Constant Fluid Research Articles

Dynamics of heat transfer in complex fluid systems: Comparative analysis of Jeffrey, Williamson and Maxwell fluids with chemical reactions and mixed convection

This study addresses the complex dynamics of heat transfer in magnetohydrodynamic (MHD) systems involving Jeffrey, Williamson, Maxwell, and Newtonian fluids, focusing on how chemical reactions, activation energy, porosity, and mixed convection impact fluid behavior. The problem is critical due to the significant influence these factors have on industrial processes and applications involving non-Newtonian fluids. The developed a mathematical model represented by partial differential equations (PDEs), which were solved using similarity transformations and the fourth order Runge-Kutta (R-K) method combined with shooting technique, with MATLAB software facilitating the solution process. The results reveal that variations in magnetic field strength, porosity, and buoyancy force significantly affect fluid velocities, while radiation, Brownian motion, and thermophoresis alter temperature profiles. Furthermore, chemical reaction rates, Schmidt number, relaxation constant, and activation energy influence fluid concentrations. Key findings include that increasing porosity and magnetic field strength generally decreases fluid velocity, while higher radiation and Prandtl numbers reduce temperature. Chemical reactions and activation energy decrease fluid concentrations, with non-Newtonian fluids showing more pronounced effects compared to Newtonian fluids. The novelty of this work lies in its comprehensive analysis of multiple interacting parameters and their combined effects on heat transfer in MHD systems, providing insights that extend beyond previous studies in the literature. This research offers valuable implications for optimizing fluid dynamics in various industrial applications, including food processing, ink formulation, and friction reduction.

Fault‐Valve Instability: A Mechanism for Slow Slip Events

AbstractGeophysical and geological studies provide evidence for cyclic changes in fault‐zone pore fluid pressure that synchronize with or at least modulate slip events. A hypothesized explanation is fault valving arising from temporal changes in fault zone permeability. In our study, we investigate how the coupled dynamics of rate and state friction, along‐fault fluid flow, and permeability evolution can produce slow slip events. Permeability decreases with time, and increases with slip. Linear stability analysis shows that steady slip with constant fluid flow along the fault zone is unstable to perturbations, even for velocity‐strengthening friction with no state evolution, if the background flow is sufficiently high. We refer to this instability as the “fault valve instability.” The propagation speed of the fluid pressure and slip pulse, which scales with permeability enhancement, can be much higher than expected from linear pressure diffusion. Two‐dimensional simulations with spatially uniform properties show that the fault valve instability develops into slow slip events, in the form of aseismic slip pulses that propagate in the direction of fluid flow. We also perform earthquake sequence simulations on a megathrust fault, taking into account depth‐dependent frictional and hydrological properties. The simulations produce quasi‐periodic slow slip events from the fault valve instability below the seismogenic zone, in both velocity‐weakening and velocity‐strengthening regions, for a wide range of effective normal stresses. A separation of slow slip events from the seismogenic zone, which is observed in some subduction zones, is reproduced when assuming a fluid sink around the mantle wedge corner.

Experimental investigation of ground-source heat pump system coupled to vertical and horizontal ground loops: A case study

This study presents an experimental investigation of the performance of the ground-source heat pump (GSHP) system coupled to both vertical and horizontal ground loops under four different scenarios in cold climate. The first two tests evaluated the performance of the two vertical ground loops coupled to the GSHP in both heating and cooling modes. The results showed that the COP of the GSHP ranged from 2.7 to 3.15 during the heating mode, while in cooling mode, the COP ranged from 3.75 to 5.4. The GSHP demonstrated a relatively constant performance during both tests, which could be attributed to a relatively stable ground temperature. The last two tests aimed to compare the performance of horizontal and vertical ground loops in cooling mode. In the first case, the ground loop flowrate was divided equally between the vertical and horizontal loops, resulting in an overall GSHP COP of 5.42. In the second case, equal Reynolds number was maintained in both loops, resulting in an overall COP of 5.36. The high COP observed in the last two cases was attributed to the constant low fluid temperature supplied by the ground loops.

