Question

The coefficient of friction \(C_{f}\) for a fluid flowing across a surface in terms of the surface shear stress, \(\tau_{w}\), is given by (a) \(2 \rho V^{2} / \tau_{w}\) (b) \(2 \tau_{w} / \rho V^{2}\) (c) \(2 \tau_{w} / \rho V^{2} \Delta T\) (d) \(4 \tau_{u} / \rho V^{2}\) (e) None of them

Short answer

Answer: (b) \(2 \tau_{w} / \rho V^{2}\)

Step-by-step solution

Identify Relevant Variables

In fluid mechanics, the coefficient of friction (\(C_{f}\)) is a dimensionless quantity that relates the shear stress (\(\tau_{w}\)) acting on a surface to the fluid properties and flow conditions. The relevant variables involved in the equation include: - Coefficient of friction (\(C_{f}\)) - Surface shear stress (\(\tau_{w}\)) - Fluid density (\(\rho\)) - Fluid velocity (\(V\)) - Optional: temperature difference (\(\Delta T\)) - only present in some choice

Determine the Dimensional Consistency

To find the correct option, we need to check the dimensional consistency of each choice. The coefficient of friction (\(C_{f}\)) is a dimensionless quantity, so the dimensions on the right side of the equation should cancel out. The dimensions of the variables involved are: - Surface shear stress (\(\tau_{w}\)): \(ML^{-1}T^{-2}\) - Fluid density (\(\rho\)): \(ML^{-3}\) - Fluid velocity (\(V\)): \(LT^{-1}\) - Optional: temperature difference (\(\Delta T\)): \([C^\circ]\) or \([F^\circ]\) (temperature unit)

Check the Dimensional Consistency for Each Option

Now, we will check each option for dimensional consistency: (a) \(2 \rho V^{2} / \tau_{w}\) : Dimensions = \([ML^{-3}] [L^2T^{-2}] / [ML^{-1}T^{-2}]\) (b) \(2 \tau_{w} / \rho V^{2}\) : Dimensions = \([ML^{-1}T^{-2}] / [ML^{-3}L^2T^{-2}]\) (c) \(2 \tau_{w} / \rho V^{2} \Delta T\) : Dimensions = \([ML^{-1}T^{-2}] / [ML^{-3}L^2T^{-2}[C^\circ]]\) (d) \(4 \tau_{u} / \rho V^{2}\) : Dimensions = \([ML^{-1}T^{-2}] / [ML^{-3}L^2T^{-2}]\) (e) None of them

Choose the Correct Option Based on Dimensional Consistency

Based on our analysis, the only option having dimensional consistency with a dimensionless quantity on the left side of the equation (Coefficient of Friction) is: (b) \(2 \tau_{w} / \rho V^{2}\)