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The maximum velocity of a one-dimensional incompressible fully developed viscous flow, between two fixed parallel plates, is 6 ms-1. The mean velocity (in ms-1) of the flow is
- a)2
- b)3
- c)4
- d)5
Correct answer is option 'C'. Can you explain this answer?
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The maximum velocity of a one-dimensional incompressible fully develop...
Concept:
Viscous flow in Plates:
Velocity equation:
where t = distance between two plates and y = distance measured from lower plate towards the upper plate.
Viscous flow in Plates:
Velocity equation:
where t = distance between two plates and y = distance measured from lower plate towards the upper plate.
Shear stress equation:
Maximum velocity happens at, y = t/2, which is given by
Ratio of Maximum velocity to the Average velocity in case of plates:
Maximum velocity happens at, y = t/2, which is given by
Ratio of Maximum velocity to the Average velocity in case of plates:
Calculation:
Given:
Parallel plates:
Umax = 6 ms-1, Uavg = ??
Given:
Parallel plates:
Umax = 6 ms-1, Uavg = ??
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The maximum velocity of a one-dimensional incompressible fully develop...
To explain why the correct answer is option 'C' (4 m/s), we need to understand the concept of fully developed viscous flow between two parallel plates.
Fully Developed Viscous Flow:
A fully developed flow refers to a flow in which the velocity profile remains constant along the flow direction. In the case of a viscous flow between two parallel plates, the flow is fully developed when the flow is no longer affected by the entrance region and reaches a steady state.
Maximum Velocity in Fully Developed Flow:
In a fully developed flow between two parallel plates, the velocity profile is parabolic, with the maximum velocity occurring at the center of the channel. The velocity at the centerline is directly related to the maximum velocity.
The Hagen-Poiseuille Equation:
The Hagen-Poiseuille equation is used to calculate the maximum velocity in a fully developed viscous flow between two parallel plates. It is given by:
Vmax = (ΔP / (2μL)) * (h²/4 - y²)
Where:
Vmax = Maximum velocity
ΔP = Pressure drop across the plates
μ = Dynamic viscosity of the fluid
L = Distance between the plates
h = Height of the channel
y = Distance from the centerline
Applying the Hagen-Poiseuille Equation:
In this question, the maximum velocity (Vmax) is given as 6 m/s. We need to find the mean velocity, which is the average velocity of the flow.
In a fully developed flow, the mean velocity (Vmean) is related to the maximum velocity (Vmax) as follows:
Vmean = (2/3) * Vmax
Substituting the given value of Vmax into the equation, we get:
Vmean = (2/3) * 6
Vmean = 4 m/s
Therefore, the correct answer is option 'C' (4 m/s).
Fully Developed Viscous Flow:
A fully developed flow refers to a flow in which the velocity profile remains constant along the flow direction. In the case of a viscous flow between two parallel plates, the flow is fully developed when the flow is no longer affected by the entrance region and reaches a steady state.
Maximum Velocity in Fully Developed Flow:
In a fully developed flow between two parallel plates, the velocity profile is parabolic, with the maximum velocity occurring at the center of the channel. The velocity at the centerline is directly related to the maximum velocity.
The Hagen-Poiseuille Equation:
The Hagen-Poiseuille equation is used to calculate the maximum velocity in a fully developed viscous flow between two parallel plates. It is given by:
Vmax = (ΔP / (2μL)) * (h²/4 - y²)
Where:
Vmax = Maximum velocity
ΔP = Pressure drop across the plates
μ = Dynamic viscosity of the fluid
L = Distance between the plates
h = Height of the channel
y = Distance from the centerline
Applying the Hagen-Poiseuille Equation:
In this question, the maximum velocity (Vmax) is given as 6 m/s. We need to find the mean velocity, which is the average velocity of the flow.
In a fully developed flow, the mean velocity (Vmean) is related to the maximum velocity (Vmax) as follows:
Vmean = (2/3) * Vmax
Substituting the given value of Vmax into the equation, we get:
Vmean = (2/3) * 6
Vmean = 4 m/s
Therefore, the correct answer is option 'C' (4 m/s).
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The maximum velocity of a one-dimensional incompressible fully developed viscous flow, between two fixed parallel plates, is 6 ms-1. The mean velocity (in ms-1) of the flow isa)2b)3c)4d)5Correct answer is option 'C'. Can you explain this answer?
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