Numerical Approaches to the Mathematical Analysis of Physical Property
Transfer in Newtonian Fluid Flow
Deepak Chaudhary1*,
Dr. Sena Pati Shukla2
1 Research Scholar, P.K. University Shivpuri,
Madhya Pradesh
dk712144@gmail.com
2 Assistant Professor, Department of Mathematics,
P K. University, Shivpuri, Madhya Pradesh
Abstract: The study of Newtonian fluid
flow is fundamental in understanding numerous physical, engineering, and
industrial processes. The transfer of physical properties such as momentum,
heat, and mass within these fluids plays a critical role in fluid dynamics.
This review focuses on the mathematical modeling of such transfer phenomena in
Newtonian fluids using various numerical methods. The paper presents an
overview of the governing equations, such as the Navier-Stokes equations and
energy conservation principles, and emphasizes the significance of numerical
techniques like Finite Difference Method (FDM), Finite Element Method (FEM),
and Finite Volume Method (FVM) in solving these complex models. Particular
attention is given to recent advances in computational fluid dynamics (CFD),
stability analyses, and convergence studies that enhance the accuracy and
efficiency of simulations. This review aims to provide a consolidated
understanding of how numerical methods contribute to the precise prediction of
physical property transport in Newtonian fluid systems.
Keywords: Newtonian Fluid, Physical
Property Transfer, Numerical Methods, Finite Difference Method, Finite Element
Method, Finite Volume Method, Computational Fluid Dynamics, Heat Transfer, Mass
Transfer, Momentum Transport
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INTRODUCTION
The mathematical modeling of Newtonian fluid flow is a critical area of fluid dynamics with wide-ranging applications in engineering, chemical processing, environmental studies, and biological systems. Newtonian fluids, which exhibit a linear relationship between shear stress and strain rate, include commonly encountered substances such as water, air, and oil. Understanding how physical properties like momentum, heat, and mass are transferred within these fluids is essential for designing efficient systems and optimizing various industrial processes.
The complexities involved in fluid flow equations make analytical solutions difficult, especially when dealing with real-world boundary conditions and geometries. Numerical methods have emerged as indispensable tools for addressing these challenges. This review provides a detailed overview of the mathematical formulation of property transfer in Newtonian fluids and the numerical strategies employed to solve these problems.
MATHEMATICAL
FRAMEWORK OF NEWTONIAN FLUID FLOW
Governing
Equations
The behavior of Newtonian fluids is described by the Navier-Stokes equations for momentum conservation, along with the continuity equation for mass conservation and the energy equation for heat transfer. These can be summarized as:
Here, 
is fluid density, 
is velocity vector, 
is pressure, μ
is dynamic viscosity, T is temperature, and S is an energy source term.
MAGNETOHYDRODYNAMICS
(MHD)
A magnetic field's effect
on fluid flow may be seen in many contexts. The use of a magnetic field
facilitates the processing of metals in liquid form. A thermonuclear fusion
reactor uses a magnetic field to separate the heated plasma from the reactor's
walls. Fluids that conduct electricity, such as electrolytes, plasma, liquid
metals, and many more, are the focus of magnetohydrodynamics (MHD). The main
idea behind MHD theory is that when a fluid is moving through a magnetic field,
it creates an electric current. This current, in turn, causes the Lorentz force
to act, altering the fluid's path. In 1832, the English physicist Michael
Faraday conducted the first ever documented investigation on MHD. The electric
current caused by the Thames River's velocity in Earth's magnetic field was
something Faraday aimed to quantify. Unfortunately, the experiment did not
succeed since galvanometers of that era had a very limited capacity for
detection. After Swedish electrical engineer Hannes Alfv'en discovered Alfv'en
waves in 1942, MHD became well-known among scholars. According to Davidson
(2016) and Hosking and Dewar (2016), Alfv'en waves are a kind of wave-like
oscillation that occur in magnetic field lines in a fluid that is very
electrically conductive. In 1970, Alfv'en was awarded the Nobel Prize in
Physics for his tremendous and priceless contributions to MHD. Among the many
fields that make use of MHD are those dealing with MHD power generators,
magnetic drug targeting, magnetic endoscopy, nuclear reactor maintenance, solar
flare and sunspot research, geomagnetic storm analysis, crude oil refineries,
and MHD pump and brake design.
