How to Determine the Stability Condition in Couette Flow?

In summary, fluid dynamics is the study of the movement and behavior of liquids and gases. Bernoulli's principle states that as fluid speed increases, pressure decreases, and this principle is used to explain phenomena like lift in airplanes. Real-world applications of fluid dynamics include designing aircraft and understanding blood flow. To solve a fluid dynamics problem, one must use mathematical equations and gather relevant data. Challenges in studying fluid dynamics include complex models, unpredictable flow, and expensive experiments.
  • #1
URIA
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Homework Statement
Hi all,
I would like to solve the attached .
Relevant Equations
Can someone help with the attached?
Dear All,
I tried to solve the attached question. it's about Couette flow, where the 2 plates move.
2023-01-13 110841.png

in fact, I have to find the stability condition. is someone familiar with this and can help?
many thanks,
uria
 
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  • #2
Do you have a general idea on how to do the stability analysis? Basically it's expanding around the un-perturbed given solution and linearizing the NS equations.
 
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  • #3
HI,
I assume I have to use the rayleigh equation because of the incompressible inviscid limit.
 

FAQ: How to Determine the Stability Condition in Couette Flow?

What is the Bernoulli equation and how is it used in fluid dynamics?

The Bernoulli equation is a principle of fluid dynamics that describes the conservation of energy in a flowing fluid. It states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline. The equation is used to predict the behavior of fluid under varying conditions of flow and height, making it essential for solving problems involving fluid motion.

How can I determine the flow rate of a fluid in a pipe?

The flow rate of a fluid in a pipe can be determined using the continuity equation, which states that the product of the cross-sectional area and the velocity of the fluid is constant along the pipe. Mathematically, it is expressed as A1V1 = A2V2, where A is the area and V is the velocity. For incompressible fluids, this helps calculate the flow rate by measuring the velocity and cross-sectional area at any two points along the pipe.

What is Reynolds number and why is it important?

Reynolds number is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. It is defined as the ratio of inertial forces to viscous forces and is given by Re = ρuD/μ, where ρ is the fluid density, u is the flow velocity, D is the characteristic length (such as diameter of a pipe), and μ is the dynamic viscosity. It helps determine whether the flow will be laminar or turbulent.

How do I solve for pressure drop in a pipe due to friction?

To solve for the pressure drop in a pipe due to friction, you can use the Darcy-Weisbach equation: ΔP = f (L/D) (ρu²/2), where ΔP is the pressure drop, f is the Darcy friction factor, L is the length of the pipe, D is the diameter of the pipe, ρ is the fluid density, and u is the flow velocity. The Darcy friction factor can be determined from the Moody chart or Colebrook equation, depending on the flow regime (laminar or turbulent).

What is the difference between laminar and turbulent flow?

Laminar flow is characterized by smooth, orderly fluid motion in parallel layers with minimal mixing between them. It typically occurs at low Reynolds numbers (Re < 2000). Turbulent flow, on the other hand, is characterized by chaotic, irregular fluid motion with significant mixing and eddies. It occurs at high Reynolds numbers (Re > 4000). The transition between laminar and turbulent flow occurs in the intermediate range of Reynolds numbers (2000 < Re < 4000).

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