Dimensionless Numbers Research Articles

A simple guideline for designing droplet microfluidic chips to achieve an improved single (bio)particle encapsulation rate using a stratified flow-assisted particle ordering method

Encapsulation of a single (bio)particle into individual droplets (referred to as single encapsulation) presents tremendous potential for precise biological and chemical reactions at the single (bio)particle level. Previously demonstrated successful strategies often rely on the use of high flow rates, gel, or viscoelastic materials for initial cell ordering prior to encapsulation into droplets, which could potentially challenge the system's operation. We propose to enhance the single encapsulation rate by using a stratified flow structure to focus and pre-order the (bio)particles before encapsulation. The stratified flow structure is formed using two simple aqueous Newtonian fluids with a viscosity contrast, which together serve as the dispersed phase. The single encapsulation rate is influenced by many parameters, including fluid viscosity contrast, geometric conditions, flow conditions and flow rate ratios, and dimensionless numbers such as the capillary number. This study focuses on investigating the influences of these parameters on the focused stream of the stratified flow, which is key for single encapsulation. The results allow the proposal of a simple guideline that can be adopted to design droplet microfluidic chips with an improved single encapsulation rate demanded by a wide range of applications. The guideline was validated by performing the single encapsulation of mouse embryonic stem cells suspended in a gelatin-methacryloyl solution in individual droplets of phosphate buffer saline, achieving a single encapsulation efficiency of up to 70%.

Colloidal droplet desiccation on a electrowetting-on-dielectric (EWOD) platform

The physics of the effects of electric field on the desiccation of colloidal droplets, comprising of dispersed negatively charged nanoparticles [2 μl, 1(w/w. %)], are studied in a standard electrowetting-on-a-dielectric configuration. The extent of contact line pinning during evaporation is found to be a function of the magnitude of the applied voltage and quantified in terms of the dimensionless electrowetting number (η). The pinned contact line led to higher particle compaction as evidenced by the characterization of dried colloidal film thicknesses. Crack formation and their dynamics have been analyzed in detail to elicit the interplay of forces near the contact line region and on the compaction front. These aspects of crack formation are elucidated in the light of magnitude and polarity of the applied electric field. It is found to influence the crack front initiation velocity, the geometry, the number of cracks, and an attempt is made to explain the same via first principle-based approaches. Therefore, this study indicates the possibility of using electrowetting as a technique to fine-tune the crack formation behavior in thin colloidal films.

Stability analysis of MHD Jeffery–Hamel flow using artificial neural network

The current research explores the useful impact of artificial neural networks (ANN) back-propagation along Levenberg-Marquardt Method (ANN-BLMM), for findng the influence of dimensionless numbers on the flow distribution pertaining magnetohydrodynamics (MHD) Jeffery–Hamel fluid amid two plates which are enclined at angles 2α. We have employed a numerical approach, ANN-BLMM to assess various aspects of our data, including testing, training, validation, Mean Square Errors (MSE), performance, and fitting. The methodology being used has been tested and validated through comparison with other results obtain numerically, showing extreme level of accuracy. Moreover, we have confirmed our findings through error histograms and regression tests. We used OHAM for the data set. Furthermore, we have also explored the influence of Reynolds number (R) on both flow and pressure distribution, visually representing our findings through graphical analysis. We have discussed about the nature and variations of velocity profiles within MHD Jefery–Hamel flow (MHDJHF), taking into account various values of Ha and Re in both convergent and divergent channels. It was discovered that a significant stabilizing influence of an increase in the magnetic field intensity was observed for both diverging and converging channel geometries and as the Hartman numbers rise, so does the fluid velocity. The absolute error is reduced to 10–2 to 10–6.

Dynamic cellular responses to gravitational forces: Exploring the impact on white blood cell(s).