HALL
CURRENT
American
physicist Edwin Herbert Hall, while completing his doctorate at the esteemed
Johns Hopkins University in Baltimore, Maryland, USA, in 1879, discovered the
Hall effect. Within a transverse magnetic field, charges on a current-carrying
conductor are redirected in an orthogonal direction to the electric current and
the magnetic field. Therefore, the Hall current flows because a secondary
potential difference is produced. When dealing with weak ionised gas densities
or strong magnetic fields, the Hall effect becomes practically relevant.
NEWTONIAN
FLUID
According
to Newton's law of viscosity, a fluid that is considered Newtonian will have a
stress that is directly proportional to the strain rate. Newtonian fluids
include items like air, water, petrol, glycerol, and mercury, among others.
NON-NEWTONIAN
FLUID
The
stress-rate relationship in non-Newtonian fluids is nonlinear and depends on
the strain rate. The Newtonian model does not apply to a wide variety of fluids
used in industry and biology, including blood, gypsum paste, adhesives,
lubricants, gels, paints, polymers, and synovial fluid. At a constant pressure
and temperature, the apparent viscosity of a non-Newtonian fluid changes
depending on the flow shape, strain rate, and deformation history (Chhabra and
Richardson, 2008). Various non-Newtonian fluid types and Newtonian fluids are
shown in Fig. 1 rheogram.
Figure 1:
Rheogram of Newtonian and non-Newtonian fluids.
The
apparent viscosity of pseudoplastic fluids, sometimes called shear-thinning
fluids, decreases as the strain rate increases. Some fluids have shear-thinning
properties; they include hair wash, paint, and ketchup. When it comes to
apparent viscosity, shear thickening/dilatant fluids show the inverse pattern.
Instances of shear-thickening fluids include cornstarch-water mixtures, wet
sand, and china clay suspensions. For a Bingham plastic fluid to behave like a
Newtonian fluid, the applied shear stress must be greater than a certain
threshold value, called the yield stress. The rheological properties of
non-Newtonian fluids are investigated in this thesis using the Casson fluid,
second-grade fluid, and Walters B′ fluid models circulation of fluids.
HEAT
TRANSFER
When
there is a difference in temperature between two objects, this is called heat
transfer. The three main processes that cause this heat exchange are radiation,
convection, and conduction, as shown in Figure 2. Intermolecular interactions
allow for the passage of heat from a substance's more energetic molecules to
its less energetic ones; this process is known as conduction.
Figure 2:
Modes of heat transfer.
Mass
transfer describes the movement of stuff from a densely populated area to an
open one. When an alien species is present in a fluid, it creates a
concentration gradient that makes mass transfer easier. One example is the
colour shift that occurs when a little quantity of potassium permanganate is
added to water, turning it purple. The scent of perfume wafts around the room
when the wearer applies it. Another example of mass transfer is the process of
dissolving sugar in lemonade, which requires a teaspoon. Absorption of
nutrients into the circulation, blood oxygenation, ion movement via osmosis,
water purification, distillation, and many other biological and chemical
processes rely on mass transfer. Both diffusion and convection play a role in
the movement of mass through fluids. The species shown in Fig. 3 undergo
diffusive mass transfer as a result of molecular mobility.
The
transmission of physical attributes such as momentum, heat, and mass in
Newtonian fluid flow involves substantial mathematical and computational
hurdles. The Navier-Stokes equations in particular are notoriously difficult to
solve analytically, especially when dealing with complex boundary conditions
and geometries, despite the fact that Newtonian fluids are simplified by the
assumption of constant viscosity. When applied to real-world scenarios,
traditional analytical approaches fail to provide adequate solutions.
Consequently, trustworthy numerical approaches must be used and developed to
adequately manage these intricacies and provide precise forecasts of fluid
behaviour and property transfer.
Figure 3: Propagation of dye molecules by
diffusion
Conversely, convective mass
transfer describes the situation where the species spreads together with the
fluid's bulk flow. The glucose produced by plants during photosynthesis is
transported to different sections of the plant by the water absorbed by the soil.
Because of its relevance in many fields, including environmental engineering,
electrical engineering, metallurgy, refrigeration, pharmaceutical drug
development, chemical constituent separation in distillation apparatus, HVAC
system design, and automobile engineering, the study of MHD flow with heat and
mass transfer has captivated many researchers.
