Boundary Layer Equations For Flow Research Articles

A numerical study of MHD nonlinear mixed convection flow over a nonlinear vertical stretching sheet with the buoyancy and suction/injection effects

The current research has a primary focus on characterizing buoyancy-driven nonlinear mixed convection. This phenomenon is studied within the context of an MHD Newtonian flow over a vertically stretching sheet with nonlinear characteristics. Flow patterns resembling those studied in this context have practical applications in various industrial sectors. Such flow patterns have practical applications in several industrial sectors, including the article and pulp industry, polymer manufacturing, electronic device cooling, solar energy collection, gas turbine plants, and nuclear power generation. The governing partial differential equations (PDEs) for nonlinear mixed convection are transformed into higher-order nonlinear ordinary differential equations (ODEs) through appropriate transformations, and these transformed ODEs are subsequently solved numerically. The software MATHEMATICA is used to solve the boundary layer flow equations, employing the semi-analytical Galerkin weighted residual method (GWRM). This approach yielded simultaneous solutions for both cases suction and injection. For both the suction and injection situations, graphical results, such as velocity, concentration, and temperature profiles are included. An augmentation in the nonlinear stretching parameter results in a reduction in velocity near the surface of the sheet. Conversely, away from the sheet, velocity increases. Moreover, an increase in nonlinear stretching parameters corresponds to higher values for both temperature and concentration. An increase in the suction/injection parameter causes a decrease in the velocity, temperature, and concentration fields. In the scenario of assisting flow, an increase in the thermal buoyancy parameter leads to higher velocity but lower temperature. Conversely, in the case of opposing flow, an increase in the thermal buoyancy parameter results in higher values for both velocity and temperature. Furthermore, a numerical examination is conducted to analyze the values of skin friction, Sherwood number, and Nusselt number.

Investigation of non-ideal effects in compressible boundary layers of dense vapors through direct numerical simulations

In this work, we present an investigation about the sources of dissipation in adiabatic boundary layers of non-ideal compressible fluid flows. Direct numerical simulations (DNS) of transitional, zero-pressure gradient boundary layer flows are performed for two fluids characterized by different complexity of the fluid molecules, namely, “air” and siloxane MM. Different sets of thermodynamic free-stream boundary conditions are selected to evaluate the influence of the fluid state on both the frictional loss and the dissipation mechanisms. The thermophysical properties of siloxane MM are calculated with a state-of-the-art equation of state. Results show that the dissipation due to both time-mean strain field, irreversible heat transfer, and turbulent dissipation differs significantly depending on both the molecular complexity of the fluid and its thermodynamic state. The dissipation coefficient calculated from the DNS results is then compared against the one obtained using a reduced-order model (ROM), which solves the two-dimensional boundary layer flow equations for an arbitrary fluid [M. Pini and C. De Servi, “Entropy generation in laminar boundary layers of non-ideal fluid flows,” in 2nd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion and Power (Springer, 2020), pp. 104–117]. Results from both the DNS and the ROM show that low values of the overall dissipation are observed in the case of fluids made of simple molecules, e.g., air, and if the fluid is at a thermodynamic state in the proximity of that of the vapor–liquid critical point.

Application of response surface methodology to optimize MHD nanofluid flow over a rotating disk with thermal radiation and joule heating

