Increasing Schmidt Number Research Articles

Effects of Ion-Slip and Hall Currents on Magnetohydrodynamic Nanofluid Flow with Thermal Diffusion Using Spectral Quasi-Linearization Method

We scrutinize and numerically investigate the behavior of magnetic nanofluid flow in stagnation region in the presence of ion-slip and Hall currents. Employing similarity technique, the governing equations modeling the boundary layer flow are switched into highly nonlinear ODEs. The resultant equations are then solved numerically by the method of spectral quasi-linearization. The effect of varying various pertinent parameters within the fluid flow are taken into account and the results are analyzed graphically. It may be noted that the velocity increases in the x- as well as z-directions with an increment in the Hall parameter. The concentration indicates a decreasing trend with increasing values of the Eckert number. The computed results also show that the volume fraction effects diminishes as the Schmidt number increases.

Velocity slip, chemical reaction, and suction/injection effects on two‐dimensional unsteady MHD mass transfer flow over a stretching surface in the presence of thermal radiation and viscous dissipation

AbstractThe purpose of this study is to analyze the impact of velocity slip, chemical reaction, and suction/injection on two‐dimensional mass transfer effects on unsteady MHD flow over a stretching surface in the presence of thermal radiation and viscous dissipation. The governing time‐dependent nonlinear partial differential equations are transformed into nonlinear ordinary differential equations by using similarity transformations. The converted equations are solved using the numerical technique with the help of Keller‐Box method. The effect of nondimensional variables is studied and graphically illustrated on velocity, temperature, concentration, friction factor, Nusselt number, and Sherwood number. Concentration and temperature profiles are enhanced and the contrasting pattern for velocity profiles as increasing the velocity slip and magnetic parameter. The concentration profile is diminished as the Schmidt number (Sc) and chemical reaction (Cr) increase. The concentration, velocity, and temperature profiles display a reversal pattern, as the suction and unsteady parameter (A) increase. The findings of this study are very well‐acknowledged with current research.

The effect of Schmidt number on gravity current flows: The formation of large-scale three-dimensional structures

The Schmidt number, defined as the ratio of scalar to momentum diffusivity, varies by multiple orders of magnitude in real-world flows, with large differences in scalar diffusivity between temperature, solute, and sediment driven flows. This is especially crucial in gravity currents, where the flow dynamics may be driven by differences in temperature, solute, or sediment, and yet the effect of Schmidt number on the structure and dynamics of gravity currents is poorly understood. Existing numerical work has typically assumed a Schmidt number near unity, despite the impact of Schmidt number on the development of fine-scale flow structure. The few numerical investigations considering high Schmidt number gravity currents have relied heavily on two-dimensional simulations when discussing Schmidt number effects, leaving the effect of high Schmidt number on three-dimensional flow features unknown. In this paper, three-dimensional direct numerical simulations of constant-influx solute-based gravity currents with Reynolds numbers 100≤Re≤3000 and Schmidt number 1 are presented, with the effect of Schmidt number considered in cases with (Re,Sc)=(100,10), (100,100), and (500, 10). These data are used to establish the effect of Schmidt number on different properties of gravity currents, such as density distribution and interface stability. It is shown that increasing Schmidt number from 1 leads to substantial structural changes not seen with increased Reynolds number in the range considered here. Recommendations are made regarding lower Schmidt number assumptions, usually made to reduce computational cost.

A macroscopic two-length-scale model for natural convection in porous media driven by a species-concentration gradient