In recent years, the allure of space exploration and human spaceflight has surged, yet the effects of microgravity on the human body remain a significant concern. Immune and red blood cells rely on hematic or lymphatic streams as their primary means of transportation, posing notable challenges under microgravity conditions. This study sheds light on the intricate dynamics of cell behavior when suspended in bio-fluid under varying gravitational forces. Utilizing the dissipative particle dynamics approach, blood and white blood cells were modeled, with gravity applied as an external force along the vertical axis, ranging from 0 to 2 g in parameter sweeps. The results revealed discernible alterations in the cell shape and spatial alignment in response to gravity, quantified through metrics such as elongation and deformation indices, pitch angle, and normalized center of mass. Statistical analysis using the Mann-Whitney U test underscored clear distinctions between microgravity (<1 g) and hypergravity (>1 g) samples compared to normal gravity (1 g). Furthermore, the examination of forces exerted on the solid, including drag, shear stress, and solid forces, unveiled a reduction in the magnitude as the gravitational force increased. Additional analysis through dimensionless numbers unveiled the dominance of capillary and gravitational forces, which impacted cell velocity, leading to closer proximity to the wall and heightened viscous interaction with surrounding fluid particles. These interactions prompted shape alterations and reduced white blood cell area while increasing red blood cells. This study represents an effort in comprehending the effects of gravity on blood cells, offering insights into the intricate interplay between cellular dynamics and gravitational forces.

Simplified Methodology for Assessing River Carrying Capacity Based on Land-Use Types

Land conversions or land use changes also become the main cause of the decline in Citarum River conditions. The relationship between river sustainability and its watershed carrying capacity plays a vital role in protecting watersheds to implement sustainable water resources management. Previous studies on river assessment for watershed management have predominantly focused on specific hydrological and/or technical results, rather than considering the process of the development of carrying capacity methodology due to land-use types. Motivated by this fact, the objective of this study is to develop a simplified carrying capacity methodology due to land-use types for sustainable river management by selecting the Upper Citarum Watershed (a main part of Citarum Watershed) located in West Java Province, Indonesia as an example of a study area. The conceptual framework development for watershed carrying capacity due to land-use types was designed the step-by-step methodology standard regarding the sustainable river management. The methodology of this study also used AHP method consisting of screening and selecting attributes, transforming and developing sub-indices, assigning weights, and formulating a runoff cumulative (Ccum) to examine standards and criteria for carrying capacity classification due to the changes of land-use types. The analysis revealed a significant change in the proportions of the various land use types of the study area. The conclusion is it integrates the measures of selected important land-use types to create a single dimensionless number of runoff cumulative and its classification, and a modified approach to communicate information on river status to the public and related policy makers. To advance future studies, it is necessary to develop a comprehensive evaluation and monitoring systems by using GIS-based spatial and temporal analysis.

Horizontal buoyant jets into viscoplastic ambient fluids

This study investigates the horizontal injection of a heavy Newtonian fluid into a lighter viscoplastic ambient fluid, in a large reservoir. The flow dynamics is experimentally captured via camera imaging, laser-induced fluorescence, and particle image velocimetry techniques. The flow parameters include various density differences, injection velocities, and ambient fluid viscoplastic properties. Our analysis identifies two key dimensionless numbers, the Froude number (Fr) and the effective viscosity ratio (m), which includes the rheology of the viscoplastic fluid. Our study also examines the effects of these dimensionless numbers on critical jet characteristics, such as bifurcation length, transition length, deviation length, and jet trajectory, and provides correlations using Fr and m, to predict these characteristic lengths. A regime classification based on the bifurcation phenomenon is also presented in the Fr−m plane.

Separate effect experimental test and theoretical investigate of pressure wave propagation and pressure rising process upon SGTR accident in advanced LFRs

Steam Generator Tube Rupture accident (SGTR) accident is one of the main safety events to the industrial application of Lead-cooled Fast Reactors (LFRs). This paper presents a separate effect test facility, called Lead-bismuth Eutectic Steam generator tube rupture Test facility (LEST), for testing high-pressure subcooled water jetting into a Lead Bismuth Eutectic (LBE) pool, and analyzes experimental data of maximum initial pressure peak, dynamic pressure attenuation together with the pressure rising process upon SGTR. The experimental analyses put forward a general equation form together with a dimensionless number of the initial pressure peak associated with a high-pressure Newtonian fluid injecting into another low-pressure Newtonian fluid, and the derivation of the general equation form is specifically applied to match a further semi-empirical equation for an initial maximum pressure peak with 5–10 MPa water injection in SGTR. The attenuation curve of dynamic pressure wave in each experiment is matched, and more weakening conditions can change the pressure peak from concentrated and non-period peak group to approximate period 1.4–3.7 s pulse group. The size relationship between pipe length L and nozzle diameter D, which L/D=12, is set to divide SGTR into small wall rupture and tube shear fracture, further three stages of pressure rising are distinguished in the tube shear fracture. This work provides experimental data to help determining the severity of SGTR and assessing the consequences of this event as part of the safety case for the industrial application of LFRs.

