Derivation of Velocity Profile

Oct 13, 2025 by windailys.com 52 views

Introduction

The study of fluid dynamics is a crucial aspect of understanding various natural phenomena and engineering applications. One of the fundamental concepts in fluid dynamics is the behavior of fluids in rotating systems, such as concentric cylinders. The velocity profile of a fluid between two concentric cylinders is a critical parameter in determining the torque experienced by the inner cylinder. However, a derivation of this velocity profile has been provided, which raises concerns about its accuracy. In this article, we will delve into the derivation and examine whether it is correct or not.

Derivation of Velocity Profile

The derivation provided assumes a steady-state, incompressible flow between two concentric cylinders. The inner cylinder is rotating with a constant angular velocity, while the outer cylinder is stationary. The velocity profile is derived using the Navier-Stokes equations, which describe the motion of fluids. The derivation involves several steps, including:

Assuming a velocity profile: The derivation assumes a velocity profile of the form:
$u(r) = \frac{\Omega r^2}{R^2 - r^2}$
where $u(r)$ is the velocity at a distance $r$ from the center of the inner cylinder, $\Omega$ is the angular velocity of the inner cylinder, and $R$ is the radius of the outer cylinder.
Applying the Navier-Stokes equations: The Navier-Stokes equations are applied to the flow between the two cylinders. The equations are:
$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + \frac{v}{r} \frac{\partial u}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{u}{r^2} \right)$ $\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial r} + \frac{v}{r} \frac{\partial v}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial \theta} + \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} + \frac{v}{r^2} \right)$
where $v$ is the velocity in the $\theta$ direction, $\rho$ is the density of the fluid, and $\nu$ is the kinematic viscosity.
Simplifying the equations: The equations are simplified by assuming a steady-state flow and neglecting the terms involving the time derivative and the velocity in the $\theta$ direction.
Solving the equations: The simplified equations are solved to obtain the velocity profile.

Analysis of Derivation

The derivation provided assumes a velocity profile of the form:

u(r) = \frac{\Omega r^2}{R^2 - r^2}

However, this velocity profile is not consistent with the Navier-Stokes equations. The Navier-Stokes equations describe the motion of fluids in terms of the velocity, pressure, and density. The velocity profile derived above does not satisfy the Navier-Stokes equations.

Torque Derivation

The torque experienced by the inner cylinder is given by:

T = \int_0^R \tau(r) 2 \pi r dr

where $\tau(r)$ is the shear stress at a distance $r$ from the center of the inner cylinder.

The shear stress is given by:

\tau(r) = \mu \frac{\partial u}{\partial r}

where $\mu$ is the dynamic viscosity of the fluid.

Substituting the velocity profile derived above, we get:

\tau(r) = \mu \frac{\partial}{\partial r} \left( \frac{\Omega r^2}{R^2 - r^2} \right)

Evaluating the derivative, we get:

\tau(r) = \mu \frac{2 \Omega r}{(R^2 - r^2)^2}

Substituting this expression for $\tau(r)$ into the torque equation, we get:

T = \int_0^R \mu \frac{2 \Omega r}{(R^2 - r^2)^2} 2 \pi r dr

Evaluating the integral, we get:

T = \frac{4 \pi \mu \Omega}{R^2}

Conclusion

The derivation provided for the velocity profile of a fluid between two concentric cylinders is incorrect. The velocity profile derived does not satisfy the Navier-Stokes equations. The torque derivation is also incorrect, as it is based on the incorrect velocity profile.

Recommendations

To derive the correct velocity profile and torque, we need to re-examine the Navier-Stokes equations and apply them correctly. We need to assume a velocity profile that satisfies the Navier-Stokes equations and then derive the torque using the correct velocity profile.

Future Work

The correct derivation of the velocity profile and torque will provide valuable insights into the behavior of fluids in rotating systems. This knowledge can be applied to various engineering applications, such as the design of pumps, turbines, and other rotating machinery.

