Flow Rate-pressure Drop Relation Research Articles

Flow of an Oldroyd-B fluid in a slowly varying contraction: theoretical results for arbitrary values of Deborah number in the ultra-dilute limit

Pressure-driven flows of viscoelastic fluids in narrow non-uniform geometries are common in physiological flows and various industrial applications. For such flows, one of the main interests is understanding the relationship between the flow rate $q$ and the pressure drop $\Delta p$ , which, to date, is studied primarily using numerical simulations. We analyse the flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, contracting channels and present a theoretical framework for calculating the $q-\Delta p$ relation. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity profile remains parabolic and Newtonian, resulting in a one-way coupling between the velocity and polymer conformation tensor. This one-way coupling enables us to derive closed-form expressions for the conformation tensor and the flow rate–pressure drop relation for arbitrary values of the Deborah number ( $De$ ). Furthermore, we provide analytical expressions for the conformation tensor and the $q-\Delta p$ relation in the high-Deborah-number limit, complementing our previous low-Deborah-number lubrication analysis. We reveal that the pressure drop in the contraction monotonically decreases with $De$ , having linear scaling at high Deborah numbers, and identify the physical mechanisms governing the pressure drop reduction. We further elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following the contraction and show that the downstream distance required for such relaxation scales linearly with $De$ .

Non-Newtonian fluid–structure interaction: Flow of a viscoelastic Oldroyd-B fluid in a deformable channel

We analyze the steady non-Newtonian fluid-structure interaction between the flow of an Oldroyd-B fluid and a deformable channel. Specifically, we provide a theoretical framework for calculating the leading-order effect of the fluid's viscoelasticity on the flow rate-pressure drop relation and on the deformation of the channel's elastic wall. We first identify the characteristic scales and dimensionless parameters governing the fluid-structure interaction in slender and shallow channels. Applying the lubrication approximation for the flow and employing a perturbation expansion in powers of the Deborah number $De$, we derive a closed-form expression for the pressure as a function of the non-uniform shape of the channel in the weakly viscoelastic limit up to $\mathrm{O}(De)$. Coupling the hydrodynamic pressure to the elastic deformation, we provide the leading-order effect of the interplay between the viscoelasticity of the fluid and the compliance of the channel on the pressure and deformation fields, as well as on the flow rate-pressure drop relation. For the flow-rate-controlled regime and in the weakly viscoelastic limit, we show analytically that both the compliance of the deforming top wall and the viscoelasticity of the fluid decrease the pressure drop. Furthermore, we reveal a trade-off between the influence of compliance of the channel and the fluid's viscoelasticity on the deformation. While the channel's compliance increases the deformation, the fluid's viscoelasticity decreases it.

Steady isothermal flow of a Carreau–Yasuda model fluid in a straight circular tube

The five-parameter version of the Carreau–Yasuda model has sufficient flexibility to fit various experimental viscosity vs. shear rate curves. Here, analytical solutions are derived for the velocity distribution and the average flow velocity for the steady isothermal flow of a Carreau–Yasuda model fluid in a straight, circular tube. The solution procedure can be applied to other incompressible generalized Newtonian fluids. The analytical expressions herein derived can be useful in validating numerical solutions. • Analytical solutions are derived for the velocity distribution and flow rate—pressure drop relation for the steady Hagen–Poiseuille flow of a Carreau-Yasuda model fluid in a pipe and between parallel plates. • The solution procedure can be applied to many other incompressible generalized Newtonian fluids. • The analytical expressions herein derived can be useful in validating numerical solutions.

Analytical predictions of concrete pumping: Extending the Khatib–Khayat model to Herschel–Bulkley and modified Bingham fluids

It has been widely reported that self-compacting concrete and other cement-based materials are shear-thickening materials. However, there is a lack of an analytical model on the prediction of flow rate–pressure drop relation in the pumping of concrete that takes into account both the presence of the lubrication layer and the shear-thickening behavior of fresh concrete. This work fills this gap by extending analytical predictions of concrete pumping in the presence of a lubrication layer to nonlinear rheological models including the Herschel–Bulkley model and the modified Bingham model. In doing so, a new method is developed to obtain the flow rate–pressure drop relation for the steady Hagen–Poiseuille coaxial flow of two immiscible, incompressible fluids. Analytical expressions are presented for the flow rate–pressure drop relation, shear rate distribution, and velocity distribution based on the two-fluid Herschel–Bulkley model and separately based on the two-fluid modified Bingham model. With those analytical expressions, volumetric flow rate vs. pressure loss curves can be readily constructed, given values for the other eight quantities: three for the rheological properties of the bulk concrete, three for the rheological properties of the fluid in the lubrication layer, pipe radius, and thickness of the lubrication layer.

