A shaft with outside diameter of 18 mm turns at 20 revolutions per second inside a stationary jo

step 1 In this question, we know that tau is given by f by a and f is given by tau A. Now going forward

A shaft with outside diameter of 18 mm turns at 20...

A shaft with outside diameter of $18 \mathrm{mm}$ turns at 20 revolutions per second inside a stationary journal bearing $60 \mathrm{mm}$ long. A thin film of oil $0.2 \mathrm{mm}$ thick fills the concentric annulus between the shaft and journal. The torque needed to turn the shaft is $0.0036 \mathrm{N} \cdot \mathrm{m}$. Estimate the viscosity of the oil that fills the gap.

Key Concepts

Lubrication Theory

Lubrication theory deals with the behavior of thin fluid films in relative motion between surfaces. It simplifies the complex equations of fluid dynamics under the assumption that one dimension (typically the film thickness) is much smaller than the others. This approximation is widely used in the analysis of journal bearings and other systems where a thin layer of lubricant controls friction and wear.

Torque Due to Fluid Viscosity

In rotating machinery or bearings, the viscous fluid in the gap between a moving shaft and its housing exerts a shear force over the surface area, which results in a resisting torque. This concept involves integrating the shear stress over the surface where the fluid contacts the moving body, and it is crucial for understanding how viscous forces affect the rotational dynamics of mechanical systems.

Viscosity

Viscosity is a property of fluids that quantifies the internal friction, or resistance to flow, under an applied shear stress. It measures how much force is needed to move one layer of fluid relative to another and is a central parameter in characterizing fluid behavior in both natural processes and engineering applications.

Newton's Law of Viscosity

This fundamental principle states that in a Newtonian fluid, the shear stress is directly proportional to the rate of strain (velocity gradient) between adjacent layers. The constant of proportionality in this relationship is the viscosity. This law is key in analyzing simple shear flows and is used to relate forces to the rate at which fluid layers slide past each other.

Couette Flow

Couette flow describes the laminar flow of a fluid between two surfaces, one of which is moving relative to the other. In many engineering applications, especially in thin film lubrication, it is assumed that the velocity in the fluid varies linearly across the gap between the surfaces. This simplification helps in analyzing shear stress distribution and, ultimately, the forces and torques generated in the fluid.

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