Problem Statement
A steel sphere of 5mm diameter falls in a glycerin at a terminal velocity of 0.05m/s. Assume Stoke’s law is applicable, determine (a) dynamic viscosity of glycerin, (b) drag force and (c) coefficient of drag. Take sp wt of steel and glycerin as 75 KN/m³ and 12.5 KN/m³ respectively.
Given Data
Solution Approach
When an object falls through a fluid and reaches terminal velocity, the net force acting on it is zero. At this point, the weight of the object is balanced by the drag force and the buoyant force. We’ll use this principle along with Stoke’s law to determine the required parameters.
Calculations
(a) Dynamic Viscosity of Glycerin
Step 1: At terminal velocity, the forces are balanced:
Therefore, Drag force:
Step 2: Calculate the weight and buoyant force:
Step 3: Apply Stoke’s law to find the dynamic viscosity:
Dynamic Viscosity (μ) = 1.7 N·s/m² = 1.7 Pa·s
(b) Drag Force
The drag force was already calculated in part (a):
Drag Force (FD) = 0.00409 N
(c) Coefficient of Drag
Step 1: First, we need to verify if Stoke’s law is applicable by calculating the Reynolds number:
Step 2: Since Re < 0.2, Stoke's law is valid. For Stoke's flow, the coefficient of drag is:
Coefficient of Drag (CD) = 133
Detailed Explanation
Terminal Velocity and Force Balance
When an object falls through a fluid, it initially accelerates due to gravity. As its velocity increases, the drag force also increases until it balances the net gravitational force (weight minus buoyancy). At this point, the object stops accelerating and falls at a constant speed called the terminal velocity.
Stoke’s Law
Stoke’s law applies to spherical particles moving through a viscous fluid at low Reynolds numbers (typically Re < 0.2). It states that the drag force is directly proportional to:
- The fluid’s dynamic viscosity (μ)
- The radius of the spherical object
- The velocity of the object
The mathematical form is FD = 6πμrV, which is equivalent to FD = 3πμDV where D is the diameter.
Reynolds Number
The Reynolds number (Re) is a dimensionless quantity that predicts the flow pattern of fluids around objects. It is defined as:
where ρ is the fluid density, V is the velocity, D is the characteristic length (diameter in this case), and μ is the dynamic viscosity.
For very low Reynolds numbers (Re < 0.2), the flow is dominated by viscous forces, and Stoke's law provides an accurate description of the drag.
Drag Coefficient
The drag coefficient (CD) is a dimensionless quantity used to quantify the drag or resistance of an object in a fluid environment. For a sphere in the Stokes flow regime, the drag coefficient is inversely proportional to the Reynolds number:
This relatively high drag coefficient (133) indicates that viscous forces are dominant in this flow scenario, which is typical for small objects moving slowly through highly viscous fluids like glycerin.
Applications
Understanding the behavior of particles in viscous fluids has numerous applications:
- Pharmaceutical industry: drug delivery systems and particle separation
- Environmental engineering: sedimentation processes and pollutant transport
- Chemical engineering: design of reactors and separation equipment
- Biomedical applications: understanding cell movement in body fluids
- Geological processes: sedimentation and erosion mechanisms
Significance of Results
The calculated dynamic viscosity of glycerin (1.7 Pa·s) is consistent with its known high viscosity at room temperature, which can range from 0.95 to 1.5 Pa·s depending on purity and temperature. This high viscosity explains why the steel sphere falls at such a slow terminal velocity despite the substantial density difference between steel and glycerin.
The verification that Re = 0.18 confirms that Stoke’s law is indeed applicable in this scenario, validating our approach and results.
