Various Values Of Prandtl Number Research Articles

A Mathematical Study on a Steady MHD Flow in Double Stratification Medium

The detailed analysis on the flow of MHD fluid in double stratification medium across a stretching sheet with exponential permeability is examined in this research. The approximate analytical solution for the governing equations in steady state is found by using a new approximate analytical method called Ananthaswamy-Sivasankari Method (ASM) and Modified Homotopy Analysis Method (MHAM). The approximate analytical expressions for the dimensionless velocity, dimensionless temperature and dimensionless concentration are derived using these methods. The analytical and numerical results (previous work) are compared and there is a good agreement between our analytical results and numerical works. The impacts of several parameters including porosity, magnetic, suction and heat source parameters are shown in graphical representation. The error table for the physical parameter namely Nusselt number for various values of Prandtl number has been provided. Both ASM and MHAM are very useful to solve some other non-linear boundary value problems especially in MHD fluid flow

Unsteady Electromagnetic Free Convection Micropolar Fluid Flow through a Porous Medium along a Vertical Porous Plate

Unsteady electromagnetic free convection flows of two-dimensional micropolar fluid through in a porous medium parallel to a vertical porous plate have been investigated numerically. Similarity analysis has been used to transform the governing equations into its non-dimensional form by using the explicit finite difference method to obtain numerical solutions. Estimated results have been gained for various values of Prandtl number, Grashof number, material parameters, micropolar parameter, electric conductivity, electric permeability, thermal relaxation time and the permeability of the porous medium. The effects of pertinent parameters on the velocity, electric induction, magnetic induction, microrotation and temperature distributions have been investigated briefly and illustrated graphically.

A scientific report on the flow of Maxwell fluid with heat transfer in vertical oscillating cylinder

The nonlinear nature of viscoelastic non-Newtonian fluids, introduce a unique challenge to physicists and mathematicians. By developing and utilizing viscoelastic models can play a special role in saving and treatments of every living species and to describe its particular characteristics. In the past three decades, viscoelastic fluid models are focused to improve its accuracy and reliability. Some rate type viscoelastic fluids include Maxwell fluid which effects in relaxation time. Such effect of relaxation time cannot be predicted by differential-type fluids. The polymers of low molecular weight are usefully described by Maxwell model. However, a keen interest of the researchers is seen in studying rate type fluids due to the fact that they incorporate both the elastic nature and memory behaviour together. In this article, viscoelastic Maxwell fluid is considered in cylindrical tube together with heat transfer due to convection caused by the buoyancy force. This problem is modelled using the classical approach and then solved for exact solution using joint the Laplace and Hankel transforms. Effects of pertinent parameters on Maxwell fluid velocity have been shown graphically. Behaviour of temperature is studied for various values of Prandtl number.

Thermal radiation effects on stagnation point flow past a stretching/shrinking sheet in a Maxwell fluid with slip condition

In this study, the numerical solution of the thermal radiation effects on a stagnation point flow past a stretching/shrinking sheet in a Maxwell fluid with slip condition is considered. The transformed boundary layer equations are solved numerically using the Runge-Kutta-Fehlberg (RKF) method. Numerical solutions are obtained for the skin friction coefficient and the wall temperature as well as the temperature and the velocity profiles. The features of the flow and the heat transfer characteristics for various values of Prandtl number, stretching/shrinking parameter, thermal radiation parameter, Maxwell parameter, dimensionless velocity slip parameter and thermal slip parameter are analyzed and discussed.

Heat flow visualization during mixed convection within entrapped porous triangular cavities with moving horizontal walls via heatline analysis

