Free Stream Boundary Conditions Research Articles

Computation of SWCNT/MWCNT-doped thermo-magnetic nano-blood boundary layer flow with non-Darcy, chemical reaction, viscous heating and Joule dissipation effects

CNTs have been shown to exhibit exceptional electrical and thermal conductivities, chemical and mechanical stability and physiochemical reliability in wide-ranging applications covering biomedicine, energy and aerospace. Motivated by emerging applications in nano-drug delivery in medicine and radiative ablation therapy, a steady-state mathematical model is established for magnetohydrodynamic boundary-layer transport of chemically reactive carbon nanotubes (CNTs)-doped electrically conducting blood along a porous stretching wall of a blood vessel. Multiple and single walls CNTs are considered using the model developed by Xue (2005). Heat generation, radiative flux, wall mass (concentration) slip, viscous heating and Joule magnetic dissipation are incorporated. Chemical reaction effects are addressed utilizing homogenous first-order model. The Darcy-Forchheimer drag force model is utilized for bulk matrix and inertial drag effects in the porous medium. Radiative heat-transport is studied considering Rosseland's diffusion model. The governing partial differential conservation equations for mass, momentum, energy and species are formulated in a two-dimensional Cartesian coordinate system with allied wall and free-stream boundary conditions. Via scaling transformations, a nonlinear boundary value problem is derived. Numerical computations are achieved through bvp4c scheme. The impact of selected parameters on dimensionless quantities (velocity, temperature, concentration, skin friction, local Nusselt and Sherwood numbers) are computed and elaborated through tables and graphs. A good correlation of the numerical solutions with earlier simpler models from the literature is included. The simulations show that with augmentation in Hartmann magnetic number and Forchheimer inertial drag number, significant damping in the nano-doped blood flow is produced for both SWCNTs and MWCNTs. Temperature is elevated with increment in dissipation effect i. e. Eckert number. Higher values of mass (solutal) slip parameter induce a depletion in the concentration. Nusselt number i. e. heat transfer rate to the blood vessel wall is improved with greater values of solid volume fraction for both CNTs. The present model is relevant to radiative ablation therapy combined with nano-medical treatments in blood vessels wherein constant mechanical loading via blood pressure and flow can elevate internal stresses (circumferential wall stress and endothelial shear stress) and induce morphological alterations in blood vessel wall (stretching) and endothelium in conjunction with biochemical reactions.

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.

A Lagrangian free-stream boundary condition for weakly compressible smoothed particle hydrodynamics

This paper proposes the first Lagrangian free-stream boundary condition for weakly-compressible smoothed particle hydrodynamics (WCSPH), significantly decreasing the domain-size effect for simulating flows around static or moving geometries and increasing computational efficiency. The boundary condition is implemented based on several numerical techniques, as follows. First, a spatio-temporal identification and an ensuring technique precisely identify surface particles without misjudgment. Second, the far-field corrections of density and velocity precisely target surface particles. Third, the improved in- and outlet boundaries guarantee the smooth inflow profile and free-outflow particles without the issue of the truncated kernel. The test cases of flows over a flat plate and around a circular cylinder have validated the accuracy and computational efficiency of the present method. In order to demonstrate its applicability, the method has also been tested with more complex examples, such as a fluid-structure interaction (FSI) problem and a 3-D flow around a sphere.

Physics-Informed Neural Networks for Low Reynolds Number Flows over Cylinder

Physics-informed neural network (PINN) architectures are recent developments that can act as surrogate models for fluid dynamics in order to reduce computational costs. PINNs make use of deep neural networks, where the Navier-Stokes equation and freestream boundary conditions are used as losses of the neural network; hence, no simulation or experimental data in the training of the PINN is required. Here, the formulation of PINN for fluid dynamics is demonstrated and critical factors influencing the PINN design are discussed through a low Reynolds number flow over a cylinder. The PINN architecture showed the greatest improvement to the accuracy of results from the increase in the number of layers, followed by the increase in the number of points in the point cloud. Increasing the number of nodes per hidden layer brings about the smallest improvement in performance. In general, PINN is much more efficient than computational fluid dynamics (CFD) in terms of memory resource usage, with PINN requiring 5–10 times less memory. The tradeoff for this advantage is that it requires longer computational time, with PINN requiring approximately 3 times more than that of CFD. In essence, this paper demonstrates the direct formulation of PINN without the need for data, alongside hyperparameter design and comparison of computational requirements.

