Calculating volume flow rate per unit width of a plate ( Fluid Mechanics )

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SUMMARY

The discussion focuses on calculating the volume flow rate per unit width of fluid flow between two parallel plates, separated by a distance of 2h. The velocity profile is defined as V(y) = V_0(1 - (y/h)^2), where V_0 represents the centerline velocity. To derive the volume flow rate, the area under the velocity profile must be integrated with respect to y, while shear stress at the wall is determined using the equation τ = μ(dV/dy), where μ is the dynamic viscosity. The participants emphasize the importance of recognizing that the flow is fully developed and steady, leading to simplifications in the equations.

PREREQUISITES
  • Understanding of fluid mechanics principles, specifically flow between parallel plates.
  • Familiarity with velocity profiles and their mathematical representations.
  • Knowledge of integration techniques for calculating area under curves.
  • Basic concepts of shear stress and its relation to velocity gradients.
NEXT STEPS
  • Study the derivation of volume flow rate in fluid mechanics, focusing on parallel plate flow.
  • Learn about the application of the Navier-Stokes equations in steady flow scenarios.
  • Explore the concept of shear stress in fluids, particularly in laminar flow conditions.
  • Investigate numerical methods for solving fluid flow problems, such as computational fluid dynamics (CFD) simulations.
USEFUL FOR

This discussion is beneficial for students studying fluid mechanics, particularly those tackling homework related to flow between parallel plates, as well as engineers and researchers involved in fluid dynamics analysis.

iwearnexus
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This is the question

The flow of a fluid between two large flat parallel plates normal to the y direction is fully developed and steady. The plates are a distance 2h apart in the y direction and the velocity profile, assuming y=0 is at the midpoint between the plate is V(y) = V_0(1-(y/h)^2) where V_0 is the centreline velocity. Derive equations for volume flow rate per unit width of the plate and the shear stress at the wall. Sketch and explain your shear stress result.


I know that d(...)/dx and d(...)/dt are zero since the flow is fully developed and steady.

also Volume flow = velocity x area.

how do i use this information to get the answer?

Im stuck.
 
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you can try integrating with respect to y giving the area under the curve, right? while setting the limits properly. I'm sure you know shear stress = mu*(dV/dy)

also, this is homework
 

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