Local Deborah Number Research Articles

A Computational Framework for Time‐Dependent Deformation in Viscoelastic Magmatic Systems

AbstractTime‐dependent ground deformation is a key observable in active magmatic systems, but is challenging to characterize. Here we present a numerical framework for modeling transient deformation and stress around a subsurface, spheroidal pressurized magma reservoir within a viscoelastic half‐space with variable material coefficients, utilizing a high‐order finite‐element method and explicit time‐stepping. We derive numerically stable time steps and verify convergence, then explore the frequency dependence of surface displacement associated with cyclic pressure applied to a spherical reservoir beneath a stress‐free surface. We consider a Maxwell rheology and a steady geothermal gradient, which gives rise to spatially variable viscoelastic material properties. The temporal response of the system is quantified with a transfer function that connects peak surface deformation to reservoir pressurization in the frequency domain. The amplitude and phase of this transfer function characterize the viscoelastic response of the system, and imply a framework for characterizing general deformation time series through superposition. Transfer function components vary with the frequency of pressure forcing and are modulated strongly by the background temperature field. The dominantly viscous region around the reservoir is also frequency dependent, through a local Deborah number that measures pressurization period against a spatially varying Maxwell relaxation time. This near‐reservoir region defines a spatially complex viscous/elastic transition whose volume depends on the frequency of forcing. Our computational and transfer function analysis framework represents a general approach for studying transient viscoelastic crustal responses to magmatic forcing through spectral decomposition of deformation time series, such as long‐duration geodetic observations.

Entropy generation analysis of Falkner–Skan flow of Maxwell nanofluid in porous medium with temperature-dependent viscosity

Entropy generation analysis in steady two-dimensional, viscous, incompressible forced convective Falkner–Skan flow of Maxwell nanofluid over a static wedge embedded in a porous medium with temperature-dependent viscosity is examined. The Buongiorno’s model has been utilised, to get the flow governing higher-order coupled nonlinear partial differential equations (PDEs) from mass, momentum, energy and concentration conservations. Suitable transformations have been done to convert governing PDEs into the coupled non-linear ODEs along with no-slip boundary conditions, which are then solved using the MATLAB programme bvp4c. The influences of diverse flow governing parameters on various flow properties and quantities of physical interest are displayed in graphical mode and discussed. It is found that entropy generation reduces only with Eckert number (Ec), while more entropy is generated for pressure gradient parameter (m), local Deborah number ( $$\beta )$$ , variable viscosity parameter ( $$\delta $$ ) and permeability parameter (K). Entropy generation due to heat transfer irreversibility is prominent with increase in m and $$\delta $$ , but it is not so for other parameters. The drag force on the wedge surface become stronger with $$\beta $$ and m, but it reduces with $$\delta $$ . Rates of heat transfer and mass transfer enhance with m and $$\delta $$ . In addition, surface drag force and heat transfer rate diminish with Brownian motion parameter (Nb) and thermophoresis parameter (Nt).

Non-linear dynamics of semi-dilute PAAm solutions in a microfluidic 3D cross-slot flow geometry

We have characterised the flow of semi-dilute PAAm solutions (3.3c* ≤ c ≤ 16.6c*) through a three-dimensional cross-slot device (width to depth aspect ratio 2: 1) over a range of Weissenberg (0.13 ≤ Wi ≤ 88.5) and Elasticity (1.5 ≤ El ≤ 58.4) numbers by means of micro-particle imaging velocimetry (μ-PIV) and pressure drop measurements. The flow regimes are mapped into a Wi−Re diagram. At intermediate Wi (12 ≤ Wi ≤ 45) we observe the formation of lip vortices in the inlet channels. At higher Wi (Wi ≥ 45) we observe the onset of an asymmetric flow regime, which occurs at a much higher Weissenberg number than previously reported in the literature. Increasing El in the asymmetric flow regime results in flow patterns that are markedly more time-dependent and unstable. The extensional rates measured along the centreline of the outlet channels are much lower than the corresponding nominal values. By means of an improved version of the numerical method that was used in our previous works, we have linked the measured velocity field with the “local Deborah number” field to quantify the spatial distribution of the deformation rate underwent by the polymer chains at high Wi flows near the stagnation point of the cross slot device. We measured the extensional viscosity and Trouton ratios of our polymer solutions over a range of 1 ≤ Wi ≤ 30. This work is of significance for the development of extensional rheometers for measuring the properties of low-viscosity fluids such as inks, blood, silk fibroin and protein solutions.

Three dimensional rotating flow of Maxwell nanofluid

Abstract This article investigates rotating flow of Maxwell liquid through submersion of nanoparticles. Flow created is because of an exponentially stretching sheet. Optimal homotopic scheme is implemented for the solution of governing model. The optimal values of auxiliary parameters are computed. The optimal solution expressions of temperature and concentration of nanoparticles are elaborated via plots by employing the various values of involved parameters. The physical quantities like Nusselt and Sherwood numbers are characterized by numerical values. Here we observed that temperature of liquid and its related thickness of boundary layer arise for the increasing values of local Deborah number while it retards for larger temperature exponent parameter.

