Number Exponent Research Articles

Entropy production in the flow of Bingham-Papanastasiou fluid having temperature and shear rate-dependent viscosity

This article comprehensively examines entropy production in a peristaltic flow of Bingham-Papanastasiou fluid in an asymmetric wavy wall channel. The viscosity of the fluid is selected to depend on the temperature in addition to the shear rate. Furthermore, viscous dissipation is considered. The equations governing momentum and energy are adjusted to incorporate the existence of a nonlinear correlation between stress and strain rate. The energy dissipation caused by viscosity in the energy equation exhibits variation between Newtonian and non-Newtonian fluids due to their distinct rheological behaviors and response to shear forces. Newton’s law of viscosity states that the viscous dissipation term for Newtonian fluids follows a simple linear relationship between shear stress and shear rate. In contrast, non-Newtonian fluids exhibit more complex rheological behaviors, with the correlation between shear stress and shear rate described by various constitutive equations, depending on the specific type of non-Newtonian behavior. Consequently, the energy equation for non-Newtonian fluid flow problems becomes more intricate and nonlinear compared to the energy equation for Newtonian fluids, resulting in a more challenging analysis. Modeling of the equations is traditionally obtained using all steps involved in analyzing peristaltic flow. Using MATLAB's bvp5c solver, the obtained coupled set of differential equations is numerically solved. The research explores how the Bingham number and stress growth exponent parameters affect different dimensionless variables, including velocity, streamlines, temperature, axial pressure gradient, entropy production, and the Bejan number.

Pulsatile pressure-driven non-Newtonian blood flow through a porous stenotic artery: A computational analysis

Blood flow through a narrowed artery, such as those caused by plaque buildup, can lead to various cardiovascular diseases. Therefore, studying this phenomenon is essential for developing better treatments and understanding the underlying mechanisms. In this study, a mathematical model is developed to simulate blood flow through stenotic arteries with porous walls. The familiar Navier-Stokes equations are executed to simulate flow, whereas the Carreau fluid model is employed to quantify blood rheology. This model illustrates both Newtonian and non-Newtonian characteristics depending on the shear rate. Other than that, the blood vessel is assumed to have a moderate level of stenosis with porous saturated walls. The solution of the governed partial differential equations is carried out using a finite difference scheme. Our research addresses the impacts of permeability and rheological parameters, stenosis size, Brinkman number, and Prandtl number on blood velocity, flow rate, resistance to flow, wall shear stress, and temperature distribution. The findings demonstrate that while flow resistance reduces, the velocity, shear stress, and flow rate increase with high permeability values. Conversely, as the stenosis amplitude increases, the magnitude of wall shear stresses, velocity, and flow rates decrease. In addition, blood rheology also has the potential to transfigure the aforementioned factors. We also found that the temperature distribution increases with the Brinkman number, stenosis size, permeability, and power law exponent. It drives down, however, by raising the values of Weissenberg number and Prandtl number. Collectively, our findings contribute to a better understanding of the complicated phenomena of blood flow in stenotic vessels with porous saturated walls. The findings of this study may be valuable in establishing improved therapies for cardiovascular disorders and formulating more precise mathematical models to forecast blood flow in vivo.

Experimental investigation of the heat transport and pressure drop in open-cell polyurethane foams

Foam-like materials have a large specific surface area and complex structure that promotes flow turbulence, which makes them useful for heat transfer. In the literature, heat transfer in metal foams (MF) with high thermal conductivity is mainly considered. In the present paper, an experimental study of the hydraulic and heat transfer characteristics of a rectangular channel filled with polyurethane foam (PUF) inserts with a thermal conductivity of 0.2 W⋅m−1⋅K−1 in the air flow was carried out. The open-cell PUF samples had 20 and 80 pores per inch (PPI), 0.97 and 0.98 porosity, 269 µm and 60 µm fiber diameter, respectively. The Reynolds number, based on the fiber diameter ranged from 0.04 to 50. From the results of hydraulic tests, friction coefficients were determined based on the fiber diameter and permeability. The Forchheimer coefficient was 0.198 and 0.318 for 20 and 80 PPI samples, respectively. The permeability was 1.889 10−7 ⋅m2 and 7.535 10−9 ⋅m2 for 20 and 80 PPI samples, respectively. The Nusselt number for both tested PUF samples was correlated with the Reynolds number based on the fiber diameter in a power law with the Reynolds number exponent and constant equal to 0.61 and 0.037 respectively. The intensity of heat transfer in the PUF samples was 7–12 times lower than that in MF. However, a significant heat transfer enhancement is still possible compared to the empty channel up to 7 times. The heat transfer performance of PUF was higher compared to the empty channel, but the thermal performance factor was lower than one. Nevertheless, in practical situations where the mass or cost of the heat exchanger is preferable, PUF can be considered as a heat transfer intensifier.

