We consider the Langevin dynamics of a Gaussian test polymer chain coupled with a surrounding matrix which can undergo the glass transition. The Martin-Siggia-Rose generating functional method and the nonpertubative Hartree approximation are used to derive the generalized Rouse equation for the test chain. It is shown that the interaction of the test chain with the surrounding matrix renormalizes the bare friction and the spring constants of the test chain in such a way that the memory function as well as the bending dependent elastic modulus appear. We find that below the glass transition temperature T G of the matrix the Rouse modes of the test chain can be frozen and moreover the freezing temperatures (or the ergodicity-nonergodicity transition temperature) T c (p) depends from the Rouse mode index p.
我们考虑了高斯测试聚合物链与周围可以发生玻璃化转变的基体耦合的朗之万动力学。利用Martin-Siggia-Rose生成泛函方法和非微扰Hartree近似导出了测试链的广义Rouse方程。结果表明，测试链与周围矩阵的相互作用使测试链的裸摩擦和弹簧常数重新正规化，从而出现记忆函数和弯曲相关弹性模量。我们发现在基体的玻璃化转变温度T G以下，测试链的劳斯模态可以冻结，并且冻结温度(或遍历-非遍历转变温度)T c (p)取决于劳斯模态指数p。
{"title":"Dynamics of a polymer test chain in a glass forming matrix: The Hartree approximation","authors":"M. Rehkopf, V. Rostiashvili, T. Vilgis","doi":"10.1051/jp2:1997196","DOIUrl":"https://doi.org/10.1051/jp2:1997196","url":null,"abstract":"We consider the Langevin dynamics of a Gaussian test polymer chain coupled with a surrounding matrix which can undergo the glass transition. The Martin-Siggia-Rose generating functional method and the nonpertubative Hartree approximation are used to derive the generalized Rouse equation for the test chain. It is shown that the interaction of the test chain with the surrounding matrix renormalizes the bare friction and the spring constants of the test chain in such a way that the memory function as well as the bending dependent elastic modulus appear. We find that below the glass transition temperature T G of the matrix the Rouse modes of the test chain can be frozen and moreover the freezing temperatures (or the ergodicity-nonergodicity transition temperature) T c (p) depends from the Rouse mode index p.","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"42 1","pages":"1469-1487"},"PeriodicalIF":0.0,"publicationDate":"1997-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86574452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new macroscopic model that describes the break-up dynamics of liquid crystalline anisotropic viscoelastic fibers. The fiber elasticity contains isotropic as well as orientation dependent surface contributions, and the anisotropic bulk viscous dissipation is described by three viscosity coefficients. For liquid crystalline fibers with molecular orientation along the fiber's axis the model predicts that capillary instabilities will break the fiber into an array of droplets, just as in the case of isotropic Newtonian fibers. The characteristic growth rate and wavelength of the instability are functions of the orientation dependent surface tension and the extensional viscosity. The liquid crystal surface elasticity tends to increase the wavelength and to decrease the growth rate of the fastest growing mode when compared to that of Newtonian fibers. Higher extensional viscosities decrease the time scale and increase the size scale of the new droplet morphology that emerges from unstable axisymmetric liquid crystalline fibers.
{"title":"Linear Stability Theory of Break-Up Dynamics of Nematic Liquid Crystalline Fibers","authors":"A. Rey","doi":"10.1051/JP2:1997166","DOIUrl":"https://doi.org/10.1051/JP2:1997166","url":null,"abstract":"We present a new macroscopic model that describes the break-up dynamics of liquid crystalline anisotropic viscoelastic fibers. The fiber elasticity contains isotropic as well as orientation dependent surface contributions, and the anisotropic bulk viscous dissipation is described by three viscosity coefficients. For liquid crystalline fibers with molecular orientation along the fiber's axis the model predicts that capillary instabilities will break the fiber into an array of droplets, just as in the case of isotropic Newtonian fibers. The characteristic growth rate and wavelength of the instability are functions of the orientation dependent surface tension and the extensional viscosity. The liquid crystal surface elasticity tends to increase the wavelength and to decrease the growth rate of the fastest growing mode when compared to that of Newtonian fibers. Higher extensional viscosities decrease the time scale and increase the size scale of the new droplet morphology that emerges from unstable axisymmetric liquid crystalline fibers.","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"17 1","pages":"1001-1011"},"PeriodicalIF":0.0,"publicationDate":"1997-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89121603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the displacement of miscible fluids of different viscosities (viscosity ratio AI) between two parallel plates, using the BGK lattice gas method. At high Peclet numbers, a symmetric interface develops in the gap between the plates. Above M +J 10, the interface becomes a well-defined finger, the reduced width of which tends to A~n = 0.56 at large AI. Assuming that a miscible displacement at high Peclet numbers is equivalent to an immiscible displacement at high capillary numbers, we extend the calculations of Reinelt & Saffman for immiscible fluids, and find the analytical shape of the finger. The result is compared to the celebrated Saffman-Taylor finger.