This paper analyzes the fluid and heat flow within entrapped porous triangular cavities involving various thermal boundary conditions (case 1: hot horizontal walls and cold inclined walls; case 2: cold horizontal walls and hot inclined walls). Finite element based numerical study has been performed for various values of Prandtl number (Prm=0.026 and 7.2), Darcy number (Dam=10-4-10-2), Reynolds number (Re=1-1000) at a high value of Grashof number (Gr=105). The upper cavity for the case 1 and the lower cavity for the case 2 show almost similar trends and vice versa based on the exact opposite thermal boundary conditions. Heat transfer is conduction dominant at Dam=10-4 involving both Prm whereas, convection heat transfer dominates at Dam=10-2 especially for Prm=7.2. Although the motion of the horizontal walls significantly influences the fluid flow field within the cavities, due to the decoupling between the fluid and thermal fields at the low Pem (Pem=0.026,0.26 and 2.6), conduction dominant heat transfer occurs. At the high Dam and Prm=7.2, a pair of symmetric streamline or heatline cells are seen for Re≤1 whereas, a bigger primary streamline or heatline cell is accompanied by a secondary streamline or heatline cell for 1⩽Re≤100. At 500⩽Re⩽1000, the single larger streamlines or heatline cells are seen especially for Prm=7.2 involving Dam=10-2. At Prm=0.026, the average heat transfer rates (Nut‾,Nub‾, Nul‾ and Nur‾) are almost constant with Re and Dam. At Prm=7.2, the larger magnitudes ofNut‾ and Nul‾ are observed for Re≤10 compared to those for Re=100 involving Dam⩾10-3 whereas, the magnitudes of Nur‾ are almost similar for all Re involving all Dam in the upper cavity (case 1) within Re⩽100. In the lower cavity (case 1), the magnitudes of Nub‾ and Nur‾ are larger for Re=100 whereas, the magnitudes of Nul‾ are larger for Re≤1 for almost entire range of Dam within Re≤100 involving Prm=7.2. The average Nusselt numbers are significantly larger at the high Re (Re=500 and 1000) involving all Dam (except for the horizontal walls involving Dam⩽8×10-4) in the cases 1 and 2 at Prm=7.2.

Study of peristaltic flow of magnetohydrodynamics Walter's B fluid with slip and heat transfer

The purpose of this paper is to investigate the effects of magnetohydrodynamics peristaltic flow of Walter's B fluid in an inclined asymmetric channel under the influence of slip conditions. The effects of heat and mass transfer are also taken into account. Analytical solutions of nonlinear coupled equations are obtained by regular perturbation method. Graphs for different flow parameters of interest are sketched and analyzed. It is observed that the absolute value of shear stress and heat transfer coefficient decreases by increasing the magnetic parameter whereas with the increase of magnetic parameter, the concentration decreases. Opposite behavior has been noted for temperature and heat transfer coefficient at upper and lower walls against the various values of Prandtl number, Eckert number, slip parameter and material constant of Walter's B fluid. Oscillatory behavior of heat transfer coefficient is observed which is due to propagation of peristaltic waves along the walls of the channel.

Stagnation point flow and heat transfer over a stretching/shrinking sheet in a viscoelastic fluid with convective boundary condition and partial slip velocity

In this study, the mathematical modeling for the stagnation point flow and heat transfer over a stretching/shrinking surface in a viscoelastic fluid (Walter’s liquid-B model) with partial slip velocity is considered. The non-linear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation before being solved numerically using the Runge-Kutta-Fehlberg method. Numerical solutions are obtained for the surface temperature, temperature gradient at the surface and the skin friction coefficient. The features of the flow and heat transfer characteristics for various values of Prandtl number, the dimensionless viscoelastic parameter, stretching parameter, constant velocity slip parameter and conjugate parameter are analyzed and discussed. It is found that the heat transfer rate is higher for Walter’s fluid compared to the classical viscous fluid and the presence of the velocity slip reduces the effects of the viscoelastic parameter on the skin friction coefficient.

Soret and Dufour effects on convective heat and mass transfer in stagnation-point flow towards a shrinking surface

A mathematical model is presented to study the Soret and Dufour effects on the convective heat and mass transfer in stagnation-point flow of viscous incompressible fluid towards a shrinking surface. Suitable similarity transformations are used to convert the governing partial differential equations into self-similarity ordinary differential equations that are then numerically solved by shooting method. Dual solutions for temperature and concentration are obtained in the presence of Soret and Dufour effects. Graphical representations of the heat and mass transfer coefficients, the dimensionless thermal and solute profiles for various values of Prandtl number, Lewis number, Soret number and Dufour number are demonstrated. With Soret number the mass transfer coefficient which is related to mass transfer rate increases for both solutions and the heat transfer coefficient (related to heat transfer rate) for both solutions becomes larger with Dufour number. The Prandtl number causes reduction in heat and the mass transfer coefficients and similarly with the Lewis number mass transfer coefficient decreases. Also, double crossing over is found in dual dimensionless temperature profiles for increasing Soret number and in dual dimensionless concentration profiles for the increase in Dufour number. Due to the larger values of Dufour number the thermal boundary layer increases and for Prandtl number increment it decreases; whereas, the solute boundary layer thickness reduces with increasing values of Prandtl number and Lewis number.

