Determination of Dynamic Viscosity of Viscous Oils using Mach Zehnder Digital Holographic Method

Corresponding Author: Rittirong Nuansri Department of Physics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand Email: kmitl.apl@gmail.com Abstract: The digital in-line holography based on Mach Zehnder interferometer (MZDIH) was setup for the determination of dynamic viscosity of the viscous oils. The spherical irons were used as the falling objects in the viscous oils. The hologram images of the iron falling along its entire path were recorded as a video file on the CMOS camera. The recorded hologram images were reconstructed on the holoviewer program via MATLAB. Then, the diameter and velocity of the iron were obtained. Finally, the dynamic viscosity of viscous oils were determined and compared with standard values. From the experimental results, it can be concluded that the MZDIH method is the simple and precise tool for determining the dynamic viscosity of fluids.


Introduction
Since the holographic principle was explained by Dennis Gabor (Gabor, 1948), this technique has become an important optical method for research and industrial application. Producing a hologram requires two identical beams from a coherent light source, such as a laser. The first wave without distortion or non-diffracted light, is called the reference beam. The second beam, which passes through the object and the wavefronts of the light diffract caused by the object, is called the object beam. The hologram patterns are recorded of both phases and amplitudes on a photographic film. In Digital Holography (DH), since the conventional hologram consumes a lot of holographic recording films, the holographic recording media has been replaced by a Charged-Couple Device Detector (CCD) (Schnars and Jüptner, 1994). Generally, DH method is categorized into two kinds, on-axis (in-line) and off-axis configurations. The characteristics of the digital in-line geometry have the simplest setup. Here, the reference beam, object beam and the recording sensor are placed in the same axis. In the other configuration, the object and reference waves are at an angle with each other in order to avoid the noise incoming from the overlapping of both wavefronts (Micó et al., 2010). Currently, the DH method has found application in various fields because this technique can be determined both phases and amplitudes accurately and clearly in three dimensions (3D). For example, the plankton and other marine organisms as well as bacteria and fruit flies have been explored using the DH method (Sun et al., 2007;Xu et al., 2001). The density fluctuations of a standing wave in a thermoacoustic oscillation (Penelet et al., 2015) and the phase compensation of spherical beam (Sánchez-Ortiga et al., 2011) have been analyzed by the hologram patterns as well. The fingerprint patterns of human can also be observed and separated by investigating their profile intensity fringes (Plaipichit and Buranasiri, 2014). A number of previous researches have used the DH method for measuring the velocity of particles flowing through liquids in microchannel (Khatmullina and Isachenko, 2016), observing the flow and mechanics of fluid (Desse et al., 2012). These groups have also characterized the fluid's properties and adjusted the size and shape of objects traveling in the liquids (Murata and Yasuda, 2001;Darakis et al., 2009).
In this research, the kinematic viscosities of the four different lubricant oils used are 27.849, 52.050, 57.181 and 70.428 centi-Stokes (Measured at 40°C followed by ASTM D445). The lubricants used were supplied by the Goods and Exhibit Analysis Office, Excise Department of Thailand. The fluids used in this study are important lubricants typically used on engine parts in vehicles as well as in industrial machinery. Choosing a suitable oil allows machinery users to protect moving engine parts in the normal operation. However, the user has to know the oil's parameters before use, particularly, the density and viscosity. The viscosity value is the most important characteristic governing the behavior and grade classification of oils. There are various ways to measure the viscosity, such as a cold cranking simulator (CCS), which is used to measure the dynamic viscosity (η, in the unit of centipoise) and the kinematic viscosity (υ, in the unit of centistoke, cSt) following the ASTM D445 standard. These conventional methods are still accurate and precise; however, they have complicated operating procedures and use expensive equipment. Therefore, in our research, we propose an alternative way, Mach Zehnder digital in-line holography (MZDIH), to find out the dynamic viscosity (η, in the unit of mPa.s) and kinematic viscosity (υ, in the unit of cSt.). This method is simpler, compact and less expensive than the conventional testing methods as mentioned above. In the experiment, the terminal velocity of a particle (v t ) moving in different viscous lubricant oils contained in a liquid chamber was measured. Furthermore, the experimental values can then be used to determine the properties of the lubricant oils and also classify the commercial grade of the oils for their use in lubricating machine parts.