Numerical Solution of Natural Convection Nanofluid Flow over aNon-Isothermal Vertical Plate

The natural convection nanofluidflow over a non-isothermal vertical plate is studied numerically. The effects of Brownian motion and thermophoresis parameters are incorporated into models used for nanofluids. The non-linear partial differential equations and boundary conditions are transformed into a set of nonlinear ordinally differential equations using the similarity transformations. The resulting system of equations are solved numerically by using shooting techniques. This solution depends on the Prandtl number (Pr), Buoyancy ratio (Nr), Brownian-motion (Nb), thermophoresis parameter (Nt), Lewis number (Le), and power-law exponent (λ). For different values of λ, Le and Pr, the influence of reduced Nusselt number with Nr, Nb and Nt is represented by correlation formulas and compared with previously published results and found to be in better agreement.

The propagation of strong cylindrical shock wave in a rotating axisymmetric non‐ideal gas with radiation heat flux

In this manuscript, we studied a quasi‐linear hyperbolic system of partial differential equations which describes the one‐dimensional adiabatic unsteady flow behind a strong cylindrical shock wave propagating in a rotating non‐ideal gas, which has a varying azimuthal fluid velocity together with a varying axial fluid velocity, with radiation heat flux, and similarity solutions are obtained by using the Lie group‐theoretic method. The basic idea of this method is that it changes the system of PDEs representing the one‐dimensional flow through the similarity variable to the system of ODEs. The flow profiles are drawn behind the shock followed by a brief discussion on the behavior of the solutions via graphs. The effects of variation in non‐ideal parameter, adiabatic index of the gas, Alfven–Mach number and ambient azimuthal velocity exponent on the flow variables are described. All numerical calculations have been performed using the “Mathematica” software.

The three-dimensional generalized Hénon map: Bifurcations and attractors.

We study dynamics of a generic quadratic diffeomorphism, a 3D generalization of the planar Hénon map. Focusing on the dissipative, orientation preserving case, we give a comprehensive parameter study of codimension-one and two bifurcations. Periodic orbits, born at resonant, Neimark-Sacker bifurcations, give rise to Arnold tongues in parameter space. Aperiodic attractors include invariant circles and chaotic orbits; these are distinguished by rotation number and Lyapunov exponents. Chaotic orbits include Hénon-like and Lorenz-like attractors, which can arise from period-doubling cascades, and those born from the destruction of invariant circles. The latter lie on paraboloids near the local unstable manifold of a fixed point.

Lattice Boltzmann Method for high Reynolds number compressible flow

We employ a Lattice Boltzmann Method (LBM) for modelling high Reynolds number (Re) compressible supersonic flow using D2Q9 lattice. In this model, a two-population thermal lattice Boltzmann (LB) is utilized in order to allow varying Prandtl number and adiabatic exponent. The equilibrium population fieq term for mass and momentum conservation is matched analytically with order of accuracy O(Ma4), while the equilibrium population gieq term for energy conservation is solved numerically through Lagrange multipliers method. By using more accurate equilibrium population gieq, the present numerical method shows the enhancement of stability, especially at high Reynolds and Mach number flows. To simulate high Reynolds and Mach number (Ma) flows efficiently, we adopt a shifted lattice scheme, stabilized by relaxation time depending on the Knudsen number. The LBM show excellent accuracy through 1D Riemann examples, capable of handling expansion shock waves and discontinuity. Our results for supersonic flow past a cylinder also find excellent agreement with experimental data in literature. Separately, we benchmark results for transonic and supersonic flows past a NACA0012 aerofoil as a demonstration of its simplicity, versatility for applications involving high Reynolds and Mach numbers (Ma∞=1.5,Re=107).

