## Questions and Answers

The Hagen-Poiseuille equation is used to describe the flow profile in microchannels.

True

The Reynolds number is a dimensionless quantity that characterizes the flow regime in microchannels.

True

In microchannels, the entrance length is negligible compared to the channel length for establishing a parabolic flow profile.

False

The characteristic length in microchannels is typically the channel radius.

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Fully developed flow in microchannels indicates that velocity profiles do not change along the channel length.

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The inviscid core concept is relevant for describing flows with high viscosity in microchannels.

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The Hagen-Poiseuille equation is used to calculate the flow rate in a cylindrical channel.

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According to the Hagen-Poiseuille equation, decreasing the radius of a cylindrical channel by a factor of 10 will result in a decrease of fluid resistance by a factor of 100.

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The fluid resistance in a cylindrical channel is directly proportional to the length of the channel.

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The Hagen-Poiseuille equation is analogous to Ohm's Law in electrical circuits.

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Fully developed flow in microchannels implies that the flow profiles remain constant as the fluid moves along the channel.

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Inviscid core in fluid mechanics refers to a region within the fluid where viscous effects dominate.

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In a microchannel, the Stokes equation is simplified for Reynolds number greater than 1.

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In microfluidics, the Hagen-Poiseuille equation describes the flow profile of pressure-driven flow.

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The boundary layer thickness increases as the Reynolds number decreases in a microchannel.

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The inviscid core in a microchannel extends across the entire cross-section of the channel.

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Fully developed flow in microchannels means that the velocity profile remains constant along the channel length.

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The pressure gradient in a microchannel with fully developed flow is constant along the channel length.

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## Study Notes

### Flow Profile of Pressure-Driven Flow in a Microchannel

- Characteristic length (Le) is a key factor in establishing a parabolic flow profile, where Le = 0.06 × Re × l, and Re is the Reynolds number.
- In a microchannel with a characteristic length of 100 µm, a flow velocity of 0.2 m/s, and water as the fluid, Le ≈ 300 µm.
- The entrance length (Le) is necessary for establishing a parabolic flow profile.

### Hagen-Poiseuille Equation

- The Hagen-Poiseuille equation describes laminar flow in a cylindrical channel, where vx(y) = -y × (∆p / (2 × η × L)).
- The equation can be rearranged to solve for ∆p: ∆p = (8 × η × L × Q) / (4 × π × r0^4).
- The Hagen-Poiseuille equation is also applicable to rectangular and slit-type channels with some modifications.
- The fluid resistance (Rfl) is analogous to Ohm's Law, where Rfl = ∆p / Q.

### Fluid Resistance

- Fluid resistance (Rfl) in a cylindrical channel is given by: Rfl = (8 × η × L) / (π × r0^4).
- In a rectangular channel, Rfl = (12 × η × L) / (w × h^3) × (1 - Σ((n × π / 2) / h) × tanh(n × π / 2)).
- In a slit-type channel, Rfl is similar to that of a rectangular channel.

### Stokes Equation

- The Stokes equation is a simplified version of the Navier-Stokes equation for Re < 1.
- The equation is given by: ∂vx / ∂x = η × ∂^2vx / ∂y^2.
- Solving the equation involves separating variables and applying boundary conditions to find the pressure profile and velocity distribution.

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## Description

Test your knowledge on the Hagen-Poiseuille equation for flow rate in a cylindrical channel, its rearrangement, and the impact of fabrication tolerances on channel geometries. Explore concepts related to electrical engineering materials and fluid dynamics.