Motion of an Object in a Viscous Fluid | Basic Physics for IIT JAM PDF Download

Substituting values into the equation for  yields

Discussion
This value is sufficiently high to imply a turbulent wake. Most large objects, such as airplanes and sailboats, create significant turbulence as they move. As noted before, the Bernoulli principle gives only qualitatively-correct results in such situations.One of the consequences of viscosity is a resistance force called viscous drag F_v that is exerted on a moving object. This force typically depends on the object’s speed (in contrast with simple friction). Experiments have shown that for laminar flow ( less than about one) viscous drag is proportional to speed, whereas for  between about 10 and 10⁶, viscous drag is proportional to speed squared. (This relationship is a strong dependence and is pertinent to bicycle racing, where even a small headwind causes significantly increased drag on the racer. Cyclists take turns being the leader in the pack for this reason.) For greater than 10⁶, drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere, F_v is proportional to fluid viscosity η , the object’s characteristic size L , and its speed v . All of which makes sense—the more viscous the fluid and the larger the object, the more drag we expect. Recall Stoke’s law F_{s = 6πrηv.} For the special case of a small sphere of radius R moving slowly in a fluid of viscosity η, the drag force F_s is given by F_{s = 6πRηv
(a) Motion of this sphere to the right is equivalent to fluid flow to the left. Here the flow is laminar with  less than 1. There is a force, called viscous drag Fv, to the left on the ball due to the fluid’s viscosity. 
(b) At a higher speed, the flow becomes partially turbulent, creating a wake starting where the flow lines separate from the surface. Pressure in the wake is less than in front of the sphere, because fluid speed is less, creating a net force to the left F′v that is significantly greater than for laminar flow. Here   is greater than 10. 
(c) At much higher speeds, where  is greater than} 10⁶ <sub>, flow becomes turbulent everywhere on the surface and behind the sphere. Drag increases dramatically.

An interesting consequence of the increase in Fv with speed is that an object falling through a fluid will not continue to accelerate indefinitely (as it would if we neglect air resistance, for example). Instead, viscous drag increases, slowing acceleration, until a critical speed, called the terminal speed, is reached and the acceleration of the object becomes zero. Once this happens, the object continues to fall at constant speed (the terminal speed). This is the case for particles of sand falling in the ocean, cells falling in a centrifuge, and sky divers falling through the air. Some of the factors that affect terminal speed. There is a viscous drag on the object that depends on the viscosity of the fluid and the size of the object. But there is also a buoyant force that depends on the density of the object relative to the fluid. Terminal speed will be greatest for low-viscosity fluids and objects with high densities and small sizes. Thus a skydiver falls more slowly with outspread limbs than when they are in a pike position—head first with hands at their side and legs together.</sub> 
Take-Home Experiment: Don’t Lose Your Marbles
By measuring the terminal speed of a slowly moving sphere in a viscous fluid, one can find the viscosity of that fluid (at that temperature). It canbe difficult to find small ball bearings around the house, but a small marble will do. Gather two or three fluids (syrup, motor oil, honey, olive oil, etc.) and a thick, tall clear glass or vase. Drop the marble into the center of the fluid and time its fall (after letting it drop a little to reach its terminal speed). Compare your values for the terminal speed and see if they are inversely proportional to the viscosities . Does it make a difference if the marble is dropped near the side of the glass?
There are three forces acting on an object falling through a viscous fluid: its weight w, the viscous drag F_v, and the buoyant force F_B.

Section Summary
1. When an object moves in a fluid, there is a different form of the Reynolds number  which indicates whether flow is laminar or turbulent.2. For N′R less than about one, flow is laminar.
3. For N′R greater than 10⁶, flow is entirely turbulent.
Conceptual Questions
What direction will a helium balloon move inside a car that is slowing down—toward the front or back? Explain your answer.
Will identical raindrops fall more rapidly in 5 C air or 25 C air, neglecting any differences in air density? Explain your answer.
If you took two marbles of different sizes, what would you expect to observe about the relative magnitudes of their terminal velocities?