Drag on Sphere - Stoke's Law

When Real Fluid past a Sphere: We will make use of different Re limits to define dragon sphere correctly.

1) For Re $\leq 0.2$ - Stoke's Law

Let $\mathbf{D}$ be the diameter of the sphere, $\mathbf{U}$ be the velocity of flow of fluid of mass density $\rho$ and viscosity $\mu$. When the velocity of flow is very small or the fluid is very viscous such that the Reynolds number is very small, being as low as $0.2$ or even less then the viscous forces are much more predominant than the intertial forces.

C.G. Stokes analysed theoretically the flow around a sphere under very low velocities, such that $\operatorname{Re}\lt0.2$.

When Real Fluid past a Sphere: We use different Re limits to define drag on sphere correctly.

I) For Re $\leq 0.2$ - Stokes found that the total drag force is given by: $\left[F_{D}=3 \pi \mu U D\right.$

He further found out that out of the total drag given by above eqn, twothirds is contributed by skin friction and one-third by the pressure difference.

Skin Friction Drag $=\frac{2}{3} F_{D}=\frac{2}{3} 3 \pi \mu U D=2 \pi \mu U D$

Pressure Drag $=\frac{1}{3} F_{D}=\frac{1}{3} 3 \pi \mu U D=\pi \mu U D$

When Real Fluid past a Sphere: We use different Re limits to define drag on sphere correctly.

I) For Re $\leq 0.2$ - Stoke's Law $\left(F_{D}=3 \pi \mu U D\right)$

However, $F_{D}=C_{D} A \frac{\rho U^{2}}{2} \quad$ Equating above two equations

$$ 3 \pi \mu U D=C_{D} A \frac{\rho U^{2}}{2} \quad \text { Where, } \mathrm{A}=\frac{\pi}{4} D^{2} $$ $$ \begin{aligned} C_{D} &=\frac{24 \mu}{\rho U D} \\ C_{D} &=\frac{24}{R_{e}} \end{aligned} $$

(Remember: Use above equation of $C d=24 / \operatorname{Re}$ or $F_{D}=3 \pi \mu U D$, both referred as Stoke's Law, only when Res $0.2$, otherwise use basic equation of drag as $\mathrm{Fd}=\mathrm{Cd} \mathrm{x} \mathrm{A} \mathrm{x} \mathrm{U}^{2} / 2$. Therefore check for $\mathrm{Re}$ is must before using the Stoke's Law).

We use different Re limits to define drag on sphere correctly.

II) For $0.2 \leq \operatorname{Re} \leq 5$ Oseen máde an improvement to the Stokes' solution by partly taking into account the effect of inertial terms (which Stokes had omitted.). $$ C_{D}=\frac{24}{R_{e}}\left(1+\frac{3}{16 R_{e}}\right) $$

III) For $5 \leq \operatorname{Re} \leq 1000$ The value of CD for Re between 5 to 1000 is equal to $0.4$

IV) For $1000 \leq \operatorname{Re} \leq 10^{5}$

With the increase in the Reynolds number the region in which the influence of viscosity is predominant is considerably reduced and is restricted only to a very small zone of boundary layer formed close to the sphere. The separation of the boundary layer however begins from the downstream stagnation point

V) For Re $\geq 10^{5}$

With a further increase in the Reynolds number, at $\operatorname{Re} \sim 3 \times 105$, the boundary layer becomes turbulent which can travel further downstream without separation.

As such in this case the points of separation shift considerably to the downstream side, which are now located at about $110^{\circ}$ from the front stagnation point Due to this, the size of the wake is reduced and the value of CD drops sharply from $0.5$ to $0.2$