Viscous Force

Unit 3 Macro Displacement

Viscous Force

Juan Juan
12 min read
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Material Layer

Viscous Force is the directed, linear resistance that arises when layers of a fluid slide past one another, opposing relative motion through internal friction. It is the Macro Displacement force in Superphysics’ Liquid Relationalities — the large-scale, translational action of the liquid medium as it resists displacement through aetherspace and the outer intraaetherspace.

Unlike surface tension (rotational/cohesive), viscous force is directional and linear — it acts tangentially along flow layers, opposing relative sliding between adjacent fluid strata.

The Equation: Viscosity Fundamentals

Newton’s Law of Viscosity (Steady-State)

τ=μdudy \tau = \mu \frac{du}{dy}

Where:

Symbol Meaning Unit
τ \tau Shear stress (force per area) Pa=N/m2 \text{Pa} = \text{N}/\text{m}^2
μ \mu Dynamic viscosity (depends on fluid, temperature) Pa⋅s \text{Pa} \cdot \text{s}
dudy \frac{du}{dy} Velocity gradient (shear rate) s−1 \text{s}^{-1}

Viscous Force on a Plate (Macro Displacement)

Fv=μAdudy F_v = \mu A \frac{du}{dy}

Where:

Symbol Meaning Unit
( F_v ) Viscous force ( \text{N} )
( A ) Area of contact between layers ( \text{m}^2 )
( \mu ) Dynamic viscosity ( \text{Pa} \cdot \text{s} )
( \frac{du}{dy} ) Velocity gradient perpendicular to flow ( \text{s}^{-1} )
ρ(∂v∂t+v⋅∇v)=−∇P+μ∇2v+f \rho \left( \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla P + \mu \nabla^2 \mathbf{v} + \mathbf{f}

Here ( \mu \nabla^2 \mathbf{v} ) is the viscous displacement term — it diffuses momentum between adjacent fluid layers, opposing relative motion.

Stokes’ Law (Viscous Drag on a Sphere)

Fd=6πμrv F_d = 6 \pi \mu r v

Where:

Symbol Meaning Unit
( F_d ) Drag force ( \text{N} )
( r ) Sphere radius ( \text{m} )
( v ) Sphere velocity ( \text{m}/\text{s} )

Kinematic Viscosity (Momentum Diffusivity)

ν=μρ \nu = \frac{\mu}{\rho}

Where ( \nu ) is kinematic viscosity, measuring how quickly momentum diffuses through the fluid.

Key Properties of Viscous Force

Property Explanation
Directional Acts tangentially, opposing relative motion between layers
Linear Proportional to velocity gradient (Newtonian fluids)
Dissipative Converts kinetic energy into heat
Temperature-dependent Liquids: higher ( T ) → lower ( \mu ); Gases: higher ( T ) → higher ( \mu )
Momentum-diffusing Transfers momentum from fast layers to slow layers
Scale-dependent Dominant at small scales; negligible at large scales
No-slip condition Fluid at boundary moves with the boundary

Reynolds Number (Viscous vs. Inertial Forces)

Re=ρvLμ Re = \frac{\rho v L}{\mu}

When ( Re \ll 1 ), viscous forces dominate (laminar flow). When ( Re \gg 1 ), inertial forces dominate (turbulent flow).

Poiseuille’s Law (Viscous Flow in a Pipe)

Q=πr4ΔP8μL Q = \frac{\pi r^4 \Delta P}{8 \mu L}

Where ( Q ) is volumetric flow rate, ( r ) is pipe radius, ( \Delta P ) is pressure difference, and ( L ) is pipe length. Viscosity directly resists displacement.

Real-Life Examples

Example How Viscous Force Works
1. Honey pouring High ( \mu ) makes honey flow slowly and resist displacement
2. Blood circulation Viscosity of blood creates resistance in arteries; affects heart workload
3. Lubricating oil Engine oil viscosity reduces friction between moving metal parts
4. Water vs. syrup Water flows easily (low ( \mu )); syrup resists flow (high ( \mu ))
5. Air resistance on vehicles Air viscosity creates drag on cars, planes, and trains
6. Sedimentation Stokes’ law governs how particles settle in viscous fluids
7. Lava flow Molten rock has extremely high viscosity, displacing slowly
8. Paint brushing Paint viscosity determines how it spreads and levels
9. Microfluidics At small scales, viscous forces dominate over inertia
10. Earth’s mantle convection Extremely high viscosity means mantle flows over geological timescales

Summary

Question Answer
What is viscous force? Linear resistance between sliding fluid layers due to internal friction
What law describes it? Newton’s Law: ( \tau = \mu \frac{du}{dy} ); Stokes’ Law: ( F_d = 6 \pi \mu r v )
What type of force? Macro Displacement (linear, Rule 1 + 3)
What drives it? Velocity gradients between adjacent fluid layers
Real examples? Honey pouring, blood flow, engine oil, lava, air drag
What’s its rotational counterpart? Surface Tension (Macro Cohesion)

Bottom line: Viscous force is the linear, momentum-diffusing resistance between sliding fluid layers — the Macro Displacement force that pairs with surface tension’s rotational cohesion in Superphysics’ Liquid Relationalities.

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