Dual solutions of magnetite/cobalt/manganese-zinc-aqueous nano-ferrofluids from a stretching sheet with magnetic induction effects: MHD stagnation flow computation and analysis

A theoretical study is presented for the steady magnetohydrodynamic (MHD) boundary layer stagnation point flow of a nano-ferrofluid along a linearly moving stretching sheet, as a simulation of functional magnetic materials processing. Due to having imerging applications in heat transfer, the nano-ferrofluids draw the attention which comprise an aqueous base fluid doped with a variety of magnetic nanoparticles, i.e. magnetite (Fe3O4), cobalt ferrite (CoFe2O4) and Manganese-zinc (Mn-Zn) ferrite. A partial differential equation mathematical model is developed for mass, momentum, magnetic field continuity (induction), and energy with appropriate wall and free stream boundary conditions. Following similarity transformations, the dimensionless resultant nonlinear ordinary differential boundary value problem is solved numerically using the robust bvp4c function in MATLAB which features very efficient 4th order optimized Runge–Kutta quadrature. Dual solutions for the upper branch and lower branch separated by a critical point are identified. Visualization of velocity, temperature, and induced magnetic field function are presented graphically including validation of solutions with previous studies. Furthermore, skin-friction coefficient and the local Nusselt number are also computed. The impact of the controlling parameters, i.e. Prandtl number nanoparticle volume fraction parameter reciprocal of magnetic Prandtl number magnetic parameter and stretching rate ratio parameter have been illustrated through graphs and evaluated when a desire heat transfer can occur. Furthermore, resistance between fluid and the plate can be increased with the growing magnetic Prandtl number values. Increment in magnetic parameter () produces an elevation in the induced magnetic field magnitudes. Skin friction and Nusselt number are found to be greater for cobalt nanoparticles when compared to magnetite and Mn-Zn ferromagnetic nanoparticles when there is an increase in reciprocal magnetic Prandtl number. The simulations provide a deeper insight into the manufacturing flows of functional nano-ferromagnetic materials of relevance to deposition and coating systems.

Mixed convection Casson polymeric flow from a nonlinear stretching surface with radiative flux and non‐Fourier thermal relaxation effects: Computation with CSNIS

AbstractThermal non‐Newtonian polymer coating flows is growing as a major area in materials processing. Inspired by new developments in this field which require more sophisticated mathematical models, the current investigation examines the laminar viscoplastic boundary layer flow and mixed convective heat transfer over a power‐law nonlinear stretching surface. To simulate thermal relaxation effects the hyperbolic Cattaneo‐Christov heat flux model is deployed. The non‐Newtonian polymer characteristics are described by employing the Casson flow model. High temperature conditions invoke thermal radiation flux which is analyzed with an algebraic flux model. Via robust similarity transformations, the primitive partial differential conservation equations for momentum and energy equations are rendered into a system of coupled non‐linear ordinary differential equations with associated wall and free stream boundary conditions. The emerging boundary value problem is solved numerically with an efficient Chebyshev Spectral Newton Iterative scheme (CSNIS), in the MATLAB platform. The resulting solutions are discussed for different emerging parameters using graphs and tables. Validation is included with special cases from the literature. With increasing power law stretching index increases, the flow is decelerated, and temperatures are reduced. Increment in mixed convection parameter boosts the velocity but suppresses temperature and thermal boundary layer thickness. Increasing non‐Fourier Deborah number, temperatures are depleted whereas with increasing radiative flux parameter they are increased. With elevation in Casson non‐Newtonian parameter, velocity is decreased whereas temperature is enhanced and Nusselt number is suppressed.

Rheology of tangent-hyperbolic (T-H) nanoliquid configured by stretchable stratified surface considering transpiration effects

This research reports the thermo-solutal mixed convective non-Newtonian (tangent-hyperbolic) fluid flow from a stretchable surface under the effect of viscous dissipation. Impermeable surface with stratifications (thermal and solutal) creates the flow. The Buongiorno nanoliquid model capturing Brownian diffusion and thermophoresis is opted for analysis. Energy expression modeling is based on heat source/sink and thermal radiation. Consideration of chemical reaction accounts for species concentration. Via relevant transformations, the flow model of nonlinear governing partial differential conservation equations and free-stream boundary conditions are extracted into coupled nonlinear ordinary differential equations which are solved analytically using homotopy technique. Comparative results ensuring the soundness of the employed technique are included. Analytical results are presented graphically for the influence of pertinent parameters on velocity, temperature, skin-friction coefficient, local Nusselt and Sherwood numbers. The obtained outcomes witness that the concentration of nanoparticles is increased with stronger values of thermophoresis and concentration Biot number while it declines with increasing values of solutal stratification variable, Brownian motion and Schmidt number.

