(451d) Deriving 2D Velocity Profile Using Streamlines Image Velocimetry (SIV)

Fluid dynamics plays a fundamental role in environmental, biological, and industrial processes ranging from mass transport to separations and heat exchange. Investigation of sand dunes, wind waves, heat transfer, fouling, and aneurysm failure are a few examples where an exact solution of the fluid velocity field is required. Current methods to investigate this velocity field are cumbersome and hence, a rapid, accurate and user friendly analysis tool is required.

Current direct velocimetry methods include Particle Imaging Velocimetry (PIV), Particle Tracking Velocimetry (PTV), and Laser Doppler Velocimetry (LDV). These methods require significant investment in equipment and computational power. For example, a commercial PIV system consists of a laser diode and an air-cooled superfast camera.

Here, we introduce a novel method termed Streamline Image Velocimetry (SIV) which allows a rapid quantification of velocity fields in arbitrary geometries. We use a single long-exposure image to capture multiple particles streaking in individual streamlines across the field. Streamlines confine streamtubes, in which the volumetric flow is constant for incompressible fluid. Using the explicit analytical solution as an initial condition, we quantify the velocity field throughout the entire field. We compiled our results into a computer program that enables the rapid derivation of velocity profiles from a single input image, which can be taken using a simple household camera. Our results show an excellent correlation to numerical simulations and are able to quantify fluid velocities across complex porous barrier, where numerical simulations are difficult at best.

Results

Analytical Solution for Rectangular Channel

The analytical solution for the velocity field in the middle of a low aspect ratio (h<<w) rectangular channel is approximated as a Fourier series in equation 1:

(1)    V_x(y)=48Q/(∏³*(w-0.63h))*Σ^∞_n,odd*n^-3*(1-cosh(n∏y/h)/cosh(n∏w/2h))*sin(n∏/2)

Where v_xis velocity vector, h is the channel height, w is the channel width, y is horizontal location inside the channel and Q is the volumetric flow rate.  

Streamline Imaging                     
To experimentally quantify the fluid velocity field we introduced fluorescent microbeads into a microfabricated channel which contained distortions. Long exposure images were taken using a fluorescent microscope, taking care to trace complete paths of multiple particles across the entire field of view. The average velocity between two streamlines is defined by volumetric flow rate divided by stream tube cross section.  Polynomials were fitted to the visualized streamlines. A mean polynomial was calculated between every two neighboring streamlines. The velocity direction was determined by the derivative of the mean polynomial. The distance between the streamlines was calculated as the length of a line perpendicular to the mean polynomial drawn from the upper polynomial to the lower one.

Computational Image Analysis

The image processing and velocity profile calculation were compiled into a single MATLAB program (code is available on request) with a graphical user interface (GUI).  The program enables to load streamlines snapshot, preliminary image processing (alignment, cropping, binary conversion, PSF deconvolution). The program is loaded with macroscopic parameters, as fluid viscosity, density and “anchor” region geometry.  Based on the entered parameters, the program can fit a set of polynomials to the streamlines, and calculate the velocity field. Data can be presented as either numerical values, or an arrow field.

SIV Analysis of Flow in Microfluidic Device

We fabricated a PDMS device with a negative arched distortion and took snapshots of the streamlines using fluorescent microscopy.  We analyzed the streamlines snapshots in MATLAB and calculated the velocity profile in cross sections. The velocities were compared to COMSOL derived numerical simulations: in the “anchor” region where the program calculates the analytical solution, in the middle of the arch where the streamlines are parallel and there is no perpendicular velocity component and in the quarter of the arch. The results show high correlation (0.97). The correlation between SIV and the numerical calculation is high (0.96) even at the quarter of the distortion where the perpendicular component of the velocity is maximal presenting a rapid change in velocity as function of distance from the channel wall.  

SIV Analysis of a Microfluidic Channel with Porous Elements    

In order to verify our method with a different flow irregularity, characterized with unknown boundary conditions, we produced the velocity field of a visualized streamlines snapshot, taken from a recent publication. The visualized streamlines were taken from microfluidic device integrated with nanoporous forests of vertically aligned carbon nanotubes. In such devices the complexity of the velocity field calculation increases due to the porous boundary conditions. However, for SIV, the boundary conditions are the streamtubes enabling the derivation of the velocity field based solely on the visualized streamlines. The results indicate that SIV produces high-precision results in complex flow regions where the fluid velocity magnitude and direction rapidly changes. SIV analysis presenting velocimetry at low cost and reduces computational resources.

Materials and Methods

Chemicals

Poly(dimethyl)siloxane (PDMS) was purchased from Dow Corning (SYLGARD). SU8-2050 was purchased from Microchem (Newton, MA). Fluorescent green microbeads, Pluronics F68, and Phosphate Buffered Saline (PBS) were purchased from Sigma Aldrich (Rehovot, Israel).

Numerical Modeling

Numerical simulations were performed using COMSOL multiphysics simulation platform v3.5A with a direct linear system solver and extra fine mesh (PARADISO). Fluid density was defined as 1x10³ kg/m³, with a dynamic viscosity of 1x10^-3Pa·s.

Microfluidic Channels

Microfluidic devices were fabricated by soft lithography. Briefly, molds were fabricated by photolithography of SU8 on silicon wafers at the Hervey Krueger Center of Nanoscience and Nanotechnology at the Hebrew University of Jerusalem. Channels were replica molded in PDMS and bonded to glass using oxygen plasma bonding. The device is 100 μm high. Main channel width is 3000 μm and the disturbance depth is 100 μm.

Streamline Imaging

Prior to the experiment the device channel was coated with Pluronic F-68 for 1 hour at room temperature to prevent non-specific adhesion of fluorescent particles. The experiments were performed with microbeads diluted in 1% Pluronic F-68 at a concentration of 2·10⁶ particles/ml. Microbeads were perfused using a Chemyx Fusion 200 syringe pump and imaged on a Zeiss Axiovert LSM700 Microscope.

Streamlines Image Analysis

Images processing was carried out in MATLAB. Image were imported and deconvolved using Gaussian filter generated from the theoretical point spread function (PSF). Streamlines were traced using MATLAB 2011b algorithm. Polynomial functions were fitted to traced streamlines and transformed into a binary 2D matrix.

Streamlines Image Velocimetry

A streamline is a curve parallel to the velocity vector which can be visualized using long exposure snapshots of fluorescent microparticles in fluid. In steady flow, the streamlines coincide with the paths of the fluid particles. A streamtube is a surface confined between streamlines. The volumetric flow rate in a streamtube is conserved; therefore, the product of the streamtube cross-section (A) and the average velocity (w) is constant as described in Equation 2:

(2)                 A₁w₁= A₂w₂=Aw

Our method relies on the fact, that in a streamtube that is formed in a simple geometry-shaped vessel, the volumetric flow rate is conserved when entering into a more complex geometry and hence, the volumetric flow rate in the complex region can be derived from flow in the simple geometry. Hence, our method requires that the analyzed vessel will include a simple geometry fluid entrance region, which will serve as an “anchor” for further calculations. Our analytical solution is suitable for rectangular cross section channels, however it can be easily modified for any other geometry.