Flow Of Incompressible Newtonian Fluid Research Articles

Computational fluid dynamics modeling of the flow in a laboratory membrane filtration cell operated at low recoveries

Scaled-down models of industrial filtration units are often used in laboratory studies of membrane processes. Knowledge of the flow field and shear stresses at the membrane surface is vital for the accurate interpretation of bench scale experiments. In this paper, we present results of computational fluid dynamics modeling of the flow within the SEPA CF flat sheet membrane filtration cell operated at low recoveries. The problem was formulated as the steady-state isothermal laminar flow of incompressible Newtonian fluid. Pressure, velocity, and shear stress distributions were computed with 1 mm resolution for different average inlet velocities. Flow was found to be unidirectional over most of the channel area with exception of the corners of the channel. Stagnation areas in dead ends of inlet and outlet tubes and in the channel areas behind duct entries as well as local regions of high shear in duct-channel transition areas were observed. The relation between the highest shear rate created in this geometry and the average inlet velocity is given.

Effect of aspect ratio and eccentricity on heat transfer from a cylinder in a cavity

This paper presents the hydrodynamic and thermal behavior of fluid that surrounds an isothermal circular cylinder in a square cavity. Simulations were carried out for four aspect ratios (defined by L/D), i.e. 2.0, 3.0, 4.0, 5.0. An incompressible flow of Newtonian fluid is considered. Prandtl number is assumed constant and equal to 1. Effect of eccentric positions (ε=−0.5 and 0.5) of the cylinder with respect to the cavity was carried out at L/D=2.0. Predicted results for eccentric cases are compared with concentric (ε=0.0) case. Grashof number is based on the diameter of the cylinder and ranges from 10 to 106. The control volume based finite volume method is used to discretize the governing equations in cylindrical coordinate. SIMPLE algorithm is used. A collocated variable arrangement is considered and SIP solver is employed to solve the system of equations. Parametric results are presented in the form of streamlines and isothermal lines for both eccentric and concentric positions. Heat transfer distribution along the perimeter of the cylinder is presented in the form of local Nusselt number. Predicted results show good agreement with the results described by Cesini et al. (1999).

Drag reduction in flow over a flat plate using active feedback control

This paper focuses on two-dimensional incompressible Newtonian fluid flow over a flat plate and studies the problem of reducing the frictional drag exerted on the plate using active feedback control. Several alternative control configurations, including both pointwise and spatially uniform control actuation and sensing, are developed and tested through computer simulations. All control configurations use control actuation in the form of blowing/suction on the plate and measurements of shear stresses along the plate. The simulation results indicate that the use of active feedback control, which employs reasonable control effort, can significantly reduce the frictional drag exerted on the plate compared to the open-loop values.

A stream function implicit finite difference scheme for 2D incompressible flows of Newtonian fluids

AbstractThe development of a new algorithm to solve the Navier–Stokes equations by an implicit formulation for the finite difference method is presented, that can be used to solve two‐dimensional incompressible flows by formulating the problem in terms of only one variable, the stream function. Two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the ADI method. An original algorithm is developed which allows a reduction from the original 11 unknowns to five and the use of the Pentadiagonal Matrix Algorithm (PDMA) in each one of the equations. An iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm. The relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the Reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes. The algorithm developed is tested by solving two classical steady fluid mechanics problems: cavity‐driven flow with Re=100, 400 and 1000 and flow in a sudden expansion with expansion ratio H/h=2 and Re=50, 100 and 200. The results obtained for the stream function are compared with values obtained by different available numerical methods, to evaluate the accuracy and the CPU time required by the proposed algorithm. Copyright © 2002 John Wiley & Sons, Ltd.

Nonlinear Control of Incompressible Fluid Flow: Application to Burgers' Equation and 2D Channel Flow

This paper proposes a methodology for the synthesis of nonlinear finite-dimensional feedback controllers for incompressible Newtonian fluid flows described by two-dimensional Navier–Stokes equations. Combination of Galerkin's method with approximate inertial manifolds is employed for the derivation of low-order ordinary differential equation (ODE) systems that accurately describe the dominant dynamics of the flow. These ODE systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed-loop system to follow the reference input asymptotically. The method is successfully used to synthesize nonlinear finite-dimensional output feedback controllers for the Burgers' equation and the two-dimensional channel flow that enhance the convergence rate to the spatially uniform steady-state and the parabolic velocity profile, respectively. The performance of the proposed controllers is successfully tested through simulations and is shown to be superior to the one of linear controllers.

