Explain drag of a Sphere.

Let us consider a case when real fluid flows past a sphere. Let D be the diameter of the sphere, v be the velocity of flow pf fluid of mass density `rho` and viscosity `mu`.

We have,

`R_e=(rho*U*D)/mu`

When velocity of flow is very small or the fluid is very viscous such that the Reynolds number is very small (`R_e<=0.2`), then the viscous forces are more predominant than the inertial force. Stokes analysed theoretically the flow around sphere under `R_e<0.2` and found that the total drag force is,

`F_D=3pimuDU`....(i)

He also found that out of total drag, two third is contributed by skin friction and one third by pressure difference.

Thus,

Skin friction drag`=2/3F_D=2pimuDU`

Pressure drag `=1/2F_d=pimuDU`

Again, the total drag is,

`F_D=C_D*(rho*U^2)/2*A`

Where `A` is the projected area of sphere.

`A=(pi*D^2)/4`

Thus,

`C_D*(rho*U^2)/2*A=3pimuDU`

`C_D=(24mu)....Show More

Let us consider a case when real fluid flows past a sphere. Let D be the diameter of the sphere, v be the velocity of flow pf fluid of mass density `rho` and viscosity `mu`.

We have,

`R_e=(rho*U*D)/mu`

When velocity of flow is very small or the fluid is very viscous such that the Reynolds number is very small (`R_e<=0.2`), then the viscous forces are more predominant than the inertial force. Stokes analysed theoretically the flow around sphere under `R_e<0.2` and found that the total drag force is,

`F_D=3pimuDU`....(i)

He also found that out of total drag, two third is contributed by skin friction and one third by pressure difference.

Thus,

Skin friction drag`=2/3F_D=2pimuDU`

Pressure drag `=1/2F_d=pimuDU`

Again, the total drag is,

`F_D=C_D*(rho*U^2)/2*A`

Where `A` is the projected area of sphere.

`A=(pi*D^2)/4`

Thus,

`C_D*(rho*U^2)/2*A=3pimuDU`

`C_D=(24mu)/(rhoUD)=24/R_e`

• For `R_e` between 0.2 and 5

• `C_D=24/R_e(1+3/(16R_e))` this is called onseen formula.

• For `R_e` between 5 and 1000, `C_D=0.4`

• For `R_e` between 1000 and 100,000, `C_D=0.5`.

• For `R_e>10^5`, `C_D=0.2`.