An Immersed Boundary Method for Calculating the Relative Viscosity of a Suspension of Rigid Particles

2014 ◽  
Author(s):  
Mohsen Daghooghi ◽  
Iman Borazjani
Keyword(s):  
Interpolation Method ◽  
Relative Viscosity ◽  
Immersed Boundary ◽  
Rigid Particles ◽  
Bulk Stress ◽  
Quadratic Interpolation ◽  
Background Grid ◽  
Triangular Elements ◽  
Quadratic Interpolation Method ◽  
Good Agreement

In this paper, we provide a numerical framework to calculate the relative viscosity of a suspension of rigid particles. A high-resolution background grid is used to solve the flow around the particles. In order to generate infinite number of particles in the suspension, a particle is placed in the center of a cubic cell and periodic boundary conditions are imposed in two directions. The flow around the particle is solved using the second-order accurate curvilinear immersed boundary (CURVIB) method [1]. The particle is discretized with triangular elements, and is treated as a sharp interface immersed boundary by reconstructing the velocities on the fluid nodes adjacent to interface using a quadratic interpolation method. Hydrodynamic torque on the particle has been calculated, to solve the equation of motion for the particle and obtain its angular velocity. Finally, relative viscosity of the suspension has been calculated based on two different methods: (1) the rate of the energy consumption and (2) bulk stress-bulk strain method. The framework has been validated by simulating a suspension of spheres, and comparing the numerical results with the corresponding analytical ones. Very good agreement has been observed between the analytical and the calculated relative viscosities using both methods. This framework is then used to model a suspension with arbitrary complex particles, which demonstrates the effect of shape on the effective viscosity.

scholarly journals Solving fractional differential equations by the ultraspherical integration method

2021 ◽  
Vol 40 ◽  
pp. 1-14
Author(s):  
Ali Khani ◽  
S. Panahi
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Optimization Problem ◽  
Integration Method ◽  
Interpolation Method ◽  
Constrained Optimization Problem ◽  
Quadratic Interpolation ◽  
Fractional Differential ◽  
Quadratic Interpolation Method ◽  
Linear Fractional Differential Equations

In this paper, we present a numerical method to solve a linear fractional differential equations. This new investigation is based on ultraspherical integration matrix to approximate the highest order derivatives to the lower order derivatives. By this approximation the problem is reduced to a constrained optimization problem which can be solved by using the penalty quadratic interpolation method. Numerical examples are included to confirm the efficiency and accuracy of the proposed method.

The stress generated in a non-dilute suspension of elongated particles by pure straining motion

1971 ◽  
Vol 46 (4) ◽  
pp. 813-829 ◽  
Author(s):  
G. K. Batchelor
Keyword(s):  
Theoretical Formula ◽  
Rigid Sphere ◽  
Cross Sectional ◽  
Outer Flow ◽  
Rigid Particles ◽  
Bulk Stress ◽  
Order Of Magnitude ◽  
Sectional Plane ◽  
Dilute Suspension ◽  
A Minor

In a pure straining motion, elongated rigid particles in suspension are aligned parallel to the direction of the greatest principal rate of extension, provided the effect of Brownian motion is weak. If the suspension is dilute, in the sense that the particles are hydrodynamically independent, each particle of length 2l makes a contribution to the bulk deviatoric stress which is of roughly the same order of magnitude as that due to a rigid sphere of radius l. The fractional increase in the bulk stress due to the presence of the particles is thus equal to the concentration by volume multiplied by a factor of order l2/b2, where 2b is a measure of the linear dimensions of the particle cross-section. This suggests that the stress due to the particles might be relatively large, for volume fractions which are still small, with interesting implications for the behaviour of polymer solutions. However, dilute-suspension theory is not applicable in these circumstances, and so an investigation is made of the effect of interactions between particles. It is assumed that, when the average lateral spacing of particles (h) satisfies the conditions b [Lt ] h [Lt ] l, the disturbance velocity vector is parallel to the particles and varies only in the cross-sectional plane. The velocity near a particle is found to have the same functional form as for an isolated particle, and the modification to the outer flow field for one particle is determined by replacing the randomly placed neighbouring particles by an equivalent cylindrical boundary. The resulting expression for the contribution to the bulk stress due to the particles differs from that for a dilute suspension only in a minor way, viz. by the replacement of log 2l/b by log h/b, and the above suggestion is confirmed. The relative error in the expression for the stress is expected to be of order (log h/b)−1. Some recent observations by Weinberger of the stress in a suspension of glass-fibre particles for which 2l/h = 7·4 and h/2b = 7·8 do show a particle stress which is much larger than the ambient-fluid stress, although the theoretical formula is not accurate under these conditions.

scholarly journals A Stress Mapping Immersed Boundary Method for Viscous Flows

2021 ◽  
Vol 87 (3) ◽  
pp. 1-20
Author(s):  
Karim M. Ali ◽  
Mohamed Madbouli ◽  
Hany M. Hamouda ◽  
Amr Guaily
Keyword(s):  
Immersed Boundary Method ◽  
Stokes Equations ◽  
Cross Flow ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Element Formulation ◽  
Navier Stokes Equations ◽  
Boundary Method ◽  
Background Grid ◽  
Dimensional Simulation