Flow and Heat Transfer of Jeffreys Non-Newtonian Fluid from Horizontal Circular Cylinder

The boundary-layer flow and heat transfer of an incompressible Jeffreys viscoelastic fluid from a permeable horizontal circular cylinder is analyzed. The surface of the cylinder is maintained at a constant temperature. The boundary-layer conservation equations, which are parabolic in nature, are normalized into nonsimilar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite difference scheme. The variation of the reduced Nusselt number and local skin-friction coefficient with Deborah number and suction parameter, for various values of Prandtl number, and ratio of relaxation to retardation times are tabulated and provided in graphical form. It is found that the velocity is reduced with increasing Deborah number, whereas the temperature is increased. Increasing the ratio of relaxation to retardation times enhances the velocity but decreases the temperature. Increasing the Deborah number is observed to decrease the Nusselt number (heat transfer rate) and the skin-friction coefficient. Increasing the ratio of relaxation to retardation times is found to increase the Nusselt number (heat transfer rate) and the local skin friction.

Closed-Form Analytical Solutions for Laminar Natural Convection on Horizontal Plates

A boundary layer based integral analysis has been performed to investigate laminar natural convection heat transfer characteristics for fluids with arbitrary Prandtl number over a semi-infinite horizontal plate subjected either to a variable wall temperature or variable heat flux. The wall temperature is assumed to vary in the form T¯w(x¯)-T¯∞=ax¯n whereas the heat flux is assumed to vary according to qw(x¯)=bx¯m. Analytical closed-form solutions for local and average Nusselt number valid for arbitrary values of Prandtl number and nonuniform heating conditions are mathematically derived here. The effects of various values of Prandtl number and the index n or m on the heat transfer coefficients are presented. The results of the integral analysis compare well with that of previously published similarity theory, numerical computations and experiments. A study is presented on how the choice for velocity and temperature profiles affects the results of the integral theory. The theory has been generalized for arbitrary orders of the polynomials representing the velocity and temperature profiles. The subtle role of Prandtl number in determining the relative thicknesses of the velocity and temperature boundary layers for natural convection is elucidated and contrasted with that in forced convection. It is found that, in natural convection, the two boundary layers are of comparable thickness if Pr ≤ 1 or Pr ≈ 1. It is only when the Prandtl number is large (Pr > 1) that the velocity boundary layer is thicker than the thermal boundary layer.

MHD forced convection boundary layer flow with a flat plate and porous substrate

The flow and heat transfer for an electrically conducting fluid with a porous substrate and a flat plate under the influence of magnetic field is considered. The magnetic field is assumed to be uniform and also along normal to the surface. The momentum and energy equations are transformed to ordinary differential equations by using suitable similarity transformation and are solved by standard techniques. But the energy equation is solved by considering two boundary layers, one in the porous substrate and the other above the porous substrate. Numerical results are presented through graphs with various values of magnetic parameter for both velocity and thermal boundary layers along with Nusselt number and for various values of Prandtl number and Eckert number in thermal boundary layer.

Similarity Theory for Forced Convection over Horizontal Plates

A similarity analysis is performed to study the fluid flow and heat transfer characteristics for the steady laminar forced convection boundary-layer flow past a semi-infinite horizontal plate in the presence of mass transfer, i.e., suction or blowing. The plate is subjected either to a nonuniform heat flux or nonuniform wall temperature. The heat flux is assumed to vary in the following form , whereas the wall temperature is assumed to vary as . Numerical results are obtained for various values of Prandtl number, suction or injection parameter , as well as for various levels of heating or . The effects of various values of Prandtl number , suction or injection parameter , and the index or on velocity profiles, temperature profiles, skin-friction coefficients, and local Nusselt number are presented. The results of the present general theory agree well with previously published results for the case of constant wall temperature (the only case with quantitative solutions that were available in the literature). Subtle new flow physics, such as a sharp reduction in the Nusselt number with increasing Prandtl number when mass injection is present and the Prandtl number is large, has been elucidated.