Investigation of the energy dissipation mechanism for centrifugal air compressor used for fuel cell based on CFD coupling with CIT

Effective evaluation indicator is the focus of air compressor structural optimization and development of control strategies, leaving to be further investigated from physical occurrence mechanism. Considering air expansion and compression, on the basis of coupling computational fluid dynamics (CFD) with classical irreversible thermodynamics (CIT), an extended entropy generation rate was established. Accordingly, various entropy generation rates for air compressor under wide operating conditions was investigated, aiming to what, where and how much the energy was dissipated. Results show that entropy generation rate related to air expansion and compression is always the largest contributor to the total entropy generation of the system under all operating conditions. A new dimensionless number R is defined to distinguish the fluid-related entropy generation rate and mechanical-related entropy generation rate. Furthermore, physical-based empirical models concerning the relationship of entropy generation rates, motor speeds and loads are proposed in this study, whose usage for guiding structural design was discussed. The work done in this paper can not only provide a new method for guiding the structural design of air compressors, but also providing a new idea for air compressor-fuel cell system’s optimal control.

Convective dissolution in a fractured porous matrix using multi-component surface reaction lattice Boltzmann method

In this paper, an improved lattice Boltzmann method-based surface reaction model for studying the convective dissolution of a multi-component system in porous matrix is propose. This model treats the generation and the consumption of components as a source term in control equation. The accuracy of the model is validated by comparing it with an analytical solution of a two-dimensional square cavity diffusion problem and achieves a good agreement. Subsequently, this model is applied to simulate the dissolution of porous medium with a triangular fissure on the surface. By investigating the aspect ratio of the fissure, the results reveal that a low base-height ratio can promote the dissolution rate. By investigating the main dimensionless number of Sc and Da, the results show that Schmidt number has a dominant influence on the dissolution process, and Da is positively correlated with the total dissolution time.

Quantifying electron transport in aggregated colloidal suspensions in the strong flow regime

Electron transport in complex fluids, biology, and soft matter is a valuable characteristic in processes ranging from redox reactions to electrochemical energy storage. These processes often employ conductor-insulator composites in which electron transport properties are fundamentally linked to the microstructure and dynamics of the conductive phase. While microstructure and dynamics are well recognized as key determinants of the electrical properties, a unified description of their effect has yet to be determined, especially under flowing conditions. In this work, the conductivity and shear viscosity are measured for conductive colloidal suspensions to build a unified description by exploiting both recent quantification of the effect of flow-induced dynamics on electron transport and well-established relationships between electrical properties, microstructure, and flow. These model suspensions consist of conductive carbon black (CB) particles dispersed in fluids of varying viscosities and dielectric constants. In a stable, well-characterized shear rate regime where all suspensions undergo self-similar agglomerate breakup, competing relationships between conductivity and shear rate were observed. To account for the role of variable agglomerate size, equivalent microstructural states were identified using a dimensionless fluid Mason number, [Formula: see text], which allowed for isolation of the role of dynamics on the flow-induced electron transport rate. At equivalent microstructural states, shear-enhanced particle-particle collisions are found to dominate the electron transport rate. This work rationalizes seemingly contradictory experimental observations in literature concerning the shear-dependent electrical properties of CB suspensions and can be extended to other flowing composite systems.

Experimental characterization of dynamics of bed‐scale liquid spreading in a trickle bed

AbstractWe report measurements performed to understand the effects of gas (QG) and liquid (QL) flow rates, surface tension (σGL), liquid viscosity (μL), and particle diameter (dp) on dynamics of local liquid spreading, pressure drop, and overall liquid holdup in a pseudo‐2D trickle bed. We show that an increase in the gas‐phase inertia leads to a decrease in the lateral liquid spreading, whereas an increase in the liquid‐phase inertia leads to an increase in the lateral liquid spreading. We also show that an increase in dp causes a reduction in the lateral liquid spreading. Using dimensionless numbers (AB and We), we propose a regime map showing contributions of different forces to the local liquid spreading. We show that the interplay between the inertia and capillary forces governs the liquid distribution near the inlet, whereas the relative contribution of gravitational force increases toward the outlet. Finally, we propose a relation between AB and We for “bed‐scale” liquid spreading.