This work introduces an immersed boundary method for two-dimensional simulation of incompressible Navier-Stokes equations. The method uses flow field mapping on the immersed boundary and performs a contour integration to calculate immersed boundary forces. This takes into account the relative location of the immersed boundary inside the background grid elements by using inverse distance weights, and also considers the curvature of the immersed boundary edges. The governing equations of the fluid mechanics are solved using a Galerkin-Least squares finite element formulation. The model is validated against a stationary and a vertically oscillating circular cylinder in a cross flow. The results of the model show acceptable accuracy when compared to experimental and numerical results.

scholarly journals A Mixed-Fidelity Numerical Study for Fan–Distortion Interaction

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Yunfei Ma ◽  
Jiahuan Cui ◽  
Nagabhushana Rao Vadlamani ◽  
Paul Tucker
Keyword(s):  
Immersed Boundary Method ◽  
Numerical Study ◽  
Immersed Boundary ◽  
Fan Performance ◽  
Inlet Distortion ◽  
Eddy Simulation ◽  
Pressure Distributions ◽  
Large Eddy ◽  
Dynamics Method ◽  
Good Agreement

Inlet distortion often occurs under off-design conditions when a flow separates within an intake and this unsteady phenomenon can seriously impact fan performance. Fan–distortion interaction is a highly unsteady aerodynamic process into which high-fidelity simulations can provide detailed insights. However, due to limitations on the computational resource, the use of an eddy resolving method for a fully resolved fan calculation is currently infeasible within industry. To solve this problem, a mixed-fidelity computational fluid dynamics method is proposed. This method uses the large Eddy simulation (LES) approach to resolve the turbulence associated with separation and the immersed boundary method (IBM) with smeared geometry (IBMSG) to model the fan. The method is validated by providing comparisons against the experiment on the Darmstadt Rotor, which shows a good agreement in terms of total pressure distributions. A detailed investigation is then conducted for a subsonic rotor with an annular beam-generating inlet distortion. A number of studies are performed in order to investigate the fan's influence on the distortions. A comparison to the case without a fan shows that the fan has a significant effect in reducing distortions. Three fan locations are examined which reveal that the fan nearer to the inlet tends to have a higher pressure recovery. Three beams with different heights are also tested to generate various degrees of distortion. The results indicate that the fan can suppress the distortions and that the recovery effect is proportional to the degree of inlet distortion.

DIFFUSION IN MULTICOMPONENT AQUEOUS SYSTEMS

1966 ◽  
Vol 44 (4) ◽  
pp. 477-485 ◽  
Author(s):  
J. E. Lane ◽  
J. S. Kirkaldy
Keyword(s):  
Aqueous Solutions ◽  
Relative Viscosity ◽  
Ionic Species ◽  
Aqueous Systems ◽  
Vacancy Model ◽  
High Concentrations ◽  
Substitutional Alloys ◽  
Good Agreement

A vacancy model, originally developed for the description of diffusion in substitutional alloys, is modified for application to aqueous solutions, including those containing ionic species. The results obtained with this model are similar to those of two recently published methods for estimating L-coefficients in dilute multicomponent liquid systems.Agreement with experimental L-coefficients at relatively high concentrations can be improved for this model by assuming that the jump probability of a diffusing species is inversely proportional to the relative viscosity of the mixture. Good agreement is then found for some systems up to combined solute concentrations of 3 M.

Direct Simulation of Dense Suspensions of Non-Spherical Particles

2011 ◽  
Author(s):  
Orest Shardt ◽  
Jos Derksen
Keyword(s):  
Reynolds Number ◽  
Red Blood Cells ◽  
Sedimentation Rate ◽  
Immersed Boundary Method ◽  
Blood Cells ◽  
Volume Fraction ◽  
Immersed Boundary ◽  
Direct Simulation ◽  
Rigid Particles ◽  
Boundary Method

We describe the direct simulation of high-solids-fraction suspensions of non-spherical rigid particles that are slightly denser than the fluid. The lattice-Boltzmann method is used to solve the flow of the interstitial Newtonian fluid, and the immersed boundary method is used to enforce a no-slip boundary condition at the surface of each particle. The surface points for the immersed boundary method are also employed for collision handling by applying repulsive forces between nearby surface points. Due to the finite number of these points, the method simulates rough surface collisions. We also discuss methods for integrating the equations of particle motion at low density ratios and propose a method with improved accuracy. Rigid particles shaped like red blood cells were simulated. Simulations of a single particle showed that the particle settles in its original orientation when the Reynolds number is low (1.2) but flips to a higher drag orientation when the Reynolds number is higher (7.3). A simulation with a 45% solids volume fraction and a low solid over fluid density ratio showed the possibility of simulating blood as it is found in the body. A simulation at a lower solids volume fraction (35%) was used to compare the results with the erythrocyte sedimentation rate (ESR), a common blood test. The sedimentation rate was estimated as 0.2 mm/hr, which is an order of magnitude lower than a typical ESR of about 6 mm/hr for a healthy adult. The most likely reasons for the discrepancy are the omission of agglomeration-inducing inter-cellular forces from the simulations and the treatment of the red blood cells as rigid particles.

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