The current study represents a sensitivity analysis of the flow of gold nanoparticles in blood over a rotating disk with the combined impacts of viscous dissipation, nonlinear thermal radiation, joule heating, and slippage using response surface methodology. We use response surface methodology to investigate the relationships between the model's parameters (the input parameters), and the response variables. Response Surface Methodology is a useful method for finding the best values for the input parameters that improve the output. It can also help researchers create a list of experiments to predict the output. We often use response surface methodology with sensitivity analysis to see how the output depends on the input parameters and to suggest the optimal values of them. After incorporating the effects of MHD, viscous dissipation, nonlinear thermal radiation, and joule heating, the boundary layer flow equations are solved using NDSolve command of Mathematica. The influences of dimensionless parameters on all flow profiles and physical quantities are presented and discussed. The arithmetic has been considered for a specific range of values for the apparent parameters in order to calculate and analyze the physical quantities of interest for various relevant parameters, for instance 2≤M≤5,0≤β≤2,0.6≤A≤1.5,0.1≤Br≤0.65,0.5≤δ≤2,0≤φ≤0.15,15≤Pr≤21,0.1≤Rd≤1.5. For sensitivity analysis, we consider skin friction coefficient and local Nusselt number for three independent input parameter namely, the radial stretching rate, the magnetic field parameter, and the thermal slip parameter. After using the statistical measures such as residual quantile – quantile plots, hypothesis testing, and adjusted R2, we concluded that our models for the skin friction coefficient and local Nusselt number are best fitted.

A study based on boundary layer and entropy generation in MHD flow of micropolar fluid with variable viscosity and thermal conductivity: A non-Darcy model approach

This paper aims to analyze the problem with the study of thermal and momentum transport with entropy generation in view of the second law of thermodynamics in Magneto hydrodynamics (MHD) micropolar fluid through porous medium under the consideration of the non-Darcy model, temperature-dependent viscosity and thermal conductivity. In practical situations at higher temperatures and high speed fluid flow, it becomes reasonable to consider variable fluid flow parameters. The governing boundary layer flow equations are first converted into a coupled system of the ordinary differential equations (ODE) under the assumption of differing plate temperatures by applying appropriate similarity transformations. A shooting method has been applied to solve ordinary differential equations numerically. The last effect of microrotation, magnetic field, variable viscosity coefficient, variable thermal conductivity, etc. on momentum and thermal transport has been depicted through various graphs. The table for skin friction coefficient and Nusselt number for ideal cases has been shown to validate the model by previous findings. It is seen that K and m enhance the velocity profile on their increment opposite to this M, [Formula: see text], F and Da have been found to reduce the velocity profile. Table 3 is constructed for numerical values of skin friction coefficient and Nusselt number for different values of parameters where it can be concluded that magnetic parameter M has a tendency to enhance the skin friction and heat transfer, while variable viscosity parameters have a tendency to decline the skin friction and heat transfer.

Dynamics of Soret and Dufour effects on oscillatory flow of couple stress fluid due to stretchable curved surface

In this theoretical study, we have discussed the impacts of Soret and Dufour on the flow of time-dependent couple stress fluid across a stretchable curved oscillatory surface with magnetic field. The to and fro motion of the surface causes the flow phenomenon. The boundary layer flow equations depicting the flow phenomena are expressed in mathematical form by means of a curvilinear coordinates system along with some suitable similarity variables. The developed flow equations in the form of partial differential equations are then solved analytically by utilizing a valuable method called the homotopy analysis method (HAM). A detailed examination is performed to evaluate the influence of various involved parameters namely, the non-dimensional radius of curvature, couple stress parameter, magnetic parameter, the relation of the parameter of the surface oscillating frequency to its stretching rate constant and Soret and Dufour numbers on velocity field, pressure distribution, concentration and temperature distribution, rate of heat transfer, coefficient of skin friction, and rate of mass transport via graphs and are explained in detail. It is confirmed that the amplitude of velocity and pressure distribution exhibits reducing behavior with higher values of the couple stress parameter. The concentration and temperature profiles also diminish with varying time instants.

Variable thermal conductivity and thermophoretic aspects of free convection flow of a micropolar fluid due to a permeable cone: Heat source/sink

Boundary layer analysis is introduced for providing a numerical treatment of steady state free convection flow of non-Newtonian fluid of micropolar type through an isothermal vertical permeable cone under impacts of variable thermal conductivity and thermophoretic force. Both heat generation or absorption and viscous dissipation are considered. The resulting altered boundary layer flow equations are numerically tackled by the help of the implicit finite difference approach. Two states are addressed: one corresponds to constant thermal conductivity, and the other is variable thermal conductivity. The skin friction, couple stress coefficients fluctuations, and Nusselt number for miscellaneous significant sundry factors are presented graphically and argued. From the calculated outcomes, the coefficient of couple stress reduces for larger values of vortex viscosity parameter R, while the reverse trend is noticed through the surface temperature exponent m factor. Augmentation in thermophoretic factor ω leads to the thermophoretic forces that encapsulates the mass transmission. As a certain case, the system flow equations of a classical Newtonian liquid are gained, by dropping the micropolar factors consequences.

