Effect Of Prandtl Number Research Articles

Significance of Activation Energy and Effective Prandtl Number in Accelerated Flow of Jeffrey Nanoparticles With Gyrotactic Microorganisms

Abstract This research addresses the interesting rheological features of Jeffrey nanofluid containing gyrotactic microorganism over an accelerated configuration. The additional consequences of activation energy and thermal radiation are also encountered in the current flow problem. The characteristics of nanofluid is utilized by using Buongiorno’s nanofluid model, while the phenomenon of bioconvection is evaluated by Kuznestov and Nield model. Unlike traditional attempts, the analysis for thermal radiation is performed by using “one parametric approach” by expressing the Prandtl number and thermal radiation parameter in combined form, namely, effective Prandtl number. The governing equations reflecting the flow problem are analytically treated with the help of homotopic algorithm. The impact of flow parameters is graphically elaborated with relevant physical significance. Further, the numerical expressions for effective local Nusselt number, local Sherwood number, and motile density number with variation of flow parameters in articulated tabular form. It is observed that magnitude of skin friction coefficient oscillates periodically with time and magnitude of oscillation increases with increment of Deborah number and mixed convection constant. It is further emphasized that the temperature distribution is enhanced with buoyancy ratio constant and bioconvection Rayleigh number. The microorganism distribution increases with buoyancy ratio constant but reverse trend has been examined for Peclet number. The observations from the reported problem can be more effective for the development of bifurcation processes, biofuels, enzymes, etc.

Natural-convection of Newtonian fluids between two concentric cylinders of a special cross-sectional form

In this paper, we performed a numerical simulation of natural-convection of Newtonian fluids between two cylinders of different cross-sectional form. The inner cylinder is supposed to be hot and the outer cylinder is assumed to be cold. The diameter of inner cylinder to the diameter of outer cylinder defines the radii ratio (RR = 2.5). The governing equations describing the physical behavior of fluid-flow and heat transfer are solved using finite volume method. The effects of Prandtl number (Pr = 0.71-100), Rayleigh number (Ra = 103-105), and inclination angle of inner cylinder (? = 0-80?) on streamlines, isotherms, and dimensionless velocity are presented and discussed. Also, the mean average Nusselt number of inner cylinder is plotted vs. the governing parameters. All present simulations are considered in 2-D for steady laminar flow regime. The obtained results showed that the flow between cylinders is more stable for the inclination angle ? = 0?. Increase in Rayleigh number increases the heat transfer rate for all values of inclination angle. Furthermore, the effect of Prandtl number on the mean average Nusselt number becomes negligible when Prandtl number is over the value 7.01. For example at Pr = 0.71 and Ra =105, increase in inclination from 0? to 40? decreases the average Nusselt number by 5.4%. A new correlation is also provided to describe the average Nusselt number as function of Prandtl number and Rayleigh number at ? = 0?.

Pulsating flow in a channel filled with a porous medium under local thermal non-equilibrium condition: an exact solution

The present work investigates analytically the problem of forced convection heat transfer of a pulsating flow, in a channel filled with a porous medium under local thermal non-equilibrium condition. Internal heat generation is considered in the porous medium, and the channel walls are subjected to constant heat flux boundary condition. Exact solutions are obtained for velocity, Nusselt number and temperature distributions of the fluid and solid phases in the porous medium. The influence of pertinent parameters, including Biot number, Darcy number, fluid-to-solid effective thermal conductivity ratio and Prandtl number are discussed. The applied pressure gradient is considered in a sinusoidal waveform. The effect of dimensionless frequency and coefficient of the pressure amplitude on the system’s velocity and temperature fields are discussed. The general shape of the unsteady velocity for different times is found to be very similar to the steady data. Results show that the amplitudes of the unsteady temperatures for the fluid and solid phases decrease with the increase in Biot number or thermal conductivity ratio. For large Biot numbers, dimensionless temperatures of the solid and fluid phases are similar and are close to their steady counterparts. Results for the Nusselt number indicate that increasing Biot number or thermal conductivity ratio decreases the amplitude of Nusselt number. Increase in the internal heat generation in the solid phase does not have a significant influence on the ratio of amplitude-to-mean value of the Nusselt number, while internal heat generation in the fluid phase enhances this ratio.