Dynamic of a Particle in Fluids
In non-compressible liquids, since the layers of the internal textures of the liquids have the relative activity to each other, the liquids have an internal resistant force. That force is called the viscous force or drag force, with direction opposite to the direction of particle motion. The drag force was determined by Sir George Stoke in 1845 (Collinson and Roper, 1995) and given by equation: where η, r, v t are dynamic viscosity, radius of the sphere and constant velocity, respectively. This law says that the force acting on a small spherical object, as it falls down through a viscous medium. Under the influence of the gravitational force: where ρ o , V o and g are density of the object, volume of the spherical object and the gravitational constant, respectively. Due to the conditions of the object specified above, the Reynolds number in the system is small, therefore the flow of liquid remains laminar (Mott, 1994). The bouyant force as described by the principle of Archimedes (Serway and Jewett, 2014), is exerted on the object in the upward direction and given by: When a small stationary spherical particle is released into a viscous liquid, the particle at a constant velocity (v t ) experiences forces both upward and downward that must be in balance by Newton's 3rd law of motion. The dynamic viscosity can be expressed by equation: where ρ 0 , ρ f , r and η are object density, fluid density, radius of the spherical object and dynamic viscosity of the fluid, respectively. Another force acting to a particle falling through the liquid in a small channel is the surface tension of the liquid in the channel. This force was recognized from the apparent stress in the surface layer of a liquid during its flow in a liquid chamber or between parallel plates.
Normally, the surface tension is an energy associated with an interface of liquid-liquid or liquid-air. Because the surface of liquids behaves like a membrane, the liquid surface may form a curvature between small channels, where the liquid drop can look more or less rounded (Hughes and Brighton, 1991). Therefore, the speed of the liquids at the center is maximum and the speed of liquid gradually decreases until it reaches a minimum at the liquid chamber surface. For this reason, the speed of a particle falling through a liquid contained in the liquid chamber is slower than moving through a liquid contained in a bigger one, because of the surface tension between the liquid and the container.

Reconstruction Method of Digital Holography
In this work, the principle of Fresnel diffraction and convolution theory of hologram reconstruction was described (Kim, 2011;Schnars and Jueptner, 2005). By assuming a parameter that propagates along the z-axis to the sensor plane (U(x,y)) represents the two dimensional (2-D) diffraction of optical fields based on the Huygens-Fresnel principle. The diffraction of light through the optical filter in the rectangular coordinates can be shown as the optical diffraction equation: where O(ξ, η) represents an optical field in two dimensions at the object plane, r equals to [z 2 +(x−ξ) 2 + (y-η) 2 ] 1/2 , which is the displacement from any point on the object plane (O(ξ, η)) to the recording or sensor plane (x, y) and z is the propagation distance of the optical field as shown in Fig. 1. The hologram function can be expressed using the convolution technique as illustrated in Equation 6: where the Kernel's equation is given by 2 exp( ) g( , , x,y) = z jkr j r ζ η λ and k represents the wavenumber of light, which equals 2π/λ. Because the size of the recording sensor is very small compared to the distance of light propagation, r 2 is reduced to z 2 . Hence, this equation can be rewritten as exp( ) g( , , x,y) = jkr jkz ζ η . By utilizing the inverse Fourier transform of the filtered Huygens-Fresnel principle, the optical field at the image plane ( ( ', ')) R ξ η can be reconstructed as: The holograms recorded by the sensor are transferred to a computer and reconstructed based on the Huygens-Fresnel principle as shown in the Equation 7. In the MZDH system, the light source was a laser diode with 635 nm wavelength at a power of 1 mW (LDM115G/633/1). A convex lens L 1 , f 1 = 200 mm, was used to collimate the laser beam that propagates in the z-axis from the light source. Then, the laser beam of about 20 mm in diameter was separated into two identical beams, the reference beam and the object beam, by a beam splitter BS 1 . The reference beam, without any distortion, is reflected at mirror M 1 , while the object beam is reflected at mirror M 2 . The object beam was a short focal length convex lens L 2 , f 2 = 10 mm and a long focal length expanded convex lens L 3 and L 4 , f 3,4 = 200 mm to cover the whole volume of liquid chamber. Then, the transmitted beam passed through the object was collimated again by concave lens L 5 , f 5 = -200 mm to generate the parallel beam. Finally, the two beams were recombined at the beam splitter BS 2 and the hologram patterns were recorded on the CMOS camera (Canon EOS 700D, Japan, 5184 × 3456 pixels, 4.29 µm. pixel pitch). The numerically reconstructed distance (R) based on the Huygens-Fresnel principle as mentioned in section 2.2 was found to be 400 mm.

Materials and Sample Preparation
Three spherical irons, each one with a diameter of about 1.50 mm and a density of 7.00 g/cm 3 were used as the object. The object size was measured by a digital Vernier caliper with 0.01 mm resolution. The average diameter of the object was found to be 1.50 ± 0.02 mm.
The liquid chamber was prepared by using two glass slides, each one having a dimension (length x width x thickness) of 76.2 × 25.4 × 1.5 mm 3 (SAIL, No. 7101). They were attached with two layers of double sided tape foam with 3.2 mm thick (3M, 4026 Natural PUR Foam) between the slides and sealed at the bottom by hot melted adhesion glue. Figure 2b shows the diagram of liquid chamber with a volume (a × b × c) of 72.6 × 3.2 × 8 mm 3 .
The viscous oils used in this study were supplied by the Goods and Exhibit Analysis Office, Excise Department of Thailand. The properties of the oils are given in Table 1. The values of density (ρ f ) at 15°C and the kinematic viscosity (v) at 40°C were obtained from ASTM D1298 and ASTM D445, respectively.

