To address this question about the Tesla turbine and the impact of design parameters like the gap between discs, disc thickness, and surface roughness ratio on torque and angular velocity, we can summarize some key insights from fluid dynamics and rotor design principles:

1. Gap Between Discs:

- Impact on Flow Behavior: The gap size affects the boundary layer interaction between the fluid and the discs. A smaller gap increases viscous shear forces, enhancing torque but possibly reducing angular velocity due to increased resistance.

- Efficiency Considerations: An optimal gap maximizes energy transfer while minimizing fluid drag and backflow losses. Very small gaps can increase the risk of clogging, particularly with non-ideal fluids.

2. Disc Thickness:

- Structural Strength vs. Weight: Thicker discs provide better structural integrity, allowing them to withstand higher rotational speeds, but they increase the overall mass, which may reduce angular velocity.

- Flow Path Influence: Disc thickness can impact the path of the fluid between discs, altering the flow profile and subsequently influencing torque generation.

3. Roughness Ratio:

- Surface Friction and Flow Separation: Higher roughness can increase friction, generating more torque. However, excessive roughness can lead to turbulent flow separation, reducing the effective energy conversion and angular velocity.

- Optimization for Different Fluids: Depending on the working fluid's properties (e.g., viscosity), optimal roughness varies to balance between laminar and turbulent flow for efficient torque generation.

Mathematical Modeling:

- In practice, these factors are modeled using equations derived from Navier-Stokes equations, with considerations for Reynolds number to account for flow regimes. Empirical models or simulations (e.g., CFD) are often used to fine-tune these parameters for specific applications.

For a precise mathematical formulation, detailed computational models are generally required as it involves complex fluid dynamics that are affected by both laminar and turbulent flow regimes.

For further research, exploring specific studies and simulations published in journals focusing on Tesla turbines or fluid dynamics would provide more concrete equations tailored to specific configurations.