Draft:Scale analysis for Couette Flow and between one fixed and one moving plate

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Couette flow

Couette flow is the laminar flow of a viscous fluid between two parallel plates, where one plate is stationary and the other moves at a constant velocity. This fundamental flow is pivotal in fluid mechanics, providing insights into shear-driven flows, velocity distributions, and viscous effects.

Couette flow is a key concept in analyzing velocity profiles in fluids confined between surfaces. Applications span across engineering fields such as lubrication systems, polymer processing, and the design of mechanical devices. Scale analysis simplifies the governing equations by estimating the order of magnitude of different terms, making complex problems more approachable.

Governing Equations

Velocity profile of Couette flow

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

${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0$

Momentum Equation (in the $x$-direction)

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

Where:

$u$ and $v$ are the velocity components in the $x$ and $y$ directions, respectively.

ρ is the fluid density.

μ is the dynamic viscosity.

$p$ is the pressure.

Physical Setup

Geometry: Two infinite parallel plates separated by a distance $h$.

Boundary Conditions:

At $y=0$ and $u=0$ (stationary plate)

At $y=h$ and $u=U$ (moving plate with velocity $U$)

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.
  • $h$: Gap between the plates (characteristic length in the $y$-direction).

• Velocity Scales:
  • $U$: Velocity of the moving plate (characteristic velocity in the $x$-direction).
  • $V$: Characteristic velocity in the $y$-direction (expected to be much smaller than $U$).

Dimensionless Variables

Introduce dimensionless variables:

$X={\frac {x}{L}},\quad Y={\frac {y}{h}},\quad U^{*}={\frac {u}{U}},\quad V^{*}={\frac {v}{V}}$

Simplifying the Continuity Equation

${\frac {\partial U^{*}}{\partial X}}+{\frac {V}{U}}{\frac {\partial V^{*}}{\partial Y}}=0$

Since $V\ll U$ and for fully developed flow ${\frac {\partial U^{*}}{\partial X}}=0$, it follows that ${\frac {\partial V^{*}}{\partial Y}}=0$, implying $V\approx 0$.

Simplifying the Momentum Equation

Neglecting inertial terms due to low Reynolds number ${\Bigl (}Re={\frac {\rho Uh}{\mu }}{\Bigr )}$ ($Re\ll 1$):

$0=-{\frac {\partial p}{\partial x}}+\mu {\frac {\partial ^{2}u}{\partial y^{2}}}$

This indicates that the pressure gradient balances the viscous forces.

Solution to the Velocity Profile

Final Equation for Couette 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 $y$:

1. First Integration:

${\frac {du}{dy}}={\frac {1}{\mu }}{\frac {dp}{dx}}y+C_{1}$

2. Second Integration:

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

Applying Boundary Conditions

1. At $y=0$, $u=0$:

$0=0+0+C_{2}\implies C_{2}=0$

1. At $y=h$, $u=U$:

$U={\frac {1}{2\mu }}{\frac {dp}{dx}}h^{2}+C_{1}h$

Solving for $C_{1}$:

$C_{1}={\frac {U-{\frac {1}{2\mu }}{\frac {dp}{dx}}h^{2}}{h}}$

Final Velocity Profile

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

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

Simplifying:

$u(y)=-{\frac {1}{2\mu }}\left({\frac {dp}{dx}}\right)(y^{2}-hy)+{\frac {U}{h}}y$

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

$u(y)={\frac {G}{2\mu }}(hy-y^{2})+{\frac {U}{h}}y$

This is the final equation for Couette flow between one fixed and one moving plate, considering a pressure gradient.

Special Cases

No Pressure Gradient ($G=0$):

$u(y)={\frac {U}{h}}y$

This represents a linear velocity profile.

Stationary Plates ($U=0$):

$u(y)={\frac {G}{2\mu }}(hy-y^{2})$

This is the classical parabolic profile of plane Poiseuille flow

Curve A: $G=0$ (No pressure gradient) — Linear profile

Curve B: $G>0$ (Adverse pressure gradient) — Flow opposes the moving plate

Curve C: $G<0$ (Favorable pressure gradient) — Flow assists the moving plate

Pressure Gradient Effects

The presence of a pressure gradient alters the velocity profile:

Positive Pressure Gradient ($G>0$): Opposes the flow due to the moving plate.

Negative Pressure Gradient ($G<0$): Assists the flow, increasing the velocity.

Applications

• Lubrication Theory: Analysis of thin lubricant films between machine components.
• Polymer Processing: Modeling flow in narrow gaps during extrusion processes.
• Geophysical Flows: Understanding glacier movements over bedrock surfaces.
• Microfluidics: Designing devices where flow occurs in narrow channels.

Conclusion

The scale analysis of Couette flow between one fixed and one moving plate simplifies the Navier-Stokes equations by focusing on dominant viscous terms and neglecting inertial effects. The resulting velocity profile demonstrates how shear-driven flow and pressure-driven flow combine, providing critical insights for engineering applications where such conditions are prevalent.

Article Prepared By

1. Saiyam Jain (Roll No.- 21135117), IIT (BHU) Varanasi
2. Amit Sharma (Roll No.- 21135015), IIT (BHU) Varanasi
3. Hemant Patel (Roll No.- 21135060), IIT (BHU) Varanasi
4. Garvit Singhal (Roll No.- 21135054), IIT (BHU) Varanasi
5. Rohith M (Roll No.-21135113), IIT (BHU) Varanasi