Radiative Magnetohydrodynamic Flow Over a Vertical Cone Filled With Convective Nanofluid

This article describes the magnetohydrodynamic radiative nano convective flow over a cone. Brownian motion and thermophoresis effects in nanofluid transport R-K method is used to compute the solutions to the governing non-dimensional partial differential conservation equations and free stream boundary conditions. Numerous thermo-physical parameters were presented, There is a detailed explanation of the simulations as well. Magnetic parameter increases, reduce velocity. According to the findings, there are numerous applications for heat exchanger technology, including geothermal energy storage and cooling, solar energy systems, and materials processing.

Thermo-solutal dual stratified convective magnetized fluid flow from an exponentially stretching Riga plate sensor surface with thermophoresis

A theoretical study is presented for the coupled thermo-solutal free convection two-dimensional boundary layer flow of a of a magnetized fluid from an exponentially stretched magnetic sensor (Riga plate) surface. Heat generation/absorption, nonlinear thermal radiation and thermophoretic body force effects are included. Furthermore, thermal and solutal stratification are also featured in the boundary layer model. The derived nonlinear partial differential equation system with associated wall (sensor surface) and free stream boundary conditions is transformed to system of ordinary differential equations (ODE) via applicable similarity variables. A numerical solution is developed with the efficient Runge–Kutta–Fehlberg (RKF) technique with a shooting numerical method, in MATLAB software using the RKF-45 method. The graphical profiles were represented to examine the impacts of physically parameterics on the important physical stream features. Streamline and isotherm plots are also included for thermal buoyancy effect (Grashof number).Validation of solutions with earlier simpler models is included. A relatively well agreement is achived between model prediction and the benchmark value. Values for skin friction factor, Nusselt number and Sherwood numbers are also tabulated to quantify momentum, heat and mass (species) transfer characteristics at the sensor surface. The results indicate that the significant depletion in temperature accompanies accumulation in magnetization and thermal stratification parameters. Significant accumulation in mixed convection and Riga plate electrode width parameter enhance the concatenation profiles. Nusselt number grows substantially as thermal stratification and radiative parameter increases. Sherwood number grows substantially as solutal stratification, thermophoresis and Schmidt parameter increases. The present study generalizes previous models to include simultaneously exponential stretching, thermophoresis, thermal and solutal (mass) stratification and heat source/sink effects.

Pool evaporation under low Grashof number downward convection

We investigate the free evaporation of water into air from a small-scale (≈0.1 m) circular pool for low Grashof (Gr) number values under dominant downward flow motion. Repeatable experiments performed at atmospheric pressure for air temperatures at 290 K, 300 K and 310 K and relative humidity values ranging from 30% to 60%, are described and compared to detailed Finite Element Method (FEM) numerical simulations. For the free evaporation regimes considered here, a downward thermally-induced flow originated at the rim of the pool overcomes the concentration-induced buoyancy, consequently forcing the far-stream warm dry air to descend into the lower temperature air-water interface. Experimental results show that a steady-state recirculation zone near the air-water interface develops for Gr numbers greater than or equal to 50. The Sherwood number (Sh) for this geometry appropriately scales with Gr1/4 for drier free-stream boundary conditions, while it approaches a constant value for smaller mass transfer potentials (i.e., more humid environments). A stable Sherwood number (Sh correlation as a function of Gr values is proposed, covering Gr values from 10 to 105.

Instability of Supersonic Radiation Mode in Hypersonic Boundary Layers

The supersonic mode in hypersonic flows is studied numerically. The radiation characteristics of this mode in a flat-plate boundary layer with cooled walls are focused on. The linear stability analysis and a marching investigation using a parabolized stability equations approach are performed. Unlike the traditional bounded freestream boundary condition, a more physical boundary condition was proposed based on the group velocity direction of the supersonic mode. Two classes of the supersonic modes, the radiation mode and the incoming mode, are distinguished according to their different group velocities in wall-normal direction. The radiation mode is more unstable than the incoming one, for which it is the dominant instability. It is also found that both the stable and unstable radiation modes can radiate slow acoustic waves into the freestream, and no coalescence occurs between the radiation mode and the wave of the slow acoustic spectrum. This process can only be predicted by the linear stability theory with the new boundary condition. The slow acoustic wave has no effect on the supersonic radiation mode.