{"title":"Fingering in 2D Parallel Viscous Flow","authors":"N. Rakotomalala, D. Salin, P. Watzky","doi":"10.1051/JP2:1997105","DOIUrl":"https://doi.org/10.1051/JP2:1997105","url":null,"abstract":"We study the displacement of miscible fluids of different viscosities (viscosity ratio AI) between two parallel plates, using the BGK lattice gas method. At high Peclet numbers, a symmetric interface develops in the gap between the plates. Above M +J 10, the interface becomes a well-defined finger, the reduced width of which tends to A~n = 0.56 at large AI. Assuming that a miscible displacement at high Peclet numbers is equivalent to an immiscible displacement at high capillary numbers, we extend the calculations of Reinelt & Saffman for immiscible fluids, and find the analytical shape of the finger. The result is compared to the celebrated Saffman-Taylor finger.","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"33 1","pages":"967-972"},"PeriodicalIF":0.0,"publicationDate":"1997-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84467946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Levelut, E. Hallouin, Dirk Bennemann, G. Heppke, D. Loetzsch
Nous avons examine la structure de la phase smectique Q dans neuf composes chiraux differents. Nous distinguons en fait quatre phases differentes, qui toutes peuvent etre considerees comme des cristaux de defauts. Les reseaux de defauts sont de symetrie quadratique pour trois de ces phases, et hexagonale pour la quatrieme ; les parametres de maille varient entre 65 et 130 A. L'une de ces phases serait un analogue - a l'echelle nanometrique - des phases bleues, car l'organisation peut se decrire comme un empilement d'agregats moleculaires a double torsion. Les autres phases sont formees par un empilement cristallin de joints de torsion entre grains smectiques, les angles de rotation etant de π/2 ou π/3.
{"title":"The smectic Q phase, a crystal of twist grain boundaries with smectic order","authors":"A. Levelut, E. Hallouin, Dirk Bennemann, G. Heppke, D. Loetzsch","doi":"10.1051/JP2:1997165","DOIUrl":"https://doi.org/10.1051/JP2:1997165","url":null,"abstract":"Nous avons examine la structure de la phase smectique Q dans neuf composes chiraux differents. Nous distinguons en fait quatre phases differentes, qui toutes peuvent etre considerees comme des cristaux de defauts. Les reseaux de defauts sont de symetrie quadratique pour trois de ces phases, et hexagonale pour la quatrieme ; les parametres de maille varient entre 65 et 130 A. L'une de ces phases serait un analogue - a l'echelle nanometrique - des phases bleues, car l'organisation peut se decrire comme un empilement d'agregats moleculaires a double torsion. Les autres phases sont formees par un empilement cristallin de joints de torsion entre grains smectiques, les angles de rotation etant de π/2 ou π/3.","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"3 1","pages":"981-1000"},"PeriodicalIF":0.0,"publicationDate":"1997-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77206544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
General analytical mathematical equations are given for n-fold saddle, tower saddle and helical tower saddle surfaces. The two-fold helical saddle tower is combined with the mathematics of the Exponential Scale to create a mathematical equation for the DNA-double helix with its bridging hydrogen bonds.