Influences of Newtonian heating and Darcy's law in micropolar fluid flow over a magnetized stretchable disk: A Bayesian analysis

A laminar two-dimensional magnetized micropolar fluid flow passed through a stretchable surface is examined. The axi-symmetric time-independent thermal flow experienced the impacts of Darcy's law, thermal conductivity, and Newtonian heating. The dimensionalized flow model is governed by similarity transformations. The resulting non-linear flow system is numerically solved through Runge-Kutta-Fehlberg (RKF-45) built in scheme. The graphics illustration of velocity and thermal fields for multiple physical parameters is presented. The numerical values at the disk surface for couple stresses, thermal rate, and shear stresses are also calculated. A Bayesian approach is used to assess the association degree amongst the under-study constraints and variables. The magnitude of the microrotational profiles is enhanced by the combined effect of micropolar parameters while an opposite effect is observed in microrotational field against magnetic and porosity parameters. The temperature field is modified through enhancing values of conjugate parameters. Moreover, such thermal field is larger for variable thermal conductivity as assumed to constant fluid property.

Dynamic Characteristics of Pressure-Balanced Metal Bellows in Fluid–Structure Interaction

As a flexible component in high-pressure vessels and pipeline systems, bellows experience significant fluid–structure interaction effects under high-speed internal fluids and external vibrations. Nevertheless, their dynamic response mechanisms coupled with fluid–structure interaction mentioned below have not yet been clarified so far. In this work, a novel pressure-balanced metal bellow (PBMB) for low-stiffness and high-pressure resistance is firstly proposed. Several fluid–structure interaction models were considered to study the dynamic response characteristics of the PBMB. An experimental platform associated with fluid–structure interaction was established to validate the effectiveness of its vibration attenuation performance. The results indicate that the PBMB has an obvious vibration attenuation effect in the range of 5–90[Formula: see text]Hz, and super-harmonic and sub-harmonic resonance phenomena occur in the range of 90–200[Formula: see text]Hz. Under constant fluid conditions, fluid density, viscosity, flow velocity, and pressure are positively correlated with the response amplitude of the PBMB. The response of the PBMB oscillates at the fluid entry point due to both pulsating flow velocity and pulsating pressure. After several cycles, the response caused by pulsating flow velocity gradually decays and stabilizes. Thus, the impact of pulsating frequency on the stability of the response of bellows is insignificant during the initial cycles.

Slip and stress in block-in-matrix shear zones: 2. numerical modelling of a serpentine-filled dilational jog

We use 2-D numerical models to explore slip and stress dynamics in a fault-bound dilational jog, informed by a field example from the Dun Mountain Ophiolite in New Zealand. The jog is hosted in a metre-scale phacoid of massive serpentinite embedded in a “block-in-matrix” creeping serpentinite shear zone. The models show how periodic exceedance of the tensile strength of the contact between the sealed jog and host rock leads to episodic opening and deposition of a new crack-seal band, with a thickness limited by the release of stress around the tips of the stepover-bounding faults. Jog stress release in the model is lower than that predicted from linear elastic fracture mechanics because additional crack opening can occur slowly due to post-failure creep on the bounding faults. For 10 km overburden and constant high fluid pressure, we predict event stress release of ca. 16 MPa and total crack opening of ∼22 μm on either side of the jog, consistent with crack-seal band widths in the field example. Because our models show that total crack width reflects both initial cracking and subsequent creep, we suggest caution when using crack-seal band widths to directly infer stress release in similar shear zone settings.

Influence of solid cylinders on fluid flow and thermal analysis in a curved channel with constant magnetic field

The current study explores the numerical simulation of constant mixed convection heat transfer and fluid flow within a channel, driven by the inlet velocity. The investigation focuses on a curved irregular channel where the presence of three differently sized circular obstacles introduces an unpredictable element to the heat transfer process. For the two-dimensional physical scenario, mathematical equations are formulated to analyze velocity and temperature. Employing appropriate dimensional variables, the coupled partial differential equations are transformed into dimensionless form. Subsequently, the Finite Element Method (FEM) is employed to simulate the results for non-dimensional physical parameters such as Reynolds number, Richardson number, Hartmann number, nanoparticles' volume fraction, and various states of the circular obstacles on heated lines, flow structure, velocities, and temperatures at mean positions generating isotherms, streamlines, velocity, and temperature profiles for various parameter increments. The analysis reveals significant influences of Reynolds number (Re) and heated circular obstacles on the temperature profile, manifesting as heated lines within the system. The analysis brings to light the substantial impacts of nanoparticles on the temperature profile at both the vertical and horizontal mean positions. An elevated Hartmann number signifies a more robust magnetic field than viscous forces. A heightened magnetic field can hinder fluid movement and modify the patterns of heat transfer. Based on the graphical results, we infer that temperature differences tend to arise between strata as the Richardson number increases, leading to a heightened and more evident stratification.

