Analytical Solution Of Unsteady Flow Research Articles

The cooling effect of blood flow during hyperthermia treatment

The amount and duration of the applied heat in hyperthermia treatment are critical for cancer survivors. The challenge is to use a mechanism dealing with the tumor cells only while keeping healthy tissues unharmed. The aim of this paper is to predict the blood temperature distribution in main dimensions during hyperthermia process by deriving a new analytical solution of unsteady flow that adequately covers the cooling factor. We adopted a separation of variable method to solve the bio-heat transfer problem of unsteady blood flow. The solution is similar to Pennes’ equation, except that it is for blood rather than tissue. We also performed computational simulations with varied flow conditions and thermal energy transports. The blood cooling effects were calculated with vessel’s diameter, tumor’s zone length, pulsating period and flow velocity. The cooling rate rises by around 133% if the tumor zone’s length is extended four times the diameter of 0.5 (mm), but it is seemingly fixed with this distance if the diameter is equal or larger than 4 (mm). Likewise, the temporal variations of temperature disappear if the blood vessel has a diameter of 4 (mm) or more. Pre-heating or post-cooling techniques perform effectively given the theoretical solution; under particular conditions, the reduction percentages of the cooling effect are between 130% and 200%, respectively.

Analytical Analysis of the Groundwater Drawdown Difference Induced by Foundation Pit Dewatering with a Suspended Waterproof Curtain

The dewatering of foundation pits with a suspended waterproof curtain causes different groundwater drawdowns inside and outside the pit, resulting in the drawdown difference between the inside and outside the pit. Maintaining a groundwater drawdown difference between the inside and outside of a foundation pit can eliminate the adverse effects of dewatering on the surrounding environment. According to previous studies on unsteady flow, an analytical solution of the groundwater drawdown with a suspended waterproof curtain under unsteady flow has been proposed. The analytical solution of unsteady flow and the formula of groundwater drawdown difference with a suspended waterproof curtain were validated by comparing pumping tests and finite-element method (FEM), in which a good agreement was observed. The magnitude of the drawdown difference generally represents the extent of surrounding groundwater affected by groundwater drawdown inside the pit. This paper also investigated the effects of sensitivity parameters on the drawdown difference for minimizing the effect of surrounding environment. During the process of dewatering with a suspended waterproof curtain, the groundwater drawdown (Sh) should not exceed the length of the waterproof curtain (L), and the optimal radius of foundation pit (Rw) and length of waterproof curtain (L) were found, i.e., Rw/H0 = 0.781 and L/H0 = 0.813 (H0 is 32 m). Beyond these values, the drawdown difference tends to be stable. The drawdown difference is also significantly affected by the dewatering time. When t < 48 h, the groundwater drawdown difference decreases rapidly; when t > 48 h, the groundwater drawdown difference stabilizes.

Analytical Solution of Unsteady Casson Fluid Flow Through a Vertical Cylinder with Slip Velocity Effect

Casson fluid is a non-Newtonian fluid with its unique fluid behaviour because it behaves like an elastic solid or liquid at a certain condition. Recently, there are several studies on unsteady Casson fluid flow through a cylindrical tube have been done by some researchers because it is related with the real-life applications such as blood flow in vessel tube, chemical and oil flow in pipelines and others. Therefore, the main purpose of the present study is to obtain analytical solutions for unsteady flow of Casson fluid pass through a cylinder with slip velocity effect at the boundary condition. Dimensional governing equations are converted into dimensionless forms by using the appropriate dimensionless variables. Dimensionless parameters are obtained through dimensionless process such as Casson fluid parameters. Then, the dimensionless equations of velocity with the associated initial and boundary conditions are solved by using Laplace transform with respect to time variable and finite Hankel transform of zero order with respect to the radial coordinate. Analytical solutions of velocity profile are obtained. The obtained analytical result for velocity is plotted graphically by using Maple software. Based on the obtained result, it can be observed that increasing in Casson parameter, time and slip velocity will lead to increment in fluid velocity. Lastly, Newtonian fluid velocity is uniform from the boundary to the center of cylinder while Casson fluid velocity is decreased when approaching to the center of cylinder. The present result is validated when the obtained analytical solution of velocity is compared with published result and found in a good agreement.

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

New Analytical Solutions for Unsteady Flow in a Leaky Aquifer between Two Parallel Streams

Analytical simulation of groundwater flow is a necessary and useful technique to predict the various behavior patterns of a groundwater system. The main aim of the present study is to derive new analytical solutions to compute the unsteady flows inside an aquifer between two parallel streams of constant and varying heads. The problems are solved by means of Laplace transform method and the solution results are verified with the results of MODFLOW. It is observed that the obtained results agreed very well with the results of MODFLOW. The solutions are carried out for two cases of ascending and descending water levels and the obtained results are compared with each other. In addition, the sensitivity of hydraulic heads to aquifer parameters and how locations of water divide change by change in aquifer parameters are investigated. In sensitivity analysis of hydraulic heads to changes in recharge rate with different values of hydraulic conductivity, thickness, and length of the aquifer, it is shown that among these parameters the length of the aquifer is the most important parameter affecting the hydraulic heads. Furthermore, the sensitivity of flow rates to recharge rates and water level change rates are analyzed.