Temperature-Dependent Viscosity and Prandtl Number Effects on Natural Convection Methanol Boundary Layers about a Vertical Plate with Injection

<p>The present study deals with the effect of temperature-dependent viscosity and Prandtl number on the steady, natural, laminar flow of methanol past a vertical porous plate with injection. The coupled nonlinear partial differential equations governing the non-similar flow have been solved numerically using an implicit finite-difference scheme along with the quasilinearization technique. Numerical results indicate that temperature-dependent viscosity and Prandtl number, both have a major role on skin friction and heat transfer parameters as well as velocity and temperature fields.</p>

Unsteady Magnetohydrodynamic Flow with Heat and Mass Transfer between Two Parallel Walls under Variable Magnetic Field

This paper presents a semi-analytical solution using homotopy perturbation method of magnetohydrodynamic unsteady flow of fluids between two inclined parallel permeable walls with the consideration of heat and mass transfer. The flow is subjected to variable magnetic field normal to the walls. The solution for the governing nonlinear partial differential equations coupled with equations of motion, concentration, and energy equation including the viscous and Joule dissipations is adopted. The effect of Prandtl and Eckart numbers, Reynolds magnetic number, Schmidt and Grashof numbers, and modified Grashof number, and magnetic field strength on velocity, temperature, concentration, and induced magnetic field are discussed. Results are represented graphically for the considered problem.

Linear instability of concentric annular flow: Effect of Prandtl number and gap between cylinders

The stability of non-isothermal Poiseuille flow in a coaxial annular domain is studied using normal mode analysis. The stably stratified flow (i.e., when buoyant force is in the direction of forced flow) is induced due to external pressure gradient and maintenance of linear variation of the temperature of the inner cylinder. We have emphasized the impact of the gap between cylinders in terms of curvature parameter (C) and Prandtl number (Pr) on the instability of the flow under axisymmetric as well as non-axisymmetric disturbances. In general, it has been found that the flow of low Pr fluids under axisymmetric disturbance is more stable in comparison with the flow under non-axisymmetric disturbance. When the gap between cylinders is relatively small (i.e. C is large), the flow under non-axisymmetric is most stable for the fluids having high Pr. It is known that the Newtonian pipe flow is linearly stable for all values of Reynolds number (Re). However, in the present study, we have seen the instability of the pipe flow with a thin rod placed at the center even for a very small value of heat source intensity (in terms of Rayleigh number) at Re=2000. Here the disturbance velocity is concentrated in the vicinity of the inner wall and instability is due to transfer of kinetic energy from the basic flow through Reynold’s stress. Depending on the value of controlling parameters, three types of instability: thermal-shear, interactive, thermal-buoyant have been observed. In general, for a fixed value of Pr, the type of instability is not affected by C and Re. The non-isothermal Poiseuille flow of low Pr (including mercury and gases) is governed by thermal-shear instability and the same of high Pr (including liquid and oil) is governed by thermal-buoyant instability.

Numerical analysis of effective Prandtl model on mixed convection flow of γAl2O3–H2O nanoliquids with micropolar liquid driven through wedge

Nanomaterials offer great potentials in enhancing the convective heat transfer performance whereas the Prandtl number creates an important responsibility in controlling the thermal and velocity boundary layers. Therefore, the impacts of models of effective Prandtl number, thermal conductivity and viscosity are collected through experimental data on mixed convection flow of micropolar γAl2O3–H2O nanoliquid from wedge are explored. The nonlinear governing PDE’s are transmuted into ODE’s through similarity variables. These equations are then solved through Nachtsheim–Swigert shooting method with Runge–Kutta Fehlberg technique. Buoyancy assisting and buoyancy opposing flows are taken into consideration. The behavior of velocity, micro rotation and temperature distribution as well as the skin friction and the Nusselt number are depicted graphically by taking different parameters. The results reveal that the multiple solutions are available in opposing buoyancy flow only. It is also scrutinized that the skin friction enlarges due to micropolar parameter in first solution and decreases in second solution, while the Nusselt number reduces in first solution. In order to verify the obtained results a correlation is provided with previous work.

Effect of Prandtl Number on Mixed Convective Heat Transfer from a Porous Cylinder in the Steady Flow Regime.

The effect of the Prandtl number (Pr) on the flow and heat transfer from a porous circular cylinder with internal heat generation in the mixed convection regime is numerically investigated. The steady flow regime is considered over the ranges of the Reynolds number (Re), Darcy number (Da), and Richardson number (Ri), varying from 5 to 40, 10−6 to 10−2, and 0 to 2, respectively. The wake structure, the temperature distribution, and the heat transfer rate are discussed. Besides precipitating the growth of the recirculating wake, the Prandtl number is found to have a significant impact on the thermal characteristics. The concave isotherms, resembling a saddle-shaped structure, occur behind the cylinder at larger Pr, resulting in swells of the isotherms pairing off at the lateral sides. These swells are found to have a negative effect on heat transfer owing to a relatively smaller temperature gradient there. Then, the heat transfer rate in terms of the local Nusselt number (Nu) and enhancement ratio (Er) is calculated, which is closely related to Pr, Re, Da, and Ri. The local minimum heat transfer rate along the cylinder surface is found at the position where the swells of the isotherms form.