CROSS-DIFUSION
BFECTS
It
is common for a concentration gradient to cause mass diffusion and a
temperature gradient to cause heat diffusion. Mass diffusion occurs as a result
of temperature gradients and vice versa when there are high concentration and
temperature gradients (Kafoussias and Williams, 1995). The Soret effect is
defined as the mass flow that results from a temperature gradient, while the
Dufour effect is defined as the heat flux that results from a concentration
gradient.
Swiss
physicist Charles Soret discovered the Soret effect in 1879. Two tubes
containing sodium chloride (NaCl) and potassium nitrate (KNO3) solutions were
placed in front of him for experimental observation. The tubes were either
straight or U-shaped. Both tubes had one end in a bath of icy water and the
other in a pot of boiling water. The experiment proved the existence of the
Soret effect by showing that the solute settled to the bottom of the tubes as
they were chilled. Species migrate from warm to cold areas as a result of the
positive Soret effect, and vice versa as a result of the negative Soret effect
(Platten, 2006). A favourable Soret effect is more often seen in heavier
species, whereas a negative Soret impact is more commonly seen in lighter
species. In 1873, the Dufour effect was discovered by a Swiss scientist called
L. Dufour. As an example, the Dufour effect would cause a temperature increase
of several degrees in a combination of hydrogen and nitrogen, which are held at
various concentrations. Rowley and Horne (1980) conducted experimental studies
on the Dufour effect in a combination of cyclohexane and carbon tetrachloride.
Gaseous mixes of intermediate molecular mass (Nitrogen and air) exhibit the
Dufour effect, according to Eckert and Drake (1972), but combinations of gases
of little molecular mass (Hydrogen and helium) exhibit the Soret effect.
Isotope separation, fuel cell design, nuclear reactor engineering, petrology,
and gas impurity removal are just a few of the many engineering fields that
make use of the Soret-Dufour phenomena, which represent a cross-diffusive
coupled heat and mass transfer in fluid flow.
CHEMICAL
REACTION AND ACTIVATION BNERGY
In
a chemical reaction, bonds in the reactant species are broken, allowing bonds
in the product species to be formed. The concentration of the species reacting
determines the pace of the chemical reaction. When the concentration of the
reactant species is directly proportional to the first power of the reaction
rate constant, we say that the reaction is of first order.
Figure 4:
Activation energy.
There
is a tight relationship between the activation energy and the reaction rate in
chemistry. Chemical reactions may only take place when the reactants reach a
certain threshold energy, according to Swedish scientist Svante Arrhenius's
1889 discovery. It was Arrhenius who first used the word "activation
energy" to describe the bare minimum of energy that molecules of the
reactants needed to start a chemical reaction. In order for the reactants to
collide and start forming product species, they must overcome this energy gap,
as depicted in Fig. 4.
The
rate constant, activation energy, and temperature are all variables in the
Arrhenius equation, which, as stated by Tencer et al. (2004), is:
Where A is the Arrhenius pre-exponential
factor, Ea is the activation energy, and kB = 8.61 × 10−5
eV/K is the Boltzmann constant. Numerous fields, including
chemical engineering, geothermal engineering, food processing, oil emulsions,
polymer manufacture, and ceramic or glassware production, benefit from studying
MHD flow in relation to chemical reaction and activation energy.
The
solid matrix of a porous material has interconnected spaces called pores. As
shown in Figure 5(a), the fluid is able to move across the medium thanks to the
web-like network formed by the linked pores. Porous media include things like
sand, wood, limestone, furnace refractory material, sponges, fabric,
fibreglass, ceramics, and even human lungs. As shown in Figure 5(b), pumice is
a very porous igneous rock that forms when lava is quickly cooled and
depressurised after being expelled during a volcanic eruption. To measure
porosity, take the volume of the pores and divide it by the volume of the medium.
The porosity value for naturally porous medium is less than or equal to 0.6
(Nield and Bejan, 2017).
Figure 5:
(a) Porous medium and (b) Pumice.
Most
flows occurring below ground have Reynolds numbers that fall within the Darcy s
range. On the other hand, in cases with steep gradients or with big solid
particles in the flow, the fluid's trajectory takes a curved shape, which
causes it to experience inertial acceleration. It follows that a non-Darcy
equation must be used, taking into account both the inertia caused by the
high-velocity flow and the typical stress caused by the distortion velocity. As
a result, a number of models that do not include Darcy have been developed. The
Darcy-Forchheimer law, which was proposed by Forchheimer (1901) and is expressed
as the quadratic form-drag in Darcy's law (Eq), is:
Where CF is the non-dimensional
form-drag coefficient. The study of convective
flow in porous media is useful in many different contexts, including but not
limited to: groundwater hydrology, magnetic-levation casting, geothermal
reservoirs, decontamination processes, alloy solidification, microporous heat
sinks, blast furnaces, and catalytic converters.
