# Torque and concentric cylinders fluids

• member 428835
In summary, the conversation discusses how to calculate torque on the outer edge of a larger cylinder that is rotating around a smaller cylinder. The equation given for shear stress is incorrect, and the correct rheological equation for a Newtonian fluid is provided. The speaker suggests looking up the components of the stress tensor online or deriving them in cylindrical coordinates. A reference is given for the derivation.
member 428835
Hi PF!

If we have two concentric cylinders with Newtonian fluid between them, and the small cylinder is at rest and the larger cylinder with radius ##R## rotates at some angular velocity ##\Omega##, how would you calculate torque ##\vec{T}## on the outer edge?

My thoughts: ##\vec{T} = \vec{r}\times \vec{F}## where ##\vec{r}=R\vec{r}##. To find ##\vec{F}##, we'll need the shear stress in the ##\theta## direction. I know in general the shear stress ##\bar{\bar\tau}## is a second order tensor defined for a Newtonian incompressible fluid as ##\bar{\bar\tau} = \mu \nabla \vec{V}##. So then ##\vec{F} = \bar{\bar\tau} \cdot \hat{\theta} = (\mu \nabla \vec{V}) \cdot \hat{\theta}##. Since I don't know ##\nabla \vec{V}## in cylindrical coordinates, I cannot proceed. Please help me out here.

If I knew ##(\mu \nabla \vec{V}) \cdot \hat{\theta}## then ##\vec{F} = \iint_S (\mu \nabla \vec{V}) \cdot \hat{\theta} \, dA## where ##S## is the boundary of the cylinder and ##dA## is an area element.

joshmccraney said:
Hi PF!

If we have two concentric cylinders with Newtonian fluid between them, and the small cylinder is at rest and the larger cylinder with radius ##R## rotates at some angular velocity ##\Omega##, how would you calculate torque ##\vec{T}## on the outer edge?

My thoughts: ##\vec{T} = \vec{r}\times \vec{F}## where ##\vec{r}=R\vec{r}##. To find ##\vec{F}##, we'll need the shear stress in the ##\theta## direction. I know in general the shear stress ##\bar{\bar\tau}## is a second order tensor defined for a Newtonian incompressible fluid as ##\bar{\bar\tau} = \mu \nabla \vec{V}##.
This equation is not correct. The shear stress is a vector and del V is a tensor. The rheological equation for a Newtonian fluid is: $$\pmb{\sigma}=-p\mathbf{I}+\mu(\pmb{\nabla}\mathbf{V}+(\pmb{\nabla}\mathbf{V})^T)$$To get the shear stress you are looking for, you would have to dot this with a unit vector in the radial direction and with a unit vector in the tangential direction.
The easiest thing to do is look up the components of the stress tensor in terms of the components of the velocity vector for cylindrical coordinates on line. Or, do you feel compelled to derive the gradient of velocity vector in cylindrical coordinates on your own?

member 428835
Chestermiller said:
The easiest thing to do is look up the components of the stress tensor in terms of the components of the velocity vector for cylindrical coordinates on line. Or, do you feel compelled to derive the gradient of velocity vector in cylindrical coordinates on your own?
I would like to see how the derivation goes, if you know a reference or would walk me through it?

joshmccraney said:
I would like to see how the derivation goes, if you know a reference or would walk me through it?
Section 1.2, BSL

Thanks!

## 1. What is torque in relation to concentric cylinders and fluids?

Torque is a measure of the force that causes an object to rotate around an axis. In the context of concentric cylinders and fluids, it refers to the force required to rotate an inner cylinder within an outer cylinder that is filled with fluid.

## 2. How is torque calculated for concentric cylinders and fluids?

The equation for torque in this situation is T = μΔω, where μ is the viscosity of the fluid, Δ is the difference in angular velocity between the inner and outer cylinders, and ω is the angular velocity of the inner cylinder.

## 3. What factors affect the torque in concentric cylinders and fluids?

The torque in this situation is affected by the viscosity of the fluid, the difference in angular velocity between the cylinders, the diameter of the cylinders, and the length of the cylinders. Additionally, the type of fluid and its temperature can also impact the torque.

## 4. How does torque affect the flow rate in concentric cylinders and fluids?

The torque applied to the inner cylinder can impact the flow rate of the fluid. A higher torque will result in a higher flow rate, while a lower torque will result in a lower flow rate. This relationship is known as the torque-speed relationship.

## 5. What is the significance of torque in practical applications of concentric cylinders and fluids?

Torque is an important concept in many practical applications involving concentric cylinders and fluids. For example, it is used in the design of pumps and turbines, as well as in the measurement of viscosity of fluids. Understanding the relationship between torque and flow rate is crucial in these applications for optimal performance.

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