Development and Investigation of Nitrate Phase Change Filler Material for Solar Process Heating Applications

Thermocline thermal energy storage system is an imperial system for solar process heating applications. The thermal energy storage system comprises filler material and heat transfer fluid. In the present study, the commercial phase change material (PCM) (filler material) solar salt (60%NaNO3+40%KNO3) is synthesized, and it is encapsulated with a 1mm thickness of stainless steel 316L encapsulation with the macro-size of 60mm; the encapsulated PCMs is considered as a single particle in the thermocline thermal energy storage system. The performance of the single encapsulated PCM is studied with a constant heat transfer fluid (Therminol-VP1) flow rate of 0.00015 kg/s during the charging process. The specific heat of the solar salt is measured in Differential Scanning Calorimeter (DSC) in the solid phase is found as 1.381 J/gK, and liquid phase is identified as 1.551 J/g-K, the enthalpy during the phase change process is found as 116.72 J/g. As well, the thermal conductivity of the solar salt is measured in Transient Heat Conduction method, and the thermal conductivity follows the linear relationship of -0.0025(T)+0.4023 (45ºC<T<100ºC) in the sensible region. The temperature variations and the phase change phenomena for the solar salt is identified and the contours are provided. The charging time for the encapsulated solar salt is identified as ~138 minutes. The energy storage cost for the single filler material (Solar Salt + SS316L encapsulation) is identified as Rs.5.93/- (USD 0.071) for the period of charging duration.

Open-source spring-driven syringe pump with 3D-printed components for microfluidic applications

The operation of microfluidic devices requires precise and constant fluid flow. Microfluidic systems in low-resource settings require a portable, inexpensive, and electricity-free pumping approach due to the rising demand for microfluidics in point-of-care testing (POCT). Open-source alternatives, employing 3D printing and motors, offer affordability. However, using motors require electrical power, which often relies on external sources, hindering the on-site use of open-source pumps. This study introduces a spring-driven, 3D-printed syringe pump, eliminating the need for an external power source. The syringe pump is operated by the flat spiral spring’s torque. By manually winding up the mainspring, the syringe pump can be operated without electricity. Various flow rates can be achieved by utilizing different syringe sizes and choosing the right gear combinations. All the parts of the syringe pump can be fabricated by 3D printing, requiring no additional components that require electricity. It operates by winding a mainspring and is user-friendly, allowing flow rate adjustments by assembling gears that modulate syringe plunger pushing velocity. The fabrication cost is $25–30 and can be assembled easily by following the instructions. We expect that the proposed syringe pump will enable the utilization of microfluidic technologies in resource-limited settings, promoting the adoption of microfluidics. Detailed information and results are available in the original research paper (https://doi.org/10.1016/j.snb.2024.135289).

Modeling and mitigation of vortex formation in ejector deep hole drilling with smoothed particle hydrodynamics

Ejector deep hole drilling achieves high-quality boreholes in production processes. High feed rates are applied to ensure a high productivity level, requiring reliable chip removal from the cutting zone for a stable process. Therefore, a constant metalworking fluid flow under high volume flow rates or high pressure is required. Experimental results show a vortex formation at the outer cutting edge. This vortex can lead to delayed chip removal from the cutting zone, and ultimately, it can lead to chip clogging and result in drill breakage due to increased torque. This paper investigates modified drill head designs using the smoothed particle hydrodynamics method. The investigated modifications include various designs of the chip mouth covering. Besides graphical analysis based on flow visualizations, flow meters are placed at the tool’s head to evaluate the impact of the modifications on the flow rate and possible increased resistance and relocation of the fluid flow from the outer cutting edge to other parts of the tool. The simulation results for the reference design show the experimentally observed vortex formation, validating the simulation model. By adding the tool’s rotation in the SPH simulation, which is not included in the experiments for observation reasons, the vortex formation is positively influenced. In addition, some designs show promising results to further mitigate the vortex formation while maintaining a sufficient fluid flow around the cutting edges.