Flow of second order fluids in curved pipes

Beginning with the work of Dean in 1927, regular perturbation methods have been used to study flows of incompressible Newtonian, generalized Newtonian and viscoelastic fluids in curved pipes of circular cross section. In these studies, the perturbation parameter is the curvature ratio: the cross sectional radius of the pipe divided by the radius of curvature of the pipe centerline. Closed form perturbation solutions for the equations of motion for second order fluids have previously been obtained by other authors for the special case when the second normal stress coefficient is zero, H.G. Sharma, A. Prakash, Ind. J. Pure Appl. Math 8 (1977) 546–557; P.J. Bowen, Ph.D. Thesis, University of Wales, (1990); P.J. Bowen, A.R. Davies, K. Walters, J. Non-Newtonian Fluid Mech. 38 (1991) 113–126. Here, we obtain closed form solutions for the perturbation equations even when the second normal stress difference is non-zero. We show that for a countable number of combinations of non-dimensional parameters a perturbation solution exists but is not unique. For other combinations of parameters a perturbation solution does not even exist. This latter result implies that for these parameter values there does not exist a steady, fully developed solution for flow in curved pipes which is a perturbation of the straight pipe solution, regardless of the magnitude of the curvature ratio. We emphasize that these singular points do not arise when the second normal stress coefficient is zero. A solution to the perturbation equations exists and is unique for values of the material constants which correspond to real polymeric fluids. For these values of the material constants, the secondary motion at zero Reynolds number is qualitatively similar to that arising in Newtonian fluids due to inertial effects.

Modelling of two-dimensional laminar flow using finite element method

In this paper, a Galerkin weighted residual finite element numerical solution method, with velocity material time derivative discretisation, is applied to solve for a classical fluid mechanics system of partial differential equations modelling two-dimensional stationary incompressible Newtonian fluid flow. Classical examples of driven cavity laminar flow and laminar flow past a cylinder are presented. Numerical results are compared with data found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.

Wavy film flows down an inclined plane: perturbation theory and general evolution equation for the film thickness.

Wavy film flow of incompressible Newtonian fluid down an inclined plane is considered. The question is posed as to the parametric conditions under which the description of evolution can be approximately reduced for all time to a single evolution equation for the film thickness. An unconventional perturbation approach yields the most general evolution equation and least restrictive conditions on its validity. The advantages of this equation for analytical and numerical studies of three-dimensional waves in inclined films are pointed out.

A numerical study of steady incompressible newtonian fluid flow over a disk at moderate reynolds numbers

AbstractThe hydrodynamic behaviour of a single thin disk settling steadily, broad face‐wise, in an incompressible Newtonian fluid has been investigated by solving the Navier‐Stokes equations numerically using an iterative procedure. The finite difference method has been used to map out the complete flow domain in terms of the values of the stream function and vorticity. Finally, the drag coefficient was evaluated by considering the energy dissipation. Extensive results on drag for a disk of thickness to diameter ratio of 0.05 show excellent agreement with the experimental, as well as, previous scant numerical results in the range 1 ≤ Re ≤ 100.

Classification of Pulsating Flow Patterns in Curved Pipes

The fully developed periodic laminar flow of incompressible Newtonian fluids through a pipe of circular cross section, which is coiled in a circle, was simulated numerically. The flow patterns are characterized by three parameters: the Womersley number Wo, the Dean number De, and the amplitude ratio beta. The effect of these parameters on the flow was studied in the range 2.19 < or = Wo < or = 50.00, 15.07 < or = De < or = 265.49 and 0.50 < or = beta < or = 2.00, with the curvature ratio delta fixed to be 0.05. The way the secondary flow evolved with increasing Womersley number and Dean number is explained. The secondary flow patterns are classified into three main groups: the viscosity-dominated type, the inertia-dominated type, and the convection-dominated type. It was found that when the amplitude ratio of the volumetric flow rate is equal to 1.0, four to six vortices of the secondary flow appear at high Dean numbers, and the Lyne-type flow patterns disappear at beta > or = 0.50.