The modelling of natural convection in porous media is receiving increased interest due to its significance in environmental and engineering problems. State-of-the-art simulations are based on the classic macroscopic Darcy–Oberbeck–Boussinesq (DOB) equations, which are widely accepted to capture the underlying physics of convection in porous media provided the Darcy number, $Da$ , is small. In this paper we analyse and extend the recent pore-resolved direct numerical simulations (DNS) of Gasow et al. (J. Fluid Mech, vol. 891, 2020, p. A25) and show that the macroscopic diffusion, which is neglected in DOB, is of the same order (with respect to $Da$ ) as the buoyancy force and the Darcy drag. Consequently, the macroscopic diffusion must be modelled even if the value of $Da$ is small. We propose a ‘two-length-scale diffusion’ model, in which the effect of the pore scale on the momentum transport is approximated with a macroscopic diffusion term. This term is determined by both the macroscopic length scale and the pore scale. It includes a transport coefficient that solely depends on the pore-scale geometry. Simulations of our model render a more accurate Sherwood number, root mean square (r.m.s.) of the mass concentration and r.m.s. of the velocity than simulations that employ the DOB equations. In particular, we find that the Sherwood number $Sh$ increases with decreasing porosity and with increasing Schmidt number $(Sc)$ . In addition, for high values of $Ra$ and high porosities, $Sh$ scales nonlinearly. These trends agree with the DNS, but are not captured in the DOB simulations.

Significance of non-similar modeling in the entropy analysis of chemically reactive magnetized flow of nanofluid subjected to thermal radiations and melting heat condition

Melting is a physical development that is associated with phase transition of materials (PCM). Melting thermal transport has fascinated researchers because of its immense usage in technological processes. In this paper, a non-similar mathematical model is established for melting aspects in the chemically reactive, radiative flow of magnetized nanofluid. The fluid flow over a vertically heated surface is triggered as a result of its linear stretching and by means of buoyancy forces. The considered setup deals with the melting thermal transport and velocity slip at the surface. The linear buoyancy in the framework of concentration and temperature is accounted for in the x-momentum equation. Frictional heating in view of viscous dissipation is convincing because of large surface velocity. An effective Buongiorno model is employed in the energy and concentration expressions with chemical reaction and magnetic and viscous dissipations. The dimensionless non-similar structure is numerically simulated by adopting local non-similarity via bvp4c. The repercussion of vital numbers on flow, entropy generation, and thermal and mass transport is discussed through graphs and tables. The graphical transport analysis suggests that the increase in buoyancy reduces the fluid flow; however, the implication of increasing velocity slip and magnetic and buoyancy ratio numbers is to enhance the fluid flow. Furthermore, the increasing radiative parameter increases the temperature in the thermal boundary layer. Concentration boundary layer analysis suggests that the impact of the increase in the Schmidt number increases the concentration and the increase in the chemical reaction decreases the concentration. The range of stable solutions for important numbers is obtained. Furthermore, the validity of results is demonstrated by comparing with the existing literature. Comparison between non-similar and local similar approximations has been made. It is finally accomplished that non-similar analysis, contrary to local similar models, is more generic and authentic in convection thermal transport analysis in the existence of buoyancy and viscous dissipation.

Inspection of convective heat transfer and KKL correlation for simulation of nanofluid flow over a curved stretching sheet

Nanofluid is treated as a smart fluid that is useful for heat and mass transfer enhancement, which is paramount in several electronics, biomedical, transportation as well as industrial applications. In view of this, in the current analysis we scrutinize the flow of nanofluid over a curved stretching sheet. The noted novelty of this work is to discuss the heat and mass transfer in nanofluid flow along with the activation energy. Further, CuO with water-based nanofluid is considered in the modelling. The viscosity and effective thermal conductivity of fluid flow suspended by nanoparticles are scrutinized by Koo–Kleinstreuer–Li (KKL) model. By employing suitable similarity transformations, the governing equations of momentum, thermal and concentration of nanoparticle are converted into ordinary differential equations and then they are solved with Runge–Kutta-Fehlberg-45 (RKF-45) process along with shooting method. The impact of pertinent non-dimensional parameters is attained and illustrated with the help of graphs. The results reveal that, the heightening of Biot number and curvature parameter heightens the thermal gradient. The mass transfer decreases as the Schmidt number and chemical reaction rate parameter increases. The upsurge in activation energy parameter declines the mass transfer.