Data-informed characterization of spatio-temporal scales in experiments of microconfined high-pressure transcritical turbulence

The spatio-temporal scales of microconfined high-pressure transcritical turbulence are characterized in relation to the resolution capabilities of present time-resolved two-dimensional μPIV technology. Utilizing CO2 as the working fluid, the physical scales are examined by considering the main dimensionless groups of the problem, which correspond to the Reynolds, Brinkman, and Stokes numbers and the mass fraction of particles in the fluid. In detail, the methodology employed leverages direct numerical simulation data to inform the estimation of hydrodynamic, thermophysical, and particle-related scales, and selects a state-of-the-art μPIV setup to describe the optical performance of the current technology. The results indicate that the temporal scales can be experimentally captured for a wide range of operating conditions. However, the scenario becomes much more complex when trying to capture the spatial scales of microconfined high-pressure transcritical turbulent flow. Particularly, the Kolmogorov and Batchelor spatial scales can be captured for bulk Reynolds numbers below O(103). Otherwise, the spatial scales can only be partially captured and/or remain completely masked due to insufficient resolution, like for example in the case of boundary layer viscous scales and fluid density variations. This limitation is not imposed by the tracer behavior of microparticles, as the Stokes number remains significantly low for all the system configurations studied. Instead, the limitation is mainly a result of the optical capabilities of present μPIV systems. Finally, given the generalizable properties of dimensionless numbers, the results and insight obtained can be extended to other experiments of μPIV-based visualization/quantification of microconfined multiscale flows involving large thermophysical variations.

Dimensionless thermal performance analysis of a closed isothermal compressed air energy storage system with spray-enhanced heat transfer

The isothermal compressed air energy storage (I-CAES) technology boasts the advantages of high theoretical round-trip efficiency and zero carbon emissions. In order to rapidly and efficiently assess the thermodynamic performance of I-CAES, and addressing the limitations of the overly complex thermodynamic model for spray heat exchanged closed-loop I-CAES (CI-CAES), this paper establishes a dimensionless evaluation model related to container size and initial state a*, spray water quantity b*, start or end spraying time c*, and the ratio of liquid level to droplet velocity u*. The expressions for evaluation indicators such as round-trip efficiency, indicated efficiency, compression exergy efficiency, and expansion exergy efficiency are summarized, revealing the impact patterns of dimensionless numbers on the thermodynamic performance of CI-CAES. The results show that the relative difference in the percentage of air enthalpy change calculated using the real gas model versus the ideal gas model during the expansion stage is 88.26 %. The indicated efficiency and round-trip efficiency of CI-CAES simulated by the real gas model were 88.18 % and 60.51 %, respectively, for the design parameters, and the variable condition operation of the hydrodynamic machinery was the main reason for the lower round-trip efficiency. The larger the values of a* and b*, the closer to the isothermal process, and the greater the effect of b* on the performance of the CI-CAES system compared to the two. When b* is 50, the polytropic indices of the compression and expansion stages reach 1.03 and 1.01, respectively. The round-trip, compression exergy efficiency and expansion exergy efficiency all vary parabolically with c*, suggesting that there is an optimal c* that optimizes the system performance. The change in pressure ratio has a greater effect on round-trip efficiency and on the percentage change in air enthalpy during compression and expansion, and u* has a lesser effect on system performance. The results of the research can provide theoretical guidance for the efficient operation of CI-CAES.

Experimental observation of parabolic wakes in thin plates

Wakes are medium perturbations created by a moving object, such as wave patterns behind boats, or wingtip vortices following an aircraft. Here, we report about an experimental study of an uncharted form of parabolic wakes occurring in media with the group velocity twice larger than the phase velocity, as opposed to the conventional case of Kelvin wakes. They are formed by moving a laser spot on a thin plate, which excites a unique wake pattern made of confocal parabolas, due to the quadratic dispersion of the zero-order flexural Lamb mode. If the spatial dimensions are rescaled by the perturbation velocity and material constant, we obtain a single universal wake with constant parabolic focal lengths. We demonstrate an evanescent regime above the critical frequency where the wave components oscillate exclusively in the direction parallel to the perturbation path, with an opening angle of 90∘. We define a dimensionless number, analogous to Froude and Mach numbers, which determines whether or not the complete parabolic wake pattern will be excited by the moving source. Finally, we generalize the physics of wake shapes to arbitrary dispersion relations. Published by the American Physical Society 2024