Magnetohydrodynamic free convection flow above an isothermal horizontal plate

In the present study a new similarity theory is developed to study the fluid flow and heat transfer characteristics for the steady laminar natural convection boundary layer flow of an incompressible and electrically conducting fluid past a semi-infinite horizontal plate subjected to a constant wall temperature under the action of transverse magnetic field. The governing parabolic boundary layer PDEs are transformed to ordinary differential equations using similarity transformation. This results in a set of three coupled, non-linear ordinary differential equations with variable coefficients (representing the interaction of the temperature and velocity fields) which are then solved by the shooting method. Asymptotic analyses and series solutions are also constructed to explore the mathematical behaviour of the solutions. The numerical results are obtained for various values of Prandtl number and magnetic field parameter ζ. The effects of various values of Prandtl number and magnetic field parameter ζ on the velocity profiles, temperature profiles, wall shear stress and heat transfer coefficients are presented. The results indicate that the wall shear stress τw decreases whereas the local Nusselt number Nux increases with increase in Prandtl number if the magnetic field parameter ζ is held constant at a particular value. On the other hand, both the wall shear stress and Nusselt number decrease with increase in the magnetic field parameter ζ for a fluid with constant Prandtl number. Generic correlations for Nux and τw have been developed in terms of the fluid Prandtl number Pr and magnetic field parameter ζ. A generic relation for free convection flow with magnetohydrodynamic effects, that is equivalent to the well-known Reynolds analogy for forced convection flow, has also been formulated in the present work.

Nanofluid Flow over a Rotating Disk with Prescribed Heat Flux

An analysis is carried out to study the problem of the steady flow and heat transfer over a rotating disk with a prescribed heat flux in nanofluid. Nanofluid considered is Copper (Cu) with water as the base fluid. The governing partial differential equations are transformed into a set of nonlinear ordinary differential equations using similarity transformation, which are then solved using the Nachtsheim-Swigert Shooting iteration technique along with the fourth order Runga Kutta method. The features of the flow and heat transfer characteristics are analyzed and discussed. The radial velocity, tangential velocity and the axial velocity for copper-water nanofluid are calculated and are represented graphically. Numerical results for dimensionless temperature, the radial skin friction coefficient and the tangential skin friction coefficient of the nanofluid flows are obtained and computations are carried out for the various values of Prandtl number. It is found that for the prescribed heat flux case (PHF case), the effect of Prandtl number is to reduce the temperature as it increases for copper-water nanofluid.

A similarity theory for natural convection from a horizontal plate for prescribed heat flux or wall temperature

An analysis is performed to study the fluid flow and heat transfer characteristics for the steady laminar natural convection boundary layer flow over a semi-infinite horizontal flat plate subjected to a variable heat flux or variable wall temperature. The heat flux qw(x¯) varies as the power of the horizontal coordinate in the form qw(x¯)=ax¯m whereas the wall temperature T¯w(x¯) is assumed to vary as T¯w(x¯)=T¯∞+bx¯n. The governing boundary layer equations are first cast into a dimensionless form and then transformed to ordinary differential equations using generalized stretching transformation to derive the appropriate similarity variables. This results in a set of three coupled, non-linear ordinary differential equations with variable coefficients (representing the interaction of the temperature and velocity fields) which are then solved by the shooting method. The numerical results are obtained for various values of Prandtl number under different levels of heating. The effects of various values of Prandtl number and the indices m and n on the velocity profiles, temperature profiles, skin friction, and heat transfer coefficients are presented. Correlation equations between Nusselt number and Grashof number, and that between skin friction coefficient and Grashof number have been derived. It is shown that when the heat flux variation is specified, Nu∧∝(GrL∗)16andcˆf∝(GrL∗)-16; when the wall temperature variation is specified, Nu∧∝(GrL)15andcˆf∝(GrL)-15. For a fixed value of m or n (including the cases of constant wall temperature or constant heat flux), the heat transfer coefficient increases whereas the local wall shear stress decreases with increasing Prandtl number. The heat transfer coefficient increases with increasing values of exponent m or n when the Prandtl number is kept constant. For a fixed Prandtl number, the local wall shear stress increases with increasing values of n, while it decreases with increasing values of m.

