The influence of the gas phase on liquid imbibition in capillary tubes

2011 ◽  
Vol 678 ◽  
pp. 600-606 ◽  
Author(s):  
MARCUS HULTMARK ◽  
JEFFREY M. ARISTOFF ◽  
HOWARD A. STONE
Keyword(s):  
Analytical Solution ◽  
Gas Phase ◽  
Excellent Agreement ◽  
Classical Theory ◽  
Capillary Tube ◽  
Viscous Resistance ◽  
Systematic Deviation ◽  
One Dimensional ◽  
Explicit Analytical Solution ◽  
Moving Front

The imbibition of liquid into a capillary tube is studied both theoretically and experimentally for sufficiently long tubes where viscous resistance from the gas phase ahead of the moving front is significant. At early times, and as the length of the tube is increased, we observe a systematic deviation from classical theory that cannot be attributed to the inertia of the liquid nor entrance effects. Instead, this behaviour is rationalized by considering the viscous resistance from the gas as it is displaced by the liquid. An explicit analytical solution for a one-dimensional description of the flow is given that accounts for viscous resistance from the displaced fluid. Excellent agreement between experiment and theory is obtained.

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

Building Vertical Walls by Deposition of Molten Metal Droplets

2005 ◽  
Author(s):  
M. Fang ◽  
S. Chandra ◽  
C. B. Park
Keyword(s):  
Surface Layer ◽  
Analytical Solution ◽  
Substrate Temperature ◽  
Layer By Layer ◽  
Droplet Temperature ◽  
Droplet Generator ◽  
One Dimensional ◽  
Metallurgical Bonding ◽  
Experimental Parameters ◽  
Vertical Walls

Experiments were conducted to determine conditions under which good metallurgical bonding was achieved in vertical walls composed of multiple layers of droplets that were fabricated by depositing tin droplets layer by layer. Molten tin droplets (0.75 mm diameter) were deposited using a pneumatic droplet generator on an aluminum substrate. The primary parameters varied in experiments were those found to most affect bonding between droplets on different layers: droplet temperature (varied from 250°C to 325°C) and substrate temperature (varied from 100°C to 190°C). Considering the cooling rate of droplet is much faster than the deposition rate previous deposition layer cooled down too much that impinging droplets could only remelt a thin surface layer after impact. Assuming that remelting between impacting droplets and the previous deposition layer is a one-dimensional Stefan problem with phase change an analytical solution can be found and applied to predict the minimum droplet temperature and substrate temperature required for local remelting. It was experimentally confirmed that good bonding at the interface of two adjacent layers could be achieved when the experimental parameters were such that the model predicted remelting.

scholarly journals Multiphase modeling of nitrate photochemistry in the quasi-liquid layer (QLL): implications for NO<sub>x</sub> release from the Arctic and coastal Antarctic snowpack

2008 ◽  
Vol 8 (16) ◽  
pp. 4855-4864 ◽  
Author(s):  
C. S. Boxe ◽  
A. Saiz-Lopez
Keyword(s):  
Gas Phase ◽  
Liquid Layer ◽  
Solar Spectrum ◽  
The Arctic ◽  
Transfer Model ◽  
Multiphase Model ◽  
Gas Phase Reactions ◽  
One Dimensional ◽  
Quasi Liquid Layer ◽  
Summer Range

Abstract. We utilize a multiphase model, CON-AIR (Condensed Phase to Air Transfer Model), to show that the photochemistry of nitrate (NO3−) in and on ice and snow surfaces, specifically the quasi-liquid layer (QLL), can account for NOx volume fluxes, concentrations, and [NO]/[NO2] (γ=[NO]/[NO2]) measured just above the Arctic and coastal Antarctic snowpack. Maximum gas phase NOx volume fluxes, concentrations and γ simulated for spring and summer range from 5.0×104 to 6.4×105 molecules cm−3 s−1, 5.7×108 to 4.8×109 molecules cm−3, and ~0.8 to 2.2, respectively, which are comparable to gas phase NOx volume fluxes, concentrations and γ measured in the field. The model incorporates the appropriate actinic solar spectrum, thereby properly weighting the different rates of photolysis of NO3− and NO2−. This is important since the immediate precursor for NO, for example, NO2−, absorbs at wavelengths longer than nitrate itself. Finally, one-dimensional model simulations indicate that both gas phase boundary layer NO and NO2 exhibit a negative concentration gradient as a function of height although [NO]/[NO2] are approximately constant. This gradient is primarily attributed to gas phase reactions of NOx with halogens oxides (i.e. as BrO and IO), HOx, and hydrocarbons, such as CH3O2.

Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model

1999 ◽  
Author(s):  
Alexander V. Kasharin ◽  
Jens O. M. Karlsson
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Planar System ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Dimensional Diffusion ◽  
Constant Diffusivity ◽  
A Cell ◽  
Cell Dehydration ◽  
The One

Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.

scholarly journals ANALYTICAL SOLUTION OF THE PROBLEM OF A CONVERGENT SHOCK IN GAS FOR ONE-DIMENSIONAL CASE

2017 ◽  
Vol 9 (4) ◽  
pp. 52-58 ◽  
Author(s):  
V.F. Kuropatenko ◽  
◽  
F.G. Magazov ◽  
E.S. Shestakovskaya ◽  
◽  
...  
Keyword(s):  
Analytical Solution ◽  
Dimensional Case ◽  
One Dimensional
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