Entropy analysis of nanofluid magnetohydrodynamic convection flow past an inclined surface: A numerical review

AbstractA numerical investigation is conducted to review the entropy study of magnetohydrodynamic (MHD) convection nanofluid flow from an inclined surface. In evaluating the thermophoresis and Brownian motion impacts, Buongiorno's model is applied to nanofluid transfer. Using Keller's implicit box technique, the governing partial differential conservation equations and wall and free stream boundary conditions are made into the dimensionless form and solved computationally. For different thermos physical parameter values, the numerical results are discussed both graphically and numerically. Verification of the present code with previous Newtonian responses is also included. To analyze the variability in fluid velocity, temperature, nanoparticle volume fraction, entropy, Bejan number, shear stress rate, wall heat, and mass transfer rates, graphical and tabulated results are reported. The study suggests applications in the manufacturing of nanomaterial fabrication, and so on.

Entropy analysis of magnetohydrodynamic nanofluid transport past an inverted cone: Buongiorno's model

AbstractThe entropy analysis of the magnetohydrodynamic (MHD) thermal convection flow of a nanofluid past an inverted cone with suction/injection is presented in this article. The Buongiorno's model is adopted for nanofluid transport, considering the Brownian motion and thermophoresis effects. The governing partial differential conservation equations and wall and freestream boundary conditions are rendered into a nondimensional form and solved computationally using the Keller‐Box finite‐difference method. The entropy analysis due to MHD fluid flow and viscous dissipation is also included. The numerical results are presented graphically for the impact of various thermophysical parameters on velocity, temperature, nanoparticle volume fraction, shear stress rate, heat transfer rate, and mass transfer. Validations with earlier solutions in the literature are also included. A comprehensive description of the simulations is included. It is observed that velocity and temperature are enhanced with the increase in Brownian motion parameter values, whereas concentration, entropy, and Bejan number are reduced. An increase in the thermophoresis parameter and buoyancy ratio parameter reduces velocity and entropy, but increases temperature, concentration, and Bejan number. An increase in the magnetic parameter is found to decrease velocity, entropy generation number, and Bejan number, but it increases temperature and concentration. Also, an increase in the suction/injection parameter is seen to reduce velocity, temperature, concentration, and Bejan number, but the entropy generation number is observed to increase. The study finds applications in heat exchangers technology, materials processing, solar energy systems, cooling and heating processes, environmental applications, geothermal energy storage, and so on.

Koopman-Based Approach to Nonintrusive Reduced Order Modeling: Application to Aerodynamic Shape Optimization and Uncertainty Propagation

A methodology for nonintrusive projection-based nonlinear model reduction originally presented by Renganathan et al. (“Koopman-Based Approach to Nonintrusive Projection-Based Reduced-Order Modeling with Black-Box High-Fidelity Models,” AIAA Journal, Vol. 56, No. 10, 2018, pp. 4087–4111) is further extended toward parametric systems with a focus on application to aerospace design. Specifically, the method is extended to address static systems with parametric geometry (that deforms the mesh) in addition to parametric freestream boundary conditions. The main idea is to first perform a transformation on the governing equations such that it is lifted to a higher-dimensional but linear underdetermined system. This enables one to extract the system matrices easily as compared to that of the original nonlinear system. The underdetermined system is closed with a set of model-dependent nonlinear constraints upon which the model reduction is finally performed. The methodology is validated on the subsonic and transonic inviscid flows past the NACA0012 and the RAE2822 airfoils with parametrized shapes. The utility of the approach is further demonstrated by applying it to two common problems in aerospace design, namely, derivative-free global optimization and parametric uncertainty quantification with Monte Carlo sampling. Overall, the methodology is shown to achieve accuracy up to 5% and computational speedup of two to three orders of magnitude relative to the full order model. Comparison against another nonintrusive model reduction method revealed that the proposed approach is more robust, accurate, and retains the consistency between the state variables.

