Analytical Solution For Flow Research Articles

Analytical Solution for Fluid Flow and Heat Transfer in a Three-Dimensional Inclined Horizontal Channel and Under The Influence of Thermal Radiation

Analytical Solution for Fluid Flow and Heat Transfer in a Three-Dimensional Inclined Horizontal Channel and Under The Influence of Thermal Radiation

Analytical Solutions for Unsteady Thin Film Flow with Internal Heating and Radiation

This study determines Lie point symmetries for differential equations that mathematically express a time-dependent thin film fluid flow with internal heating and thermal radiation to construct invariants. These invariants are used in the derivation of similarity transformations for reducing the flow equations into systems of equations that possess only one independent variable. The homotopy analysis method is employed to analytically solve the reduced system of equations. The new similarity transformations and the corresponding analytical solutions comprehensively consider flow dynamics and heat transfer under multiple physical conditions. These solutions are presented graphically to demonstrate the effects of variations in the radiative heat flux with internal heating on the flow dynamics and heat transfer properties. Moreover, the variations in fluid dynamics are described graphically using the obtained analytical homotopy solution under different values of the unsteadiness parameter and Prandtl number.

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 Solutions of Peristalsis Flow of Non-Newtonian Williamson Fluid in a Curved Micro-Channel under the Effects of Electro-Osmotic and Entropy Generation

In the current effort, the effects of entropy generation and electro-osmotic driven peristaltic flow of non-Newtonian Williamson liquid in a curved micro-channel is investigated. Formulation of the problem is conducted in a wave frame of reference. Due to the complexity of non-Newtonian fluid, the analytical solutions of non-linear coupled equations are not easy to obtain and are very rarely found in the literature. For analytical solutions, the governing equations are reduced in the form of the Bessel function. The electric double layer is employed as a result of a zeta potential of about 25 mV. The low Reynolds number and long wavelength approximations are taken into account. Graphical analysis has been carried out for velocity, temperature and entropy for physical parameters. It is noted that the Brinkmann number enhances the temperature. The results of this model will be extremely helpful in designing electro-peristaltic pumps for thermal systems.

Exact analytical solutions of stagnation point flow over a heated stretching cylinder: A phase flow nanofluid model

The stagnation point flow over a heated stretching cylinder is mathematically modeled under the assumptions of phase flow approach. The interaction of impinging stagnation flow with the flow developed due to stretching surface of cylinder is thoroughly interpreted. To consider the underlying phase flow approach, water is used as the base fluid with multi-wall carbon nanoparticles. Exact analytical solutions are computed for this interesting problem and the results are evaluated graphically as well. A descriptive heat transport analysis with graphical interpretation of Nusselt number (Nu) is incorporated as well. Streamlines are included to analyze the flow visualization for the dominant impact of stretching developed flow or the dominance of impinging stagnation flow due to oncoming pressure. A more powerful or fully developed impinging stagnation flow is observed for the higher values of oncoming flow pressure parameter β when stretching cylinder has lower stretching velocity A. We can say that the stagnation flow is dominant over stretching flow if higher oncoming pressure flow values are taken. Most of the literature work provided on stagnation point flow covers the topic through numerical techniques, however in our study we aim to present some exact solution for the problem under consideration.

Adoption of Homotopy Perturbation Method (HPM) for non-Darcy Flow in Porous Media

For non-Darcy flow in coarse porous media, an analytical solution based on the Homotopy Perturbation Method (HPM) is proposed. Subsurface water profile data of a laboratory model are used for six different inlet discharges in both rounded and crushed coarse porous media types. The model slope is S = 0.00001 close to the horizon. Different upstream and downstream water level boundary conditions are considered. The results of the analytical solution of non-Darcy flow by the HPM method are compared with experimental data. The normal objective function (NOF) is used for better comparison between the results of analytical solutions and experimental data. Results depict that the HPM method provides acceptable solutions in both rounded and crushed media types. The analytical results of flow rates q = 26.25 lit/s with NOF of 0.000294586 percent in rounded porous media and q = 30 lit/s with NOF of 0.00028660 percent in crushed one are the most consistent with the experimental data. The proposed HPM solution performs very well, particularly at higher flow rates, in both rounded and crushed porous media.