NANOFLUID AND HYBRID NANOFLUID
The
use of effective coolants is essential for the upkeep of complex machinery and
industrial equipment. Power generation, chemical manufacture, food processing,
automobile engines, nuclear reactors, textile production, and microelectronics
are just a few examples of industries that have a significant need for effective
cooling techniques. The rate of heat transfer between the solid surface and the
fluid also has a significant impact on the produced product's quality. Metals,
oxides, and carbon nanotubes (CNTs) often have a much higher thermal
conductivity than more traditional base fluids such as water (H2O), engine oil
(EO), and ethylene glycol (EG). Recent developments in nanotechnology have made
it possible to enhance the base fluid's heat transmission properties by adding
nanoparticles. One kind of nanoparticle is colloidally suspended in the base
fluid of a nanofluid, making it an increased heat transfer fluid. The
groundbreaking study on the improvement of heat transmission in Cu − H2O
nanofluid was conducted by Choi and Eastman (1995), who also established the
term "nanofluid". The single-phase and double-phase models are the
main tools for studying nanofluid flow. According to the single-phase model,
the base fluid and nanoparticles move at the same speed because of thermal
equilibrium. According to Albojamal and Vafai (2017), the nanofluid's heat
transfer properties may be altered by focussing on the two phases of the model:
the solid nanoparticle phase and the fluid phase. The rate of heat transfer in
the cavity flow of Cu − H2O nanofluid was investigated by Tiwari and Das
(2007) using the single phase model. As a function of the volume percentage of
nanoparticles, the authors have given a detailed list of the thermo-physical
characteristics of nanofluid.
Researchers
have created a new kind of fluid called a hybrid nanofluid, which has several
kinds of nanoparticles mixed with the base fluid, to improve the thermo
physical characteristics of nanofluids. For the thermo-physical characteristics
of hybrid nanofluid, the correlations have been supplied by Devi and Devi
(2016). Due to their diverse applications in nano-drug administration, heat
exchangers, refrigeration, solar collectors, supercomputers, autos,
cryosurgery, cancer treatment, transformers, heat pumps, photoelectric
equipment, textile industries, geothermal power extraction, and nuclear reactor
maintenance, nanofluid and hybrid nanofluid flow research has recently
attracted the interest of many researchers.
CATTANEO-CHRISTOV DOUBLE DIFUSION MODEL
The Fourier slaw expresses
the heat flux (⃗q) as:
Whereas the mass flux (⃗j) is represented by the Fick s law as:
The
energy equation becomes parabolic when Fourier's heat flux and Fick's mass flow
are both included, and the species concentration equation becomes parabolic as
well. This means that an initial disturbance is instantly distributed over the
whole domain. In other words, the "paradox of heat conduction" holds
true, and the heat disturbance travels at an unlimited pace. Cattaneo (2019)
offered the Maxwell-Cattaneo (MC) law as a solution to this unrealistic
occurrence; this law incorporates the thermal relaxation time component into
the Fourier's heat flow and is expressed as:
Where λT is the thermal
relaxation time, which signifies the time lag for acquiring steady-state heat
conduction in a fluid element exposed to a sudden temperature gradient. Most
materials have a thermal relaxation time that is negligible, measured in
picoseconds at the most. As to Chandrasekharaiah (1998), components such as
sand (21s), glass ballotini (11s), ion exchanger (54s), H− acid (25s),
and NaHCO3 (29s) cannot be disregarded.
By
using the MC rule, the energy equation takes on a hyperbolic form, suggesting
that heat disturbances travel at a limited wave speed. Christov (2009) made an
additional adjustment by substituting the Oldroyd upper convected derivative
for the partial time derivative; this latter function generalises the MC rule
to a frame-independent extent. The
concurrent heat and mass diffusion represented via the Cattaneo-Christov theory
is termed the Cattaneo-Christov double diffusion model, and the respective
expressions for ⃗q and ⃗j
are (Shankar and Naduvinamani, 2019):
Where λC denotes the mass
relaxation time. According to Rehman et al. (2023), the
Cattaneo-Christov double diffusion scheme accurately represents the heat and
mass transport processes caused by concentrated and high-temperature gradients
as well as by fields of changing concentration and temperature.