Diameter Measurement of Spherical Iron
Among four oils used in this work, the lubricant oil is transparent. Therefore, it was used for the diameter measurement of spherical iron. It was filled in the liquid chamber which placed in the object arm of the interferometer as shown in Fig. 2(a). The first spherical iron object was dropped into the liquid chamber. The hologram images of the object falling along its entire path were recorded as a video file on the CMOS camera (Canon EOS 700D, Japan, 5184×3456 pixels, 4.29 µm. pixel pitch). Then, the recorded hologram images were reconstructed on the holoviewer program via MATLAB by using the reconstructing distance of 400 mm. Figs.  3(a) and 3(b) show the typical recorded hologram images and the reconstructed images of a falling spherical iron at any position in the lubricant oil. However, it is noted that all the experiments in this work were performed at the room temperature (27°C).
The holoviewer program via MATLAB was used to analyze the image as shown in Figure 3(d) and the diameter was determined. The experiments were carried for three times and the results are given in Table 2. The average diameter of the first spherical iron was found to be 1.51251 mm. The similar experiments were performed for the second and third spherical irons and the results are also given in Table 2. It is seen that the diameter of spherical iron obtained by DIH method agree with that measured by the Vernier caliper. In addition, it should be pointed out that DIH method can be used for diameter measurement with very high accuracy (5 decimal points).

Determination of Spherical Iron Velocity
To determine the velocity of spherical irons, each one was dropped into the liquid chamber and the hologram images were recorded as a video file on the CMOS camera (Canon, EOS 700D, Japan, 5184 × 3456 pixels, 4.29 µm. pixel pitch). The video files of a spherical iron falling in the liquid chamber were captured as an image at different positions (P) along the liquid chamber of 10, 20, 30, 40, 50 and 60 mm as denoted by P 10 , P 20 ,…, P 60 , respectively. Then, the falling times (t) at any measured position were obtained and the velocity of spherical iron was determined. Figs. 4(a) and 4(b) show the recorded and reconstructed hologram images of a spherical iron falling in liquid chamber filled with the lubricant oil. The typical values of measured velocity and time at different positions of three spherical irons falling in the lubricant oil are given in Table 3.
Similar experiments were performed with transformer oil, gasoline engine oil and cold lubricant oil but the results are not given here. However, the relationship between velocity and distance for all spherical irons in lubricant oil, transformer oil, gasoline engine oil and cold lubricant oil are illustrated in Fig. 5. It is clearly observed that the velocity is constant in the falling path for all spherical irons in all viscous oils. Hence, the average velocity values are assumed to be terminal velocity for spherical irons which will be used for the determination of dynamic viscosity as shown in Table 4.

Determination of Dynamic Viscosity of the Oils
The dynamic viscosity of the oils was determined from Equation (4) Table 1) r = Radius of the iron (obtained from Table 2) g = Gravitational constant = 9.80665 m/s 2 v t = Terminal velocity of an iron falling in the oil (obtained from Table 3)           Then, the dynamic viscosity of all four viscous oils using three spherical irons were determined and the results are given Table 4.
The relationship between dynamic viscosity (η, in mPa.s unit) and kinematic viscosity (υ, in cSt unit) is given by: Therefore, the kinematic viscosity of all four viscous oils were calculated and the results are given Table 4.
From the dynamic viscosity (η) given in Table 1, for example the lubricant oil, the η as obtained from three spherical irons are 31.925, 31.928 and 32.370 mPa.s, respectively, which are in good agreement. The dynamic viscosity for lubricant oil, transformer oil, gasoline engine oil and cold lubricant oil are approximately 31.925, 50.574, 58.773 and 76.008 mPa.s, respectively. Therefore, the cold lubricant oil gives the highest dynamic viscosity.
To compare the results obtained in this work with those of standard values of ASTMD445 as given Table 1, it is necessary to convert the dynamic viscosity (η) into the kinematic viscosity (υ) by Equation (7) as mentioned above. The kinematic viscosity (υ) of all four viscous oils obtained from three spherical irons are displayed in Fig. 6.
From Fig. 6, it is clearly observed that the kinematic viscosity of all four viscous oils obtained from the first, second and third spherical irons are approximately equal.
Furthermore, it is seen that the kinematic viscosity values obtained in this work are higher than those of standard values as given in Table 1. The difference in kinematic viscosity values are due to the temperature difference of the viscous oils. Since all the experiments in this work were performed at 27°C while the standard measurements were carried out at 40°C. However, it is observed that the higher viscous oil temperature results in the lower kinematic viscosity.

Conclusion
The digital in-line holographic based on Mach Zehnder interferometer was setup for the determination of dynamic viscosity of four viscous oils. In the system, the liquid chamber filled with the viscous oil to be studied was developed. Three spherical irons of about the same diameter were used as the falling objects in the liquid chamber. The hologram images of the spherical iron falling along its entire path were recorded as a video file on the CMOS camera. The recorded hologram images were reconstructed on the holoviewer program via MATLAB. Then, the diameter and velocity of the spherical iron were obtained. Finally, the dynamic viscosity (η) of all four viscous oils were determined. It can be concluded that this method is the simple and precise tool for the determination of dynamic viscosity of fluids.