Prediction of Boiling Heat Transfer Coefficient for Micro-Fin Using Mini-Channel

The prediction of the heat transfer coefficient commonly relies on the development of new empirical prediction equations when operating conditions and refrigerants change from the existing equations. Creating new prediction equations is expensive and time-consuming; therefore, recent attention has been given to machine learning approaches. However, machine learning requires a large amount of data, and insufficient data can result in inadequate accuracy and applicability. This study showed that using mini-channel data as highly relevant data for the micro-fin heat transfer coefficient yields high prediction accuracy, even when the experimental dataset of interest is limited. In the proposed method, we added dimensionless numbers assumed to significantly influence heat transfer coefficients calculated from experimental data to the training dataset. This allowed efficient learning of the characteristics of thin liquid films present in mini-channels and micro-fins. By combining distinctive physical mechanisms related to heat transfer coefficients with DNN/GPR/Fine-tuning, the proposed method can predict 96.7% of the data points within ±30% deviation. In addition, it has been confirmed that the dryout quality and post-dryout heat transfer coefficients were predicted with high accuracy. Additionally, we utilized visualization techniques to investigate the contents of the black-box machine learning models.

Impact pressure coefficient and object mobilization length in mass flows

As the impact force of a flow overcomes the frictional resistance of an object, the object moves as long as the pressure exerted by the flow is greater than its shear resistance. With this concept, here, we develop simple analytical models for the mechanical description of the coefficient of impact pressure Cp0 and the mobilization length L when a mass flow impacts a movable object. We invented the obstacle mobilization number Cp0, a dimensionless number, which is expressed as the ratio of the obstacle shear resistance to the kinetic energy of the flow, per unit length. We also relate it to the obstacle Froude number Fr0. When the object moves through the length L against the basal friction, after consuming its kinetic energy, the object eventually stops. This balance between the kinetic energy and friction yields an analytical model for the mobilization length of the object. Equivalently, if the mobilization length is known, the impact velocity can be calculated. This is very important in estimating the flow velocity of any large-scale natural mass flows. Our model predicts that Cp0 decreases strongly to weakly non-linearly with the impact velocity, density ratio between the granular particle and the mobile object, and the area ratio between the grain covered area of the object and its base. Moreover, Cp0 increases linearly with the basal friction coefficient. However, the dynamic response of Cp0 to the shape of the object may vary immensely between objects of different shapes. Our model also predicts that the mobilization length L of the mobile object varies with the square of the impact velocity of flowing mass, but decreases with basal friction of the obstacle and the component of gravitational acceleration in the direction normal to the flow depth. We conduct several laboratory chute experiments with different native Nepalese complex fruit seeds and food grains to validate the physical significance and scope of Cp0 and L. Our simple analytical models very well describe the coefficient of impact pressure and the mobilization length of the mobile object impacted by the laboratory granular flows. We discuss the applicability of Cp0 and L to real flow situations in designing protective civil defense structures as well as mitigating from natural disasters, and industrial transports of complex granular flows.

Centrifugal failure mechanism analysis for ferrofluid seals based on a bidirectional coupled multi-physics model

In this paper, we propose a bidirectional coupled multi-physics (BCM) model to study the centrifugal failure mechanism of ferrofluid seals, employing a hybrid finite volume–finite element method. In this model, the evolution of the flow field is described by the two-phase incompressible Navier-Stokes (NS) equations with Kelvin force. The volume of fluid (VOF) method is used to capture the gas-ferrofluid interface. Magnetic scalar potential is introduced to solve for the magnetostatic field. We conducted secondary development on the OpenFOAM and Elmer software packages, which are based on the finite volume method (FVM) and finite element method (FEM), respectively, to calculate the flow field and magnetic field in ferrofluid seals. By developing interpolation methods between FEM and FVM grids based on the parallel bidirectional coupler EOF-Library[1], field information exchange between the flow field and magnetic field is achieved. The BCM model executes iterative computations on the flow and magnetic fields in an alternating way, continuously synchronizing the information of each, thus facilitating bidirectional coupling between the FVM and FEM frameworks. We employ the BCM model to investigate the failure mechanisms of ferrofluid seals under different conditions, including various load pressure and rotational velocity. The results indicate that the ferrofluid will leave the surface of the shaft along the pole teeth under the action of centrifugal forces, with the highest radial velocity occurring at the boundary of the liquid film. Under varying conditions of rotational velocity and pressure drop, the centrifugal failure modes of ferrofluid seals exhibit distinct differences. Moreover, two dimensionless numbers are proposed to quantitatively evaluate the effects of load pressure and centrifugal force, thereby augmenting the efficacy of seal design.