References

[1] White, F. M. (2011). Viscous Fluid Flow. McGraw-Hill.
[2] Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2007). Transport Phenomena. Wiley.
[3] Schlichting, H. (1960). Boundary Layer Theory. McGraw-Hill.

Appendix

The following appendix provides additional information and derivations that are not included in the main text.

A.1 Derivation of Velocity Profile

The velocity profile is derived using the Navier-Stokes equations. The equations are:

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + \frac{v}{r} \frac{\partial u}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{u}{r^2} \right)

\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial r} + \frac{v}{r} \frac{\partial v}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial \theta} + \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} + \frac{v}{r^2} \right)

Simplifying the equations, we get:

u \frac{\partial u}{\partial r} + \frac{v}{r} \frac{\partial u}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{u}{r^2} \right)

u \frac{\partial v}{\partial r} + \frac{v}{r} \frac{\partial v}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial \theta} + \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} + \frac{v}{r^2} \right)

Assuming a steady-state flow and neglecting the terms involving the time derivative and the velocity in the $\theta$ direction, we get:

u \frac{\partial u}{\partial r} = -\frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{u}{r^2} \right)

u \frac{\partial v}{\partial r} = -\frac{1}{\rho} \frac{\partial p}{\partial \theta} + \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} + \frac{v}{r^2} \right)

Solving these equations, we get:

u(r) = \frac{\Omega r^2}{R^2 - r^2}

This is the velocity profile derived in the main text.

A.2 Derivation of Torque

The torque experienced by the inner cylinder is given by:

T = \<br/> **Is this torque derivation of velocity profile of fluid between concentric cylinders wrong?** =========================================================== **Q&A** ------ **Q: What is the velocity profile of a fluid between two concentric cylinders?** A: The velocity profile of a fluid between two concentric cylinders is a critical parameter in determining the torque experienced by the inner cylinder. However, the derivation provided assumes a velocity profile of the form: $$u(r) = \frac{\Omega r^2}{R^2 - r^2}

Q: Is this velocity profile correct? A: No, this velocity profile is not consistent with the Navier-Stokes equations. The Navier-Stokes equations describe the motion of fluids in terms of the velocity, pressure, and density. The velocity profile derived above does not satisfy the Navier-Stokes equations.

Q: What is the correct velocity profile? A: The correct velocity profile can be derived by re-examining the Navier-Stokes equations and applying them correctly. The correct velocity profile will provide valuable insights into the behavior of fluids in rotating systems.

Q: How is the torque experienced by the inner cylinder related to the velocity profile? A: The torque experienced by the inner cylinder is given by:

T = \int_0^R \tau(r) 2 \pi r dr

where $\tau(r)$ is the shear stress at a distance $r$ from the center of the inner cylinder.

Q: What is the relationship between the shear stress and the velocity profile? A: The shear stress is given by:

\tau(r) = \mu \frac{\partial u}{\partial r}

where $\mu$ is the dynamic viscosity of the fluid.

Q: How can the correct torque be derived? A: The correct torque can be derived by using the correct velocity profile and applying the Navier-Stokes equations correctly. This will provide a more accurate understanding of the behavior of fluids in rotating systems.

Q: What are the implications of this derivation for engineering applications? A: The correct derivation of the velocity profile and torque will provide valuable insights into the behavior of fluids in rotating systems. This knowledge can be applied to various engineering applications, such as the design of pumps, turbines, and other rotating machinery.

Q: What are the limitations of the current derivation? A: The current derivation assumes a steady-state flow and neglects the terms involving the time derivative and the velocity in the $\theta$ direction. This simplification may not be accurate for all cases, and a more detailed analysis may be necessary to obtain a more accurate understanding of the behavior of fluids in rotating systems.

Q: How can the correct derivation be obtained? A: The correct derivation can be obtained by re-examining the Navier-Stokes equations and applying them correctly. This will require a more detailed analysis of the flow and a more accurate understanding of the behavior of fluids in rotating systems.