Study of differential transform technique for transient hydromagnetic Jeffrey fluid flow from a stretching sheet

AbstractThis article investigates the time-dependent MHD heat transfer flow of Jeffrey fluid from a stretching sheet, the topic significance to non-Newtonian viscoelastic material processing. Using similarity transformations, the governing coupled non-linear PDE’s are remodel into ODE’s with suitable free stream and wall boundary conditions. The developed non-dimensional non-linear problem is revealed to be analysed by several key thermosphysical and rheological parameters, namely, Jeffrey fluid parameter (λ), Deborah number (β), Prandtl number (Pr), buoyancy parameter (ξ), magnetic parameter (M) and unsteadiness parameter (A). The semi-exact differential transform technique is applied to elucidate the coupled nonlinear governing equation of non-Newtonian Jeffrey fluid problem. Also, the solution is validated with numerical results attained via the MATLAB bvp4c function. Excellent accurateness is attained through the DTM approach. Further validation with available consequences from the existing literature is incorporated. The results indicate that fluid velocity and temperature are boosted with increasing Deborah number and stretching parameter however it shows a decreasing trend with Jeffrey fluid parameter and convection parameter. It also shows when augmenting the magnetic parameter which reduces the flow and increases the thickness of the boundary layer.

COMPUTATION OF EYRING-POWELL MICROPOLAR CONVECTIVE BOUNDARY LAYER FLOW FROM AN INVERTED NON-ISOTHERMAL CONE: THERMAL POLYMER COATING SIMULATION

Thermal coating of components with non-Newtonian materials is a rich area of chemical and process mechanical engineering. Many different rheological characteristics can be simulated for such coatings with a variety of different mathematical models. In this work we study the steady-state coating flow and heat transfer of a non-Newtonian liquid (polymer) on an inverted isothermal cone with variable wall temperature. The Eringen micropolar and three-parameter Eyring-Powell models are combined to simulate microstructural and shear characteristics of the polymer. The governing partial differential conservation equations and wall and free stream boundary conditions are rendered into dimensionless form and solved computationally with the KellerBox finite difference method (FDM). Validation with earlier Newtonian solutions from the literature is also included. Graphical and tabulated results are presented to study the variations of fluid velocity, micro-rotation (angular velocity), temperature, skin friction, wall couple stress (micro-rotation gradient) and wall heat transfer rate. With increasing values of the first Eyring-Powell parameter temperatures are elevated, micro-rotation is suppressed and velocities are enhanced near the cone surface but reduced further into the boundary layer. Increasing values of the second Eyring-Powell parameter induce strong flow deceleration, decrease temperatures but enhance micro-rotation values. An increase in non-isothermal power law index suppresses velocities, temperatures and micro-rotations i.e. all transport characteristics are maximum for the isothermal case (n =0). Increasing Eringen vortex viscosity parameter significantly enhances temperatures and also micro-rotations. The present numerical simulations find applications in thermal polymer coating operations and industrial deposition techniques and provide a useful benchmark for more general computational fluid dynamics (CFD) simulations.

Adomian decomposition method simulation of von Kármán swrling bioconvection nanofluid flow

The study reveals analytically on the 3-dimensional viscous time-dependent gyrotactic bioconvection in swirling nanofluid flow past from a rotating disk. It is known that the deformation of the disk is along the radial direction. In addition to that Stefan blowing is considered. The Buongiorno nanofluid model is taken care of assuming the fluid to be dilute and we find Brownian motion and thermophoresis have dominant role on nanoscale unit. The primitive mass conservation equation, radial, tangential and axial momentum, heat, nano-particle concentration and micro-organism density function are developed in a cylindrical polar coordinate system with appropriate wall (disk surface) and free stream boundary conditions. This highly nonlinear, strongly coupled system of unsteady partial differential equations is normalized with the classical Von Karman and other transformations to render the boundary value problem into an ordinary differential system. The emerging 11th order system features an extensive range of dimensionless flow parameters i.e. disk stretching rate, Brownian motion, thermophoresis, bioconvection Lewis number, unsteadiness parameter, ordinary Lewis number, Prandtl number, mass convective Biot number, Peclet number and Stefan blowing parameter. Solutions of the system are obtained with developed semi-analytical technique i.e. Adomian decomposition method. Validation of the said problem is also conducted with earlier literature computed by Runge-Kutta shooting technique.

Numerical study of self-similar natural convection mass transfer from a rotating cone in anisotropic porous media with Stefan blowing and Navier slip

