Controlling viscoelastic flow by tuning frequency during occlusions R. Collepardo-Guevara 1,2 and E. Corvera Poiré 1,3, * 1 Departamento de Física y Química Teórica, Facultad de Química, UNAM Ciudad Universitaria, México D.F. 04510, Mexico 2 Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom 3 Departament ECM, Facultat de Física, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain Received 2 April 2007; revised manuscript received 28 June 2007; published 2 August 2007 We study the flow of a viscoelastic fluid flowing in an occluded tube due to either central or peripheral obstructions. We show that, by driving the fluid with a dynamic pressure gradient at the frequency that maximizes the dynamic permeability of the obstructed system, the magnitude of the flow can partially be recovered without the removal of the obstruction. We compare the results obtained for the two types of occlusions studied and find that flow recovering is larger in the case of central occlusions. DOI: 10.1103/PhysRevE.76.026301 PACS numbers: 47.50.-d, 47.85.L-, 47.63.-b I. INTRODUCTION Occlusions of tubes have always represented a problem. From engines and filters to arteries and bronchia we can find a countless amount of systems in which the lack of move- ment of a fluid due to the presence of an obstacle results in the partial or total failure of a process. In particular, the occlusion of biotubes in the human body represents an im- portant issue in many diseases. For instance, during the oc- clusion of arteries, blood decreases its velocity and, in criti- cal cases, is effectively unable to flow through the remaining space. Such a lack of movement prevents irrigation and, in many cases, results in the eventual death of tissues. Recent experimental and theoretical work on viscoelastic fluids 1–6 have found that the dynamic permeability can increase orders of magnitude at certain frequencies. The dy- namic permeability is an intrinsic property of the system viscoelastic fluid-confining media and determines the system response to different signals of the pressure gradient. It can be considered as a measure of the resistance to flow, the larger the dynamic permeability, the less the resistance to flow. The increase of the dynamic permeability at certain frequencies suggests that the magnitude of the flow might be increased by driving the fluid with a pressure gradient that contains the frequency that maximizes the dynamic perme- ability. We have indeed verified that by imposing a periodic pressure gradient at the frequency that maximizes the dy- namic permeability, the magnitude of the flow of a viscoelas- tic fluid flowing in a tube can largely be increased. This implies that the dynamics for the pressure gradient with a properly chosen frequency provides a way of controlling the magnitude of the flow. We present analytical results for the simple case of a pressure gradient consisting of a single sinu- soidal mode in order to show that the maximum value of the flow magnitude depends on two things: the real part of the dynamic permeability and the cross-sectional area available for flow. When an obstruction occurs, it is clear that if one recovers the value of the real part of the dynamic permeabil- ity by driving the fluid at the proper frequency, one elimi- nates one of the two factors that provoke the dramatic de- crease of flow. Having established that, we model two types of occlusions and show that, by driving the fluid with a pe- riodic pressure gradient at the frequency that maximizes the permeability of the obstructed system, the flow can partially be recovered without the removal of the obstruction. We compare the results obtained for the two types of occlusions studied, namely central occlusions and peripheral occlusions and find that flow recovering is larger in the case of central occlusions. We also compare the results for two different dynamics of the pressure gradient and find that even though the dynamics does make a difference in the magnitude of the flow, it does not make a big difference when it comes to the percentage of flow that can be recovered. II. MODEL We model the viscoelastic fluid, by means of the linear- ized Maxwell model, which is the simplest hydrodynamic model to describe viscoelastic behavior, i.e., t r   2 v  t 2 +   v  t =- t r  p  t - p +  2 v . 1 Here t is time, v is velocity, p is pressure, and t r , , and  are the Maxwell relaxation time, density, and viscosity of the viscoelastic fluid. Equation 1 is nothing but the linearized momentum balance equation of hydrodynamics together with the constitutive relation of the Maxwell model. This one is built in such a way that in the limit t r → 0 reduces to the constitutive equation of a Newtonian fluid, and in the limit t r →  reduces to the constitutive equation of an elastic solid. The Maxwell relaxation time is defined as t r   / G where G is the elastic modulus of the fluid. Equation 1 should be solved for a particular geometry which in turn depends on the type of occlusion considered. We consider occlusions that result from the partial obstruction of flow in two different ways. The first type of obstruction is one in which the walls of a tube have been internally engrossed and the fluid circu- lates through a tube that has effectively a smaller radius. We call this peripheral occlusion and we model the space of flow as the one inside a cylinder with a radius smaller than the one of the unobstructed tube. The second type of obstruc- tion considered is one in which the fluid must flow between *Corresponding author. eugenia.corvera@gmail.com PHYSICAL REVIEW E 76, 026301 2007 1539-3755/2007/762/0263017 ©2007 The American Physical Society 026301-1