A Framework for the Magnetic Dipole Effect on the Thixotropic Nanofluid Flow Past a Continuous Curved Stretched Surface

The magnetic dipole effect for thixotropic nanofluid with heat and mass transfer, as well as microorganism concentration past a curved stretching surface, is discussed. The flow is in a porous medium, which describes the Darcy–Forchheimer model. Through similarity transformations, the governing equations of the problem are transformed into non-linear ordinary differential equations, which are then processed using an efficient and powerful method known as the homotopy analysis method. All the embedded parameters are considered when analyzing the problem through solution. The dipole and porosity effects reduce the velocity, while the thixotropic nanofluid parameter increases the velocity. Through the dipole and radiation effects, the temperature is enhanced. The nanoparticles concentration increases as the Biot number and curvature, solutal, chemical reaction parameters increase, while it decreases with increasing Schmidt number. The microorganism motile density decreases as the Peclet and Lewis numbers increase. Streamlines demonstrate that the trapping on the curved stretched surface is uniform.

MHD Powell\u2013Eyring dusty nanofluid flow due to stretching surface with heat flux boundary condition

A steady MHD boundary layer flow of Powell–Eyring dusty nanofluid over a stretching surface with heat flux condition is studied numerically. It is assumed that the fluid is incompressible and the impacts of thermophoresis and Brownian motion are taken into regard. In addition, the Powell–Eyring terms are considered in the momentum boundary layer and thermal boundary layer. The dust particles are seen as to be having the same size and conform to the nanoparticles in a spherical shape. We obtain a system of ordinary differential equations that are suitable for analyzed numerically using the fourth-order Runge–Kutta method via software algebraic MATLAB by applying appropriate transformations to the system of the governing partial differential equations in our problem. There is perfect compatibility between the bygone and current results when comparing our numerical solutions with the available data for values of the selected parameters. This confirms the validity of the method used here and thus the validity of the results. The influence of some parameters on the boundary layer profiles (the velocity and temperature for the particle phase and fluid phase, and nanoparticle concentration) is discussed. The results of this study display that the profiles of the velocity for particle and fluid phases increase with increasing Powell–Eyring fluid parameter, but reduce with height in magnetic field values. Mass concentration of the dust particles decreases the temperature of both the particle and fluid phases. The results also indicate the concentration of nanoparticle contraction as Schmidt number increases.

Complete theory of the elastic wall jet: A new flow geometry with revisited two-phase nanofluids

In the present research, it is solved in an innovative manner the essentially Non-Similar nanofluids jet discharged over a generalized non-linearly stretching wall. The Non-Similar set-up is interpreted as a combination of wall jet flow at the leading edge and non-linearly stretching sheet flow far away from it. Nanofluids are simulated employing Buongiorno’s two-phase model together with a modified boundary condition for nanoparticles transport equation. The compositely-transformed partial differential equations (PDE) of Non-Similar forms were solved numerically after employing a new mapping function to abridge the horizontal coordinate of an infinite length into a finite one. The numerical algorithm contains quasi-linearization technique together with an implicit algorithm of tridiagonal form. It should be pointed out that such a new Non-Similar setup is brought into account for the first time in the literature. In this respect, we summarize the outstanding novel findings of the present work as: •[I.] Accounting a modified boundary condition for nanoparticles concentration at the wall, the so-called and widely-used Brownian motion parameter, namely Nb, will have absolutely NO impact on Nusselt number or more precisely, on Energy Equation. In this respect, it is presented a new way of defining dimensionless variables.•[II.] As the stretching ratio increases, it is observed a discrepancy in boundary layer thickness; it expands close to the jet area, shrinks in the transition region and adapts the far-field condition, either by expanding or shrinking, depending on the stretching ratio.•[III.] As the stretching ratio increases, Nusselt number decreases slightly shortly after the jet origin and then increases dramatically to match the stretching region.•[IV.] As the stretching ratio increases, nanoparticles at the wall become more concentrated at the downstream and hence, close to the jet region as well as the transition region, the concentration is sparse.•[V.] As thermophoresis parameter, namely Nt, increases, convective heat transfer coefficient drops non-uniformly, from the leading edge to downstream, as well as nanoparticles concentration at the wall.•[V1.] As Schmidt number increases, convective heat transfer drops; however, nanoparticles concentration at the wall increases over the entire spatial domain.

