Answer: 965.03 N/m^2
Explanation:
To solve this problem, we'll use the Reynolds equation for the hydrodynamic pressure in a slider bearing. The equation is given by:
P = (6 * η * U * h) / L^2
where P is the hydrodynamic pressure, η is the dynamic viscosity of the lubricant, U is the relative velocity between the shaft and the sleeve, h is the film thickness (the gap between the shaft and the sleeve), and L is the length of the bearing.
Given:
Inside diameter of the sleeve (Ds) = 2.6 cm
Diameter of the shaft (D) = 2.54 cm
Length of the sleeve (L) = 5.08 cm
Relative velocity (U) = 6.1 m/s
Maximum total force on the sleeve (F_max) = 2.2 N
First, we need to calculate the film thickness (h):
h = (Ds - D) / 2 = (2.6 - 2.54) / 2 = 0.03 cm = 0.0003 m
Next, we need to find the bearing area (A):
A = π * D * L = π * 0.0254 * 0.0508 ≈ 0.004079 m^2
Now, we can find the maximum allowable pressure (P_max):
P_max = F_max / A = 2.2 / 0.004079 ≈ 539.17 Pa
Now, we can solve for the viscosity of the grease (η):
η = (P_max * L^2) / (6 * U * h) = (539.17 * (0.0508)^2) / (6 * 6.1 * 0.0003) ≈ 0.04746 Pa·s
The viscosity of the grease should be approximately 0.04746 Pa·s to limit the total force on the sleeve to less than 2.2 N.
Now, let's calculate the magnitude of the momentum flux in the gap. The momentum flux (also known as shear stress) is given by:
τ = η * (du/dy)
where τ is the shear stress, du is the change in velocity across the gap, and dy is the thickness of the gap.
The velocity gradient (du/dy) in the gap can be calculated as:
du/dy = U / h = 6.1 / 0.0003 ≈ 20333.33 s^(-1)
Now, we can find the shear stress (τ):
τ = η * (du/dy) = 0.04746 * 20333.33 ≈ 965.03 N/m^2
The magnitude of the momentum flux in the gap is approximately 965.03 N/m^2. The direction of the momentum transport is along the direction of the motion of the shaft, which is axially along the sleeve