A mathematical model is presented for laminar, steady natural convection mass transfer in boundary layer flow from a rotating porous vertical cone in anisotropic high-permeability porous media. The transformed boundary value problem is solved subject to prescribed surface and free stream boundary conditions with a Maple 17 shooting method. Validation with a Chebyshev spectral collocation method is included. The influence of tangential Darcy number, swirl Darcy number, Schmidt number, rotational parameter, momentum (velocity slip), mass slip and wall mass flux (transpiration) on the velocity and concentration distributions is evaluated in detail. The computations show that tangential and swirl velocities are enhanced generally with increasing permeability functions (i.e., Darcy parameters). Increasing spin velocity of the cone accelerates the tangential flow, whereas it retards the swirl flow. An elevation in wall suction depresses both tangential and swirl flow. However, increasing injection generates acceleration in the tangential and swirl flow. With greater momentum (hydrodynamic) slip, both tangential and swirl flows are accelerated. Concentration values and Sherwood number function values are also enhanced with momentum slip, although this is only achieved for the case of wall injection. A substantial suppression in tangential velocity is induced with higher mass (solutal) slip effect for any value of injection parameter. Concentration is also depressed at the wall (cone surface) with an increase in mass slip parameter, irrespective of whether injection or suction is present. The model is relevant to spin coating operations in filtration media (in which swirling boundary layers can be controlled with porous media to deposit thin films on industrial components), flow control of mixing devices in distillation processes and also chromatographical analysis systems.

Computational study of unsteady couple stress magnetic nanofluid flow from a stretching sheet with Ohmic dissipation

To provide a deeper insight of the transport phenomena inherent to the manufacturing of magnetic nano-polymer materials, in the present work a mathematical model is developed for time-dependent hydromagnetic rheological nano-polymer boundary layer flow and heat transfer over a stretching sheet in the presence of a transverse static magnetic field. Joule heating (Ohmic dissipation) and viscous heating effects are included since these phenomena arise frequently in magnetic materials processing. Stokes’ couple stress model is deployed to simulate non-Newtonian microstructural characteristics. The Tiwari–Das nanoscale model is adopted which permits different nanoparticles to be simulated (in this article, both copper–water and aluminium oxide–water nanofluids are considered). Similarity transformations are utilized to convert the governing partial differential conservation equations into a system of coupled, non-linear ordinary differential equations with appropriate wall and free stream boundary conditions. The shooting technique is used to solve the reduced non-linear coupled ordinary differential boundary value problem via MATLAB symbolic software. Validation with published results from the literature is included for the special cases of non-dissipative and Newtonian nanofluid flows. Fluid velocity and temperature profiles for both copper and aluminium oxide (Al2O3) nanofluids are observed to be enhanced with greater non-Newtonian couple stress parameter and magnetic parameter, whereas the opposite trend is computed with greater values of unsteadiness parameter. The boundary layer flow is accelerated with increasing buoyancy parameter, elastic sheet stretching parameter and convection parameter. Temperatures are generally increased with greater couple stress rheological parameter and are consistently higher for the aluminium oxide nanoparticle case. Temperatures are also boosted with magnetic parameter and exhibit an overshoot near the wall when magnetic parameter exceeds unity (magnetic force exceeds viscous force). A decrease in temperatures is induced with increasing sheet stretching parameter. Increasing Eckert number elevates temperatures considerably. With greater nanoparticle volume fraction, both skin friction and Nusselt number are elevated, and copper nanoparticles achieve higher magnitudes than aluminium oxide.

Application of differential transform method to unsteady free convective heat transfer of a couple stress fluid over a stretching sheet

AbstractIn the present article, the transient rheological boundary layer flow over a stretching sheet with heat transfer is investigated, a topic of relevance to non‐Newtonian thermal materials processing. Stokes couple stress model is deployed to simulate non‐Newtonian characteristics. Similarity transformations are utilized to convert the governing partial differential equations into nonlinear ordinary differential equations with appropriate wall and free stream boundary conditions. The nondimensional boundary value problem emerging is shown to be controlled by a number of key thermophysical and rheological parameters, namely the rheological couple stress parameter (), unsteadiness parameter (), Prandtl number (Pr), buoyancy parameter . The semi‐analytical differential transform method (DTM) is used to solve the reduced nonlinear coupled ordinary differential boundary value problem. A numerical solution is also obtained via the MATLAB built‐in solver “bvp4c” to validate the results. Further validation with published results from the literature is included. Fluid velocity is enhanced with increasing couple stress parameter, whereas it is decreased with unsteadiness parameter. Temperature is elevated with couple stress parameter, whereas it is initially reduced with unsteadiness parameter. The flow is accelerated with increasing positive buoyancy parameter (for heating of the fluid), whereas it is decelerated with increasing negative buoyancy parameter (cooling of the fluid). Temperature and thermal boundary layer thickness are boosted with increasing positive values of buoyancy parameter. Increasing Prandtl number decelerates the flow, reduces temperatures, increases momentum boundary layer thickness, and decreases thermal boundary layer thickness. Excellent accuracy is achieved with the DTM approach.