Chapter 7, Problem 7.15P

Interpretation Introduction

Interpretation: The maximum sinking speed of the ram and rack due to presence of gravity and viscous forces is to be calculated.

Concept introduction:

The frictional resistance is the force which inhibits the motion of the fluid/particle through the substance on which it is moving.

The frictional resistance is given as,

  $\text{F}\text{=}\text{τ}\text{×}\text{A}$ ……. (1)

F = Frictional resistance

A = Area of the cylinder

  $\text{τ}$ = Shear stress developed

The area of the cylinder is given as,

  $\text{A}\text{=}{\text{πD}}_{\text{i}}\text{L}$ ……. (2)

D_i = Inside diameter of the cylinder

L = Length of the Ram

The shear stress is given by the Newton’s law of viscosity in which the shear stress is directly proportional to the shear rate and the proportionality constant is viscosity.

The equation governing the above statement is,

  $\text{τ}\text{=}\text{μ}\frac{\text{du}}{\text{dy}}\text{=}\text{μ}\frac{\text{v}}{\text{t}}$ ……. (3)

  $\text{μ}$ = Viscosity of the fluid

v = Velocity of the ram

t = Gap width due to ram

The gap width is given as,

  $\text{t}\text{=}\frac{{\text{D}}_{\text{o}}\text{-}{\text{D}}_{\text{i}}}{\text{2}}$ ……. (4)

D_O = Outside diameter due to ram

The kinematic viscosity is given as,

  $\text{ν}\text{=}\frac{\text{μ}}{\text{ρ}}$ ……. (5)

The force acting on the ram due to the gravity is given as,

  $\text{<msup> <mo>F</mo> <mo>′</mo></msup>}\text{=}\text{mg}$ ……. (6)

  $\text{<msup> <mo>F</mo> <mo>′</mo></msup>}$ = Force due to gravity

m = Mass of the combined rack and ram

g = Acceleration due to gravity