Temperature-Transient Analysis for Matrix Linear Flow Toward Infinite-Conductivity Hydraulically Fractured Wells

SummaryThis work presents novel analytical temperature-transient solutions for matrix linear flow toward an infinite-conductivity hydraulically fractured well producing under specified constant rate (CR) or constant bottomhole pressure (CBHP) production. The solutions apply to the single-phase flow of a slightly compressible fluid with constant viscosity (e.g., undersaturated oil reservoirs with irreducible water saturation or liquid-dominated geothermal reservoirs). They include the effects of conduction, convection, the Joule-Thomson (J-T) expansion of fluids and adiabatic expansion of the total rock and fluid system, and fluid loss fracture damage and assume constant rock and fluid physical and thermophysical properties with pressure and temperature. They are obtained by using Laplace (for CR) and Boltzmann (for CBHP) transformations. To validate the analytical solutions, an in-house numerical solution is also developed. It solves the mass and thermal energy balance equations coupled simultaneously and accounts for the variation of rock and fluid properties with pressure and temperature. The in-house simulator was validated by using a commercial reservoir simulator. Results indicate that the fracture surface temperature is decreasing with a square root of time for CR production but is constant for CBHP production. The temperature responses for both modes of production are controlled by the adiabatic expansion of the rock and fluid properties and the thermal diffusivity of the rock. The effect of thermal conductivity plays a significant role in both production modes as the matrix permeability decreases. The fracture damage has different signatures on temperature transients depending on the mode of production. The approximate analytical solutions show the information content of temperature-transient data acquired from an infinitely conductive hydraulically fractured well under matrix linear flow. They are simple and can be used to perform matrix linear-flow analysis jointly with pressure and rate transient data to estimate the physical and thermophysical properties of the rock and fluids. We also propose a simple correction to fluid viscosity variation as input for the analytic solutions to accurately estimate the physical and thermophysical parameters in case drawdowns are significant.

Analytical solution of supersonic axisymmetric flow around a sharp convex corner

Supersonic axisymmetric expansion flow is a typical and fundamental issue in gas dynamics. It plays a vital role in the high-speed external and internal flow fields regarding the contour design and performance evaluation of supersonic/hypersonic vehicles and their propulsion systems. The supersonic two-dimensional (2D) planar expansion flow is dominated by the well-known Prandtl–Meyer (P–M) theory. However, no similar explicit relation exists for the supersonic axisymmetric expansion flow, and only the computational fluid dynamics results could be employed at present. Therefore, this work focuses on developing the analytical solution of supersonic axisymmetric flow around a sharp convex corner on the basis of the generic gasdynamic functions in a newly established coordinate system for addressing the aforementioned issue. Theoretical derivations and numerical results prove that the flow deflection angle and Mach number in supersonic axisymmetric flow around a sharp convex corner obey the identical law to the 2D planar situation, that is, the P–M theory, while the local axisymmetric expansion fan is not the simple wave flow despite the conical flow. Meanwhile, the method of characteristics is employed to further explicate the intrinsic connection and difference between the 2D and axisymmetric sharp convex corner flow. The equivalence of sharp corner and curved surface flows with the identical deflection angle is discussed, and three limitations of the proposed analytical solution are clarified.

