Draft:Hagen-Poiseuille flow

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Declined by I dream of horses 38 minutes ago. Last edited by Citation bot 5 seconds ago. Reviewer: Inform author.

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Comment: Written in a way that's hard to understand. I dream of horses (Hoofprints) (Neigh at me) 22:59, 8 October 2024 (UTC)

Scale analysis for fully developed Hagen-Poiseuille flow between two fixed plates Hagen-Poiseuille flow describes the laminar flow of a viscous fluid through a channel formed by two stationary, parallel plates. This flow is essential for understanding pressure-driven fluid motion, providing insight into velocity profiles and viscous effects, which are vital in fluid mechanics and related fields.

Introduction

Understanding Hagen-Poiseuille flow is crucial for analysing pressure-driven flows in confined geometries. This flow is prevalent in applications such as microfluidics, blood flow through vessels, and lubrication systems. Scale analysis helps simplify complex governing equations by evaluating the relative importance of various terms, making the study of pressure-driven flows more manageable.

Governing Equations

The flow is assumed to be:

Steady: No changes with time
Incompressible: Constant density (ρ)
Laminar: Smooth flow without turbulence
Fully Developed: Velocity profile does not change along the flow direction

Continuity Equation

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$ ^[1]^[2]^[3]

Momentum Equation (in the x-direction)

$\rho \left(u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}\right)=-{\frac {\partial p}{\partial x}}+\mu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)$ ^[1]^[2]^[4]

Momentum Equation (in the y-direction)

$\rho \left(u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}\right)=-{\frac {\partial p}{\partial y}}+\mu \left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right)$

Where:

u and v are the velocity components in the x and y directions, respectively.
ρ (rho) is the fluid density.
μ (mu) is the dynamic viscosity.
p is the pressure.

Physical Setup

Geometry: Two infinite parallel stationary plates separated by a distance D

Boundary Conditions:

At y = D/2 and u = 0 (stationary plate)

At y = -D/2 and u = 0 (stationary plate)

No-slip condition applies at both plates.

Scale Analysis

Scale analysis estimates the relative magnitudes of terms in the governing equations to simplify them.

Characteristic Scales

Length Scales:

L: Characteristic length in the x-direction. x~L.
D: Gap between the plates (characteristic length in the y-direction). y~D.

Velocity Scales:

U: Average flow velocity in the x-direction (characteristic velocity in the x-direction). u~U.
V: Characteristic velocity in the y-direction (expected to be much smaller than U ). v~V.

1.Simplifying the Continuity Equation

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$ $\implies {\frac {U}{L}}+{\frac {V}{D}}\sim 0$ $\implies {V}\sim {\frac {DU}{L}}$

We can then think of the fully developed region as that section of the duct flow situated far enough from the entrance that the scale of v is negligible. Based on this definition, the mass continuity equation requires a fully developed flow limit.

$\implies {v}=0$ and ${\frac {\partial u}{\partial x}}=0$

This implies that u is a function of y only, ${\frac {\partial u}{\partial y}}={\frac {du}{dy}}$ ^[5]

2.Simplifying the y-Momentum Equation

Using results of continuity equation analysis, y momentum equation reduces to: ${\frac {\partial p}{\partial y}}=0$

This implies that p is a function of x only $\implies {\frac {\partial p}{\partial x}}={\frac {dp}{dx}}$

3.Simplifying the x-Momentum Equation

Neglecting inertial terms due to low Reynolds number

$0=-{\frac {\partial p}{\partial x}}+\mu \left({\frac {\partial ^{2}u}{\partial y^{2}}}\right)$ ^[1]^[2]^[5]

${\frac {dp}{dx}}=\mu {\frac {d^{2}y}{dx^{2}}}$ = constant

Each term must be equal to the same constant because P is a function of x and u is a function of y. ^[5]

This indicates that the pressure gradient balances the viscous forces.

Solution to the velocity profile

Final equation for Poiseuille flow

Considering a general pressure gradient, the simplified momentum equation becomes:

$\mu {\frac {d^{2}u}{dy^{2}}}={\frac {dp}{dx}}$

Integrating twice with respect to y:

First Integration :

${\frac {du}{dy}}={\frac {1}{\mu }}\left({\frac {dp}{dx}}\right)y+c_{1}$

Second Integration :

$u(y)={\frac {1}{2\mu }}\left({\frac {dp}{dx}}\right)y^{2}+c_{1}y+c_{2}$

Applying Boundary Conditions

1. At y = +D/2, u=0

$0={\frac {1}{2\mu }}\left({\frac {dp}{dx}}\right)\left({\frac {D^{2}}{4}}\right)+c_{1}{\frac {D}{2}}+c_{2}$ ...(1)