Laminar flow of power law fluids in concentric annuli

Ranger (1994) demonstrated recently that it is possible to achieve higher volumetric flow rate for the laminar flow of incompressible Newtonian fluids in a hollow annular pipe than that in an annulus with the solid core for equal flow areas and pressure gradients. His calculations reveal enhancements of up to 250% in flow rate for certain combinations of the sizes of pipe involved in it. Aside from the theoretical interest in this problem (e.g. Shivakumar and Ji, 1993), this result is of considerable pragmatic importance when different fluids need to be transported between the same points. However, no physical explanation was presented to explain this phenomenon. This note demonstrates that greater enhancements in flow rate are possible for the laminar flow of incompressible power law fluids under the same circumstances. Representative numerical results are presented for a range of combinations of the flow behavior index and geometric parameters to elucidate the interplay between the nonlinear fluid characteristics and the geometrical parameters. The observed trends are qualitatively explained by considering limiting cases and in terms of the results for known geometries

Steady Developing Laminar Flow in Helical Pipes with Finite Pitch

Steady developing laminar flow of incompressible Newtonian fluids in helical pipes having a constant circular cross-section and a finite pitch was numerically studied. The governing equations were fully parabolized and were written in the orthogonal helical coordinate system. A loose coiling analysis was employed to obtain the axial length scale and the parabolic behavior of the developing flow. The numerical simulations were carried out without invoking the loose coiling approximation. The results for the axial wall shear rate and axial velocity agree with published data for the developing flow in a torus. The flow was found to be oscillatory in the developing region. The hydrodynamic developing length was correlated with the flow Dean number (Dn = Reγ½). Here Re is the flow Reynolds number and γ is the dimensionless curvature ratio. When torsion is small, its effect is more pronounced in the developing flow region. This is specially true for 0.1 < γ < 0.3. Here γ ( = η γ−½ Dn −½) is the flow pattern transition parameter for high Dean number flows and η is the dimensionless torsion. When the torsion is large, the friction factor and the developing length approach the limits corresponding to a straight pipe flow as torsion is increased.

Unsteady flow in a rigid 3-D model of the carotid artery bifurcation.

In the present study, finite element calculations are performed of blood flow in the carotid artery bifurcation under physiological flow conditions. The numerical results are compared in detail with laser-Doppler velocity measurements carried out in a perspex model. It may be concluded that the numerical model as presented here is well capable in predicting axial and secondary flow of incompressible Newtonian fluids in rigid-walled three-dimensional geometries. With regard to the flow phenomena occurring, a large region with reversed axial flow is found in the carotid sinus opposite to the flow divider. This region starts to grow at peak systole, has its maximal shape at minimal flow rate and totally disappears at the start of the acceleration phase. C-shaped axial velocity contours are formed in the deceleration phase, which are highly influenced by secondary flows. These latter flows are mainly induced by centrifugal forces, flow branching, and tapering of the carotid sinus. Lowering the sinus angle, the angle between the main branch and the carotid sinus, results in a smaller region with reversed axial flow.

P-Version least squares finite element formulation for three-dimensional, isothermal, Newtonian fluid flow

A p-version least squares finite element formulation (LSFEF) is presented for three-dimensional Navier-Stokes equations for isothermal, incompressible Newtonian fluid flow. The Navier-Stokes equations are cast into a set of first order partial differential equations using pressure-velocity-stress form where the stresses are auxiliary variables. The pressure, velocities and stresses are interpolated using equal order p-version hierarchical approximation functions derived from the Lagrange family of interpolation function. Newton's method with line search is used to find a solution vector {δ} which satisfies necessary and sufficient conditions resulting from least squares minimization procedure. Numerical examples are presented to demonstrate convergence characteristics and accuracy of the formulation. Numerical results are compared with analytical solutions and the results reported by other researchers.

Peristaltic transport in a tapered tube

The axisymmetric peristaltic flow of a viscous incompressible Newtonian fluid through a circular tube of varying cross section has been studied when the wave propagating along the wall of the tube is sinusoidal and the initial flow is Hagen-Poiseuille. A perturbation technique has been employed to analyze the problem where an attempt has been made to account for the nonlinear convective acceleration terms. Two particular cases have been studied in a greater detail. These are 1. (i) when the Reynolds number ( wave speed × radius / coefficient of kinematic viscosity ) is small and 2. (ii) when the wave number (2 π × radius / wave length ) is small. The analysis is based upon second order approximations. The derived analytical expressions have been computed in order to have an in-depth study of an important physiological problem, viz. spermatic flow through the vas deferens, in which peristaltic motion is quite dominant.