Magnetohydrodynamic unsteady natural convective flow in a porous medium over a moving infinite cylinder

AbstractThe effect of the magnetic field on the unsteady natural convective one‐dimensional laminar flow of an incompressible viscous and slightly electrically conducting fluid over a vertical cylinder is investigated analytically. The cylinder is assumed to be embedded in a porous medium, and it moves with a uniform velocity along the vertically upward direction. The boundary layer flow equations are converted to a non‐dimensional form using appropriate non‐dimensional parameters, and the closed‐form solutions are obtained using the Laplace transform method. Fluid velocity depends on radial distance, time, magnetic parameter, Grashof number, permeability parameter, and Prandtl number. Graphical and tabulated results of the fluid velocity profile and shear stress based on the analytical solutions are presented and discussed. The analytical solution reveals that the unsteady fluid velocity tends to a steady state at a large time. The present study demonstrates that increased porosity leads to enhancement in the fluid velocity, whereas the transverse magnetic field resists the fluid velocity. Furthermore, the absolute value of shear stress decreases with a decrease in porosity of the medium or an increase in magnetic field effect.

Hybrid Nanofluid Flow in a Porous Medium with Second-Order Velocity Slip, Suction and Heat Absorption

The foremost objective of this study is to reflect the behaviour of hybrid nanofluid towards a permeable porous medium, with consideration of second-order velocity slip and heat absorption impacts on the fluid flow. Two distinct fluids of copper (Cu) and aluminium oxide (Al2O3) are reviewed in this study to work out as a hybrid nanofluid flow. The equations of boundary layer flow in the form of partial differential equations are reduced to a system of ODEs by conducting a similarity transformation technique, and the findings that obtained from this study are presented in the respective tables and figures. The effects of involving parameters such as suction, porous medium permeability, heat absorption and second order velocity slip are perceived, as well as our intention in observing the impact of traditional nanofluid and hybrid nanofluid on the fluid flow. Our findings revealed that the hybrid Cu-Al2O3/water nanofluid performs well on the fluid flow behaviour against the mono Al2O3/water nanofluid. Moreover, the participating parameters of porous medium permeability, suction and nanoparticle volume fraction are said to improve the boundary layer thickness, while second-order velocity slip parameter is deemed to decay the fluid flow.

Nonlinear thermally radiative heat transport for brinkman type micropolar nano-material over an inclined surface with motile microorganisms and exponential heat source

The novel aim of current analysis is to identify the characteristics of Arrhenius activation energy and nonlinear thermal radiation in bio-convective brinkman type micropolar nanofluid flow with gyrotactic motile microorganisms and melting phenomena through a stretched surface. The temperature dependent heat source/sink and exponential space-based thermal source are considered. Further, concentration of nano-materials and thermal fields are considered in the occurrence of Brownian motion and thermophoresis diffusion. The narrating boundary layer flow equations of brinkman type micropolar fluid model are reframed on the form of ordinary ones with the help of appropriate similarity transformation. The bvp4c collocation technique is used in the transformed model with shooting scheme to obtain a numerical solution of the original problem. The numerical data for the velocity, thermal and solutal fields, and microorganism's field is graphically designed and impacts of the prominent parameters are described in detail. It is notice from the current study velocity of fluid has diverse nature against magnetic parameter. Growing behavior of velocity field is depicted for higher estimation of mixed convection parameter. The microorganism field is boosted up by increasing melting parameter. The current model is applicable in nanotechnology, biotechnology, drug delivery, protein chips, DNA chips, food processing, cancer treatment, smart sensors and in many more fields