Effect of unipolar charge injection direction on the onset of Rayleigh-Bénard convection: a lattice Boltzmann study

This paper numerically investigates the effect of unipolar charge injection on the onset of Rayleigh-Bénard convection (RBC) in a layer of insulating liquid contained between two parallel plates. A recent developed unified model on the basis of the mesoscopic lattice Boltzmann method (LBM) is employed to solve the entire macro control equations. Two different cases of charge injection from above and from below are considered. It is interesting to find that the charge injection advances the onset of convection regardless of the injection direction. Besides, the decrease of critical Rayleigh number, along with the increase of flow intensity and the enhancement of heat transfer resulting from charge injection remain the same for different injection directions. However, the results show that the charge injection directions have significant effect on temperature and charge density distribution, as well as the evolution of flow motion, especially for the case of larger electric Rayleigh number. In addition, the effects of Prandtl number, the mobility number and the injection number are also studied.

Thermal radiation on oscillatory flow past a moving vertical plate in a time varying gravity field

Thermal radiation on oscillatory flow of a viscous incompressible fluid of an optically dense medium past a moving hot vertical plate under the influence of a time varying gravity field by using Rosseland approximation is investigated. Since the flow is characterised by a time varying gravity field, a g-jitter force is associated with microgravity field in space. In a g-jitter driven flow, the empty space above the leading edge of the medium corresponds to a crystal growth in space; the net mass flow rate in the system is zero. In a time varying gravity field, the flow is characterised by the oscillating velocity near the plate surface with reference to a phase angle and to increase the fluid velocity with increase in phase angle due to impulsive onset into motion. The effects of radiation parameter, Prandtl number and Grashof number on the velocity field are analysed. The effect of Nusselt number at the plate is discussed.

Entropy optimized dissipative flow of effective Prandtl number with melting heat transport and Joule heating

Melting heat in mixed convective magnetohydrodynamic flow of viscous material bounded by a stretchable plate is examined. Thermal expression consists of Joule heating and heat generation. Here (γAl2O3 − H2O and γAl2O3 − C2H6O2) nanofluids are analyzed. Boundary layer flows is determined for both Prandtl number and effective Prandtl number. Melting heat is accounted. Second law of thermodynamics is utilized to determine entropy generation. Entropy generation is minimized and discussed graphically through different parameters. Furthermore the drag forces and heat transfer rates have been examined through tabulated values.

Thermally developed unsteady viscoelastic micropolar nanofluid with modified heat/mass fluxes: A generalized model

The main concern of current investigation is to develop an unsteady mathematical model for viscoelastic micropolar nanofluid flow caused by periodically accelerating surface. The generalized heat and mass flux relations are utilized in the energy equation. Further, the energy equation is altered by employing thermal radiation features derived from famous theory of Rosseland’s approximation. The Prandtl number and radiation constants are premeditated as new parameter called here as effective Prandtl number. The solution of formulated dimensionless equations has been stimulated with the help of homotopy analysis method. A theoretical based observation for each engineering parameters is carried out with detailed physical properties. The involved physical quantities namely effective Nusselt number and Sherwood number are evaluated numerically. Numerous contributions have been advised for flow of micropolar fluid but no attempt has been made regarding unsteady viscoelastic micropolar nanofluid with Cattaneo–Christov model which may eventually reduced to the traditionally viscous, viscoelastic and micropolar fluid results simultaneously as limiting cases.

NUMERICAL SOLUTIONS FOR AXISYMMETRIC NON-NEWTONIAN STAGNATION ENROBING FLOW, HEAT, AND MASS TRANSFER WITH APPLICATION TO CYLINDRICAL PIPE COATING DYNAMICS

Heat and mass transfer in variable thermal conductivity micropolar axisymmetric stagnation enrobing flow on a cylinder is studied. Numerical solutions are obtained with an optimized variational finite element procedure and also a finite difference method. Graphical variations of velocity, angular velocity, temperature and concentration are presented for the effects of Reynolds number, viscosity ratio, curvature parameter, Prandtl number and Schmidt number. Excellent agreement is obtained for both finite element method (FEM) and finite difference method (FDM) computations. Further validation is achieved with a Chebyshev spectral collocation method (SCM). Skin friction is elevated with greater Reynolds number whereas it is suppressed with increasing micropolar parameter. Heat transfer rate decreases with an increase in the thermal conductivity parameter. Temperature and thermal boundary layer thickness is reduced with increasing thermal conductivity parameter and Reynolds number. Greater Reynolds number accelerates the micro-rotation values. Higher Schmidt number reduces the mass transfer function (species concentration) values. The mathematical model is relevant to polymeric manufacturing coating (enrobing) flows.

