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 represents the viscous resistance offered by a flat plate to the flow of fluid over it. It is crucial in fluid dynamics problems for optimizing the performance and energy efficiency of various engineering systems. The friction coefficient is related to the drag force acting on the plate through the surface shear stress. To calculate the drag force, the friction coefficient multiplied by the dynamic pressure is integrated across the plate's surface area.

Step-by-step solution

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will be used to elucidate this concept step by step.

Definition of Friction Coefficient

The friction coefficient, denoted by \(C_f\), is a dimensionless parameter that represents the resistance offered by a surface to the flow of a fluid over it. It quantifies the intensity of the viscous forces acting on the surface as a result of the fluid flow.

Importance of Friction Coefficient in Fluid Flow

The friction coefficient is an essential parameter in fluid dynamics problems, particularly for engineering applications like vehicle aerodynamics, flow over aerofoils, and heat exchange in surfaces. It helps in determining the viscous drag force acting on the surface of the objects, which is crucial for optimizing the performance and energy efficiency of these systems.

Derivation of the Drag Force Relationship

To relate the friction coefficient to the drag force acting on the flat plate, we first need to calculate the shear stress acting on the plate due to fluid flow. The shear stress \(\tau\) is given by: $$ \tau = \mu\frac{du}{dy} $$ where \(\mu\) is the dynamic viscosity of the fluid, and \(\frac{du}{dy}\) is the velocity gradient perpendicular to the plate at the wall (y = 0). Now, we can define the friction coefficient \(C_f\) as: $$ C_f = \frac{\tau}{\frac{1}{2}\rho u_\infty^2} $$ where \(\rho\) is the fluid density, and \(u_\infty\) is the free-stream velocity of the fluid. The drag force \(F_d\) acting on the plate can be computed by integrating the shear stress over the entire surface area of the plate: $$ F_d = \int_A \tau dA $$

Expression for Drag Force in terms of Friction Coefficient

Combining the expressions for friction coefficient and drag force, we have: $$ F_d = \int_A \left(C_f \times \frac{1}{2}\rho u_\infty^2\right) dA $$

Conclusion

In conclusion, the friction coefficient represents the viscous resistance offered by a flat plate to the flow of fluid over it. It is related to the drag force acting on the plate through the surface shear stress. To calculate the drag force, the friction coefficient multiplied by the dynamic pressure is integrated across the plate's surface area. The friction coefficient is crucial in fluid dynamics problems for optimizing the performance and energy efficiency of various engineering systems.