Flow rate–pressure drop relations for new configurations of slender compliant tubes arising in microfluidics experiments

We investigate the steady-state fluid–structure interaction between a Newtonian fluid flow and a deformable microtube in two novel geometric configurations arising in recent microfluidics experiments. The first configuration is a cylindrical fluidic channel surrounded by an annulus of soft material with a rigid outer wall, while the second one is a cylindrical fluidic channel extruded from a soft rectangular slab of material. In each configuration, we derive a mathematical theory for the nonlinear flow rate–pressure drop relation by coupling lubrication theory for the flow with linear elasticity for the inner tube wall’s deformation. Using the flow conduit’s axial slenderness and its axisymmetry, we obtain an analytical expression for the radial displacement in each configuration from a plane-strain configuration. The predicted displacement field, and the resulting closed-form flow rate–pressure drop relation, are each validated against three-dimensional direct numerical simulations via SimVascular’s two-way-coupled fluid–structure interaction solver, svFSI, showing good agreement. We also show that weak flow inertia can be easily incorporated in the derivation, further improving the agreement between theory and simulations for larger imposed flow rates.

Flow rate-pressure drop relation for deformable channels via fluidic and elastic reciprocal theorems

Viscous flows through compliant conduits apply forces at the solid-liquid interface, leading to deformation of the cross-section, which affects the flow rate--pressure drop relation. Conventionally, calculating this relation requires solving the coupled elastohydrodynamic problem of flow and deformation. Instead, we employ reciprocal theorems for Stokes flow and linear elasticity to derive a closed-form expression for the flow rate--pressure drop relation in deformable channels, using a domain perturbation expansion that only requires the fluid flow solution and the elastic deformation due to the fluid stress distribution in an undeformed channel.

Reciprocal theorem for calculating the flow rate–pressure drop relation for complex fluids in narrow geometries

A key aspect in understanding pressure-driven flows of non-Newtonian fluids in narrow and confined geometries is the relationship between the flow rate and pressure drop. Using the Lorentz reciprocal theorem, we derive a closed-form expression for the flow rate-pressure drop relation of complex fluids in narrow channels of arbitrary shape, which holds for a wide class of viscoelastic and shear-thinning constitutive models. For the weakly non-Newtonian limit, our theory provides the first-order non-Newtonian correction for the flow rate-pressure drop relation solely using the corresponding Newtonian solution, eliminating the need to solve the non-Newtonian flow problem.

Flow rate–pressure drop relation for shear-thinning fluids in narrow channels: approximate solutions and comparison with experiments

Abstract

Revisiting steady viscous flow of a generalized Newtonian fluid through a slender elastic tube using shell theory

AbstractA flow vessel with an elastic wall can deform significantly due to viscous fluid flow within it, even at vanishing Reynolds number (no fluid inertia). Deformation leads to an enhancement of throughput due to the change in cross‐sectional area. The latter gives rise to a non‐constant pressure gradient in the flow‐wise direction and, hence, to a nonlinear flow rate–pressure drop relation (unlike the Hagen–Poiseuille law for a rigid tube). Many biofluids are non‐Newtonian, and are well approximated by generalized Newtonian (say, power‐law) rheological models. Consequently, we analyze the problem of steady low Reynolds number flow of a generalized Newtonian fluid through a slender elastic tube by coupling fluid lubrication theory to a structural problem posed in terms of Donnell shell theory. A perturbative approach (in the slenderness parameter) yields analytical solutions for both the flow and the deformation. Using matched asymptotics, we obtain a uniformly valid solution for the tube's radial displacement, which features both a boundary layer and a corner layer caused by localized bending near the clamped ends. In doing so, we obtain a “generalized Hagen–Poiseuille law” for soft microtubes. We benchmark the mathematical predictions against three‐dimensional two‐way coupled direct numerical simulations (DNS) of flow and deformation performed using the commercial computational engineering platform by ANSYS. The simulations show good agreement and establish the range of validity of the theory. Finally, we discuss the implications of the theory on the problem of the flow‐induced deformation of a blood vessel, which is featured in some textbooks.

Theory of the flow-induced deformation of shallow compliant microchannels with thick walls.

