Exact Solution to the One-Dimensional Inverse-Stefan Problem in Nonideal Biological Tissues

1995 ◽  
Vol 117 (2) ◽  
pp. 425-431 ◽  
Author(s):  
Y. Rabin ◽  
A. Shitzer
Keyword(s):  
Steady State ◽  
Phase Change ◽  
Stefan Problem ◽  
Blood Perfusion ◽  
Steady State Solution ◽  
Biological Tissues ◽  
Initial Condition ◽  
Quasi Steady State ◽  
Single Temperature ◽  
The One

A new analytic solution of the inverse-Stefan problem in biological tissues is presented. The solution, which is based on the enthalpy method, assumes that phase change occurs over a temperature range and includes the thermal effects of metabolic heat generation, blood perfusion, and density changes. As a first stage a quasi-steady-state solution is derived, defined by uniform velocities of the freezing fronts and thus by constant cooling rates at those interfaces. Next, the fixed boundary condition leading to the quasi-steady state is calculated. It is shown that the inverse-Stefan problem may not be solved exactly for a uniform initial condition, but rather for a very closely approximating exponential initial condition. Very good agreement is obtained between the new solution and an earlier one assuming biological tissues to behave as pure materials in which phase change occurs at a single temperature. A parametric study of the new solution is presented taking into account property values of biological tissues at low freezing rates typical of cryosurgical treatments.

scholarly journals Qualitative analysis of an autocatalytic chemical reaction model with decay

2014 ◽  
Vol 144 (2) ◽  
pp. 427-446 ◽  
Author(s):  
Jun Zhou ◽  
Junping Shi
Keyword(s):  
Steady State ◽  
Chemical Reaction ◽  
Reaction Diffusion ◽  
Steady State Solution ◽  
Reaction Model ◽  
Existence Of A Solution ◽  
Positive Steady State ◽  
Chemical Reaction Model ◽  
The One ◽  
Necessary And Sufficient

In this paper, we revisit a reaction—diffusion autocatalytic chemical reaction model with decay. For higher-order reactions, we prove that the system possesses at least two positive steady-state solutions; hence, it has bistable dynamics similar to the system without decay. For the linear reaction, we determine the necessary and sufficient condition to ensure the existence of a solution. Moreover, in the one-dimensional case, we prove that the positive steady-state solution is unique. Our results demonstrate the drastic difference in dynamics caused by the order of chemical reactions.

A SIMPLIFIED CIRCUIT OF THE SEISMIC ELECTRIC METHOD AND ITS STEADY‐STATE SOLUTION

Geophysics ◽  
1936 ◽  
Vol 1 (3) ◽  
pp. 336-339 ◽  
Author(s):  
M. M. Slotnick
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
State Solution ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Steady State Response ◽  
State Response ◽  
In Series ◽  
The One ◽  
Electric Effect

The Seismic Electric Effect gives rise to the problem of finding the steady state response of a circuit consisting of an inductance and a response of a circuit consisting of an inductance and a resistance of the form R+A cos cot (R>A) in series with a D.C. input. In this paper a solution is given, other than the one usually obtained by the method of successive approximations.

An Efficient Algorithm for Computing the Steady-State Dynamic Response of High-Speed Mechanisms

1994 ◽  
Author(s):  
Tyler J. Selstad ◽  
Kambiz Farhang
Keyword(s):  
Steady State ◽  
High Speed ◽  
Steady State Solution ◽  
Time Varying ◽  
Initial Condition ◽  
Steady State Response ◽  
Periodicity Condition ◽  
Varying Coefficients ◽  
State Response ◽  
Time Varying Coefficients

Abstract An efficient method for obtaining the steady-state response of linear systems with periodically time varying coefficients is developed. The steady-state solution is obtained by dividing the fundamental period into a number of intervals and establishing, based on a fourth-order Rung-Kutta formulation, the relation between the response at the start and end of the period. Imposition of periodicity condition upon the response facilitates computation of the initial condition that yields the steady-state values in a single pass; i.e. integration over only one period. Through a practical example, the method is shown to be more accurate and computationally more efficient than other known methods for computing the steady-state response.

Combined Solution of the Inverse Stefan Problem for Successive Freezing/Thawing in Nonideal Biological Tissues

1997 ◽  
Vol 119 (2) ◽  
pp. 146-152 ◽  
Author(s):  
Y. Rabin ◽  
A. Shitzer
Keyword(s):  
Phase Transition ◽  
Heat Generation ◽  
Thermal Effects ◽  
Blood Perfusion ◽  
Biological Tissues ◽  
Initial Condition ◽  
Freezing Front ◽  
Metabolic Heat ◽  
Metabolic Heat Generation ◽  
Combined Solution

