Fluid resistance is mainly divided into two types: frictional resistance (viscous drag) and pressure difference resistance (pressure drag). Frictional resistance originates from the viscosity of the fluid; when the fluid flows over the surface of an object, due to the viscosity, shear stress is generated, forming resistance. Pressure difference resistance, on the other hand, arises from the uneven pressure distribution in front and behind the object, which may cause vortices at the tail of the object, leading to pressure differences.

　　I. Basic Concepts



In fluid mechanics, the magnitude of resistance is usually represented by the drag coefficient $ C_D $, which is defined as follows:



$$



C_D = \frac{2F_D}{\rho v^2 A}



$$



Where:



- $ F_D $: Resistance (unit: Newton N);



- $ \rho $: Fluid density (unit: kg/m³);



- $ v $: The incoming flow velocity (unit: m/s);



- $ A $: The reference area of the object (unit: m²).



Resistance coefficients for different shapes of objects can be obtained through experiments or numerical simulations, and actual resistance can be calculated accordingly.



II. Methods of Resistance Calculation



1. Empirical formula method



For some common geometric shapes (such as spheres, cylinders, plates, etc.), there is a large amount of experimental data to support the use of empirical formulas to calculate resistance. For example, the resistance of a sphere at low Reynolds number can be given by the Stokes' law:



$$

　　F_D = 6\pi \mu r v



$$

　　Where $ \mu $ is the fluid dynamic viscosity, $ r $ is the sphere radius, and $ v $ is the velocity. However, when the Reynolds number is high, it is necessary to introduce a resistance coefficient versus Reynolds number chart for estimation.



2. Numerical simulation method (CFD)



With the development of computer technology, Computational Fluid Dynamics (CFD) has become an important tool for resistance calculation. By establishing fluid control equations (such as the Navier-Stokes equations) and using finite element or finite volume methods to numerically solve the flow field, resistance distribution under complex geometries can be accurately simulated.



3. Wind tunnel or water tunnel experimental method



Experimental methods are still an important means to verify theories and numerical models. By applying a fluid of known velocity to the model and measuring the resistance it experiences, resistance data under real-world conditions can be obtained.



III. Factors Affecting Resistance



- Reynolds number Re: Determines the flow state (laminar or turbulent) and has a significant impact on the drag coefficient.



- Object shape: The more streamlined the shape, the smaller the resistance.

　　- Surface roughness: Rough surfaces will increase frictional resistance.



- Fluid properties: Factors such as viscosity and density will also affect the magnitude of resistance.



IV. Application Examples



Taking car design as an example, designers continuously optimize the body shape through wind tunnel tests and CFD simulations to minimize the drag coefficient $ C_D $ as much as possible, thereby improving fuel efficiency and driving stability. The $ C_D $ value of modern sedans is usually between 0.25 and 0.3.



V. Summary



The calculation of resistance in fluid mechanics involves various methods and factors, ranging from classical theories to modern numerical simulations. The advancement of technology enables us to predict and optimize resistance more accurately. Whether in aeronautics and aerospace, transportation, or environmental engineering, mastering the principles and methods of resistance calculation is crucial for achieving efficient, energy-saving, and environmentally friendly designs.