{"title":"Saddle, tower and helicoidal surfaces","authors":"M. Jacob","doi":"10.1051/JP2:1997169","DOIUrl":"https://doi.org/10.1051/JP2:1997169","url":null,"abstract":"General analytical mathematical equations are given for n-fold saddle, tower saddle and helical tower saddle surfaces. The two-fold helical saddle tower is combined with the mathematics of the Exponential Scale to create a mathematical equation for the DNA-double helix with its bridging hydrogen bonds.","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"51 1","pages":"1035-1044"},"PeriodicalIF":0.0,"publicationDate":"1997-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76292972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The method of multiple scales is used to analyse the nonlinear propagation of waves on the interface between a liquid and a subsonic gas in the presence of magnetic field taking into account surface tension. The evolution of the amplitude is governed by a nonlinear Schr6dinger equation which gives the criterion for modulational instability. Numerical results are given in the graphical form. considerable attention during the last decade. We consider in this paper, the weakly nonlinear physical system, the interaction of capillary gravity waves with a subsonic flow moving uni- formly parallel to the undisturbed liquid surface in the presence of magnetic field. The same problem without magnetic field when the liquid is of finite depth has been investigated earlier by Nayfeh and Hassan (1). The inclusion of nonlinear terms results in amplitude modulation. In various problems of interest, it has been shown that the long-time slow modulation of wave amplitude is governed by the nonlinear Schr0dinger equation. In recent years, evolution of wave packets on the surface of an electrically conducting fluid has been investigated by a number of workers. El Shehawey (2) discussed the nonlinear condi- tions of stability and instability of electro-hydrodynamic Kelvin-Helmholtz mechanisms in the presence of a normal field in the absence of surface charges on the interface. Pusri and Malik (3) investigated the propagation of wave packets on the surface of an electrically conducting fluid of uniform depth in the presence of a tangential magnetic field in three dimensions by extending the analyses of Djordjevic and Redekopp (4) and Ablowitz and Segur (5) to incorporate magne- tohydrodynamic effects. The method of multiple scales was very successfully used by Hasimoto and Ono (6) to derive a single equation describing the long-time evolution of the envelope of a packet of plane finite amplitude gravity waves. In this presentation, by the multiple scale method, we plan to develop the nonlinear Schrodinger equation describing the evolution of the finite amplitude wave packet on the liquid surface in the presence of normal magnetic field with
{"title":"Nonlinear Interaction of Capillary-Gravity Waves with a Subsonic Gas in the Presence of Magnetic Field","authors":"Doo-sung Lee","doi":"10.1051/JP2:1997170","DOIUrl":"https://doi.org/10.1051/JP2:1997170","url":null,"abstract":"The method of multiple scales is used to analyse the nonlinear propagation of waves on the interface between a liquid and a subsonic gas in the presence of magnetic field taking into account surface tension. The evolution of the amplitude is governed by a nonlinear Schr6dinger equation which gives the criterion for modulational instability. Numerical results are given in the graphical form. considerable attention during the last decade. We consider in this paper, the weakly nonlinear physical system, the interaction of capillary gravity waves with a subsonic flow moving uni- formly parallel to the undisturbed liquid surface in the presence of magnetic field. The same problem without magnetic field when the liquid is of finite depth has been investigated earlier by Nayfeh and Hassan (1). The inclusion of nonlinear terms results in amplitude modulation. In various problems of interest, it has been shown that the long-time slow modulation of wave amplitude is governed by the nonlinear Schr0dinger equation. In recent years, evolution of wave packets on the surface of an electrically conducting fluid has been investigated by a number of workers. El Shehawey (2) discussed the nonlinear condi- tions of stability and instability of electro-hydrodynamic Kelvin-Helmholtz mechanisms in the presence of a normal field in the absence of surface charges on the interface. Pusri and Malik (3) investigated the propagation of wave packets on the surface of an electrically conducting fluid of uniform depth in the presence of a tangential magnetic field in three dimensions by extending the analyses of Djordjevic and Redekopp (4) and Ablowitz and Segur (5) to incorporate magne- tohydrodynamic effects. The method of multiple scales was very successfully used by Hasimoto and Ono (6) to derive a single equation describing the long-time evolution of the envelope of a packet of plane finite amplitude gravity waves. In this presentation, by the multiple scale method, we plan to develop the nonlinear Schrodinger equation describing the evolution of the finite amplitude wave packet on the liquid surface in the presence of normal magnetic field with","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"284 1","pages":"1045-1055"},"PeriodicalIF":0.0,"publicationDate":"1997-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82830248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Schiller, K. Camara, G. Pelzl, S. Diele, F. Zeitler
Some smectic liquid crystal phases exhibit a long range bond-orientational order. In hexatic smectic phases with molecules tilted towards the layers director rotations are combined with distortions of the bond net. The coupling of the director to the bond field influences strongly the threshold behaviour of the Freedericksz transition of planar oriented smectic I films sandwiched between parallel plates. In contrast to nematic and smectic C films, the threshold voltage depends on the film thickness. Investigating this thickness dependence several material parameters can be obtained by adjusting the experimental data to the threshold formula.