Numerical approach of micropolar dust-particles natural convection fluid flow due to a permeable cone with nonlinear temperature

This contribution devotedly analyzes the behaviour of momentum and heat transfer of dust particles micropolar fluid flow due to a vertical permeable cone with suction and variable wall temperature. Homogeneously suspended dust nano-particles is suggested. Non-similar solutions for governing equations are noticed to be exist when the surface temperature varies as a power law with the length measured from the leading edge. Via suitable coordinate transformation the flow governing equations transformed into non-similar boundary layer flow equations. The subsequent nonlinearly system of micropolar liquid and the dust particle phases are solved numerically via an efficient, tridiagonal, iterative, implicit finite difference technique using FORTRAN software. Aspects of distinct emerging variables on the computations (carrier and dust-particle phases of velocity and temperature) are illustrated through tables and drawn graphical, also Nusselt number and shear and couple stresses factor are given as functions of suction variables. Results point out that an increment in the suction variable or the relative density parameter tends to enhance the local Nusselt number. The present computations are validated by favorable comparisons with earlier published data.

Magneto-hydro dynamic squeezed flow of Williamson fluid transiting a sensor surface

The present article reports the combined effects of radiation and heat origination on the electro-kinetically induced hydromagnetic squeezed flow of a pseudoplastic fluid. The fluid is passing over a microcantilever sensor surface positioned in the superficial free stream. Microcantiliver sensor can detect the flow rate and the variance in the temperature of the fluid. The thermal conductivity and fluid viscosity are assumed as a function of temperature. Boundary layer approximations are considered to construct a pseudoplastic fluid flow model. The governing system is then resolved into a non-dimensional form with the assistance of an appropriate set of control parameters. The solution to these non-dimensional equations has calculated with the assistance of familiar numerical techniques i.e. Shooting technique. The results specify that flow of fluid, temperature, and velocity profiles are remarkably influenced by the radiation parameter, fluid parameter, heat generation parameter, thermal relaxation parameter, magnetic parameter, and the squeezing number. A comprehensive graphical and tabular study is constructed to check the convergence of the obtained results. One can detect that the temperature curve is changing slightly for the Christov-Cattaneo heat transfer model as compared to classical Fourier's law of heat transfer. Further, the physical quantities, i.e. free stream velocity, variable viscosity, thermal conductivity, Weissenberg number, and Prandtl number have strong impacts on the boundary layer flow equations. It is perceived that the fluid velocity profile rises for the growing value of the magnetic parameter, but reduces for squashed flow index b. Also, a positive variation is found in the temperature profile for rising values of β and Q.

Analytical Investigation on CNT Based Maxwell Nano-fluid with Cattaneo–Christov Heat Flux Due to Thermal Radiation

At first, we derive a series form solution of the coupled highly nonlinear equations, which includes various conditions. Then, via the method of directly defined inverse mapping with the series form solution firstly reported in this paper, we can obtain theoretical and approximate analytical analysis about the transfer of heat as well as the magnetohydrodynamic flow of Maxwell nanofluid by the influence of convective heating with effects of thermal radiation. In the energy equation, heat flux model is adopted to develop the equations for viscoelastic relaxation over boundary layer flow. For this investigation, we considered base liquid as engine oil and other forms of carbon nanotubes such as single walled nanotubes and multi-walled nanotubes. Suitable similarity transformations are applied for transformation of given boundary layer flow equations. Results are compared numerically by Keller–Box method. It is found that for both singled and multi-walled carbon nanotube based nanofluids the thermal relaxation time and temperature function are inversely proportional. More interestingly it is noted that for the two types of nanofluids, fluid relaxation parameter exactly coordinates with heat transfer rate as well as skin friction investigated. Also shown that the base functions of solutions are highly convergence.