Modified decomposition solution of convex fins of different profile index with all power law dependent thermal properties

The present work investigates the non-linear mathematical model of one regular and two irregular convex fins where all thermo physical parameters are power law dependent. The extended length of regular convex fins varies parabolically along its length with a power exponent of ½ and two irregular convex fins with different power exponent ¼ and 9/10 respectively. The energy equations of one regular and two irregular convex fins are three different singular value equation but with a common non-linearity. The three different singular value equations are solved separately by the theory of modified Adomian decomposition method (MADM). The effects of power dependent thermal conductivity parameter, effect of conduction-convection parameter, effect of power exponent of heat transfer co-efficient, effect of power exponent of surface emissivity parameter, and effect of power exponent of heat generation number on the temperature distribution of three different convex fins are analyzed and compared separately.

Lie group of invariance technique for analyzing propagation of strong shock wave in a rotating non‐ideal gas with azimuthal magnetic field

In this manuscript, the propagation of strong shock wave in a non‐ideal gas in the presence of azimuthal magnetic field with rotational effect is examined by using the Lie group of invariance technique with an assumption of uniform density in the undisturbed medium, whereas the azimuthal component of the fluid velocity and magnetic field are supposed to vary. The main advantage of this method is to reduce the number of independent variables by one. We obtained a special class of self‐similar solutions to the problem, and the flow profiles are depicted graphically behind the shock followed by a brief discussion on the behavior of solutions through graphs. The effects of variation in non‐ideal parameter, adiabatic index of the gas, Alfven–Mach number and ambient azimuthal velocity exponent on the flow variables are presented. All numerical calculations have been done using the “Mathematica”software.

Numerical Study of Thermophoresis on Mass Transfer from Natural Convection Flow over a Vertical Porous Medium with Variable Wall Heat Fluxes

This study investigates heat and mass transfer under natural convection flow along a vertical permeable surface with variable wall heat fluxes through a porous medium. The non-Darcian model is employed for the medium. The effects of suction/blowing, inertia, buoyancy ratio, exponent of heat flux, position parameter, Schmidt number, and thermophoresis are considered. The governing equations of continuity, momentum, energy, and concentration are solved by adopting similarity transformation and Runge–Kutta integration with a shooting technique. Results of interest, such as velocity, temperature, and concentration profiles related to local Nusselt and Sherwood numbers, are obtained for the selected buoyancy ratio at different magnitudes of the thermophoretic effect. The numerical solutions help us to realize the gas diffusion phenomena and control the transport technology.

Linear Algebraic Approach to Formulate A New Recurrence Relation for Bernoulli Numbers from the Power-Sum of Natural Numbers with Experiments on Pedagogy

In this article, a new recurrence relation formula for Bernoulli numbers have been derived, and sum of integer exponents of natural numbers has been revisited from this novel perspective. Some interesting pedagogical experiments on wording and presentation of mathematical derivation have been attempted, and development from first principle have been undertaken in line with this experimental approach.

Experimental study of boulder concentration effect on flow resistance in gravel bed channels

Previous studies demonstrated that for open channel flows the power velocity profile can be integrated to obtain the flow resistance law. In this paper the relationship between Γ coefficient of the power velocity distribution, the channel slope and Froude number (Eq. (9)) is firstly presented. Then the measurements carried out in a laboratory flume covered by hemispheric elements for two different hydraulic conditions (partially inundated and inundated) are used to calibrate this relationship. These elements are placed with a square arrangement and have a concentration range 4–64%. For the two investigated hydraulic conditions, this analysis demonstrates that the relationship to estimate the coefficient of the velocity profile is characterized by the same exponent of slope and Froude number while a different scale factor a has to be used. The effect of roughness element concentration Ch on flow resistance is also studied by the calibration of Eq. (9) for each experimental series characterized by a single value of Ch. For the partially inundated condition and values of the roughness element concentration less than 25%, the coefficient a increases with Ch while for Ch values ranging from 25 to 64% a decreases with concentration. For the submerged condition, the coefficient a increases with Ch and results almost constant for concentration values greater than 49%. The variability of the coefficient a with Ch is the effect of the variability of the reference plane for the water depth measurement with the concentration. Finally, the obtained relationships to estimate Γ function and the theoretical flow resistance law are verified by 205 measurements carried out in gravel bed rivers. In conclusion, the developed analysis demonstrates that an accurate estimate of the friction factor can be obtained for the investigated gravel surfaces by the proposed flow resistance equation and the obtained relationship to estimate Γ function.