{"title":"Reorientation of the director and the bonds in a hexatic smectic phase","authors":"P. Schiller, K. Camara, G. Pelzl, S. Diele, F. Zeitler","doi":"10.1051/JP2:1997167","DOIUrl":"https://doi.org/10.1051/JP2:1997167","url":null,"abstract":"Some smectic liquid crystal phases exhibit a long range bond-orientational order. In hexatic smectic phases with molecules tilted towards the layers director rotations are combined with distortions of the bond net. The coupling of the director to the bond field influences strongly the threshold behaviour of the Freedericksz transition of planar oriented smectic I films sandwiched between parallel plates. In contrast to nematic and smectic C films, the threshold voltage depends on the film thickness. Investigating this thickness dependence several material parameters can be obtained by adjusting the experimental data to the threshold formula.","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"90 1","pages":"1013-1022"},"PeriodicalIF":0.0,"publicationDate":"1997-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82382401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study theoretically the change of the curvature moduli of a surfactant membrane due to the adsorption of a polymer solution. Using a mean field theory of polymer adsorption, we study both cases of reversible and irreversible polymer adsorption in good and θ solvents. The curvature moduli of the adsorbed polymer layers are dominated by the short loops that the polymer forms on the membrane. The polymer contribution to the membrane bending modulus is always negative and the polymer contribution to the Gaussian curvature modulus is always positive.
{"title":"Curvature Elasticity of an Adsorbed Polymer Layer","authors":"F. Clément, J. Joanny","doi":"10.1051/JP2:1997108","DOIUrl":"https://doi.org/10.1051/JP2:1997108","url":null,"abstract":"We study theoretically the change of the curvature moduli of a surfactant membrane due to the adsorption of a polymer solution. Using a mean field theory of polymer adsorption, we study both cases of reversible and irreversible polymer adsorption in good and θ solvents. The curvature moduli of the adsorbed polymer layers are dominated by the short loops that the polymer forms on the membrane. The polymer contribution to the membrane bending modulus is always negative and the polymer contribution to the Gaussian curvature modulus is always positive.","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"4 1","pages":"973-980"},"PeriodicalIF":0.0,"publicationDate":"1997-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78898259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nous avons calcule la relation de dispersion des ondes de surface dans une couche de ferrofluide d'epaisseur et de viscosite quelconques, soumise a un champ magnetique normal a sa surface (instabilite de pics de Rosensweig). Cette relation montre que le champ magnetique critique et le vecteur d'onde critique de l'instabilite dependent de l'epaisseur de la couche de fluide. La relation de dispersion a ete simplifiee pour quatre regimes asymptotiques: couche epaisse ou mince et comportement visqueux ou inertiel. Nous avons calcule les valeurs critiques de l'instabilite dans ces quatre cas. Nous montrons qu'un parametre typique du ferrofluide permet de savoir dans quel regime, visqueux ou inertiel, se situe le ferrofluide pres du seuil de l'instabilite.