P-VERSION LEAST-SQUARES FINITE-ELEMENT FORMULATION FOR AXISYMMETRIC INCOMPRESSIBLE NEWTONIAN AND NON-NEWTONIAN FLUID FLOW WITH HEAT TRANSFER

Abstract This article presents a p-version least-squares finite-element formulation for axisymmetric incompressible Newtonian and non-Newtonian fluid flow with heat transfer. The dimension-less form of the equations describing the fluid motion and heat transfer are cast into a set of first-order coupled partial differential equations involving non-Newtonian stresses and heat fluxes as auxiliary variables. The pressure, velocities, stresses, and heat fluxes are interpolated using equal-order, C° continuity, p-version hierarchical approximation functions. The least-squares error functional is constructed using the system of coupled first-order nonlinear partial differential equations without linearization, approximations, or assumptions. The minimization of this least-squares error functional results in a solution vector {δ} that makes the partial derivatives of the error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. The article presents impl...

P-Version least squares finite element formulation for two-dimensional incompressible Nnewtonian and non-Newtonian non-isothermal fluid flow

This paper presents a p-version least squares finite element formulation (LSFEF) for two-dimensional incompressible Newtonian and non-Newtonian fluid flow with heat transfer. The dimensionless form of the differential equations describing the fluid motion and heat transfer are cast into a set of first order differential equations using non-Newtonian stresses and heat fluxes as auxiliary variables. The velocities, pressure and temperature, as well as stresses and heat fluxes, are interpolated using equal order, C 0, p-version hierarchical approximation functions. The application of the least squares finite element procedure to the set of coupled first order partial differential equations results in finding a solution vector which makes partial derivatives of the error functional with respect to , a null vector. This is accomplished by using Newton's method with a line search. The paper presents implementation of the power law model for the non-Newtonian viscosity. The fluid properties are also considered to be a function of temperature. Two numerical examples (couette shear flow problem and the 4:1 symmetric sudden expansion) are used to present numerical results for non-isothermal Newtonian and power law fluid flow. The numerical examples demonstrate the convergence characteristics and the accuracy of the formulation.

P-Version least squares finite element formulation for axisymmetric incompressible Newtonian fluid flow

This paper presents a p-version least squares finite element formulation for axisymmetric incompressible Newtonian fluid flow. The partial differential equations describing the fluid motion are cast into a set of first order coupled partial differential equations involving viscous stresses as auxiliary variables. The pressure, velocities (primary variables) and the viscous stresses (auxiliary variables) are interpolated over an element using equal order, C 0, p-version hierarchical approximation functions. The p-version least squares finite element formulation is constructed using the set of first order coupled partial differential equations which results in finding a solution vector {δ} for which the partial derivatives of the error functional (integrated sum of squares of the errors resulting from individual equations for the entire discretization) with respect to the nodal degrees of freedom {δ} become zero. This is accomplished by using Newton's method with a line search. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.

Boundary element solution for steady and unsteady Stokes flow

Abstract This paper represents a boundary element formulation for steady and unsteady slow viscous flow of incompressible Newtonian fluid. The steady flow completely ignores the inertial effect, while the unsteady flow retains the local acceleration term. The boundary integral equations and fundamental solutions are presented in two and three dimensions. A Laplace transform and numerical inversion approach is used for the unsteady problem. Two-dimensional internal flow problems are solved to illustrate the capability of the developed algorithms.

Laminar flow in an annulus between two concentric rotating porous spheres

An analysis is presented for the steady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric porous spheres with injection/suction at their boundaries. The inner sphere rotates with constant angular velocity about its own fixed axis, while the outer sphere is stationary. A solution of the Navier-Stokes equations is obtained by employing a regular perturbation technique. The solution obtained is in the form of a power series expansion in terms of the rotational Reynolds number Re, and an injection/suction Reynolds number Rew, and is valid for small values of these parameters. Results for the velocity distributions, streamlines, and viscous torques for various values of the flow parameters Re, Rew, and radius ratios λ are presented. Viscous torques at the inner and outer spheres are compared with those obtained from the numerical solution of the Navier-Stokes equations, in order to find the range of Re and Rew for which this solution is accurate.