z you can carry out the dot product with the outwardly pointing unit normal of the surface n and then do the integral to get the flow rate. Once you have the flow rate, the average velocity is this flow rate divided by the area of the surface, which is the equation you asked about below.
I hope that helps to clarify the usefulness of the Navier-Stokes equations in calculating forces on walls and flow rates.
Sincerely,
Dr. Morrison
----- "Brian Ricchi" <bjricchi@mtu.edu> wrote:
> From: "Brian Ricchi" <bjricchi@mtu.edu>
> To: "Faith Morrison" <fmorriso@mtu.edu>
> Cc: "bjricchi" <bjricchi@mtu.edu>
> Sent: Saturday, October 1, 2011 9:11:29 PM GMT -05:00 US/Canada Eastern
> Subject: Navier-Stokes Equation
>
> Good evening Dr. Morrison,
>
> Please excuse my late email but I have a couple questions regarding
> the Navier-Stokes equation. I have been working on Homework #2 and
> doing other problems as well and don't really understand what the
> Navier-Stokes equation tells us. In the problems I've done, I have
> gotten one of the three components to equal zero, one that relates to
> the pressure of the system and one relating to the velocity profile.
> With these known, how can I determine the force on the surface?
> Wouldn't the force just be the pressure on the surface at any given
> point? Also, with the velocity profile, I can determine the
> volumetric flow rate through the equation <v> = Q/Area? If you could
> please help me with this confusion it would be much appreciated.
>
> Thank you,
>
> Brian Ricchi
> Chemical Engineering
> Michigan Technological University
> American Institute of Chemical Engineers