Numerical analysis of Sakiadis flow problem considering Maxwell nanofluid

This article investigates the flow of Maxwell nanofluid over a moving plate in a calm fluid. Novel aspects of Brownian motion and thermophoresis are taken into consideration. Revised model for passive control of nanoparticle volume fraction at the plate is used in this study. The formulated differential system is solved numerically by employing shooting approach together with fourth-fifth-order-Runge-Kutta integration procedure and Newton?s method. The solutions are greatly influenced with the variation of embedded parameters which include the local Deborah number, the Brownian motion parameter, the thermophoresis parameter, the Prandtl number, and the Schmidt number. We found that the variation in velocity distribution with an increase in local Deborah number is non-monotonic. Moreover, the reduced Nusselt number has a linear and direct relationship with the local Deborah number.

Numerical solution for Sakiadis flow of upper-convected Maxwell fluid using Cattaneo-Christov heat flux model

Present work studies the well-known Sakiadis flow of Maxwell fluid along a moving plate in a calm fluid by considering the Cattaneo-Christov heat flux model. This recently developed model has the tendency to describe the characteristics of relaxation time for heat flux. Some numerical local similarity solutions of the associated problem are computed by two approaches namely (i) the shooting method and (ii) the Keller-box method. The solution is dependent on some interesting parameters which include the viscoelastic fluid parameter β, the dimensionless thermal relaxation time γ and the Prandtl number Pr. Our simulations indicate that variation in the temperature distribution with an increase in local Deborah number γ is non-monotonic. The results for the Fourier’s heat conduction law can be obtained as special cases of the present study.

Simulations for Maxwell fluid flow past a convectively heated exponentially stretching sheet with nanoparticles

This article addresses steady flow of Maxwell nanofluid induced by an exponentially stretching sheet subject to convective heating. The revised model of passively controlled wall nanoparticle volume fraction is taken into account. Numerical solutions of the arising non-linear boundary value problem (BVP) are obtained by using MATLAB built-in function bvp4c. Simulations are performed for various values of embedded parameters which include local Deborah number, Prandtl number, Biot number, Brownian motion parameter and thermophoresis parameter. The results are consistent with the previous studies in some limiting cases. It is found that velocity decreases and temperature increases when the local Deborah number is increased. Moreover the influence of Brownian diffusion on temperature and heat transfer rate is found to be insignificant.

Sakiadis flow of Maxwell fluid considering magnetic field and convective boundary conditions

In this paper we address the flow of Maxwell fluid due to constantly moving flat radiative surface with convective condition. The flow is under the influence of non-uniform transverse magnetic field. The velocity and temperature distributions have been evaluated numerically by shooting approach. The solution depends on various interesting parameters including local Deborah number De, magnetic field parameter M, Prandtl number Pr and Biot number Bi. We found that variation in velocity with an increase in local Deborah number De is non-monotonic. However temperature is a decreasing function of local Deborah number De.

A quantitative analysis of spatial extensional rate distribution in nonlinear viscoelastic flows

A numerical method is proposed to calculate local extensional rate along arbitrary streamlines measured experimentally. It leads to the calculation of a “local Deborah number” (De) field, which characterizes heterogeneous flows of polymer solutions in a more insightful way than the commonly used nominal Weissenberg number (Wi), defined by an average shear rate. The spatial distribution of non-equilibrium polymer conformation across flow domain could be visualized through an image of the De field for flows of aqueous semi-dilute polyacrylamide (PAAm) solutions through the 4:1:4, 8:1:8 and 16:1:16 planar contraction geometries, over a range of Wi(1.4⩽Wi⩽131.7) and Elasticity numbers (El) (13.3⩽El⩽476.8). The time evolution of these De fields and their profiles near the contraction plane are presented. The correlation between the maximum local De and the nominal Wi is analyzed. The results show that the vortex formation near the contraction entry coincides with shifting the position of the maximum local De away from the centerline. Its actual location is sensitive to flow rate and strongly depends on dynamic flow regime.

Flow and Heat Transfer of Jeffrey Fluid Over a Continuously Moving Surface With a Parallel Free Stream

Abstract This communication studies the flow and heat transfer characteristics over a continuously moving surface in the presence of a free stream velocity. The Jeffrey fluid is treated as a rheological model. The series expressions of velocity and temperature fields are constructed by applying the homotopy analysis method (HAM). The influence of emerging parameters such as local Deborah number (β), the ratio of relaxation and retardation times (λ2), the Prandtl number (Pr), and the Eckert number (Ec) on the velocity and temperature profiles are presented in the form of graphical and tabulated results for different values of λ. It is found that the boundary layer thickness is an increasing function of local Deborah number (β). However, the temperature and thermal boundary layer thickness decreases with the increasing values of local Deborah number (β).

