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

In summary, the conversation discusses the fully developed and steady flow of a fluid between two large parallel plates, with a distance of 2h apart and a velocity profile of V(y) = V_0(1-(y/h)^2). The equations for volume flow rate per unit width and shear stress at the wall are derived, with the information that d(...)/dx and d(...)/dt are zero. The volume flow rate is calculated by multiplying velocity and area, while the shear stress is equal to mu*(dV/dy). The conversation also mentions integrating with respect to y and setting limits properly. It is noted that this is a homework question.
  • #1
iwearnexus
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0
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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  • #2
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
 

1. What is the formula for calculating volume flow rate per unit width of a plate?

The formula for calculating volume flow rate per unit width of a plate is Q = v * w * h, where Q is the volume flow rate, v is the velocity of the fluid, w is the width of the plate, and h is the height of the fluid. This formula is also known as the continuity equation in fluid mechanics.

2. How is the velocity of the fluid determined in this calculation?

The velocity of the fluid can be determined using various methods such as using a flow meter or by measuring the rate of change of pressure in the fluid. It can also be calculated using Bernoulli's equation, which takes into account the fluid's density, height, and pressure at two different points.

3. Is this formula applicable for all types of fluids?

Yes, this formula can be used for both liquids and gases. However, the density of the fluid must be taken into consideration, as it may affect the accuracy of the calculation. In cases where the fluid is compressible, the ideal gas law can be used to account for changes in density.

4. Can this formula be used for irregularly shaped plates?

No, this formula is specifically designed for calculating the volume flow rate per unit width of a flat and rectangular plate. For irregularly shaped plates, other equations such as the Navier-Stokes equation or the momentum equation may be used.

5. Are there any units associated with this calculation?

Yes, the units for volume flow rate per unit width are typically cubic meters per second per meter (m^3/s/m) in SI units. However, other units such as cubic feet per minute per foot (ft^3/min/ft) may also be used. It is important to ensure that all units are consistent in the calculation for accurate results.

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