On the Numerical Analysis of Unsteady MHD Boundary Layer Flow of Williamson Fluid Over a Stretching Sheet and Heat and Mass Transfers

Influence of magnetic field-dependent viscosity on Casson-based nanofluid boundary layers: A comprehensive analysis using Lie group and spectral quasi-linearization method

This study examines the effects of magnetic-field-dependent (MFD) viscosity on the boundary layer flow of a non-Newtonian sodium alginate-based F e 3 O 4 nanofluid over an impermeable stretching surface. The non-Newtonian Casson and homogeneous nanofluid models are utilized to derive the governing flow and heat transfer equations. Applying Lie group transformations to dimensional partial differential equations yields nondimensional ordinary differential equations, which are then numerically solved using the spectral quasi-linearization technique. The analysis primarily focuses on the impacts of the MFD viscosity parameter, nanoparticle volume fraction of F e 3 O 4 , and magnetic parameters on the flow and heat transfer characteristics. The local skin friction and heat transfer rate behaviors influenced by viscosity changes due to the magnetic field are discussed. It is found that MFD viscosity significantly impacts flow and thermal energies, enhancing skin friction coefficients and reducing Nusselt numbers in the boundary layer region.

Axisymmetric boundary layer slip flow with heat transfer over an exponentially stretching bullet-shaped object: A numerical assessment

The slip flow and thermal transfer inside the boundary layer are extremely significant for various problems in aerodynamics, wing stall, skin friction drag on an entity, high-level velocity aircraft, etc. The current research investigated the effect of the slip factor and shape factor on the axisymmetric bullet-shaped object by taking the viscous dissipation parameter and location parameter. The analysis is conducted for both fixed and moving bullet-shaped objects due to thinner and thicker surfaces. The governing equations are transformed into a system of ordinary differential equations using suitable local axisymmetric similarity transformations and solved by applying the spectral quasi-linearization method. A new correlation analysis is made for velocity and temperature gradients. It is observed that the boundary layer structure has no defined shape due to a thicker bullet-shaped object instead it forms a steep angle with the axis which is contradictory to the formation of the boundary layer. A negative correlation is observed for the parameters M, Ec, Q*, and s but a positive correlation is observed for the parameters such as Pr, P, λ, and ε. The surface thickness and stretching ratio significantly affect the fluid flow and heat transfer processes. It is also noticed that the thinner bullet-shaped object performs as a better cooling conductor compared to a thicker one. The skin friction is reduced in the case of a thinner bullet-shaped object compared to a thicker one. The present analysis reveals that the heat transfer rate and the friction factor may be helpful in industrial sectors for controlling the cooling rate and quality of the final product. This research brings forward to increase in the rate of heat transfer inside the boundary layer region. The result may help to design the various types of moving objects in the automobile engineering sector when the objects pass through the fluid.

Study the Impact of an Exponentially Stretching Rate and Shape Factor of the Axisymmetric Bullet-Shaped Object on the Mixed Convection Boundary Layer Flow and Heat Transfer with Stream-Wise Coordinate and Viscous Dissipation