Analysing flow and heat transfer of a viscoelastic fluid over a semi-infinite horizontal moving flat plate

This paper presents a numerical study of the flow and heat transfer of an incompressible homogeneous second grade type fluid above a flat plate moving with constant velocity U. Such a viscoelastic fluid is at rest and the motion is created by the sheet. The effects of the non-Newtonian nature of the fluid are governed by the local Deborah number K (the ratio between the relaxation time of the fluid and the characteristic time of the flow). When K = ( 33 - 1 ) / 8 , a new analytical solution for this flow is presented and the effects of fluid's elasticity on flow characteristics, dimensionless stream function and its derivatives are analysed in a wide domain of K. A novel result of the analysis is that a change in the flow solution's behaviour occurs when the dimensionless stream function at the edge of the boundary layer, f ∞ , equals 1.0. It is found that velocity at a point decreases with increase in the elasticity of the fluid and, as expected, the amount of fluid entrained diminishes when the effects of fluid's elasticity are augmented. In our heat transfer analyses we assume that the surface temperature has a power-law variation. Two cases are studied, namely, (i) the sheet with prescribed surface temperature (PST case) and (ii) the sheet with prescribed heat flux (PHF case). Local similarity heat-transfer solutions are given for PST case when s = 2 (the wall temperature parameter) whereas when s = - 1 2 a similarity solution takes place in the case of prescribed wall heat flux. The numerical results obtained are fairly in good agreement with the aforementioned analytical ones.

NATURAL CONVECTION FLOW OF SECOND-GRADE FLUID ALONG A VERTICAL HEATED SURFACE WITH POWER-LAW TEMPERATURE

In this study we investigated the effects of free convection flow of a viscoelastic second-grade fluid along a vertical flat surface with power-law temperature distribution. The boundary layer equations for the momentum and the energy transport have been reduced to local similarity equations using appropriate transformations. Solutions of the reduced equations are obtained employing the local non-similarity method as well as the implicit finite difference method against ξ (the local Deborah number) in the range [0, 10] for fluids having Prandtl numbers of 10, 50, and 100. The regular perturbation solutions are also been obtained for smaller values of ξ together with the Padé approximation. Results thus obtained are discussed in terms of the local skin friction and local rate of heat transfer for different values of the physical parameters, like n and Pr. Effect of the Deborah number, De, on the velocity and temperature profiles has also been shown graphically. It is observed that both the local skin friction and heat transfer coefficients decrease with increase in the value of De for given values of n and Pr.

Polyelectrolyte spin assembly: Influence of ionic strength on the growth of multilayered thin films

AbstractLayer‐by‐layer (LbL) assembly of polymer electrolyte multilayers is now a well‐established method for the fabrication of thin films by sequential adsorption of alternating layers of oppositely charged polyelectrolytes. Most commonly, such adsorptions have been from quiescent solutions of varying ionic strength and pH. Here, we report results on an alternative processing route for the achievement of polymeric multilayer assemblies of poly(sodium‐4‐styrene sulfonate) and poly(allylamine hydrochloride) that utilizes conventional spin coating. We investigated and describe herein the dependence of multilayer film buildup on solution ionic strength for comparison with similar dependence in quiescent adsorption. Using UV‐Vis spectroscopy we monitored the growth of the multilayered films, while with Atomic Force Microscopy (AFM) we examined the surface features and measured coating thicknesses at different salt concentrations. AFM and UV‐Vis data reveal two regimes of behavior with increasing salt: strong salt‐dependence at low salt contents, and weak salt‐dependence for high salt contents. To explain this observation, we introduce the relevance of the dimensionless groupDe=τ, the local Deborah Number, to the problem. As ionic strength increases, τ increases so that spin‐assembly flow influences adsorbed conformation, and thus LbL growth rate. Our results indicate the ability to design and control polyelectrolyte multilayered structures prepared via spin assembly by varying solution properties that influence the conformation of deposited polymer chains. Additionally, our studies reveal the need for study of the fundamental mechanisms of polyelectrolyte adsorption within complex flow fields. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3654–3666, 2004

Local similarity solution for the flow of a “second-grade” viscoelastic fluid above a moving plate

The boundary layer flow of a viscoelastic fluid of the second-grade type over a rigid continuous plate moving through an otherwise quiescent fluid with constant velocity U is studied. Assuming the flow to be laminar and two-dimensional, local similarity solution is found with fluid's elasticity and plate's withdrawal speed as the main variables. Results are presented for velocity profiles, boundary layer thickness, wall skin friction coefficient and fluid entrainment in terms of the local Deborah number. A marked formation of boundary layer is predicted, even at low Reynolds numbers, provided the Deborah number is sufficiently large. The boundary layer thickness and the wall skin friction coefficient are found to scale with fluid's elasticity—both decreasing the higher the fluid's elasticity. The amount of fluid entrained is also predicted to decrease whenever a fluid exhibits elastic behavior.