Transient free convection heat and mass transfer of Casson nanofluid over a vertical porous plate subjected to magnetic field and thermal radiation

In this present study, the transient magnetohydrodynamics free convection heat and mass transfer of Casson nanofluid past an isothermal vertical flat plate embedded in a porous media under the influence of thermal radiation is studied. The governing systems of nonlinear partial differential equations of the flow, heat and mass transfer processes are solved using implicit finite difference scheme of Crank-Nicolson type. The numerical solutions are used to carry out parametric studies. The temperature as well as the concentration of the fluid increase as the Casson fluid and radiation parameters as well as Prandtl and Schmidt numbers increase. The increase in the Grashof number, radiation, buoyancy ratio and flow medium porosity parameters causes the velocity of the fluid to increase. However, the Casson fluid parameter, buoyancy ratio parameter, the Hartmann (magnetic field parameter), Schmidt and Prandtl numbers decrease as the velocity of the flow increases. The time to reach the steady state concentration, the transient velocity, Nusselt number and the local skin-friction decrease as the buoyancy ratio parameter and Schmidt number increase. Also, the steady-state temperature and velocity decrease as the buoyancy ratio parameter and Schmidt number increase. Also, the local skin friction, Nusselt and Sherwood numbers decrease as the Schmidt number increases. However, the local Nusselt number increases as the buoyancy ratio parameter increases. It was established that near the leading edge of the plate), the local Nusselt number is not affected by both buoyancy ratio parameter and Schmidt number. It could be stated that the present study will enhance the understanding of transient free convection flow problems under the influence of thermal radiation and mass transfer as applied in various engineering processes.

Direct numerical simulations of nanoparticle formation in premixed and non-premixed flame–vortex interactions

Direct numerical simulations (DNSs) of nanoparticle formation in reactive flows are challenging, and only greatly simplified DNS test-cases are possible, which help clarify the turbulence–particle–dynamics interaction and guide the necessary modeling efforts. As a basis for such studies, a new DNS database is introduced, which resolves the smallest relevant scales of the nanoparticle concentration field to obtain insights into the statistics of nanoparticle formation in reactive flows. Formation and evolution of iron oxide nanoparticles in premixed and non-premixed flames wrapped-up by a vortex have been investigated using the sectional model and direct chemistry. The DNSs capture the “engulfing” and local dilution of the particle fields. Different zones of high particle number concentration have been found in every flame, and it was shown that the thickness of these zones decreases with increasing Schmidt number, which confirms that in simulations of nanoparticle-forming turbulent reacting flows, the grid resolution has to be very fine to resolve the characteristic scale for high sections. The contributions to the change in particle concentration due to diffusion, coagulation, and nucleation have been analyzed in detail, and dominant contributions across the particle number concentration layers and across the flames have been identified. This analysis has also been carried out in terms of flat, concave, and convex iso-surface geometries, induced by the flame–vortex interaction and characterized by the curvature of the particle number concentration fields and also by the flame curvature. The results demonstrate that the flame curvature effects cannot be ignored in modeling strategies. The probability density functions for the particle number concentrations have been analyzed and quantified in terms of Shannon information entropy, which illustrates the effect of fast diffusion (and entropy production) of the smaller particles and slow diffusion (and entropy production) of the largest particles with high Schmidt numbers. In addition, the unclosed filtered or averaged agglomeration term was evaluated as a basis for future modeling efforts, showing that agglomeration rates will be underestimated by orders of magnitude unless suitable models are developed.

Effect of Thermal Radiation and Chemical Reaction on MHD Flow of Blood in Stretching Permeable Vessel

This paper focuses on the theoretical analysis of blood flow in the presence of thermal radiation and chemical reaction under the influence of time dependent magnetic field intensity. Unsteady non linear partial differential equations of blood flow consider time dependent stretching velocity, the energy equation also accounts time dependent temperature of vessel wall and the concentration equation includes the time dependent blood concentration. The governing non linear partial differential equations of motion, energy and concentration are converted into ordinary differential equations using similarity transformations solved numerically by applying ode45. The effect of physical parameters, viz., the permeability parameter, unsteadiness parameter, Prandtl number, Hartmann number, thermal radiation parameter, chemical reaction parameter and Schmidt number on flow variables, viz., velocity of blood flow in vessel, temperature and concentration of blood, has been analyzed and discussed graphically. From the simulation study the following important results are obtained: velocity of blood flow increases with the increment of both permeability and unsteadiness parameter. The temperature of blood increases at the vessel wall as the Prandtl number and Hartmann number increase. Concentration of blood decreases as time dependent chemical reaction parameter and Schmidt number increases.

