|[cm3110-l] Re: Navier-Stokes Equation
||Sunday, October 02, 2011
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
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
In my description here I will refer to equations that are in my
equation handout, which is at this link:
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
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.
----- "Brian Ricchi" <firstname.lastname@example.org> wrote:
> From: "Brian Ricchi" <email@example.com>
> To: "Faith Morrison" <firstname.lastname@example.org>
> Cc: "bjricchi" <email@example.com>
> Sent: Saturday, October 1, 2011 9:11:29 PM GMT -05:00 US/Canada
> Subject: Navier-Stokes Equation
> Good evening Dr. Morrison,
> Please excuse my late email but I have a couple questions
> the Navier-Stokes equation. I have been working on Homework
> doing other problems as well and don't really understand what the
> Navier-Stokes equation tells us. In the problems I've done,
> gotten one of the three components to equal zero, one that relates
> the pressure of the system and one relating to the velocity
> 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
> 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