A new combined solution of the one-dimensional inverse Stefan problem in biological tissues is presented. The tissue is assumed to be a nonideal material in which phase transition occurs over a temperature range. The solution includes the thermal effects of blood perfusion and metabolic heat generation. The analysis combines a heat balance integral solution in the frozen region and a numerical enthalpy-based solution approach in the unfrozen region. The subregion of phase transition is included in the unfrozen region. Thermal effects of blood perfusion and metabolic heat generation are assumed to be temperature dependent and present in the unfrozen region only. An arbitrary initial condition is assumed that renders the solution useful for cryosurgical applications employing repeated freezing/thawing cycles. Very good agreement is obtained between the combined and an exact solution of a similar problem with constant thermophysical properties and a uniform initial condition. The solution indicated that blood perfusion does not appreciably affect either the shape of the temperature forcing function on the cryoprobe or the location and depth of penetration of the freezing front in peripheral tissues. It does, however, have a major influence on the freezing/thawing cycle duration, which is most pronounced during the thawing stage. The cooling rate imposed at the freezing front also has a major inverse effect on the duration of the freezing/thawing.

scholarly journals An Accurate Approximation of the Two-Phase Stefan Problem with Coefficient Smoothing

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1924
Author(s):  
Vasily Vasil’ev ◽  
Maria Vasilyeva
Keyword(s):  
Phase Change ◽  
Stefan Problem ◽  
Error Function ◽  
Discontinuous Coefficients ◽  
Two Phase ◽  
Spatial Interval ◽  
The One ◽  
Transfer Problems ◽  
The Mathematical Model ◽  
The Given

In this work, we consider the heat transfer problems with phase change. The mathematical model is described through a two-phase Stefan problem and defined in the whole domain that contains frozen and thawed subdomains. For the numerical solution of the problem, we present three schemes based on different smoothing of the sharp phase change interface. We propose the method using smooth coefficient approximation based on the analytical smoothing of discontinuous coefficients through an error function with a given smoothing interval. The second method is based on smoothing in one spatial interval (cell) and provides a minimal length of smoothing calculated automatically for the given values of temperatures on the mesh. The third scheme is a convenient scheme using a linear approximation of the coefficient on the smoothing interval. The results of the numerical computations on a model problem with an exact solution are presented for the one-dimensional formulation. The extension of the method is presented for the solution of the two-dimensional problem with numerical results.

The Exact Solutions of Some Stefan Problems With Prescribed Heat Flux

1981 ◽  
Vol 48 (4) ◽  
pp. 732-736 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Heat Flux ◽  
Phase Change ◽  
Exact Solutions ◽  
Stefan Problem ◽  
Supercooled Liquid ◽  
Freezing Temperature ◽  
Similarity Solutions ◽  
Initial Condition ◽  
Stefan Problems ◽  
Before And After

The Stefan problem of a semi-infinite material with arbitrarily prescribed initial and flux conditions is studied. When the surface temperature is initially different from the freezing temperature, there exists a presolidification or a premelting period prior to the occurrence of a phase change. The exact solutions for both periods, before and after the appearance of a phase change, are established. This indicates that the initial condition of the Stefan problem with a prescribed heat flux at the surface cannot be assumed to be constant. Possibilities of similarity solutions of the problem are also examined. A similarity solution exists only when the heat flux is proportional to t−1/2 and the initial and boundary conditions satisfy an inequality. The solidification of a supercooled liquid is also investigated. The exact solution is obtained.

scholarly journals Recovering the initial condition in the one-phase Stefan problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chifaa Ghanmi ◽  
Saloua Mani Aouadi ◽  
Faouzi Triki
Keyword(s):  
Stefan Problem ◽  
Synthetic Data ◽  
Unique Continuation ◽  
Parabolic Type ◽  
Stability Estimate ◽  
Initial Condition ◽  
Unique Continuation Property ◽  
Ill Posed ◽  
The One ◽  
Representation Techniques

<p style='text-indent:20px;'>We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.</p>

Comparison of Computational and Quasi-Static Solutions of Phase Change With Heat Generation

2007 ◽  
Author(s):  
John Crepeau ◽  
Ali Siahpush ◽  
Blaine Spotten
Keyword(s):  
Steady State ◽  
Phase Change ◽  
Analytical Model ◽  
Heat Generation ◽  
Static Solution ◽  
Steady State Solution ◽  
State Solution ◽  
Computational Results ◽  
Solid Liquid ◽  
Volumetric Heat Generation

In this paper, we investigate the effects that the volumetric heat generation has on the movement and steady-state location of a solid-liquid phase change front in melting and freezing processes. Volumetric heat generation enhances melting and impedes freezing. This phenomenon occurs in nuclear, geologic, cryogenic and material processing applications. We compare the results from a FLUENT computational model with analytical results of a quasi-static solution of the governing equations. These models are applied for constant surface temperature boundary conditions and various volumetric heat generation values in cylindrical plane wall and spherical geometries. The quasi-static method results in an exact steady-state solution which shows that the location of the phase change front is inversely proportional to the square root of the volumetric heat generation. This method is valid for Stefan numbers less than one and the computational results for this regime give excellent agreement with the analytical model, thereby validating the technique and solutions. For the sake of comparison, we also plot the analytical model and computational results for Stefan numbers of one and greater. The quasi-static analytical solution converges more rapidly to the steady-state value than the computational solution. As expected, at longer time intervals, both the analytical and computational solutions converge to the exact steady-state solution.

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