2. At y = -D/2, u=0

$0={\frac {1}{2\mu }}\left({\frac {dp}{dx}}\right)\left({\frac {D^{2}}{4}}\right)-c_{1}{\frac {D}{2}}+c_{2}$ ...(2)

On solving (1) and (2) we get,

$c_{1}$ = 0

$c_{2}$ = $-{\frac {1}{2\mu }}\left({\frac {dp}{dx}}\right)\left({\frac {D^{2}}{4}}\right)$

Final velocity profile

Substituting $c_{1}$ and $c_{2}$ back into the velocity equation:

$u(y)={\frac {1}{2\mu }}\left({\frac {dP}{dx}}\right)\left(y^{2}-{\frac {D^{2}}{4}}\right)$

Let $G=-{\frac {dp}{dx}}$ (pressure gradient),then :

$u(y)={\frac {G}{2\mu }}\left({\frac {D^{2}}{4}}-y^{2}\right)$

This is the final equation for Poiseuille flow between two fixed plates considering a pressure gradient.

This is the classical parabolic profile of plane Poiseuille flow.

Applications of Poiseuille Flow

Fluid Transport in Pipes: Analysis of laminar flow in pipes for designing pipelines in water supply systems, oil transport, and chemical processing.
Medical Devices: Understanding blood flow in capillaries and arteries, aiding in the design of medical devices like blood pumps and IV drips.
Microfluidics: Applications in lab-on-a-chip technologies for controlling fluid flow in small-scale devices for diagnostics and drug delivery.
Hydraulics: Design of hydraulic systems and machinery where controlled fluid flow is essential, such as in brakes and lifting devices.
Food Industry: Optimization of fluid flow in processing and packaging, ensuring efficient transport of liquids like juices and sauces.
Chemical Engineering: Modelling the behavior of viscous fluids in reactors and pipelines during chemical manufacturing.
Research: Used in various experimental setups to study fluid dynamics, viscosity, and related properties in physics and engineering studies.
Environmental Engineering: Understanding groundwater flow through porous media for aquifer studies and contamination transport.

Results

Flow rate is directly proportional to the pressure gradient and pipe diameter, and inversely proportional to fluid viscosity.
Provides insights into the behavior of the flow in different geometries and conditions.

Authors

Anmol Shukla (Roll no. - 21135020), IIT(BHU) Varanasi
Dhananjay Khandelwal (Roll no. - 21135051), IIT(BHU) Varanasi
Mohit Marathe (Roll no. - 21135083), IIT(BHU) Varanasi
Nishant Bansal (Roll no. - 21135088), IIT(BHU) Varanasi
Varad Kharade (Roll no. - 21134012), IIT(BHU) Varanasi

^ ^a ^b ^c White, Frank M. (2006). Viscous fluid flow. McGraw-Hill series in mechanical engineering (3. ed.). Boston, Mass.: McGraw-Hill. ISBN 978-0-07-240231-5.
^ ^a ^b ^c Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R.; Tryggvason, Gretar (2016). Fluid mechanics (Sixth ed.). Amsterdam Boston Heidelberg London: Elsevier, Academic Press. ISBN 978-0-12-405935-1.
^ Munson, Bruce Roy, ed. (2013). Fundamentals of fluid mechanics (7. ed.). Hoboken, NJ: Wiley. ISBN 978-1-118-11613-5.
^ Schlichting, Hermann; Gersten, Klaus (2017). Boundary-Layer Theory. doi:10.1007/978-3-662-52919-5. ISBN 978-3-662-52917-1.
^ ^a ^b ^c Bejan, Adrian (2017). Convection Heat Transfer. Wiley. doi:10.1007/978-3-662-52919-5. ISBN 978-1118021340. {{cite book}}: Check |isbn= value: checksum (help); Check date values in: |year= / |date= mismatch (help)

[:0-1] White, Frank M. (2006). Viscous fluid flow. McGraw-Hill series in mechanical engineering (3. ed.). Boston, Mass.: McGraw-Hill. ISBN 978-0-07-240231-5.

[:1-2] Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R.; Tryggvason, Gretar (2016). Fluid mechanics (Sixth ed.). Amsterdam Boston Heidelberg London: Elsevier, Academic Press. ISBN 978-0-12-405935-1.

[3] Munson, Bruce Roy, ed. (2013). Fundamentals of fluid mechanics (7. ed.). Hoboken, NJ: Wiley. ISBN 978-1-118-11613-5.

[4] Schlichting, Hermann; Gersten, Klaus (2017). Boundary-Layer Theory. doi:10.1007/978-3-662-52919-5. ISBN 978-3-662-52917-1.

[:2-5] Bejan, Adrian (2017). Convection Heat Transfer. Wiley. doi:10.1007/978-3-662-52919-5. ISBN 978-1118021340. {{cite book}}: Check |isbn= value: checksum (help); Check date values in: |year= / |date= mismatch (help)

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