Solving the Nonlinear Boundary Layer Flow Equations with Pressure Gradient and Radiation

The physical problem under consideration is the boundary layer problem of an incompressible, laminar flow, taking place over a flat plate in the presence of a pressure gradient and radiation. For the mathematical formulation of the problem, the partial differential equations of continuity, energy, and momentum are taken into consideration with the boundary layer simplifications. Using the dimensionless Falkner–Skan transformation, a nonlinear, nonhomogeneous, coupled system of partial differential equations (PDEs) is obtained, which is solved via the homotopy analysis method. The obtained analytical solution describes radiation and pressure gradient effects on the boundary layer flow. These analytical results reveal that the adverse or favorable pressure gradient influences the dimensionless velocity and the dimensionless temperature of the boundary layer. An adverse pressure gradient causes significant changes on the dimensionless wall shear parameter and the dimensionless wall heat-transfer parameter. Thermal radiation influences the thermal boundary layer. The analytical results are in very good agreement with the corresponding numerical ones obtained using a modification of the Keller’s-box method.

Nonlinear convective boundary layer flow of micropolar‐couple stress nanofluids past permeable stretching sheet using Cattaneo‐Christov heat and mass flux model

AbstractThe present work aims to examine the effects of viscous dissipation and unsteadiness parameters on nonlinear convective laminar boundary layer flow of micropolar‐couple stress nanofluid past a permeable stretching sheet with non‐Fourier heat flux model in the presence of suction/injection variable. The unsteadiness in the flow, temperature, and concentration profile is caused by the time‐dependence of the stretching velocity, surface temperature, and surface concentration of the boundary layer flow. Similarity transformation is applied to transform the time‐dependent boundary layer flow equations into the corresponding highly nonlinear coupled ordinary differential equations with appropriate boundary conditions. The robust numerical technique called Galerkin finite element method is used to solve the obtained dimensionless governing equations of the flow. The effects of Eckert number, unsteadiness parameter, suction/injection parameter, mixed convection parameter, material parameter, Schmidt number, and couple stress parameter on linear velocity, angular velocity, temperature, concentration, local skin friction coefficient, local wall couple stress, local Nusselt number, and local Sherwood number is analyzed with the help of graphical and tabular form. Under special conditions, the present result is compared with the existing literature and revealed good agreement. Our result shows that as unsteadiness parameter boost, both heat and mass transfer rate rises. The present study has a significant application in material processing technology.

Radiation and Magnetic Field Impacts on Time-Dependent Mixed Convection Flow and Heat Transmission of Maxwellian Fluid Past A Stretching Sheet

The current study was devoted to explicating the impacts of heat transmission in an unsteady mixed convection flow of an upper convected Maxwell (UCM) fluid passing over a continuously stretching surface under the influence of radiation and magnetic field. Appurtenant similarity transmutations were adopted in order to express the constitutive boundary layer Equations of flow and heat transmission in non-dimensionalized form. The reduced system of partial differential Equations was solved by implementing the implicit finite difference method (IFDM). Our center of attention was to scrutinize the behavior of influential flow parameters on some significant features of flow and heat transmission, which were briefly examined, discussed, and presented in both graphical and tabular formats. Finally, a comparison was established with existing literature in limiting cases to support the present results, and a good agreement was found, corroborating our work. It was predicted that the thermal diffusion rate could be controlled by varying the Prandtl number. Moreover, a rise in radiation and magnetic field parameters reduced the skin friction coefficient and led to enhance the heat transmission rate at the surface. The outcomes of the study might have viable implementations in order to improve the quality of industrial products.

The influence of magnetized couple stress heat, and mass transfer flow in a stretching cylinder with convective boundary condition, cross-diffusion, and chemical reaction

This paper addresses an analysis for axisymmetric boundary layer couple stress fluid flow by stretching cylinder under the influence of the magnetic field, chemical reaction, Soret, and Dufour. The leading partial differential equations of boundary layer flow are transformed into nonlinear differential equations by applying similarity transformations then solved by the Bvph2.0 based on Homotopy analysis method (HAM). The effect of pertinent parameters shown graphically and observed that velocity decreases when Reynold number and magnetic parameters are increased. The velocity, temperature, and concentrations are enhanced with curvature parameter rise. The dimensionless skin shear stress (skin friction coefficient), heat and mass transfer rates are computed and its convergence analysis also studied. It is assumed that the axial velocity decelerates for the couple stress parameter while the thermal and solutal profile reverse situation is observed.

