Problem Statement
A metallic ball of diameter 5mm drops in a fluid of sp.gr. 0.8 and viscosity 30 poise. The sp.gr. of metallic ball is 9.0. Find (a) the force exerted by the fluid on the ball, (b) the pressure drag and skin friction drag, and (c) the terminal velocity of the ball in the fluid.
Given Data
Solution Approach
To solve this problem, we’ll analyze the forces acting on the ball as it falls through the fluid. At terminal velocity, the ball reaches equilibrium where the net force is zero. The forces involved are:
- Weight of the ball (downward)
- Buoyant force (upward)
- Drag force (upward)
For low Reynolds number flows (Re < 0.2), where viscous forces dominate, we can use Stokes' law to determine the drag force.
Calculations
(a) Force Exerted by the Fluid on the Ball
Step 1: At terminal velocity, the forces on the ball are in equilibrium:
Therefore, the force exerted by the fluid on the ball (drag force) is:
Step 2: Calculate the weight of the ball:
Step 3: Calculate the buoyant force:
Step 4: Calculate the drag force:
Force exerted by the fluid on the ball (FD) = 0.00526 N
(b) Pressure Drag and Skin Friction Drag
Step 1: For low Reynolds number flows (Stokes flow), the pressure drag and skin friction drag have a fixed ratio:
Step 2: The skin friction drag is:
Pressure Drag = 0.00175 N
Skin Friction Drag = 0.00351 N
(c) Terminal Velocity of the Ball
Step 1: For low Reynolds number flows, Stokes’ law gives the drag force as:
Where V is the terminal velocity we need to find.
Step 2: Rearranging to solve for the terminal velocity:
Step 3: Verify that the flow is within the Stokes regime by calculating the Reynolds number:
Since Re < 0.2, the flow is within the Stokes regime, confirming that our approach is valid.
Terminal Velocity of the Ball (V) = 0.0372 m/s
Detailed Explanation
Physical Interpretation
When a ball falls through a fluid, it experiences three main forces:
- Weight: The gravitational force pulling the ball downward
- Buoyant force: The upward force due to the displaced fluid
- Drag force: The resistance force from the fluid opposing the motion
Initially, the ball accelerates downward. As its velocity increases, the drag force also increases until it balances the net gravitational force (weight minus buoyancy). At this point, the ball reaches its terminal velocity and continues to fall at a constant speed.
Stokes’ Law and Low Reynolds Number Flow
This problem involves a small ball moving slowly through a viscous fluid, which is characterized by a low Reynolds number (Re = 0.05). In such flows, viscous forces dominate over inertial forces, and Stokes’ law applies.
Stokes’ law states that the drag force on a sphere moving through a viscous fluid is directly proportional to:
- The velocity of the sphere
- The radius of the sphere
- The viscosity of the fluid
The drag force is composed of two components:
- Pressure drag (form drag): Due to pressure differences between the front and rear of the ball, accounting for 1/3 of the total drag in Stokes flow
- Skin friction drag: Due to viscous shearing stresses along the surface of the ball, accounting for 2/3 of the total drag in Stokes flow
Applications and Significance
This type of analysis is important in many practical applications:
- Sedimentation processes in water treatment
- Understanding particle movement in the atmosphere
- Design of separation processes in chemical engineering
- Study of cellular and molecular movement in biological systems
- Analysis of lubrication processes in mechanical systems
Factors Affecting Terminal Velocity
The terminal velocity of the ball depends on several factors:
- The density difference between the ball and fluid (greater difference leads to higher terminal velocity)
- The diameter of the ball (larger diameter increases terminal velocity)
- The viscosity of the fluid (higher viscosity reduces terminal velocity)
- The shape of the object (spheres have lower drag compared to irregular shapes)
In this case, the high density of the metallic ball (9000 kg/m³) compared to the fluid (800 kg/m³) creates a significant driving force, but the high viscosity of the fluid (3 Pa·s) results in substantial resistance, leading to a moderate terminal velocity of 0.0372 m/s.
