Question

What does the friction coefficient represent in flow over a flat plate? How is it related to the drag force acting on the plate?

Short answer

Answer: The friction coefficient (C_f) is a dimensionless quantity representing the ratio of the wall shear stress to the dynamic pressure of the fluid flowing over a surface. It characterizes the fluid's interaction with the surface and the resistance to its flow. In the context of flow over a flat plate, the friction coefficient is related to the drag force acting on the plate through the equation: \(F_{drag} = C_f \cdot \frac{1}{2}\rho U^2 \cdot A\), where \(\rho\) is the fluid density, \(U\) is the fluid velocity, and \(A\) is the total surface area of the plate exposed to the flow. Higher friction coefficients indicate increased drag due to greater interaction between the fluid and the surface of the plate.

Step-by-step solution

1. Definition of Friction Coefficient

The friction coefficient, often denoted as 'C_f', is a dimensionless quantity that represents the ratio of the wall shear stress to the dynamic pressure of the fluid flowing over a surface. The wall shear stress is the force exerted by the fluid parallel to the surface (i.e. in the direction of the flow), and the dynamic pressure is the pressure created by the fluid's motion. In other words, the friction coefficient characterizes how effectively the fluid interacts with the surface, creating resistance to its flow.

2. Boundary Layer and Flow over a Flat Plate

When fluid flows over a flat plate, a thin layer of fluid adjacent to the plate's surface, called the boundary layer, is formed. Within this layer, the fluid experiences viscous resistance, which results in the development of a velocity gradient (a change in fluid velocity across the boundary layer). The friction between the fluid and the surface of the plate generates a force acting on the plate, which is known as the wall shear stress.

3. Drag Force on a Flat Plate

The drag force acting on a flat plate is the resistance force experienced by the plate due to the flow of fluid over its surface. The drag force can be divided into two components: (i) pressure drag, resulting from the pressure difference between the front and back of the plate, and (ii) friction drag, caused by the frictional force between the fluid and the surface of the plate. In the case of flow over a flat plate with a streamlined shape, the friction drag is the dominant factor contributing to the total drag force.

4. Relationship between Friction Coefficient and Drag Force

Now that we understand the meaning of friction coefficient and its role in flow over a flat plate, we can establish the relationship between the friction coefficient and the drag force. The drag force acting on the plate can be calculated as the product of the surface shear stress (frictional force per unit area) and the total surface area of the plate exposed to the flow: \(F_{drag} = \tau_w \cdot A\) Since the friction coefficient is the ratio of the wall shear stress to the dynamic pressure, we can express the wall shear stress as: \(\tau_w = C_f \cdot \frac{1}{2}\rho U^2\) Where \(\rho\) is the fluid density and \(U\) is the fluid velocity. Combining these equations, we can write the expression for the drag force on the plate in terms of the friction coefficient: \(F_{drag} = C_f \cdot \frac{1}{2}\rho U^2 \cdot A\) This equation shows how the friction coefficient is related to the drag force acting on a flat plate in a fluid flow. The friction coefficient plays a significant role in determining the magnitude of the drag force, with higher friction coefficients indicating increased drag due to greater interaction between the fluid and the surface of the plate.