## 1 Introduction  
At the micro or nano scale, fluid flow is subjected to high drag with decreasing dimensions and increasing surface-to-volume ratio in confined systems. Reducing hydrodynamic drag is a crucial issue in many applications of micro-/nano-fluidic channels, including micro-/nano-electromechanical systems (MEMS/NEMS), micro-/nano-fluidic systems, confined biological systems, etc., and has been of wide scientific interest [1]. It is obvious that boundary conditions at the solid-liquid interface have a direct influence on hydrodynamic drag, as published studies show that boundary slip is believed to inhibit hydrodynamic drag [2; 3; 4; 5]. Slippage of fluid flow at the solid-liquid interface, the so-called boundary slip, is of great importance in hydrodynamics. Boundary slip is characterized by slip length according to Navier's slip condition [6], via \(v=b\left.\frac{dv}{dy}\right|_{y=0}\), where \(b\) is the slip length, \(v\) is the tangential velocity, and \(y\) is the axis perpendicular to the wall. At the micro or nano scale, slip lengths of various surfaces immersed in liquid have been investigated in theoretical and experimental studies [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. For the measurement technique,slip length can be derived via measurement of surface force or hydrophobic force using surface force apparatus (SFA) or atomic force microscopy (AFM). [PERSON] [18; 19; 20] investigated the drainage of a thin liquid film confined between two spheres, where hydrophobic force could apply to obtain slip length by employing Navier's slip condition for liquid flow near the surface. [PERSON] and [PERSON] [21] presented a reformulation of the hydrodynamic conditions based on [PERSON]'s work, taking into account the boundary curvature effects of spheres. The Reynolds equation for hydrodynamic lubrication is generalized to obtain analytic expressions of the pressure and drag force for arbitrary values of slip length and for arbitrary radii of the approaching sphere, which has been widely used. [PERSON] and [PERSON] [22] studied boundary slip on chemically modified octadecyltrichlorosilane (OTS) and octadecyltriethoxysilane (OTE) samples by using SFA. The [PERSON] group [23] utilized AFM to measure the slip length of deionized (DI) water on silicon wafers roughened by treatment with KOH. The [PERSON] group [24; 25] studied boundary slip on SiO\({}_{2}\) particle composite-coated surfaces with varying degrees of roughness by using AFM. Roughness is believed to have an effect on measurement. For the technique using hydrodynamic forces to derive slip length on a rough surface, the measured slip length should be replaced by the effective slip length. However, previous studies have not included the effect of roughness on measurement and quantification, and the positions of the \"reference surface,\" i.e., the surface from which the slip velocity is determined, in boundary slip measurement are ambiguous. For the technique using hydrodynamic forces to derive the slip, different choices of reference surface can lead to a shift of the fitting force curve and result in different calculated values of slip length, as pointed out by the [PERSON] group [20] and [PERSON] group [26]. Studies have focused on smooth surfaces, and set the reference surface at the peak of the surface. If the reference surface is located on the peak of the rough surface, the calculated value of the measured slip length will tend to be larger than the true value, as the liquid molecules can still flow between the peaks and valleys. Thus, studies on the boundary slip of surfaces with various degrees of roughness must locate the position of the reference surface. [PERSON] and [PERSON] [27] investigated slippage lengths on randomly generated surfaces with Gaussian distribution heights, and reported that slippage was independent of detailed surface profile, and they suggested that the location of the reference surface was related to \(R_{a}\) roughness.