Computational analysis of the bioconvective flow of Williamson–Sutterby fluid with microorganisms and activation energy subject to Cattaneo–Christov double-diffusion effects

This research aims to develop a mathematical model and outline the implications of the free convection flow within a non-Newtonian Williamson–Sutterby type nonfluid through a stretching surface placed horizontally with magnetic dipole effect. The present attempt establishes the influence of multiple factors, including double diffusion, and heat flux. Utilizing the Cattaneo–Christov heat mass flux model approximations, to shape the thermal and concentration balance equations, incorporating various generalized transport laws. Furthermore, the transportation mechanism incorporates Arrhenius activation energy and binary chemical reactions. The impact of bioconvection resulting from self-propelled microorganisms is incorporated into the fluidic model. A nonlinear set of partial differential equations (PDEs) are appropriately formulated, employing boundary layer approximations theory, to comprehensively describe the convective flow. The PDEs are converted to ordinary differential equations via similarity transformation, and then computationally computed via a well-structured RKF45 technique with a shooting algorithm. To validate the results, a comparison is made with published findings in limiting cases. It is observed that the velocity profiles exhibit an escalation in tandem with ascending values of the electric parameter and mixed convection. Conversely, the velocity diminishes with the augmentation of the magnetic factor, ferrohydrodynamic interaction parameter, porosity parameter, and Sutterby fluid parameter. Temperature profiles elevate in response to escalating values of the Eckert number, radiation parameter, and Brownian motion parameter while diminishing with the augmentation of the thermal relaxation constant, Prandtl number, and Sutterby fluid parameter. The concentration distribution exhibits an augmentation with the increasing magnitude of the thermophoresis parameter and activation energy while experiencing a reduction with the improvement in the behavior of Schmidt number and Brownian parameter. The ongoing investigation encompasses a wide spectrum of applications within the field of applied sciences, placing particular emphasis on thermal oil recovery, geothermal reservoirs, chemical engineering, and the cooling processes pertinent to nuclear reactors.

Computing Relative Permeability and Capillary Pressure of Heterogeneous Rocks Using Realistic Boundary Conditions

Relative permeability and capillary pressure are key parameters in multiphase flow modelling. In heterogeneous porous media, flow direction- and flow-rate dependence result from non-uniform saturation distributions that vary with the balance between viscous, gravitational, and capillary forces. Typically, relative permeability is measured using constant inlet fractional-flow—constant outlet fluid pressure conditions on samples mounted between permeable porous plates to avoid capillary end-effects. This setup is replicated in numeric experiments but ignores the extended geologic context beyond the sample size, impacting the saturation distribution and, consequently, the upscaled parameters. Here, we introduce a new workflow for measuring effective relative permeability and capillary pressure at the bedform scale while considering heterogeneities at the lamina scale. We harness the flexibility of numeric modelling to simulate continuum-REV-scale saturation distributions in heterogeneous rocks eliminating boundary artefacts. Periodic fluid flux boundary conditions are applied in combination with arbitrarily oriented, variable-strength pressure gradient fields. The approach is illustrated on a periodic model of cross-bedded sandstone. Stepping saturation while applying variable-strength pressure-gradient fields with different orientations, we cover the capillary-viscous force balance spectrum of interest. The obtained relative permeability and capillary pressure curves differ from ones obtained with traditional approaches highlighting that the definition of force balances needs consideration of flow direction as an additional degree of freedom. In addition, we discuss when the common viscous and the capillary limits are applicable and how they vary with flow direction in the presence of capillary interfaces.

Experimental and Numerical Investigation of Single-Phase Liquid Cooling for Heterogeneous Integration Multichip Module

Abstract The variety of new electronic packaging technologies has grown significantly over the last 20 years as a result of market demands for higher device performance at lower costs and in less space. Those demands have pushed for heterogeneous packaging, where computer chips with different stack heights are closely packed, creating nonuniform heat flux and temperature and additional challenges for thermal management. Without implementing an appropriate thermal management strategy for heterogeneous packages, large temperature gradients can be observed within the package, which would increase the thermal stresses on the chip and raise reliability issues. To mimic this real-life scenario of such packaging, an experimental setup was designed and built. The design of the new experimental setup consists of four identical 1.2 cm × 1.2 cm ceramic heaters, each of which is connected to a separate power supply and can reach a heat flux of 140 W/cm2. Accordingly, this mock package is capable of delivering different power levels to mimic different multicore microprocessor conditions. To give the heater the ability to move precisely in the x-, y-, and z-directions, each heater is mounted to an XYZ linear stage. Deionized water (DI) was used as the working fluid, and a pin-fin heat sink was used to run the initial steady-state tests on the experimental rig. The tests showed how different flow rates at a constant fluid temperature and input power affect the temperatures of the heaters and the thermohydraulic performance of the heat sink. In addition, a three-dimensional numerical model has been developed and validated with experimental data in terms of heat sink pressure drop and the temperatures of the heaters.