Viscous Flow over Nonlinearly Stretching Sheet with Effects of Viscous Dissipation

The flow and heat transfer characteristics of incompressible viscous flow over a nonlinearly stretching sheet with the presence of viscous dissipation is investigated numerically. The similarity transformation reduces the time‐independent boundary layer equations for momentum and thermal energy into a set of coupled ordinary differential equations. The obtained equations, including nonlinear equation for the velocity field f and differential equation by variable coefficient for the temperature field θ, are solved numerically by using the fourth order of Runge‐Kutta integration scheme accompanied by shooting technique with Newton‐Raphson iteration method. The effect of various values of Prandtl number, Eckert number and nonlinear stretching parameter are studied. The results presented graphically show some behaviors such as decrease in dimensionless temperature θ due to increase in Pr number, and curve relocations are observed when heat dissipation is considered.

Implementation of diffuse reflection boundary conditions in a thermal lattice Boltzmann model with flux limiters

We discuss three new implementation versions of diffuse reflection boundary conditions in a thermal lattice Boltzmann model. Their accuracy is investigated in the case of Couette flow by considering the slip regime. The best results are recovered with versions 2 and 3, which rely on outgoing fluxes to express the particle distribution functions in the ghost nodes outside the flow domain. Version 2 is found to be more economical since it involves no interpolation procedure. This version was thereafter used to investigate the temperature profile in Couette flow for various values of Prandtl number, as well as the capability of the thermal LB model to capture the Knudsen minimum in Poiseuille flow.

Free Convection over a Vertical Plate in a Micropolar Fluid Subjected to a Step Change in Surface Temperature

The Keller box method has been employed for free convection over a vertical plate subjected to a step change in surface temperature in micropolar fluid. Numerical results presented include the reduced angular velocity profiles, development of wall shear stress or skin friction and development of the rate of change of the gyration component at the wall for various values of Prandtl numbers and temperature ratios. The study shows that the present results obtained in micropolar fluids, when temperature ratio 0w2=1, agree very well with the previous study without temperature change.

Effect of the Temperature Difference Aspect Ratio on Natural Convection in a Square Cavity for Nonuniform Thermal Boundary Conditions

The present numerical investigation deals with steady natural convection flow in a closed square cavity when the bottom wall is sinusoidal heated and vertical walls are linearly heated, whereas the top wall is well insulated. In the nonuniformly heated bottom wall maximum temperature TH attains at the center of the bottom wall. The sidewalls are linearly heated, maintained at minimum temperature Tc at top edges of the sidewalls and at temperature Th at the bottom edges of the sidewalls, i.e., Tc≤Th≤TH. Nonlinear coupled PDEs governing the flow have been solved by the penalty finite element method with biquadratic rectangular elements. Numerical results are obtained for various values of Prandtl number (Pr)(0.01≤Pr≤10) and temperature difference aspect ratio A=[(Th−Tc)∕(TH−Tc)](0≤A≤1) for higher Raleigh number Ra=105. Results are presented in the form of streamlines, isotherm contours, local Nusselt number, and the average Nusselt number as a function of temperature difference aspect ratio A. The overall heat transfer process is shown to be tuned efficiently with suitable selection of A.

Lie group analysis of unsteady MHD three dimensional by natural convection from an inclined stretching surface saturated porous medium

Lie group method is investigated for solving the problem of heat transfer in an unsteady, three-dimensional, laminar, boundary-layer flow of a viscous, incompressible and electrically conducting fluid over inclined permeable surface embedded in porous medium in the presence of a uniform magnetic field and heat generation/absorption effects. A uniform magnetic field is applied in the y-direction and a generalized flow model is presented to include the effects of the macroscopic viscous term and the microscopic permeability of porous medium. The infinitesimal generators accepted by the equations are calculated and the extension of the Lie algebra for the problem is also presented. The restrictions imposed by the boundary conditions on the generators are calculated. The investigation of the three-independent-variable partial differential equations is converted into a two-independent-variable system by using one subgroup of the general group. The resulting equations are solved numerically with the perturbation solution for various times. Velocity, temperature and pressure profiles, surface shear stresses, and wall-heat transfer rate are discussed for various values of Prandtl number, Hartmann number, Darcy number, heat generation/absorption coefficient, and surface mass-transfer coefficient.