Q: What are the benefits of obtaining the correct derivation? A: The correct derivation will provide a more accurate understanding of the behavior of fluids in rotating systems. This knowledge can be applied to various engineering applications, such as the design of pumps, turbines, and other rotating machinery.

Q: How can the correct derivation be used in practice? A: The correct derivation can be used in practice by applying the Navier-Stokes equations correctly and using the correct velocity profile. This will provide a more accurate understanding of the behavior of fluids in rotating systems and can be applied to various engineering applications.

Q: What are the challenges in obtaining the correct derivation? A: The challenges in obtaining the correct derivation include re-examining the Navier-Stokes equations and applying them correctly. This requires a more detailed analysis of the flow and a more accurate understanding of the behavior of fluids in rotating systems.

Q: How can the correct derivation be verified? A: The correct derivation can be verified by comparing the results with experimental data or numerical simulations. This will provide a more accurate understanding of the behavior of fluids in rotating systems and can be applied to various engineering applications.

Conclusion

The derivation of the velocity profile and torque of a fluid between two concentric cylinders is a complex problem that requires a detailed analysis of the flow. The current derivation assumes a steady-state flow and neglects the terms involving the time derivative and the velocity in the $\theta$ direction. This simplification may not be accurate for all cases, and a more detailed analysis may be necessary to obtain a more accurate understanding of the behavior of fluids in rotating systems. The correct derivation can be obtained by re-examining the Navier-Stokes equations and applying them correctly. This will provide a more accurate understanding of the behavior of fluids in rotating systems and can be applied to various engineering applications.

Recommendations

To obtain the correct derivation, the following steps can be taken:

Re-examine the Navier-Stokes equations and apply them correctly.
Use the correct velocity profile and apply the Navier-Stokes equations correctly.
Compare the results with experimental data or numerical simulations to verify the correct derivation.

Future Work

The correct derivation of the velocity profile and torque of a fluid between two concentric cylinders will provide valuable insights into the behavior of fluids in rotating systems. This knowledge can be applied to various engineering applications, such as the design of pumps, turbines, and other rotating machinery. Future work can include:

Applying the correct derivation to various engineering applications.
Verifying the correct derivation using experimental data or numerical simulations.
Extending the correct derivation to more complex flows, such as those involving multiple cylinders or rotating systems.

References

[1] White, F. M. (2011). Viscous Fluid Flow. McGraw-Hill.
[2] Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2007). Transport Phenomena. Wiley.
[3] Schlichting, H. (1960). Boundary Layer Theory. McGraw-Hill.

Appendix

The following appendix provides additional information and derivations that are not included in the main text.

A.1 Derivation of Velocity Profile

The velocity profile is derived using the Navier-Stokes equations. The equations are:

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + \frac{v}{r} \frac{\partial u}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{u}{r^2} \right)

\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial r} + \frac{v}{r} \frac{\partial v}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial \theta} + \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} + \frac{v}{r^2} \right)

Simplifying the equations, we get:

u \frac{\partial u}{\partial r} + \frac{v}{r} \frac{\partial u}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{u}{r^2} \right)

u \frac{\partial v}{\partial r} + \frac{v}{r} \frac{\partial v}{\partial \theta} = -\frac{1}{\rho} \frac{\partial p}{\partial \theta} + \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} + \frac{v}{r^2} \right)

Assuming a steady-state flow and neglecting the terms involving the time derivative and the velocity in the $\theta$ direction, we get:

u \frac{\partial u}{\partial r} = -\frac{1}{\rho} \frac{\partial p}{\partial r} + \nu \left( \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{u}{r^2} \right)

u \frac{\partial v}{\partial r} = -\frac{1}{\rho} \frac{\partial p}{\partial \theta} + \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} + \frac{v}{r^2} \right)

Solving these equations, we get:

u(r) = \frac{\Omega r^2}{R^2 - r^2}

This is the velocity profile derived in the main text.

A.2 Derivation of Torque

The torque experienced by the inner cylinder is given by:

T = \int_0^R \tau(r) 2 \pi r