Understanding Dynamic Viscosity Dimensions

Let's figure out the dimensions of dynamic viscosity. Dynamic viscosity, often represented by the symbol $\mu$ (mu), is a measure of a fluid's resistance to flow. It relates the shear stress ($\tau$) applied to a fluid to the rate of shear or velocity gradient ($du/dy$) within the fluid.

The fundamental relationship between shear stress and velocity gradient in a Newtonian fluid is given by Newton's law of viscosity:

\(\tau = \mu \frac{du}{dy}\)

Where:

• \(\tau\) is the shear stress
• \(\mu\) is the dynamic viscosity
• \(\frac{du}{dy}\) is the velocity gradient (rate of change of velocity with respect to distance perpendicular to the flow direction)

To find the dimensions of dynamic viscosity ($\mu$), we can rearrange this equation:

\(\mu = \frac{\tau}{\frac{du}{dy}}\)

Determining Dimensions of Each Term

We need to find the dimensions of shear stress ($\tau$) and velocity gradient ($\frac{du}{dy}$).

Dimensions of Shear Stress ($\tau$)

Shear stress is defined as force per unit area. The dimensions of force are derived from Newton's second law (\(F = ma\)), which are [Mass] × [Acceleration].

• Dimension of Mass (M) = \([M]\)
• Dimension of Acceleration (a) = \([L][T^{-2}]\) (Length per time squared)
• Dimension of Force (F) = \([M][L][T^{-2}]\)

The dimension of area (A) is \([L^2]\) (Length squared).

So, the dimensions of shear stress (\(\tau\)) are:

\(\text{Dimensions of } \tau = \frac{\text{Dimensions of Force}}{\text{Dimensions of Area}} = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{1-2}][T^{-2}] = [M][L^{-1}][T^{-2}]\)

Dimensions of Velocity Gradient (\(\frac{du}{dy}\))

The velocity gradient is the change in velocity ($du$) divided by the change in distance ($dy$).

• Dimension of Velocity (u) = \([L][T^{-1}]\) (Length per unit time)
• Dimension of Distance (y) = \([L]\)

So, the dimensions of velocity gradient (\(\frac{du}{dy}\)) are:

\(\text{Dimensions of } \frac{du}{dy} = \frac{\text{Dimensions of Velocity}}{\text{Dimensions of Distance}} = \frac{[L][T^{-1}]}{[L]} = [L^{1-1}][T^{-1}] = [L^0][T^{-1}] = [T^{-1}]\)

Calculating Dimensions of Dynamic Viscosity

Now, substitute the dimensions of shear stress and velocity gradient into the equation for dynamic viscosity:

\(\text{Dimensions of } \mu = \frac{\text{Dimensions of } \tau}{\text{Dimensions of } \frac{du}{dy}} = \frac{[M][L^{-1}][T^{-2}]}{[T^{-1}]}\)

To simplify, we bring the denominator dimensions to the numerator by changing the sign of the exponents:

\(\text{Dimensions of } \mu = [M][L^{-1}][T^{-2}] \times [T^{1}] = [M][L^{-1}][T^{-2+1}] = [M][L^{-1}][T^{-1}]\)

Thus, the dimensions of dynamic viscosity are \([M][L^{-1}][T^{-1}]\).

Comparing with Options

Let's compare the calculated dimensions with the given options:

1. \([M][L][T^{-1}]\) - Incorrect
2. \([M][L^{-1}][T^{-1}]\) - Correct
3. \([M][L^{-1}][T^{-2}]\) - Incorrect (These are dimensions of shear stress/pressure)
4. \([M^{-1}][L^{-1}][T^{-1}]\) - Incorrect

The calculated dimensions \([M][L^{-1}][T^{-1}]\) match option 2.

Revision Table: Dynamic Viscosity Dimensions

Reviewing the steps to determine the dimensions of dynamic viscosity:

• Start with the definition relating shear stress and velocity gradient: \(\tau = \mu \frac{du}{dy}\)
• Isolate dynamic viscosity: \(\mu = \frac{\tau}{du/dy}\)
• Determine dimensions of shear stress (\(\tau\)): \([M][L^{-1}][T^{-2}]\)
• Determine dimensions of velocity gradient (\(\frac{du}{dy}\)): \([T^{-1}]\)
• Substitute and simplify: \(\mu = \frac{[M][L^{-1}][T^{-2}]}{[T^{-1}]} = [M][L^{-1}][T^{-1}]\)

Additional Information: Understanding Viscosity

Viscosity is a crucial property of fluids in fluid mechanics. It quantifies the internal friction within a fluid that resists flow. Think of it as the "thickness" or "stickiness" of a fluid. High viscosity fluids (like honey) flow slowly, while low viscosity fluids (like water) flow easily.

• Dynamic Viscosity ($\mu$): This is the absolute viscosity or dynamic viscosity. It measures the fluid's resistance to shearing flow when an external force is applied. Its standard SI unit is the Pascal-second (Pa·s) or Newton-second per square meter (\(N \cdot s/m^2\)). In the CGS system, the unit is the Poise (P). 1 Pa·s = 10 Poise.
• Kinematic Viscosity ($\nu$): This is the ratio of dynamic viscosity to the fluid density ($\rho$). \(\nu = \mu / \rho\). It is useful in analyzing fluid flow under the influence of gravity because the density is already accounted for. Its SI unit is square meters per second (\(m^2/s\)), and its CGS unit is the Stokes (St). 1 \(m^2/s\) = \(10^4\) Stokes.

Understanding dimensions helps in verifying equations and converting units in physics and engineering.