Heat transfer through converging-diverging channels using Adomian decomposition method

This study consists of proposing a new mathematical method to develop a new model for evaluating thermal distributions throughout convergent-divergent channels between non-parallel plane walls in Jeffery Hamel flow. Subsequently, dimensionless equations that govern temperature fields and velocity are numerically tackled via the Runge–Kutta-Fehlberg approach based on the shooting method. Additionally, an analytical study is performed by applying an effective computation technique named Adomian Decomposition Method. Determining the effect of Reynolds and Prandtl numbers on the heat transfer and fluid velocity inside converging/diverging channels can be mentioned as the fundamental purpose of this research. Based on the results obtained for dimensionless velocity and thermal distributions, a supreme match can be observed between numerical and analytical results indicating the adopted ADM method is valid, applicable, and has great precision.

Experimental investigation of turbulent Rayleigh-Bénard convection of water in a cylindrical cell: The Prandtl number effects for Pr > 1

We report an experimental study of turbulent Rayleigh-Bénard convection in a cylindrical cell of aspect ratio unity, focusing on the effects of the Prandtl number (Pr). Purified water was used as the convecting fluid. Five different Pr between 3.58 and 9.40 were achieved by changing the mean temperature of water, and the measurements were carried out over the Rayleigh number range 2.63 × 108 ≤ Ra ≤ 3.89 × 1010. Over the present parameter range, the measured Nusselt number Nu is found to scale as Nu ∼ Raβ with β = 0.30 and to be independent of Pr. Based on the oscillation period of the measured temperature, the Reynolds number Re scales as Re ∼ Ra0.47Pr−0.72. The local temperature fluctuations at the cell center and near the cell’s sidewall were measured, and their relations with Ra and Pr were studied. Our results further reveal that the non-Oberbeck-Boussinesq effects of water have a relatively small influence on the measured scaling relation Nu ∼ Raβ.

LBM simulation of piezo fan in square enclosure

PurposeThis paper aims to investigate the use of a piezo fan in an enclosure on wall heat transfer and thermal boundary layer profile in constant wall temperature situation.Design/methodology/approachThe governing partial differential equations of mass, momentum and energy in addition to boundary conditions are solved by lattice Boltzmann method. The problem is solved numerically using D2Q9 population's model and Bhatnagar–Gross–Krook collision model with a code written in MATLAB.FindingsThe effects of Prandtl number (Pr) and the frequency of piezo fan vibrations are critically investigated on the hydrothermal characteristics of the square cavity. The mesh independency study and the validation of the proposed model are accomplished with numerical results of Ghia et al. (1982) and analytical solution of pure conduction very good agreement is found between present results and benchmark findings. Generally, with increasing beam frequency, the heat removal from heat source increased. It is found that, for all Prandtl numbers, wall Nusselt number will increase with the increase of the beam frequency. This enhancement is more intense in higher Prandtl number.Originality/valueBased on these results, the use of piezo fan in an enclosure can be classified as standalone as well as heat sink integrated cooling solution.

Integrating thermal‐concentration smoothed profile with lattice Boltzmann methods for simulating sedimentation of nonisothermal circular particles

SummaryA thermal‐concentration smoothed profile‐lattice Boltzmann method is proposed to study the effect of the concentration field on the dynamic behavior of nonisothermal cylindrical particles during the sedimentation process. The velocity, temperature, and concentration equations are solved using the lattice Boltzmann method. Moreover, the smoothed profile method is employed to enforce the nonslip boundary condition as well as constant temperature and constant concentration boundary conditions at the particles surfaces. Moreover, the Boussinesq approximation is used to couple the velocities, temperatures, and concentrations fields. The proposed combined method is validated by comparing the present numerical results with those found in the literature, showing good consistency. Then, the effect of the concentration buoyancy on the behavior of nonisothermal particles is discussed. In addition, the effect of Prandtl, Schmidt, and thermal Grashof numbers on the settling process is investigated. The results show that, by adding the effect of concentration, the maximum settling velocity of hot particles is reduced more relative to the cold ones; accordingly, the cold particles are settled faster than the hot ones. Finally, the sedimentation of two particles in a container at high thermal Grashof is investigated. It is shown that, at high thermal Grashof, there is an intense competition between the buoyancy force and gravity for the hot particles. The buoyancy flow generated leads to the reversal of the drafting‐kissing‐tumbling motion of the hot particles, making the particles move upward.