Eyring–Powell nanofluid flow with nonlinear mixed convection: Entropy generation minimization

Background: Entropy is the amount of energy which is lost during any irreversible process. Here our main focus is that how can we reduce this energy loss to enhance the capability of our system. Blood is an example of Eyring–Powell fluid. Many strategies are used to rise the capacity of heat transport. Heat transport can be enhanced by intensifying the materials thermal conductivity through nanoparticles. Thermal conductivity of the material can be enhanced by adding nanoparticles in base fluid. The objective of this work is to discuss entropy generation in MHD Eyring–Powell nanofluid flow. The flow is generated by a linear stretchable surface. Current analysis includes the effects of viscous dissipation, nonlinear mixed convection and Joule heating. Nanoparticles analyzed the consequences of Brownian motion and thermophoresis effects.Method: The boundary layer flow equations are solved for series solutions by applying homotopic technique.Results and conclusion: Graphical results of involved quantities like entropy generation, velocity, concentration and thermal fields are presented. Skin friction, Sherwood and Nusselt number are numerically scrutinized.

Aspects of homogeneous-heterogeneous reactions on natural convection flow of micropolar fluid past a permeable cone

Boundary layer solutions are introduced in order to provide a numerical treatment of nonlinear steady natural convection flow of non-Newtonian micropolar fluid due to an isothermal vertical permeable cone with the impact of homogeneous-heterogeneous reactions. The resulting transformed boundary layer flow equations are solved numerically by the aid of implicit finite difference technique. Additionally, the fluctuations of skin friction, couple stress coefficients and Nusselt number for various interesting sundry parameters are depicted graphically and discussed. From the obtained results, the coefficient of couple stress increases for larger values of surface temperature exponent n whereas reverse trend is clarified through vortex viscosity parameter K. Once again, it is found from current contribution that concentration of the micropolar fluid is decreases function by increasing the strength of homogeneous and heterogeneous reactions. The code of numerical technique is validated by comparing our data with previous published contributions and a good agreement is given.

Numerical Solution of MHD Casson Fluid Flow Due to a Moving Extensible Surface with Second-Order Velocity Slip and Carbon Nanotubes

This theoretical research work deals with the effect of aligned magnetic field flow and heat transfer of carbon nanotubes towards a nonlinear stretching sheet. In addition, we have considered two kinds of carbon nanotubes, namely SWCNTs and MWCNTs, used with water as the base fluid. The governing boundary layer flow equations narrating partial differential equations are transformed into a system of ordinary differential equations with the assistance of similarity transformation. Obtained coupled non-linear differential equations are solved by fourth-order Runge-Kutta (R-K) method along with shooting technique. A comparative study of the formerly published results and the present results for a special case shows that all these results are in an excellent agreement.

The thermal and laminar boundary layer flow over prolate and oblate spheroids

This paper is concerned with the thermal and laminar boundary layer flow over rotating axisymmetric bodies in an otherwise still fluid. The thermal boundary layer flow equations for two types of spheroids (prolate and oblate) are formulated. Similarly to the previous work of other researchers, the laminar flow equations for two types of spheroids are also reproduced. Furthermore, the series solutions are derived for the thermal and laminar flow equations for each body.The series solutions are numerically calculated and the laminar flow profiles for each spheroid along with its heat transfer profile are visualized in detail. It is shown that for increasing eccentricity, the convection effects are increasing at higher latitudes of prolate spheroids, while the conduction effects are dominated at these higher latitudes of oblate spheroids. The results for spheres are reproduced in the case of zero eccentricity.By using commercial routines from NAG, the solutions for rotating spheres in the case of laminar flow are also mentioned as in the previous existing works. In particular, the results from the transformation technique of solutions are in good agreement with those obtained from NAG in the case of spheres, particularly at lower latitudes for all eccentricities.