Long, shallow microchannels embedded in thick, soft materials are widely used in microfluidic devices for lab-on-a-chip applications. However, the bulging effect caused by fluid-structure interactions between the internal viscous flow and the soft walls has not been completely understood. Previous models either contain a fitting parameter or are specialized to channels with plate-like walls. This work is a theoretical study of the steady-state response of a compliant microchannel with a thick wall. Using lubrication theory for low-Reynolds-number flows and the theory for linearly elastic isotropic solids, we obtain perturbative solutions for the flow and deformation. Specifically, only the channel's top wall deformation is considered, and the ratio between its thickness t and width w is assumed to be (t/w)2≫1. We show that the deformation at each stream-wise cross section can be considered independently, and that the top wall can be regarded as a simply supported rectangle subject to uniform pressure at its bottom. The stress and displacement fields are found using Fourier series, based on which the channel shape and the hydrodynamic resistance are calculated, yielding a new flow rate-pressure drop relation without fitting parameters. Our results agree favourably with, and thus rationalize, previous experiments.

Non-Newtonian fluid–structure interactions: Static response of a microchannel due to internal flow of a power-law fluid

We study fluid–structure interactions (FSIs) in a long and shallow microchannel, conveying a non-Newtonian fluid, at steady state. The microchannel has a linearly elastic and compliant top wall, while its three other walls are rigid. The fluid flowing inside the microchannel has a shear-dependent viscosity described by the power-law rheological model. We employ lubrication theory to solve for the flow problem inside the long and shallow microchannel. For the structural problem, we employ two plate theories, namely Kirchhoff–Love theory of thin plates and Reissner–Mindlin first-order shear deformation theory. The hydrodynamic pressure couples the flow and deformation problem by acting as a distributed load onto the soft top wall. Within our perturbative (lubrication theory) approach, we determine the relationship between the flow rate and the pressure gradient, which is a nonlinear first-order ordinary differential equation for the pressure. From the solution of this differential equation, all other quantities of interest in non-Newtonian microchannel FSIs follow. Through illustrative examples, we show the effect of FSI coupling strength and the plate thickness on the pressure drop across the microchannel. Through direct numerical simulation of non-Newtonian microchannel FSIs using commercial computational engineering tools, we benchmark the prediction from our mathematical theory for the flow rate–pressure drop relation and the structural deformation profile of the top wall. In doing so, we also establish the limits of applicability of our perturbative theory.

Flow rate–pressure drop relation for deformable shallow microfluidic channels

Laminar flow in devices fabricated from soft materials causes deformation of the passage geometry, which affects the flow rate–pressure drop relation. For a given pressure drop, in channels with narrow rectangular cross-section, the flow rate varies as the cube of the channel height, so deformation can produce significant quantitative effects, including nonlinear dependence on the pressure drop (Gervaiset al.,Lab on a Chip, vol. 6, 2006, pp. 500–507). Gervaiset al. proposed a successful model of the deformation-induced change in the flow rate by heuristically coupling a Hookean elastic response with the lubrication approximation for Stokes flow. However, their model contains a fitting parameter that must be found for each channel shape by performing an experiment. We present a perturbation approach for the flow rate–pressure drop relation in a shallow deformable microchannel using the theory of isotropic quasi-static plate bending and the Stokes equations under a lubrication approximation (specifically, the ratio of the channel’s height to its width and of the channel’s height to its length are both assumed small). Our result contains no free parameters and confirms Gervaiset al.’s observation that the flow rate is a quartic polynomial of the pressure drop. The derived flow rate–pressure drop relation compares favourably with experimental measurements.

Static response of deformable microchannels: a comparative modelling study

We present a comparative modelling study of fluid–structure interactions in microchannels. Through a mathematical analysis based on plate theory and the lubrication approximation for low-Reynolds-number flow, we derive models for the flow rate-pressure drop relation for long shallow microchannels with both thin and thick deformable top walls. These relations are tested against full three-dimensional two-way-coupled fluid–structure interaction simulations. Three types of microchannels, representing different elasticity regimes and having been experimentally characterized previously, are chosen as benchmarks for our theory and simulations. Good agreement is found in most cases for the predicted, simulated and measured flow rate-pressure drop relationships. The numerical simulations performed allow us to also carefully examine the deformation profile of the top wall of the microchannel in any cross section, showing good agreement with the theory. Specifically, the prediction that span-wise displacement in a long shallow microchannel decouples from the flow-wise deformation is confirmed, and the predicted scaling of the maximum displacement with the hydrodynamic pressure and the various material and geometric parameters is validated.