Powder-bed 3D printing by selective activation of magnesium phosphate cement: Determining significant processing parameters and their effect on mechanical strength

The present work addresses powder bed binder jetting additive manufacturing by selective magnesium phosphate cement activation. Despite the potential of this technology to aid the digitalization of the construction industry, the effect of processing parameters on the mechanical performance of printed materials has not yet been studied to generate a guideline for the further development of the technology. Statistical methodologies were used to screen the effect of four printing process parameters (printing speed, layer thickness, raster angle, and build direction on flexural and compressive strength). As the exploited technology works with constant fluid pressure, the physical interpretation of the effect of each factor can be considered taking into account the interactions between the binder materials in the powder bed. Analysis of variance (ANOVA) indicated that printing speed and layer thickness significantly affect mechanical performances. Furthermore, the layout of samples for the printing process is preferable to be parallel the printhead movement. An anisotropic behavior was observed, and the samples subjected to compressive forces parallel to the layer plane possessed lower strength values. This effect can be interpreted as a result of a weak area of low density in between layers, leading to a pronounced delamination under compression.Even though the strength of the printed material is not suitable for a structural concrete, it can be marginally improved by design of experiment and optimized for non-structural applications, such as for porous artificial stone. Design of experiment coupled with ANOVA methods can be used in the future to support the development of novel material mixtures, thus expanding the fields of application of this novel additive manufacturing technology.

FULL FACTORIAL EXPERIMENTAL DESIGN AND COMPUTER SIMULATION APPLIED TO SHELL AND TUBE HEAT EXCHANGER SETUP

In the operation of heat exchangers there are some variables to be controlled, making it difficult to found optimized parameters. The aim of this study was to compare the experimental and simulated values of the outlet temperatures, as well as to understand the influence of operating variables for the equipment. It was used a shell and tube heat exchanger didactic module, with a constant cold fluid flow rate equal to 1.4 L.min 1. The experiments were carried out on the basis of a 2² full factorial experimental design with central points, as well as computer simulations in the steady state and transient regime. The higher values for heat exchange overall heat transfer coefficient determined was around 250 W.m 2.K 1. Thus, the flow regime affects the evaluated response. In addition, the computer simulation in the permanent regime presented less relative deviation. Therefore, it can be seen that although the simulations show results close to the experimental ones, there are still associated errors that should be studied and minimized, since factors such as bubble formation were not considered in the simulations. Thus, it was found that computer simulations can be used to understand the operation of heat exchangers, but they are limited to real phenomena that are not considered in theoretical mathematical models. Therefore, this study elucidates the application of statistical and computer-assisted methods as a tool to comprehend heat exchangers behavior for industrial and didactic purposes.

Effects of particle creation rate in two-fluid interacting cosmologies

ABSTRACT In this work, a two-fluid interacting model in a flat FLRW universe has been studied considering particle creation mechanism with a particular form of particle creation rate $\Gamma =\Gamma _0 H+\frac{\Gamma _1}{H}$ from different aspects. Statistical analysis with a combined data set of SNe Ia (Supernovae Type Ia) and Hubble data is performed to achieve the best-fitting values of the model parameters, and the model is compatible with current observational data. We also perform a dynamical analysis of this model to get an overall qualitative description of the cosmological evolution by converting the governing equations into a system of ordinary differential equations considering a proper transformation of variables. We find some non-isolated sets of critical points, among which some usually are normally hyperbolic sets of points that describe the present acceleration of the universe dominated by dark energy mimicking cosmological constant or phantom fluid. Scaling solutions are also obtained from this analysis, and they can alleviate the coincidence problem successfully. Statefinder diagnosis is also carried out for this model to compare it with the ΛCDM, and any other dark energy models byfinding various statefinder parameters. Finally, the thermodynamic analysis shows that the generalized second law of thermodynamics is valid in an irreversible thermodynamic context.