Analytical solutions for vortex flow at the tangential inlet of a vertical dropshaft

Vertical dropshafts with tangential intake structures are often used in urban drainage systems to route surface flow into underground systems. Vortex flow driven by a tangential inlet is the typical flow pattern in the vertical dropshaft. However, analytical solutions of the key hydraulic transition from chute flow to vortex flow have not been investigated, and theoretical guidelines of the vortex dropshaft are not available. In the present study, a series of theoretical analytical solutions are proposed to determine the circumferential flow generated at the vertical dropshaft inlet. The contraction ratio of the tapering chute, dropshaft size, and flow depth at the joint section determines the vortex flow performance in the vertical dropshaft. Based on the free surface drop height, theoretical criteria are proposed to describe the spiral movement intensity in the vertical dropshaft. Systemic experiments for different vortex dropshaft models are tested, and the experimental observations agree with the theoretical analysis. The present investigation provides a general reference for a tangential dropshaft inlet design that can smoothly route rotational flow down a vertical dropshaft.

Physically-based reduction function to model unsaturated flow associated with plant transpiration

Vegetation plays an important ‘hydrological’ role in stabilising geostructures. Soil water is extracted by the roots due to transpiration, this increases soil suction and, hence, soil shear strength. Transpiration occurs in two different regimes, energy-limited and the water-limited regimes respectively. These two regimes are reflected in the two branches of the transpiration reduction function used to model the hydraulic boundary conditions for vegetated ground. The water-limited branch accounts for the reduced transmissivity of the soil-root system when the degree of saturation and, hence, the hydraulic conductivity declines. The water-limited branch of existing reduction functions (e.g., Feddes function) is defined in purely phenomenological fashion with parameters that have no clear link with the complex interaction between soil hydraulic properties and root architecture. A paradigm shift can be achieved through physically-based reduction functions. These require analytical closed-form solutions of radial water flow at the soil-root interface that, in turn require introducing simplifying assumptions, i.e., steady-state flow and a simplified hydraulic conductivity function. This paper explores the implications of these assumptions by i) benchmarking the water-limited branch of the reduction function derived analytically against the one derived numerically for more realistic hydraulic behaviour and ii) assessing the steady-state assumption.

Analytical Solutions of Steady Flow Toward a Partially Penetrating Well in a Rectangular Leaky-Confined Aquifer

Analytical Solutions of Steady Flow Toward a Partially Penetrating Well in a Rectangular Leaky-Confined Aquifer

There is No Such Thing as a Quick Fix: Travel Times to Subsurface Drains

Highlights Bromide tracer was applied at different distances from subsurface drains to assess travel times and preferential flow. Some bromide arrived at the drain outlet with the first water flow after application, indicating preferential flow. Total time for “cleanout” of bromide from the plots varied with drain spacing and relative distance from the drain. When changing management practices, the full impact of the new system will not be reflected until several years later. Abstract. Subsurface “tile” drains are an important water management practice for many productive agricultural soils, but drains also deliver nitrate-nitrogen and other soluble chemicals from the bottom of the rootzone to surface ditches and streams. Many studies have focused on determining different ways to reduce the delivery of nitrate and other chemicals to subsurface drains while still providing adequate drainage for crop production. One important consideration in such studies that is often overlooked is the response time of drainage outflow to changes in management at the soil surface. Drains integrate flow from distances immediately adjacent to the drain all the way to the midplane between parallel drains, which means changes in management will not be fully reflected in drainage waters for months to several years, depending on the drainage intensity, soil characteristics, and precipitation. This bromide tracer study was conducted at the Southeast Purdue Agricultural Center (SEPAC) in Indiana, USA. Bromide tracer was applied at the beginning of the drainage season (November) at different distances from the drains on multiple plots of different drain spacings. The bromide concentration in the drainflow was monitored over the next five years to determine breakthrough curves and the amount of time required to flush the bromide from the system. The initial breakthrough of bromide occurred during the first appearance of drainage after application, regardless of the location of the bromide strip, suggesting some preferential flow was occurring. This was further investigated with comparison to curves of theoretical travel time following Kirkham’s analytic solution for flow to a tile drain. The full transport of the bulk of the chemical took 2-3 years, but varied greatly among the different treatments, reflecting the longer travel times for locations further from the drain. Keywords: Keywords., Drainage, Preferential flow, Streamlines, Subsurface drainage, Travel times.