Stably stratified exact coherent structures in shear flow: the effect of Prandtl number

We examine how known unstable equilibria of the Navier-Stokes equations in plane Couette flow adapt to the presence of an imposed stable density difference between the two boundaries for varying values of the Prandtl number $Pr$, the ratio of viscosity to density diffusivity, and fixed moderate Reynolds number, $Re=400$. In the two asymptotic limits $Pr \to 0$ and $Pr \to \infty$, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the $Pr \to 0$ limit, density variations away from a constant stable density gradient become vanishingly small as diffusion of density dominates over advection, allowing equilibria to exist for bulk Richardson number $Ri_b \lesssim O(Re^{-2}Pr^{-1})$. Alternatively, at high Prandtl numbers, density becomes homogenised in the interior by the dominant advection which creates strongly stable stratified boundary layers that recede into the wall as $Pr\to\infty$. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing $Ri_b$. An asymptotic analysis is presented in the passive scalar regime $Ri_b \lesssim O(Re^{-2})$, which reveals $O(Pr^{-1/3})$-thick stratified boundary layers with $O(Pr^{-2/9})$-wide eruptions, giving rise to density fingers of $O(Pr^{-1/9})$ length and $O(Pr^{-4/9})$ width that invade an otherwise homogeneous interior. Finally, increasing $Re$ to $10^5$ in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.

Overshooting in simulations of compressible convection

Context.Convective motions that overshoot into regions that are formally convectively stable cause extended mixing.Aims.We aim to determine the scaling of the overshooting depth (dos) at the base of the convection zone as a function of imposed energy flux (ℱn) and to estimate the extent of overshooting at the base of the solar convection zone.Methods.Three-dimensional Cartesian simulations of hydrodynamic compressible non-rotating convection with unstable and stable layers were used. The simulations used either a fixed heat conduction profile or a temperature- and density-dependent formulation based on Kramers opacity law. The simulations covered a range of almost four orders of magnitude in the imposed flux, and the sub-grid scale diffusivities were varied so as to maintain approximately constant supercriticality at each flux.Results.A smooth heat conduction profile (either fixed or through Kramers opacity law) leads to a relatively shallow power law withdos∝ ℱn0.08for low ℱn. A fixed step-profile of the heat conductivity at the bottom of the convection zone leads to a somewhat steeper dependency ondos∝ ℱn0.12in the same regime. Experiments with and without subgrid-scale entropy diffusion revealed a strong dependence on the effective Prandtl number, which is likely to explain the steep power laws as a function of ℱnreported in the literature. Furthermore, changing the heat conductivity artificially in the radiative and overshoot layers to speed up thermal saturation is shown to lead to a substantial underestimation of the overshooting depth.Conclusions.Extrapolating from the results obtained with smooth heat conductivity profiles, which are the most realistic set-up we considered, suggest that the overshooting depth for the solar energy flux is about 20% of the pressure scale height at the base of the convection zone. This is two to four times higher than the estimates from helioseismology. However, the current simulations do not include rotation or magnetic fields, which are known to reduce convective overshooting.

Determination of Heat Transfer Coefficient and Thermal Dispersion of a Representative Porous Structure Based on Pore Level Simulations

In the present work, pore level simulations were performed to calculate heat transfer coefficient and thermal dispersion coefficient of a consolidated porous structure. A three-dimensional cubic unit cell structure, made up of six cuboid struts was selected to represent the idealized porous structure. To analyze fluid flow and heat transfer through the porous structure, governing fluid flow and energy equations were solved using the finite-volume method. A parametric study was carried out to investigate the effects of Reynolds number, Prandtl number, and porosity on heat transfer coefficient of the considered structure. A separate parametric study explored the variation of stream-wise dispersion coefficient with Reynolds number, Prandtl number, thermal conductivity ratio of the solid and the fluid phase, and porosity. Finally, generalized correlations for both non-dimensional heat transfer coefficient and stream-wise dispersion coefficients were proposed. It is observed that the obtained heat transfer coefficient follows two distinct trend lines for both porosities considered in the present investigation. For the considered operating conditions, the stream-wise dispersion coefficient is found to be proportional to square of Peclet number.