BNTROPY GENERATION
Entropy
is created during an irreversible process, according to the second rule of
thermodynamics. Every time energy is transformed into usable work, some of that
energy is lost, reducing the system's efficiency. Because of the inherent
irreversibilities in thermodynamic processes, entropy formation occurs in
direct proportion to this energy deterioration. Entropy production in convective
heat transmission was the subject of Bejan's seminal research (1979). The
analysis takes into account the effects of conduction heat transfer and viscous
dissipation on the entropy development rate. When heat and mass transmission
are occurring at the same time, Mourad et al. (2006) examined how entropy is
generated. So far as we can tell from Mourad et al. (2006), the SG rate at the
local level is:
In where R stands for the
ideal gas constant. Here, heat transfer by conduction (first term), viscous dissipation
(second term), diffusive mass transfer due to concentration gradient alone
(third term), and cross-diffusive mass transfer due to temperature and
concentration gradients (fourth term) all contribute to entropy generation.
Improving the system's performance relies heavily on analysing entropy
creation, which aids in reducing the elements that degrade energy. Numerous
fields rely on entropy analysis, including biological systems, electronic
cooling, geothermal energy generation, solar collectors, magnetic
refrigeration, and solid-state physics.
BOUNDARY LAYER THEORY
German
physicist Ludwig Prandtl proposed the boundary layer hypothesis in 1904.
Because there is no frictional loss when a thin layer of a fluid with a low
viscosity runs over a solid surface that is not moving, the flowing fluid will
stick to the solid. Subsequent fluid layers are retarded by this stationary
layer, and the process continues thereafter in the same manner. But as we go
further away from the bounding surface in a normal direction, the retarding
effect on the fluid layers decreases. Consequently, a thin layer known as the
boundary layer forms close to the boundary surface, causing the fluid velocity
to increase from zero at the boundary surface and eventually approach the free
stream velocity (U∞(x)).
Rotation
and viscous forces have a major impact inside the boundary layer. The outer
layer is the area outside the boundary layer where the flow is inviscid and
irrotational. A high Reynolds number (Re) is required for a boundary layer to
form in a fluid. Since the viscous forces would be noticeable over the whole
flow domain if Re is minimal, the boundary layer would not be present. The
boundary layers for concentration and heat are described similarly when mass
and heat transfer are taking place at the boundary. Unlike in the situation of
inviscid Euler equations, boundary layer theory allows one to determine the
viscous drag force by specifying the no-slip condition at the solid border. In
addition, by analysing the components in terms of order of magnitude, the
unnecessary ones may be removed from the boundary layer flow Navier-Stokes
equations, making them simpler. Schlichting and Gersten provide a comprehensive
analysis of boundary layer theory and its practical applications (2017). For a
Newtonian fluid in two dimensions, the Prandtl boundary layer equations for a
steady incompressible flow without body forces are (Bansal, 1977):
SIMILARITY TRANSFORMATION
In a two-dimensional
boundary layer flow with velocity field V⃗
= (u(x, y), v(x, y)), the non-dimensional velocity component u/U∞(x) varies with only
one variable η =
y/g(x), where g(x) is proportional to the
bondary layer thickness (δ(x)), and η is called the similarity
variable (Som, et al., 2012; Bansal, 1977; Kundu and Cohen, 2004). This
change, known as the similarity transformation, converts the dimensional system
of PDEs into a non-dimensional system of ODEs that are conceptually identical.
German physicist Heinrich Blasius first proposed the similarity transformation in
1908 as a solution to the boundary layer flow issue across a flat plate. A
Newtonian fluid's continuous incompressible flow across a semi-infinite flat
plate with a constant free stream velocity (U∞) was something that
Blasius took into consideration.
SUCTION/INJECTION
To
manage the boundary layer's development, suction/injection applications are
critical. To keep the boundary layers from separating and to increase the lift
coefficient, suction is an essential tool in aerodynamic engineering. The
boundary layer flow, which counteracts the negative pressure gradient close to
the bordering surface, is drawn in by the slowed fluid molecules via suction.