MHD Free-Convective Couette flow in a Vertical Porous Microchannel Using Non-Linear Boussinesq Approximation

An analytical solution for free convection flow of an electrically conducting fluid in a vertical micro-porous-channel, in the existence of transversely applied magnetic-field and nonlinear Boussinesq approximation is carried out in this article. The governing equations representing stated objective are obtained and solved analytically using method of undetermiend coefficients and direct integration. Pictorial and tabularlar representions of solutions obtained are carried out, so as to ascertain the role of various governing parameters entering flow formation. During the course of numerical simulation of results, it is found that the volumetric flow rate increases with increase in Couette flow parameter, asymmetric heating parameter and suction/injection parameter but decreases with increase in nonlinear Boussinesq approximation parameter.

Numerical and analytical evaluation of airship configuration

Today, airships have many uses and have been considered because they combine the benefits of a ship and an aircraft and the low cost of energy consumption. In this research, with the help of numerical solutions and analytical solutions in Datcom, three common configurations of an airship at various angles of attack are investigated. The results of numerical and analytical solutions are compared. Flow condition was considered at an altitude of 4 km above the ground. The same results are not obtained in numerical and analytical solutions. Certainly, numerical results can be cited more than analytical solutions because Datcom software has computational errors and has been simplified. It has been developed for the analytical flow solution on the slender body. However, despite the existing error, this software can be used to extract computational processes and sensitization some parameters. The results show that the approach of the tail fins to the tip of the airship increases the lift-to-drag ratio, and with increasing the length-to-diameter ratio, the aerodynamic performance also improves. In all variants, converting the fins to a cross-type variant improves the vehicle's aerodynamic performance.

Flow-based method demonstrates improved accuracy for calculating wall shear stress in arterial flows from 4D flow MRI data

Four-dimensional flow magnetic resonance imaging (i.e., 4D flow MRI) has become a valuable tool for the in vivo assessment of blood flow within large vessels and cardiac chambers. As wall shear stress (WSS) has been correlated with the development and progression of cardiovascular disease, focus has been directed at developing techniques to quantify WSS directly from 4D flow MRI data. The goal of this study was to compare the accuracy of two such techniques – termed the velocity and flow-based methods – in the setting of simplified and complex flow scenarios. Synthetic MR data were created from exact solutions to the Navier-Stokes equations for the steady and pulsatile flow of an incompressible, Newtonian fluid through a rigid cylinder. In addition, synthetic MR data were created from the predicted velocity fields derived from a fluid–structure interaction (FSI) model of pulsatile flow through a thick-walled, multi-layered model of the carotid bifurcation. Compared to the analytical solutions for steady and pulsatile flow, the flow-based method demonstrated greater accuracy than the velocity-based method in calculating WSS across all changes in fluid velocity/flow rate, tube radius, and image signal-to-noise (p < 0.001). Furthermore, the velocity-based method was more sensitive to boundary segmentation than the flow-based method. When compared to results from the FSI model, the flow-based method demonstrated greater accuracy than the velocity-based method with average differences in time-averaged WSS of 0.31 ± 1.03 Pa and 0.45 ± 1.03 Pa, respectively (p <0.005). These results have implications on the utility, accuracy, and clinical translational of methods to determine WSS from 4D flow MRI.

Modify homotopy perturbation method using yang transform to study the effect of magnetic field with mass and heat transfer on pulsatile blood flow in tapered stenosis arteries

AbstractThe effect of a magnetic field on two‐dimensional pulsatile blood flow in tapered stenosis arteries was studied by employing the Yang transform homotopy perturbation method (YTHPM). The mathematical model which was solved by the researchers Liu and Liu, was successfully developed by adding the effect of a magnetic field for the blood flow in addition to the effect of mass and heat transfer on it. The impact of the angle of tapering, the Grashof number, the solutal Grashof number, and the magnetic field on the axial velocity, the wall shear stress, flow resistance, and the volumetric flow rate was investigated in two cases absence and presence of the magnetic field. The results prove that YTHPM is efficacious and highly accurate in finding the analytical approximate solution for pulsatile blood flow in tapered stenosis arteries under the influence of a magnetic field. The convergence of the new solutions is also discussed in the absence and presence of the magnetic field. Further, the graphs of these new solutions show the validity of, a necessity for, and utility of YTHPM, and are in line with results put forward by other studies.