Investigation of heat and mass transfer of oldroyd – 8 constant fluid in wire coating process employing response surface methodology

Wire coating is widely used for electrical insulation, shock protection, preventing leakage, and facilitating smooth current flow. This study aims to evaluate the applicability of Oldroyd-8 constant fluid for wire applications in a magnetic field using a pressure-type die. Based on the Buongiorno model with a constant pressure gradient and temperature-dependent thermophysical properties, governing equations are developed. These equations are transformed into dimensionless form and solved numerically. The study thoroughly investigates key features, including momentum, thermal distribution, nanoparticle volume fraction, and shear stress rate. Also, this study employs Response Surface Methodology to optimize heat transfer and shear stress rates in coated wire. Quadratic models, developed through a central-composite design, determine optimal levels for the Oldroyd-8 constant fluid parameter and nanofluid parameters, ensuring optimal heat transfer and shear stress rates for the melt. Nanofluid parameters contribute to an improvement in the thermal distribution profile. Optimal heat transfer and shear stress rate occur when the nanofluid parameters are minimized, and non – Newtonian fluid parameter is maximized.

Non-Fourier Heat Flux Model for the Magnetohydrodynamic Casson Nanofluid Flow Past a Porous Stretching Sheet using the Akbari-Gangi Method

The Casson fluid flow with porous material in magnetohydrodynamics is examined in this work. Additional semi-analytical results are investigated using the Silver-Water nanofluid. The Akbari-Ganji Method (AGM) is used to solve the semi-analytical Cattaneo-Christov heat flux model after taking thermal radiation into account. With the use of appropriate parameters, such as the relaxation time parameter, Prandtl number, radiation parameter, magnetic parameter, and so on, the normalized shear stress at the wall, temperature profile, and rate of heat flux may be examined. This issue has numerous industrial applications and technical procedures, such as the extrusion of rubber sheets and the manufacture of glass fiber. The main physical application is the discovery that a rise in the thermal relaxation parameter and Prandtl number maintains a constant fluid temperature.

Path and Slip Dependent Behavior of Shallow Subduction Shear Zones During Fluid Overpressure

AbstractElevated pore fluid pressure is proposed to contribute to slow earthquakes along shallow subduction plate boundaries. However, the processes that create high fluid pressure, disequilibrium compaction and dehydration reactions, lead to different effective stress paths in fault rocks. These paths are predicted by granular mechanics frameworks to lead to different strengths and deformation modes, yet granular mechanics do not predict their effects on fault stability. To evaluate the role of fluid overpressure on shallow megathrust strength and slip behavior, we conducted triaxial shear experiments on chlorite and celadonite rich gouge layers. Experiments were conducted at constant temperature (130 and 100°C), confining pressure (130 and 140 MPa), and pore fluid pressures (between 10 and 120 MPa). Fluid overpressure due to disequilibrium compaction was simulated by increasing confining and pore fluid pressure synchronously without exceeding the target effective pressure, whereas overpressure due to dehydration reactions was simulated by first loading the sample to a target isotropic effective pressure and then increasing pore fluid pressure to reduce the effective pressure. We find that the effects of fluid pressure and stress path on the mechanical behavior of the chlorite and celadonite gouges can generally be described using the critical state soil mechanics (CSSM) framework. However, path effects are more pronounced and persist to greater displacements in chlorite because its microstructure is more influenced by stress path. Due to its effects on microstructure, the stress path also imparts greater control on the rate‐dependence of chlorite strength, which is not predicted by CSSM.

Determination of the diameter variation peculiarities along the path of distribution drainage pipe-line

This article presents a methodology for calculating the hydraulic characteristics and structural parameters of variable (decreasing) cross-sectional distribution pressure drainage pipelines. The approach is based on analyzing a system of differential equations describing fluid flow with a variable flow rate in the investigated pipes. The study explores the most general case of distributor operation, considering a horizontal groundwater level above the pipeline. The analysis of the original equations includes a scenario where a constant average fluid flow rate is maintained along the length of the pressure channel. The velocity magnitude is influenced by the granulometric features of the surrounding soil and the filtration characteristics of the pipeline's side walls material. The investigation employs the concept of a conditional distribution infinitely long pipeline or a pipeline with limited length and infinite permeability of the side walls surface. This allows assuming a pressure close to zero at the pipeline's final cross-section, which minimally impacts the obtained results. The analysis is presented in a dimensionless form, and the resulting computational formulas are both simple and convenient. The article includes corresponding graphs to illustrate the presented dependencies.