Dynamic response characteristics of microchannel evaporator in a mechanically pumped two-phase loop under hypergravity environment

The unique hypergravity environment and severe heat-dissipation requirements of modern fighter aircrafts pose serious challenges to the potential utilization of mechanically pumped two-phase loops (MPTLs), which are an efficient cooling technology. The microchannel evaporator is a key component of the MPTL system, and understanding its startup and operation characteristics is of great significance for design and optimization. In this context, an experimental MPTL system is built and fixed on a rotating platform, which is designed to generate hypergravity ranging from 0 g0 to 15.0 g0 (g0 = 9.8 m·s−2). The dynamic-response characteristics of the temperature, pressure, and flow rates during the startup and operation of the microchannel evaporator under various hypergravity conditions are investigated. The results indicate that different startup modes, namely gradual startup under low hypergravity and overshoot startup under high hypergravity, are achieved on the microchannel evaporator, even if the flow rate and heat flux are not changed, which is completely different from that under normal gravity. These two startup modes can be quantitatively recognized by two dimensionless numbers: the boiling number (Bo) and Froude number (Fr). A larger Bo or 1/Fr, which corresponds to a larger heat flux or hypergravity; both conditions contribute to the startup mode of the microchannel evaporator shifting from gradual to overshoot startup. During the operation of a microchannel evaporator, the competition between the elevated saturation temperature of the fluid and the induced Coriolis and centrifugal forces caused by hypergravity determines the wall-temperature variation of the evaporator. The dynamic wall temperature decreases under hypergravity below 2.5 g0, is stable with small fluctuations under hypergravity from 4.7 g0 to 7.5 g0, and increases for hypergravity larger than 11.0 g0.

On more insightful dimensionless numbers for computational viscoelastic rheology

Abrupt contraction flows involving viscoelastic fluids represent a longstanding computational challenge within the field of non-Newtonian fluid mechanics. Despite the apparent simplicity of the geometry, these flows have given rise to intricate discussions in the study of viscoelastic phenomena. This study aims to re-examine the numerical solutions for flows through abrupt contractions, offering a fresh interpretation through the lens of reformulated dimensionless numbers. These numbers are designed to consider the characteristic shear rate of the problem, providing a more comprehensive understanding of the underlying dynamics.When investigating models with intermediate levels of complexity, such as the Giesekus and Phan-Thien-Tanner constitutive equations, the usual comparison with the corresponding Oldroyd-B model becomes inadequate because it tends to rely on the nominal relaxation time (λ) and the nominal total viscosity (η) instead of their effective counterparts when defining the Reynolds number (Re), the Weissenberg number (Wi) and the ratio of solvent to total viscosities (β) (β plays a role only in rheological models involving a solvent contribution). If these dimensionless numbers are tailored to account for the characteristic shear rate specific to the problem under investigation, the choice of the corresponding Oldroyd-B flow, at the adequate values of Re, Wi, and β allows for significantly better quantification of the correct effects of nonlinear viscoelasticity of the original model.We show the conventional approach tends to overemphasize the role of the nonlinear parameter in nonlinear constitutive equations, like the Giesekus and PTT models, when examining standard abrupt contraction flow outputs such as the Couette correction and vortex size. This overestimation occurs because the conventional method does not allow the Reynolds and Weissenberg numbers (and possibly β) to carry the portion of the nonlinear effect that can potentially be captured by the linear Oldroyd-B model through the use of characteristic shear rate-based values. We believe the present approach provides a better perspective of the role played by the nonlinear parameter and its extension to more general flows is also discussed.

Pressurized membranes between walls: Thermodynamic process changes force and stiffness

Pressurized solids are ubiquitous in nature. Mechanical properties of biological tissues arise from cell turgor pressure and membrane elasticity. Flat contact between cells generate nonlinear forces. In this work, cells are idealized as pressurized elastic membranes in frictionless contact with one another. Contact forces are experimentally measured on rubber-like membranes and computed using finite element analysis (FEA). FEA matches experimental force-indentation relationships from small to large indentations. With the chosen dimensionless numbers, data gather on a master curve. The isobaric force exhibits a 4/3 power law over 1.5 decades of indentation. Forces for other thermodynamic processes (adiabatic, isothermal/osmotic and isochoric) are interpolated from isobaric data. Regarding stiffness, the isochoric process is superlinear contrary to the sublinear isobaric stiffness. Simple force-indentation relationships are given for each process.