Perturbation-based simplified models for unsteady incompressible microchannel flows

A fast simplified model for a pressure-driven, unsteady and incompressible flow in two dimensional microchannels with a variable cross section is presented. The model relies on a perturbation analysis of the equations of mass and momentum under the assumptions that the channels have a small aspect ratio of height to length and that they have a slowly varying cross-section. A key advantage of this model is that it provides (quasi-) analytic expressions for the velocity, pressure distribution and mass flow rate in terms of the pressure drop along the channel, which makes the evaluation of such expressions very fast, compared to simulating the full Navier–Stokes equations. This is especially important in applications where the flow rate-pressure drop relation can be cast into a circuit, to be potentially integrated in a more complex system. The model turns out to be fairly accurate for a wide range of non-negligible Reynolds and Strouhal numbers and a fairly large class of channels. Furthermore, the model offers a speed factor of at least 500 over classical CFD simulations.

Hemodynamic consequences of stenosis remodeling during coronary angioplasty.

Quantitative hemodynamic assessment during various endovascular interventions including balloon angioplasty is lacking. Translesional pressure drops measured by angioplasty catheters can cause flow blockage and thus lead to inaccurate estimates of preintervention and postintervention flow rates. A new analytical model of the flow rate-pressure drop relation across vascular stenoses is utilized that is nonlinear yet relatively simple in principle, easily applicable in vivo, and compatible with the presence of catheters. The model incorporates in vitro experimental evidence, angiographic data on the dimensions and shapes of coronary arterial stenoses before and after balloon angioplasty, reported translesional pressure gradients, and measurements of coronary flow reserve. Reasonable estimates of mean coronary artery flow rates and translesional pressure drops in the absence of angioplasty catheters are obtained. Prior to angioplasty significant flow restriction across a 68% diameter stenosis exists during hyperemic flow conditions. Following successful balloon dilation, increased minimal cross-sectional area (residual 40% diameter stenosis) results in an improved flow rate-pressure drop relation. Despite minimal flow restriction during hyperemic conditions following angioplasty remodeling, residual luminal constriction leads to elevated wall shear stress levels within the entry region of the stenosis. The flow analysis described may be of clinical utility in evaluating the hemodynamic significance of the anatomic severity of stenoses in coronary and peripheral arteries before and after endovascular therapeutic interventions.

Flow rate-pressure drop relation in coronary angioplasty: catheter obstruction effect.

Quantitative methods to measure the hemodynamic consequences of various endovascular interventions including balloon angioplasty are limited. Catheters measuring translesional pressure drops during balloon angioplasty procedures can cause flow blockage and thus inaccurate estimates of pre- and post-intervention flow rates. The purpose of this investigation was to examine the influence of the presence and size of an angioplasty catheter on measured mean pressure gradients across human coronary artery stenoses. Analytical flow modeling and in vitro experimental evidence, coupled with angiographic data on the dimensions and shape of stenotic vessel segments before and after angioplasty, indicated significant flow blockage effects with the catheter present.

Extensional flow of semidilute suspensions of rod-like particles through an orifice

This paper deals with the extensional flow of semidilute suspensions of non-Brownian rod-like particles through an orifice. The experimental setup allows simultaneously visualizations of the flow pattern and measurements of the flow rate-pressure drop relation. The flow pattern in the converging region is characterized by the appearance of a vortex ring which grows with the volume fraction of the particles and the aspect ratio. At the same time, the pressure drop experienced by the suspension as it flows through the orifice increases. The experimental results are interpreted in terms of a competition between the shear stress due to the solvent and the extensional stress generated by the particles. This interpretation is supported by the scaling of the vortex size and of the pressure drop with the volume fraction and the aspect ratio of the particles.

Improved pressure drop/flow rate equation for non-Newtonian fluids in laminar flow

Most fluids used in the oil industry show some degree of non-Newtonian behavior. To obtain reasonable pressure drop calculations, therefore, one must take this into account. Examples of non-Newtonian fluids are drilling muds, cements, waxy crude oils, polymer-water solutions and so on. This paper uses a three-parameter rheological model, the Collins-Graves model, to derive a flow rate-pressure drop relation for non-Newtonian fluids. The objective is to provide a relatively accurate but simple tool to estimate pressure drop for tubing of any size. The input data are rheological constants obtained from a viscosimeter.