To reduce the likelihood of boundary layer separation, fast-moving fluid
molecules counteract the slow-moving ones. In addition, the flow separation is
hindered by the kinetic energy delivered by the tangential injection of fluid
into the boundary layer. The impact of suction on an airfoil's aerodynamic lift
coefficient was studied by Shoejaefard et al. (2005). The findings show that
injection lowers the skin friction coefficient, whereas suction increases the
lift coefficient. Spacecraft re-entry, rocket engine propellants, ablative and
heat sink cooling, and transpiration cooling the process of sucking or injecting
fluid through porous surfaces have industrial and technical uses.
DIMENSIONAL ANALYSIS
Researching
flow behaviour on a full-scale model is a laborious, costly, and complex
process. For this reason, researchers often choose to test the prototype's flow
behaviour on a smaller scale before making any firm predictions about the
prototype's actual performance. The non-dimensional factors ensure that the
model flow and the prototypical flow are identical (White, 2011; Cengel and
Cimbala, 2014). It is common for fluid dynamics issues to have four main
dimensions: mass (M), length (L), time (t), and temperature (T). These
dimensions are used to construct the secondary dimensions. The dynamic
viscosity dimension, for instance, is given by 
An essential tool for non-dimensionalizing the
governing equations is dimensional analysis. By combining the issue's
dimensional parameters into a collection of non-dimensional parameters, we may
simplify the problem and do fewer trials. Adopting the Buckingham Pi theorem,
established by Buckingham (1914), yields the non-dimensional parameters.
Dimensional analysis is useful for a number of reasons, including deciphering
experimental results, tackling issues that are too complex for straightforward
techniques to handle mathematically, and determining the relevance of a
particular physical occurrence.
BOUSSINESQ APPROXIMATION
The
buoyant force components in the equations that regulate natural convective flow
are approximated using the Boussinesq method. According to Incropera et al.
(2006), the momentum equation is the only one that takes into account the
fluctuation of fluid density in incompressible natural convective flows.
BOUNDARY CONDITIONS
Boundary
conditions are set at the edge of the flow domain and are used to solve the
system of differential equations regulating the flow. Common boundary
conditions seen in fluid dynamics include Dirichlet, Neumann, and Robin's. At y
= 0, the circumstances at the solid surface are given, whereas at y →
∞, the ambient conditions are given.
NUMERICAL
METHODS FOR NEWTONIAN FLUID FLOW
Finite
Difference Method (FDM)
The finite difference method discretizes the partial differential equations using approximate difference equations. It is commonly applied to simple geometries and is widely used due to its straightforward implementation.
Finite
Element Method (FEM)
The finite element method provides flexibility in handling complex geometries and boundary conditions. It divides the fluid domain into smaller elements and applies variational principles to derive discrete equations.
Finite
Volume Method (FVM)
The finite volume method is highly popular in computational fluid dynamics. It ensures conservation of mass, momentum, and energy within each discrete control volume, making it suitable for both structured and unstructured grids.
HEAT AND
MASS TRANSFER IN NEWTONIAN FLUIDS
The simultaneous transfer of heat and mass is essential in processes like chemical reactors, heat exchangers, and environmental flows. Numerical models often incorporate convection-diffusion equations to simulate these phenomena. Studies show that coupling momentum, heat, and mass transfer equations improves the accuracy of predictions in mixed convective flows and phase change problems.
ADVANCES
IN COMPUTATIONAL FLUID DYNAMICS (CFD)
The development of CFD software and high-performance computing has significantly expanded the capability of numerical fluid analysis. Techniques such as adaptive meshing, turbulence modeling, and parallel computation are now routinely used to solve large-scale Newtonian fluid flow problems with enhanced precision.
STABILITY
AND CONVERGENCE ANALYSIS
Ensuring the numerical stability and convergence of the solutions is critical. Techniques like the von Neumann stability analysis and grid independence studies are essential to validate numerical models. Improved time-stepping schemes and iterative solvers have also contributed to more stable and accurate solutions.
CONCLUSION
The mathematical study of physical property transfer in Newtonian fluids is essential for advancing engineering solutions and optimizing industrial processes. Numerical methods such as FDM, FEM, and FVM have revolutionized this field by providing accurate and computationally efficient tools to solve complex flow problems. With the continued development of computational resources and numerical algorithms, future studies are expected to offer even more precise insights into the dynamics of Newtonian fluids across various applications.
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