Fractional Casson Blood Flow with Gold Nanoparticles (AuNPs) in a Slip Velocity Cylinder

The technology of nanofluid in the blood flow has attracted researchers to further study theoretically and experimentally due to its importance in treating the tumour and delivering the drugs effectively. Due to the lack of analytical study on nano-bloodfluid, this present paper aims to obtain an analytical solution of the Casson blood flow with gold nanoparticles in the cylinder with the free convection flow and slip velocity effect. The dimensionless governing equations are modelled with the Caputo-Fabrizio fractional derivative approach. Next, the joint methods of the Laplace transform and finite Hankel transform are used to obtain the analytical solution. The velocity profile increased as a fractional parameter, slip velocity parameter, nanoparticles volume fraction parameter and Grashof number are increased. Meanwhile, it decreased as the Casson parameter and Prandtl number increased. The temperature profile increased as the nanoparticle volume fraction parameter increased. However, it decreased as the Prandtl number increased. The obtained results are beneficial for the accuracy checking of the numerical methods. Besides, these results are significant to study the human blood flow behaviour with nanoparticles that help diagnose and treat the tumour cell.

Fractal Analytical Solutions for Nonlinear Two-Phase Flow in Discontinuous Shale Gas Reservoir

The paths of a two-phase flow are usually non-linear and discontinuous in the production of shale gas development. To research the influence mechanism between shale gas and water, several integer two-phase flow models have been studied but few analytical solutions have been obtained on shale gas and water pressure. This study first developed a local fractional mathematical model for gas and water two-phase flow in shale gas production. The model thus created considers the effects of capillary pressure, the fractal dimension of the flow pipe, and the discontinuity of the flow path. Second, the local fractional traveling wave method and variational iteration method were applied to this model for the development of iterative analytical solutions. Both shale gas and water pressure were analytically derived. Third, the depressurization process of the shale gas and water was analyzed, and a parametric study was conducted to explore the impacts of fractional dimension, entry capillary pressure, and travel wave velocity on shale gas pressure. Finally, our conclusions are drawn, based on the results of these studies.

Analytical Solution of Transient Flow of Fractional Oldroyd-B Fluid between Oscillating Cylinders

This paper investigates the fractional Oldroyd-B fluid flow across two interminable coaxial cylinders, where the fluid’s motion is generated by the oscillatory movement of cylinders and the oscillating pressure gradient. The profile of velocity and shear stress of the flow is derived with the assistance of Caputo fractional derivative utilizing an analytical technique, finite Hankel transform and Laplace transforms. The semi-analytical solution is then displayed as generalized functions of G and R satisfying fundamental constitutive equations and all initial and boundary conditions. To validate the results, some limitations have been imposed on the determined equations and the results have been contrasted against previous results. Moreover, the influence of various parameters on the flow of fractionalized Oldroyd-B fluid is investigated and depicted graphically.

Analytical solutions of slip flow and H1 heat transfer in elliptical microchannels

The slip flow and heat transfer problems inside a micro- or nano-channel are active research topics. Understanding the velocity and temperature distributions in these channels is essential in both theory and applications. In this paper, the analytical solutions of the fully developed slip flow and heat transfer under the H1 boundary condition in elliptical microchannels are obtained with the velocity slip and temperature jump. The separation of variables method is employed to solve the momentum and energy equations, and the coefficient of the velocity profile is calculated using the binomial series twice. Based on the result of the velocity profile, the analytical solution of the temperature profile is also derived. A new model is deduced to predict the Poiseuille number in elliptical microchannels. The close-form solution of the Nusselt number is first obtained and can serve as a benchmark for other approaches.