To derive Stoke's law [tex]\( F_d = 6 \pi \eta r_t v \)[/tex] using the dimensional analysis method, let's first identify and analyze the dimensions of the involved quantities:
1. Drag force [tex]\( F_d \)[/tex]:
- The dimension of force is given by [tex]\( [F_d] = M L T^{-2} \)[/tex].
2. Dynamic viscosity [tex]\( \eta \)[/tex]:
- The dimension of dynamic viscosity is given by [tex]\( [\eta] = M L^{-1} T^{-1} \)[/tex].
3. Radius of the sphere [tex]\( r_t \)[/tex]:
- The dimension of radius is given by [tex]\( [r_t] = L \)[/tex].
4. Velocity [tex]\( v \)[/tex]:
- The dimension of velocity is given by [tex]\( [v] = L T^{-1} \)[/tex].
Stokes’ law in its given form is:
[tex]\[ F_d = 6 \pi \eta r_t v \][/tex]
We need to verify the dimensional consistency of this equation. We start by checking the dimensions on both sides of the equation.
### Dimensions on the Right-Hand Side (RHS):
The RHS of the equation is [tex]\( 6 \pi \eta r_t v \)[/tex].
Since [tex]\( 6 \pi \)[/tex] is a dimensionless constant, it does not affect the dimensions. We will consider the dimensions of [tex]\( \eta \)[/tex], [tex]\( r_t \)[/tex], and [tex]\( v \)[/tex]:
[tex]\[ [6 \pi \eta r_t v] = [\eta][r_t][v] \][/tex]
Now, substituting the respective dimensions:
[tex]\[ [\eta] = M L^{-1} T^{-1} \][/tex]
[tex]\[ [r_t] = L \][/tex]
[tex]\[ [v] = L T^{-1} \][/tex]
Multiply these dimensions together:
[tex]\[ [\eta][r_t][v] = (M L^{-1} T^{-1})(L)(L T^{-1}) \][/tex]
Thus,
[tex]\[ [\eta][r_t][v] = M L^{-1} T^{-1} \cdot L \cdot L T^{-1} \][/tex]
Combining the dimensions,
[tex]\[ [6 \pi \eta r_t v] = M L^{-1} T^{-1} \cdot L \cdot L T^{-1} = M \cdot L^{-1 + 1 + 1} \cdot T^{-1 + -1} = M L T^{-2} \][/tex]
### Dimensions on the Left-Hand Side (LHS):
[tex]\[ [F_d] = M L T^{-2} \][/tex]
### Conclusion:
Since the dimensions on both sides of the equation match:
[tex]\[ M L T^{-2} = M L T^{-2} \][/tex]
This dimensional analysis confirms that [tex]\( F_d = 6 \pi \eta r_t v \)[/tex] is dimensionally consistent and therefore valid.
Stokes' law [tex]\( F_d = 6 \pi \eta r_t v \)[/tex] correctly describes the drag force experienced by a spherical object moving through a viscous fluid.