Homotopy solution for MHD mixed convective unsteady flow of a Powell-Eyring fluid in a vertical porous space with oscillating wall temperature

In this paper, unsteady magnetohydrodynamic (MHD) flow of Powell-Eyring fluid in a vertical channel with diffusion thermo and thermal diffusion effects in the presence of heat source have been studied. The governing nonlinear, coupled partial differential equations are reduced to ODEs using oscillating parameter and then solved by homotopy analysis method (HAM). The influence of emerging parameters on heat and mass transfer characteristics of the flow are studied. It is found that the velocity of a Non-Newtonian fluid is found to decrease in comparison with Newtonian fluid. Also, it is observed that thermal parameters and Soret number are lead to promote the temperature of the fluid significantly. Increasing magnetic parameter tends to produce oscillating behaviour on volume flow rate whereas uniform depreciation can be noticed with increasing Schmidt number.

Homotopy solution for MHD mixed convective unsteady flow of a Powell-Eyring fluid in a vertical porous space with oscillating wall temperature

In this paper, unsteady magnetohydrodynamic (MHD) flow of Powell-Eyring fluid in a vertical channel with diffusion thermo and thermal diffusion effects in the presence of heat source have been studied. The governing nonlinear, coupled partial differential equations are reduced to ODEs using oscillating parameter and then solved by homotopy analysis method (HAM). The influence of emerging parameters on heat and mass transfer characteristics of the flow are studied. It is found that the velocity of a Non-Newtonian fluid is found to decrease in comparison with Newtonian fluid. Also, it is observed that thermal parameters and Soret number are lead to promote the temperature of the fluid significantly. Increasing magnetic parameter tends to produce oscillating behaviour on volume flow rate whereas uniform depreciation can be noticed with increasing Schmidt number.

Metachronal propulsion of a magnetised particle-fluid suspension in a ciliated channel with heat and mass transfer

Biologically-inspired pumping systems are of great interest in modern engineering since they achieve enhanced efficiency and circumvent the need for moving parts and maintenance. Industrial applications also often feature two-phase flows. In this article, motivated by these applications, the pumping of an electrically-conducting particle-fluid suspension due to metachronal wave propulsion of beating cilia in a two-dimensional channel with heat and mass transfer under a transverse magnetic field is investigated theoretically. The governing equations for mass and momentum conservation for fluid- and particle-phases are formulated by ignoring the inertial forces and invoking the long wavelength approximation. The Jeffrey viscoelastic model is employed to simulate non-Newtonian characteristics. The normalised resulting differential equations are solved analytically. Symbolic software is employed to evaluate the results and simulate the influence of different parameters on flow characteristics. Results are visualised graphically with carefully selected and viable data. With increasing wave number (β) fluid velocity is accelerated in the core region whereas it is decelerated near the channel wall, for the Newtonian case. With increasing eccentricity of cilia elliptic path (α), a similar response is computed as for the wave number. The size of the bolus is enhanced (and quantity of boluses is reduced) with increasing eccentricity of the cilia elliptic path (α) and Hartmann (magnetic) number (M) whereas bolus size is decreased (and quantity of boluses is increased) with increasing wave number (β) and particle volume fraction (C). It is also noted that increasing Schmidt number (Sc) and Soret number (Sr) diminish the concentration magnitudes. Furthermore, Brinkman number (which represents viscous heating effects) significantly boosts the temperature magnitudes. The current analysis provides a useful benchmark for more general computational simulations.