{"title":"Dispersion relation in a ferrofluid layer of any thickness and viscosity in a normal magnetic field ; asymptotic regimes","authors":"B. Abou, G. N. D. Surgy, J. Wesfreid","doi":"10.1051/jp2:1997178","DOIUrl":"https://doi.org/10.1051/jp2:1997178","url":null,"abstract":"Nous avons calcule la relation de dispersion des ondes de surface dans une couche de ferrofluide d'epaisseur et de viscosite quelconques, soumise a un champ magnetique normal a sa surface (instabilite de pics de Rosensweig). Cette relation montre que le champ magnetique critique et le vecteur d'onde critique de l'instabilite dependent de l'epaisseur de la couche de fluide. La relation de dispersion a ete simplifiee pour quatre regimes asymptotiques: couche epaisse ou mince et comportement visqueux ou inertiel. Nous avons calcule les valeurs critiques de l'instabilite dans ces quatre cas. Nous montrons qu'un parametre typique du ferrofluide permet de savoir dans quel regime, visqueux ou inertiel, se situe le ferrofluide pres du seuil de l'instabilite.","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"27 1","pages":"1159-1171"},"PeriodicalIF":0.0,"publicationDate":"1997-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74991304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stability of the flow of a fluid adjacent to a polymer gel is studied using a linear stability analysis. The system consists of a Newtonian fluid of density rho, viscosity eta and thickness R flowing adjacent to a polymer gel of density rho, modulus of elasticity E, viscosity eta(g) and thickness HR. The base flow in the fluid is a plane Couette flow. The Navier-Stokes equations for the fluid and the elasticity equations for the gel are solved numerically, and the characteristic equation is obtained using the boundary conditions at the interface. The characteristic equation for the growth rate is a non-linear equation, so analytical solutions cannot be obtained in general. Numerical solutions are obtained by analytic continuation using the exact solutions at zero Reynolds number as the starting guess. The growth rate depends on the parameter Sigma = (rho ER2/eta(2)), the ratio of thickness H, the ratio of viscosities eta(r) = (eta(g)/eta) and wave number k. For eta(r) = 0, it is found that the perturbations become unstable when the Reynolds number is increased beyond a transition value Re-t for all Sigma and k. The critical Reynolds number Re-c, which is the minimum of the Re-t - K curve, increases proportional to Sigma for Sigma much less than 1, and it shows a scaling behavior Re-c proportional to Sigma(beta) for Sigma much greater than 1, where 0.75 less than or equal to beta less than or equal to 0.8. Re-c decreases with increase in the ratio of thickness of gel to fluid H, but the scaling behavior remains unchanged. A variation in the ratio of viscosities eta(r) qualitatively changes the stability characteristics. For relatively low values of 1 less than or equal to Sigma less than or equal to 10(3), it is found that the transition Reynolds number decreases as eta(r) is increased, indicating that an increase in the gel viscosity has a destabilizing effect. For relatively higher values of 10(4) less than or equal to Sigma less than or equal to 10(5), the transition Reynolds number increases as eta(r) is increased and goes through a turning point. In this case, perturbations are unstable only when eta(r) is less than a maximum value eta(max), and there is no instability for eta(r) > eta(max).
{"title":"Flow Induced Instability of the Interface between a Fluid and a Gel","authors":"L. Srivatsan, V. Kumaran","doi":"10.1051/JP2:1997164","DOIUrl":"https://doi.org/10.1051/JP2:1997164","url":null,"abstract":"The stability of the flow of a fluid adjacent to a polymer gel is studied using a linear stability analysis. The system consists of a Newtonian fluid of density rho, viscosity eta and thickness R flowing adjacent to a polymer gel of density rho, modulus of elasticity E, viscosity eta(g) and thickness HR. The base flow in the fluid is a plane Couette flow. The Navier-Stokes equations for the fluid and the elasticity equations for the gel are solved numerically, and the characteristic equation is obtained using the boundary conditions at the interface. The characteristic equation for the growth rate is a non-linear equation, so analytical solutions cannot be obtained in general. Numerical solutions are obtained by analytic continuation using the exact solutions at zero Reynolds number as the starting guess. The growth rate depends on the parameter Sigma = (rho ER2/eta(2)), the ratio of thickness H, the ratio of viscosities eta(r) = (eta(g)/eta) and wave number k. For eta(r) = 0, it is found that the perturbations become unstable when the Reynolds number is increased beyond a transition value Re-t for all Sigma and k. The critical Reynolds number Re-c, which is the minimum of the Re-t - K curve, increases proportional to Sigma for Sigma much less than 1, and it shows a scaling behavior Re-c proportional to Sigma(beta) for Sigma much greater than 1, where 0.75 less than or equal to beta less than or equal to 0.8. Re-c decreases with increase in the ratio of thickness of gel to fluid H, but the scaling behavior remains unchanged. A variation in the ratio of viscosities eta(r) qualitatively changes the stability characteristics. For relatively low values of 1 less than or equal to Sigma less than or equal to 10(3), it is found that the transition Reynolds number decreases as eta(r) is increased, indicating that an increase in the gel viscosity has a destabilizing effect. For relatively higher values of 10(4) less than or equal to Sigma less than or equal to 10(5), the transition Reynolds number increases as eta(r) is increased and goes through a turning point. In this case, perturbations are unstable only when eta(r) is less than a maximum value eta(max), and there is no instability for eta(r) > eta(max).","PeriodicalId":14774,"journal":{"name":"Journal De Physique Ii","volume":"72 1","pages":"947-963"},"PeriodicalIF":0.0,"publicationDate":"1997-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84498752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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