A wet paint film of uniform thickness, δ, is painted on a vertical wall. The wet paint can be ap

step 1 In this equation, we have to find the maximum thickness of the paint theme before flow occurred.

A wet paint film of uniform thickness, δ, is painted on a...

A wet paint film of uniform thickness, $\delta$, is painted on a vertical wall. The wet paint can be approximated as a Bingham fluid with a yield stress, $\tau_{y},$ and density, $\rho .$ Derive an expression for the maximum value of $\delta$ that can be sustained without having the paint flow down the wall. Calculate the maximum thickness for lithographic ink whose yield stress $\tau_{y}=40$ Pa and density is approximately $1000 \mathrm{kg} / \mathrm{m}^{3}$

Key Concepts

Yield Stress

Yield stress is the critical stress level that must be reached or exceeded for a material exhibiting Bingham behavior to start flowing. It defines the transition point between solid-like behavior and fluid-like flow, and thus directly determines the maximum force or stress the material can sustain without movement.

Static Equilibrium and Force Balance

Static equilibrium in a fluid film is achieved when the shear stress resulting from gravitational forces is exactly balanced by the material's yield stress. By equating the maximum gravitationally induced shear stress to the yield stress, one can derive the maximum allowable film thickness that can be maintained without the fluid flowing under gravity.

Shear Stress Distribution in a Fluid Film

For a fluid film on a vertical surface, the gravitational force produces a shear stress that typically varies with depth, reaching a maximum at the interface where the fluid contacts the wall. This distribution is important in figuring out whether the entire film will remain stationary, as the maximum shear stress must not exceed the yield stress of the fluid.

Bingham Fluid

A Bingham fluid is a non?Newtonian fluid that behaves like a solid until a specified yield stress is exceeded, after which it flows like a viscous fluid. This characteristic is central to analyzing the conditions under which the fluid, such as wet paint, will begin to flow or will remain stationary under applied stress.

Gravitational Force

Gravitational force acts on the fluid due to its density and earth's gravitational acceleration. In the context of a film on a vertical wall, gravity generates a body force that creates a shear stress within the fluid, driving potential flow down the wall.

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