An Analytical Solution of Unsteady Flow and Heat Transfer of Rotating Fluid with Suspended Parasitical

An Analytical Solution of Unsteady Flow and Heat Transfer of Rotating Fluid with Suspended Parasitical

Analytical Solutions for Unsteady Flow of a Fourth-Grade Fluid Arising in the Metallic Wire Coating Process Inside a Cylindrical Roll Die

The unsteady flow of a non-Newtonian fluid in a metallic wire coating process inside a cylindrical roll die is considered. The constitutive equations of the fluid are modeled for a fourth-grade non-Newtonian fluid with non-homogenous boundary conditions. Analytical solutions for the axial velocity field have been obtained in explicit forms by modified homotopy perturbation transform method. These solutions are written as the sum between the permanent and the transient solutions. However, for large values of t, both of these solutions become identical. The obtained velocity field is compared with the existing exact solution of the same flow problem for a second grade fluid. Also, graphs representing the solutions are discussed and appropriate conclusions drawn.

A density-diffusivity approach for the unsteady state analysis of natural gas reservoirs

The traditional approach employed in the analytical analysis of unsteady state responses for natural gas reservoirs is firmly based on recasting the non-linear pressure diffusivity equation in terms of pseudo variables or pseudofunctions—namely, pseudo-pressure and pseudotime—able to render the equation amenable to analytical treatment. Because the pseudofunction-based approach succeeds in linearizing the diffusivity equation and generating the required benchmark analytical solutions, the implementation of pseudo-variables is understood to be essential to unsteady state analysis of natural gas reservoirs. In this study, the authors propose an approach for the analytical solution of unsteady flow in natural gas reservoirs based on the use of the density-diffusivity equation. To demonstrate its validity and range of application, the authors discuss analytical arguments and perform numerical simulations to corroborate its performance for some of the most widely used inner boundary conditions (BCs): constant pressure and constant rate production. By implementing the concept of depletion-driven dimensionless parameters, the proposed density-diffusivity approximation is able to successfully describe reservoir unsteady behavior while keeping all variables involved in the analysis firmly grounded on physical intuition. In addition, it is shown that the proposed approach also provides a new avenue for the interpretation of liquid type-curves in gas analysis and the iteration-free estimation of original gas in place based on the analysis of boundary-dominated production data.

An Analytical Solution for Unsteady Flow in a Phreatic Aquifer in the Case of Continuous Rise

In this study, we examine the classical problem of unsteady flow in a phreatic aquifer, induced by continuous rise of the water flux and head on its boundary. A closed-form analytical solution for the governing Boussinesq equation is derived for a semi-infinite aquifer.

Variational iteration method for unsteady flow of gas through a porous medium using He’s polynomials and Pade approximants

In this paper, we apply the variational iteration method using He’s polynomials for finding the analytical solution of unsteady flow of gas through a porous medium. The proposed method is an elegant combination of He’s variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. The diagonal Pade approximants are effectively used in the analysis to capture the essential behavior of the solution and to determine the initial slope. A clear advantage of this technique over the decomposition method is that no calculation of Adomian’s polynomials is needed.

The Use of the Dimensionless Womersley Number to Characterize the Unsteady Nature of Internal Flow

Dimensionless numbers are very useful in characterizing mechanical behavior because their magnitude can often be interpreted as the relative importance of competing forces that will influence mechanical behavior in different ways. One dimensionless number, the Womersley number (Wo), is sometimes used to describe the unsteady nature of fluid flow in response to an unsteady pressure gradient; i.e., whether the resulting fluid flow is quasi-steady or not. Fluids surround organisms which themselves contain fluid compartments; the behaviors exhibited by these biologically-important fluids (e.g. air, water, or blood) are physiologically significant because they will determine to a large extent the rates of mass and heat exchange and the force production between an organism and its environment or between different parts of an organism.In the biological literature, the use of the Womersley number is usually confined to a single geometry: the case of flow inside a circular cylinder. We summarize the evidence for a broader role of the Womersley number in characterizing unsteady flow than indicated by this geometrical restriction. For the specific category of internal flow, we show that the exact analytical solution for unsteady flow between two parallel walls predicts the same pattern of fluid behavior identified earlier for flow inside cylinders; i.e., a dichotomy in fluid behavior for values ofWo<1 andWo>1. WhenWo<1, the flow is predicted to faithfully track the oscillating pressure gradient, and the velocity profiles exhibit a parabolic shape such that the fluid oscillating with the greatest amplitude is farthest from the walls (“quasi-steady” behavior). WhenWo>1, the velocity profiles are no longer parabolic, and the flow is phase-shifted in time relative to the oscillating pressure gradient. The amplitude of the oscillating fluid may either increase or decrease asWo>1, as described in the text.

UNSTEADY FLOW TO A LARGE-DIAMETER WELL IN A FINITE AQUIFER

ABSTRACT Using the analytical solution derived by Muskat and Kuiper for unsteady flow to a well with negligible storage in a finite confined aquifer, an analytical solution for unsteady flow to a large-diameter well in a finite confined aquifer during pumping and recovery is obtained. Existence of a barrier boundary at a finite distance from the well has little influence on contribution of the well storage to pumping. The barrier boundary affects drawdown significantly. Well storage is not completely recovered in finite aquifer and there is a permanent drawdown in the initial piezometric surface consequent to an event of pumping.