Entropy generation optimisation in the nanofluid flow of a second grade fluid with nonlinear thermal radiation

The flow of a second grade fluid by a rotating stretched disk is considered. Brownian motion and thermophoresis characterise the nanofluid. Entropy generation in the presence of heat generation / absorption, Joule heating and nonlinear thermal radiation is discussed. Homotopic convergent solutions are developed. The behaviour of velocities (radial, axial, tangential), temperature, entropy generation, Bejan number, Nusselt number, skin friction and concentration is evaluated. The radial, axial and tangential velocities increase for larger viscoelastic parameters while the opposite trend is noted for temperature. Concentration decreases when Schmidt number and Brownian diffusion increase. Entropy generation increases when the Bejan number increase while the opposite is true for the Brinkman number and the magnetic parameter.

Wavelet analysis of stagnation point flow of non-Newtonian nanofluid

The wavelet approach is introduced to study the influence of the natural convection stagnation point flow of the Williamson fluid in the presence of thermophysical and Brownian motion effects. The thermal radiation effects are considered along a permeable stretching surface. The nonlinear problem is simulated numerically by using a novel algorithm based upon the Chebyshev wavelets. It is noticed that the velocity of the Williamson fluid increases for assisting flow cases while decreases for opposing flow cases when the unsteadiness and suction parameters increase, and the magnetic effect on the velocity increases for opposing flow cases while decreases for assisting flow cases. When the thermal radiation parameter, the Dufour number, and Williamson’s fluid parameter increase, the temperature increases for both assisting and opposing flow cases. Meanwhile, the temperature decreases when the Prandtl number increases. The concentration decreases when the Soret parameter increases, while increases when the Schmidt number increases. It is perceived that the assisting force decreases more than the opposing force. The findings endorse the credibility of the proposed algorithm, and could be extended to other nonlinear problems with complex nature.

Phase separation in binary fluid mixtures with symmetric and asymmetric Schmidt numbers: A DPD study.

We investigate the effect of the Schmidt number (Sc) on phase separation dynamics of two immiscible fluids in a two-dimensional periodic box using dissipative particle dynamics. The range of Sc investigated spans liquid-liquid separation processes. Phase separation is characterized by a domain size r(t), which typically follows a power law tβ in time t, where β is a characteristic exponent corresponding to the coarsening mechanism at play. The phase separation dynamics is studied for strongly (deep quench) separating mixtures. We consider cases of critical (ϕ ∼ 0.5) and off-critical (ϕ < 0.5) mixtures of fluids A and B for both ScA = ScB and ScA ≠ ScB. In all cases, the growth dynamics slow down with the increasing Schmidt number of either fluid. We observe the power law exponent β = 0.5 for symmetric (ScA = ScB) critical mixtures and β = 0.33 for all other cases. However, for off-critical mixtures, the exponent is 0.33 irrespective of the Schmidt number ratio of the two fluids. We explain these results from an analysis of the competition between diffusive effects vis-á-vis dynamical forces.

Influence of Cattaneo–Christov Heat Flux on MHD Jeffrey, Maxwell, and Oldroyd-B Nanofluids with Homogeneous-Heterogeneous Reaction

This research article deals with the determination of magnetohydrodynamic steady flow of three combile nanofluids (Jefferey, Maxwell, and Oldroyd-B) over a stretched surface. The surface is considered to be linear. The Cattaneo–Christov heat flux model was considered necessary to study the relaxation properties of the fluid flow. The influence of homogeneous-heterogeneous reactions (active for auto catalysts and reactants) has been taken in account. The modeled problem is solved analytically. The impressions of the magnetic field, Prandtl number, thermal relaxation time, Schmidt number, homogeneous–heterogeneous reactions strength are considered through graphs. The velocity field diminished with an increasing magnetic field. The temperature field diminished with an increasing Prandtl number and thermal relaxation time. The concentration field upsurged with the increasing Schmidt number which decreased with increasing homogeneous-heterogeneous reactions strength. Furthermore, the impact of these parameters on skin fraction, Nusselt number, and Sherwood number were also accessible through tables. A comparison between analytical and numerical methods has been presented both graphically and numerically.

High-Dimensional Entanglement in States with Positive Partial Transposition.

Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen etal. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i)invariant under partial transposition on the smallest of their two subsystems, or (ii)absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.