Kinetic Theory of Gas question involving viscosity

[rabbit44]

Homework Statement

PLEASE help! I really should be able to do this but am so stuck:



Two plane disks, each of radius 5 cm, are mounted coaxially with their adjacent surfaces
1 mm apart. They are in a chamber containing Ar gas at S.T.P. (viscosity
2.1×10−5 Nsm−2) and are free to rotate about their common axis. One of them rotates
with an angular velocity of 10 rad s−1. Find the couple which must be applied to
the other to keep it stationary.

Homework Equations

I know there's an equation of F/A = viscosity*d<ux>/dz

Where <ux> is the average x-velocity at a given z (z is along the axis).

The Attempt at a Solution

I tried taking ux to be the component of a particle's velocity along the theta direction (perpendicular to the radius of the disc), and replacing it with rw. I then tried to integrate that equation up there wrt z, so that the RHS gives viscosity*r*(w - w0) where w0 is the value given for angular velocity and w is the value at the other disc.

However I am confused as to what F is. I think it's the force exerted on the spinning disc, but if so how do I know what it is? I thought about using Idw0/dt = torque, but w0 is constant. I also thought about centripetal force acting on parts of the disc, but I don't understand how that could be right.

I'm very confused!

Thanks!

 

[anathema]

Dear student,

I understand that you are struggling with this problem. Let's break it down step by step.

First, let's consider the equation F/A = viscosity*d<ux>/dz. This equation is known as the Navier-Stokes equation and it describes the relationship between the force per unit area (F/A) and the rate of change of the average velocity in the x-direction (d<ux>/dz) due to the presence of a fluid with a certain viscosity. In this case, the fluid is Ar gas at S.T.P. and the viscosity is given to be 2.1×10−5 Nsm−2.

Next, let's consider the disks. Since they are mounted coaxially, we can treat them as one system with two surfaces. One disk is rotating with an angular velocity of 10 rad s−1, while the other is stationary. We want to find the couple (torque) that must be applied to the stationary disk in order to keep it stationary.

To do this, we can use the equation for torque, T = r x F, where r is the distance from the axis of rotation to the point where the force is applied and F is the force itself. In this case, the force we are interested in is the force exerted by the rotating disk on the stationary one.

To find this force, we can use the Navier-Stokes equation. Since the disks are very close together (1 mm apart), we can assume that the average velocity in the x-direction is the same for both disks. This means that we can use the same value for <ux> in the equation for both disks.

Now, we need to determine the distance r. Since the disks are mounted coaxially, the distance from the axis of rotation to the point where the force is applied is simply the radius of the disks (5 cm).

Putting it all together, we get T = r x (F/A) = r x (viscosity*d<ux>/dz). Plugging in the given values, we get T = (0.05 m) x (2.1×10−5 Nsm−2) x (10 rad s−1) = 1.05×10−5 Nm.

So, the couple (torque) that must be applied to the stationary disk is 1.05×10−5 Nm.

I hope this helps you understand the problem better. Good luck with

 

[thomate1]

Hello,

The problem you are trying to solve involves the Kinetic Theory of Gases, specifically the concept of viscosity. Viscosity is a measure of a gas's resistance to flow, and it is caused by the interaction between gas molecules and the walls of the container they are in. In this case, the gas in the chamber is Ar (argon) at standard temperature and pressure.

To solve this problem, we can use the formula you provided: F/A = viscosity*d<ux>/dz. Here, F is the force exerted on the disc, A is the area of the disc, and <ux> is the average x-velocity at a given z (distance along the axis). This formula relates the force exerted on the disc to the velocity gradient (change in velocity with respect to distance) of the gas molecules near the disc's surface.

To find the force exerted on the disc, we can use the concept of torque. Torque is the rotational equivalent of force, and it is equal to the force multiplied by the distance from the point of rotation. In this case, the force exerted by the gas molecules on the disc will create a torque, causing the disc to rotate. The torque can be calculated by multiplying the force (F) by the radius (r) of the disc (since the force is acting at a distance from the center of rotation).

To keep the other disc stationary, we need to apply an equal and opposite torque to counteract the torque caused by the rotating disc. This can be done by applying a couple (two equal and opposite forces acting at a distance from each other) to the other disc. The value of this couple can be calculated by using the formula: torque = force * distance. In this case, the distance would be the radius of the disc, and the force can be found using the formula F/A = viscosity*d<ux>/dz.

I hope this helps you understand the problem better. Please let me know if you have any further questions or need clarification on any of the concepts. Good luck with your homework!

 

[baba89]

The problem at hand involves the Kinetic Theory of Gases and the concept of viscosity. Viscosity is a measure of a fluid's resistance to flow, and in this case, it is the resistance of the Ar gas to flow between the two rotating disks. To find the couple required to keep one disk stationary, we need to consider the forces acting on the gas molecules between the two disks.

From the given information, we can calculate the area of contact between the two disks (A = πr^2) and the velocity gradient (∂<ux>/∂z = (w-w0)/d). Using the equation you mentioned, F/A = viscosity*∂<ux>/∂z, we can then solve for the force (F) acting on the gas molecules between the two disks.

Next, we need to consider the torque acting on the disk due to this force. The torque is given by the formula T = Fr, where r is the distance from the axis of rotation to the point where the force is applied. In this case, r is equal to the radius of the disk (5 cm). The torque acting on one disk due to the force from the gas molecules can then be calculated.

Since the two disks are coaxial, the torque acting on one disk will be equal in magnitude but opposite in direction to the torque acting on the other disk. This means that to keep the second disk stationary, a couple with equal magnitude but opposite direction must be applied. This couple can be calculated using the formula T = F*r, where r is again equal to the radius of the disk.

In summary, to find the couple required to keep one disk stationary, we need to calculate the force acting on the gas molecules between the two disks, and then use this force to calculate the torque acting on one disk. This torque can then be used to determine the required couple to keep the second disk stationary. I hope this helps!