A computational model for hybrid nanofluid flow on a rotating surface in the existence of convective condition

The current work aims to study heat transfer properties within the flow. A newly developed idea regarding hybrid nanofluid is utilized to obtain the desired outcomes. This study focused on hybrid base nano liquid flow past an exponentially stretching rotating surface in the presence of the convective condition. Ethylene glycol-water (50%–50%) mixture is taken as base fluid. Thermophysical properties are discussed. The probe of emerging parameters indicates that hybrid base nanofluid gives a preferable rating as compared to nanofluid. Appropriate choice of composition plays a vital role. Physical characteristics of essential parameters such as Prandtl number, convective parameter, stretching ratio, rotation parameter, and temperature exponent are evaluated. Graphical depiction reveals that the velocity curve decline by increasing rotation parameter for CuO-hybrid base fluid and TiO2 hybrid base fluid. By expanding the values of the stretching ratio the velocity profile p′(η) is decreasing while the velocity curve q′(η) increased for CuO as well as TiO2 hybrid base fluid. Concentration boundary layer thickness growing by enlarging rotation parameters for both types of nanofluid. Moreover, the influence of various parameters on skin friction factor and Nusselt numbers are also evaluated.

Influence of Different Gases on the Design Point of an Industrial Axial Compressor and Deduced Aerodynamical Rematching Methodology

This paper presents a numerical investigation of the effects of compressing various gases, for example, carbon dioxide (CO2) and methane (CH4), on an eight-stage axial air compressor. Several adaptation methods are applied to achieve a similar operating point as for air. Theoretically, the operating point depends on Mach number, flow angles, Reynolds number and isentropic exponent. Numerical results show mismatch effects which arise in the parameters using a non-adapted geometry. A rematching procedure is described, including deduced speed adjustments, in order to achieve Mach number equality at compressor inlet. Only shroud modifications are performed to rematch the flow angles of the air simulation. Although Reynolds and Mach number are kept constant at compressor inlet, an inevitable deviation in downstream flow causes mismatches in efficiency and pressure ratio. Both analytical and numerical methodologies show that the scale of shroud adjustments, as well as the size of mismatch in Mach and Reynolds number, can be correlated to the isentropic gas exponent. In summary, the main impact on gas behavior in an axial air compressor is attributable to the change in isentropic exponent. Derivations of shroud adaptation and analyses of inevitable aerodynamic mismatch are therefore developed depending on the isentropic exponent.

Primes of the form 2p+1

Arithmetic congruences are derived for the exponents of composite Mersenne numbers. It is known that the 2 p −1 is composite if p is a Sophie Germain prime which is congruent to 3 modulo 4. After verifying the equality of estimates of the density of these primes through refined sieve and proabablistic methods, a set of the arithmetic sequences for the exponents are listed. The coefficients in these sequences generally have a nontrivial common factor, and a shift in the number of doubling cycles for a given number of partitions is found to yield divisible Mersenne numbers. Several sequences with relatively prime coefficients have terms that are congruent to 3 modulo 4. Furthermore, a linear recursion relation for the exponents would have a non-zero density of solutions representing positive integers in the natural numbers by the Skolem-Mahler-Lech theorem, thereby predicting the infinite extent of the prime exponents of composite Mersenne numbers of this kind and the Sophie Germain primes.

Exponential Computing Digital Circuit Design Developed for FPGA-based Embedded Systems

Digital systems consist of thousands of digital circuit blocks operating in the background, working in their simplest form such as addition, subtraction, multiplication, division. In exponential expressions like square roots and cube roots, just like these circuits, it is found in many digital systems and performs tasks. Although these processes seem to be used only in circuits carrying out mathematical operations, they actually take an active role in solving many engineering problems. In this study, a digital circuit design that computes both the integer and a floating point exponent of a 32-bit floating-point number has been realized. This digital circuit, which is coded with VHDL language, can be used from beginner to advanced level in FPGA based systems. This digital circuit, which is coded with VHDL language, can be used from beginner to advanced level in FPGA based systems. In addition, three floating IP cores - logarithm, multiplication and exponent - were used in this digital circuit, and results were obtained with a total of five finite state machines in sixty-six clock pulse time.

