1.4. Stokes Law

So, we now proceed to the another concept called Stokes law. So, we have to, you know, if we consider the flow of viscous flow past a sphere. So, this is the sphere and the, the flow is, there is a flow of velocity V coming to the sphere, the diameter of the sphere is D, mu is the kinematic fluid viscosity, then we will, you know, see that when the Reynolds number rho V D / mu is very small. When this happens, then the viscous forces are dominant over the initial forces.

Stokes Law applies to the flow of viscous fluids around small spherical objects. According to this law, when the Reynolds number is low (which indicates that viscous forces dominate over inertial forces), the fluid behaves in a predictable manner. The Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. A small Reynolds number (Re < 1) indicates a laminar flow where the fluid flows in smooth layers, resulting in a predictable drag force. In such cases, the viscous forces largely influence how an object moves through the fluid.

Imagine stirring honey with a spoon. The honey is very thick (viscous), and as you move the spoon, the honey flows smoothly around it in layers without mixing vigorously. This is akin to Stokes flow where the viscosity greatly influences the fluid motion compared to the inertia of the spoon. Conversely, if you were to stir water quickly, you may see chaotic motion and swirls due to higher velocities and lower viscosity, leading to turbulent conditions.

So, Stokes derived an equation for drag force on the sphere in creeping flow. […] 2 pi mu VD is due to viscous forces and 1 / 3rd contribution is due to the pressure forces.

Stokes derived a significant result that quantifies the drag force experienced by a sphere moving through a viscous fluid. This drag force, mathematically represented as F_d = 6πμRV, takes into account the sphere's radius (R), fluid viscosity (μ), and velocity (V). In this equation, two-thirds of the drag experienced is due to viscous forces, while the remaining one-third is attributable to pressure forces acting on the sphere. Understanding this equation is crucial for applications in fields like microbiology, sediment transport, and industrial applications where small particles move through fluids.

Think of a small ball bearing being dropped in a thick fluid like syrup. As it moves, most of the resistance it experiences is due to the syrup’s stickiness (viscous forces). The ball's weight tries to pull it down while the syrup exerts an upward drag that opposes this motion. The ball eventually reaches a constant speed called terminal velocity, at which the drag force equals the weight of the ball.

So, for the sake of easiness, I have put a figure here, on the right side top corner, where you can see the sphere. So, what is terminal velocity? It is the maximum velocity that can be attained by an object as it falls through a fluid. […] the fluid, weight of the particle and this is equal to density multiplied by volume is mass, and g is acceleration due to gravity.

Terminal velocity refers to the constant speed an object eventually reaches when the force of gravity pulling it down is balanced by the drag force pushing it up due to the fluid. For a sphere, when it falls through a fluid, two main forces act upon it: the downward force of gravity (weight) and the upward force of drag from the fluid. At terminal velocity, these two forces are equal, resulting in zero acceleration and a constant falling speed. Therefore, understanding terminal velocity is vital in predicting how objects interact with fluids, especially in environmental contexts like sedimentation in water bodies.

Consider a parachutist jumping from an airplane. Initially, they accelerate downwards due to gravity, but as they gain speed, they encounter increasing air resistance (drag). Eventually, they reach terminal velocity, where the downward force of gravity is equal to the upward drag force, resulting in a steady fall rather than an increase in speed. This illustrates how understanding forces at play influences safety in skydiving.

So, as all the other problems what we are going to do is, we are going to have a white sheet and start writing the things that are given first. […] and we would have applied the terminal velocity.

To calculate terminal fall velocity, we typically use the formula derived from balancing forces: V_t = (D²(ρ_particle - ρ_fluid)g) / (18μ). This equation incorporates the diameter of the particle, the difference in density between the particle and fluid, gravitational acceleration (g), and fluid viscosity (μ). By substituting known values into the equation, we can find the terminal velocity. This calculation is essential when dealing with sediment transport or understanding the settling rates of particles in different fluids.

Picture a small cotton ball being dropped into a swimming pool. Because it is light and has a large surface area relative to its weight, it takes a while to sink. It slowly reaches a steady speed owing to the upward drag force of the water. By applying the principles from Stokes law, we can determine how quickly the cotton ball will settle to the bottom, allowing scientists to model similar behaviors in natural water bodies.

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