Lecture 6 Flashcards
(13 cards)
What is the fundamental assumption that justifies using the lubrication approximation in flow problems?
The lubrication approximation assumes a large separation of length scales: the flow domain is much longer in one direction than the other, i.e., ℓ2/ℓ1≪1. This leads to nearly unidirectional flow and negligible inertia compared to viscous effects.
In the lubrication approximation, what does the condition ℓ2/ℓ1≪1 imply about the velocity field?
It implies that the flow is nearly parallel (unidirectional), typically with a dominant velocity component in the long direction and negligible velocity variation in the narrow dimension.
What is the dominant force balance in lubrication flows?
Viscous forces dominate over inertial forces due to the thin geometry and slow flow, leading to a balance primarily between pressure gradients and viscous stresses.
How does one typically rescale the velocity and coordinates when deriving the lubrication approximation?
Let
𝑋=𝑥/ℓ1,
𝑌=𝑦/ℓ2,
𝑈=𝑢/𝑢𝑐,
𝑉=𝑣/𝑣𝑐
, and 𝑇=𝑡/𝑡𝑐.
Choose scales such that continuity and the momentum equations become dimensionally balanced under the limit 𝜖=ℓ2/ℓ1≪1.
In the x-momentum equation, what is the estimated pressure scale?
The pressure scale is Δ𝑃∼𝜇𝑢_𝑐ℓ1/ℓ2, derived from balancing ∂𝑝/∂𝑥 with the dominant viscous term 𝜇∂2𝑢/∂𝑦2.
What simplification is made to the y-momentum equation in the lubrication approximation?
Since inertial terms are negligible and pressure variation in y is small, the equation reduces to ∂𝑝/∂𝑦=0, meaning pressure is constant across the narrow dimension.
Derive the expression for pressure-driven velocity in a narrow gap using the lubrication assumption.
From 𝜇 ∂^2𝑢/∂𝑦^2=𝑑𝑝/𝑑𝑥, integrate twice with no-slip at 𝑦=0 and 𝑦=ℎ, giving:
𝑢(𝑦)=1/2𝜇 𝑑𝑝/𝑑𝑥 (𝑦2−𝑦ℎ)
What does the volumetric flow rate 𝑄 per unit width become in a lubrication flow?
Q=∫ 0 to h (u(y)dy=−h^3/12μ dp/dx+Uh/2
The first term is pressure-driven, the second is shear-driven.
What happens to 𝑑𝑝/𝑑𝑥 as the film thickness ℎ decreases?
dp/dx∝h^−2 ; thus, pressure gradient increases dramatically as ℎ gets smaller.
In a slider bearing with linearly varying height ℎ(𝑥), what is the total load supported?
∫ 0 to L p(x)dx This requires solving the lubrication equation for 𝑝(𝑥) and integrating.
Why is the flow quasi-steady in lubrication?
Because inertial effects are negligible compared to viscous effects due to thin geometry, and pressure gradients adjust quickly.
What does it mean physically when we say lubrication flows are “nearly parallel”?
Streamlines are aligned with the long axis of the domain, and velocity components perpendicular to that axis are small.