Nonlinear vibrations and chaotic phenomena of functionally graded material truncated conical shell subject to aerodynamic and in-plane loads under 1:2 internal resonance relation

This paper focuses on the nonlinear dynamic responses of a functionally graded material (FGM) truncated conical shell under 1:2 internal resonance relation. The FGM truncated conical shell is subjected to the in-plane load and the aerodynamic load along the meridian direction. According to a power-law distribution, the material properties are assumed to be modified along the thickness direction smoothly and continuously and the material properties are temperature dependent. The aerodynamic load is obtained by the first-order piston theory with the curvature correction term. According to von Karman type nonlinear geometric relations, first-order shear deformation shell theory, Hamilton principle, the nonlinear equations of motion for the FGM truncated conical shell are established. Furthermore, the nonlinear equations of motion are reduced into a system of the ordinary differential equations by utilizing Galerkin procedure. The multiple scales method is used to obtain the averaged equations for the FGM truncated conical shell under the relations of 1:2 internal resonance and 1/2 subharmonic resonance. The frequency–response curves, time history diagrams, phase portraits, Poincare maps and bifurcation diagrams with different parameters are yielded by employing numerical calculations. The influences of exponent of volume fraction, Mach number, damping coefficient and in-plane load on the nonlinear resonance behaviors of the FGM truncated conical shell are investigated. The chaotic and periodic motions of the FGM truncated conical shell have been discussed in detail.

Supergravitational turbulent thermal convection.

High-Rayleigh number convective turbulence is ubiquitous in many natural phenomena and in industries, such as atmospheric circulations, oceanic flows, flows in the fluid core of planets, and energy generations. In this work, we present a novel approach to boost the Rayleigh number in thermal convection by exploiting centrifugal acceleration and rapidly rotating a cylindrical annulus to reach an effective gravity of 60 times Earth's gravity. We show that in the regime where the Coriolis effect is strong, the scaling exponent of Nusselt number versus Rayleigh number exceeds one-third once the Rayleigh number is large enough. The convective rolls revolve in prograde direction, signifying the emergence of zonal flow. The present findings open a new avenue on the exploration of high-Rayleigh number turbulent thermal convection and will improve the understanding of the flow dynamics and heat transfer processes in geophysical and astrophysical flows and other strongly rotating systems.

Modified Newton identities

The sums of powers with identical exponents of natural, real, or complex numbers, considered as roots of algebraic equation, are expressed directly through the products of coefficients of that equation, starting from the well-known Newton identities. The final Eq. (6) includes the same power of sum of all numbers ± a sum over all partitions of the exponent. Each term of the last sum is the equation coefficients product with the net power keeping the “dimensionality” of the exponent and having a numerical factor, equal to a proper polynomial coefficient, built of exponents of equation coefficients entering the product. The revers Eq. (43) for equation coefficients is also a sum over all partitions of the same exponent with known numerical coefficients. The entering products are built of “commutators-anticommutators of power of sum and sum of powers” (C-A) of the initial sum addends. The numerous identities Eq. (44) for a C-A with an exponent, exceeding the number of C-A sum terms by 2, and similar C-A-s with lower exponents are established.

Marangoni flow and mass transfer of power-law non-Newtonian fluids over a disk with suction and injection

We scrutinize the approximate analytical solutions by the optimal homotopy analysis method (OHAM) for the flow and mass transfer within the Marangoni boundary layer of power-law fluids over a disk with suction and injection in the present paper. Concentration distribution on the surface of a disk varies in a power-law form. The non-Newtonian fluid flow is due to the surface concentration gradient without considering gravity and buoyancy. According to the conservation of mass, momentum and concentration, the governing partial differential equations are established, and the appropriate generalized Kármán transformation is found to reduce them to the nonlinear ordinary differential equations. OHAM is used to access the approximate analytical solution. The influences of Marangoni the number, suction/injection parameters and power-law exponent on the flow and mass transfer are examined.