The current work has been investigated the influence of the exponentially stretching rate and shapes factor of the axisymmetric bullet-shaped object on the mixed convection magnetohydrodynamic boundary layer flow and heat transfer with viscous dissipation, stream-wise coordinate, and internal heat generation. The main goal of this problem is to discuss the effect of the surface shape and size, stream-wise coordinate, and also the exponential stretching factor of the bullet-shaped object on the fluid flow distribution. The novelty of the present work involved in the area of recently developed numerical method to solve these highly nonlinear differential equations. The present analysis has been performed for both of the fixed (ε = 0) and moving (ε > 0) bullet-shaped object in the two cases of thinner (0 < s < 0.1) and thicker (s ≥ 0.3) surfaces of the bullet-shaped object. It is noted that when ε = 0 means for a fixed bullet-shaped object in a moving fluid and while a moving bullet-shaped object in a fixed fluid represents when ε > 1. The governing equations have been converted into a system of ODEs by using suitable local axisymmetric transformations and solved by applying the spectral quasi-linearization method. This method helps to identify the accuracy, validity, and convergence of the present numerical computations. The computations have been investigated by the effects of different parameters on the flow field, wall friction, and heat transfer. The investigation depicts that the flow field and temperature do not converge the free stream condition asymptotically in the case of a thicker bullet-shaped object instead it intersects the axis with a steep angle which is contradictory with the boundary layer theory and the boundary layer structure has no defined shape whereas in the case of a thinner bullet-shaped object (0 < s < 0.1) the ambient condition satisfies asymptotically and formed a definite boundary layer structure. Heat transfer rate at the bullet-shaped object is negatively correlated with the magnetic parameter, Eckert number, heat generation parameter, and surface thickness parameter but positively correlated with the Prandtl number, location parameter, mixed convection parameter, and stretching ratio parameter. The investigation represents that surface thickness parameter (shape and size) and stretching ratio parameter have a prominent effect on fluid flow properties and cannot be neglected. It is also noticed that the thinner bullet-shaped object acts as a good cooling conductor compared to thicker bullet-shaped object and the wall friction can be reduced much when a thinner bullet-shaped object is used rather than the thicker bullet-shaped object in both types of moving or static bullet-shaped object (ε = 0.0 and ε = 0.2). The present analysis reveals that the heat transfer and the friction factor will be helpful in industrial sectors such as a cooling device in nuclear reactors, automotive engineering, electronic engineering, biomedical engineering, control the cooling rate and quality of the final product.

Influence of Shape Factor and Non-Linear Stretching of the Bullet-Shaped Object on the Mixed Convection Boundary Layer Flow and Heat Transfer with Viscous Dissipation and Internal Heat Generation

The two-dimensional incompressible axisymmetric mixed convection magnetohydrodynamic fluid flow and energy transfer over a bullet-shaped object with a non-linear stretching surface have been investigated. The main goal of this problem is to discuss the effect of the shape and size of the bullet-shaped object on the fluid velocity and temperature distributions. The present analysis has been performed in about two cases ε=0.0 and 2.0. Therefore, fluid velocity and temperature distributions have been investigated in two types of flow geometries such as the thicker surface (s ≥ 2) and the thinner surface (0 < s < 2) of the bullet-shaped object. The equations for momentum and heat transfer have been converted into ODEs by using suitable local similarity transformations. These equations have been performed with a recently developed spectral quasi-linearization method (SQLM). This method helps to identify the accuracy, validity, and convergence of the present solution. The novelty of the present work has been applying the recently developed numerical method to solve these highly nonlinear differential equations. The investigation shows that in the case of a thicker bullet-shaped object (s ≥ 2) the velocity and temperature profiles do not converse the far-field boundary condition asymptotically but cross the axis with an upright angle and the boundary layer structure has no definite shape whereas in the case of a thinner bullet-shaped object (0 < s < 2) the velocity profile converge the ambient condition asymptotically and the boundary layer structure has a definite shape. The innovation of this current work lies in the unification of relevant physical parameters into the governing equations and trying to explain how the flow properties are affected by these parameters.

Computational Analysis for Bioconvection of Microorganisms in Prandtl Nanofluid Darcy-Forchheimer Flow across an Inclined Sheet

The two-dimensional boundary layer flow of a Prandtl nanofluid was explored in the presence of an aligned magnetic field over an inclined stretching/shrinking sheet in a non-Darcy permeable medium. To transform the PDEs of the leading equations into ODEs, a coupled boundary value problem was formed and suitable similarity functions were used. To obtain numerical answers, an efficient code for the Runge–Kutta technique with a shooting tool was constructed with a MATLAB script. This procedure is widely used for the solution of such problems as it is efficient and cost-effective with a fifth-order accuracy. The significance of immersed parameters on the velocity, temperature, concentration, and bioconvection is shown through figures. Furthermore, the physical parameters of the skin friction coefficient and the Nusselt numbers are demonstrated in tables. The declining behavior of the flow velocity against the porosity parameter Kp and the local inertia co-efficient Fr is shown, and the both parameters of the Darcy resistance and Darcy–Forchheimer resistance are responsible for slowing the fluid speed. The increasing values of the Schmidt number Sc decrease the concentration of the nano entities.

Impact of Bioconvection and Chemical Reaction on MHD Nanofluid Flow Due to Exponential Stretching Sheet

Thermal management is a crucial task in the present era of miniatures and other gadgets of compact heat density. This communication presents the momentum and thermal transportation of nanofluid flow over a sheet that stretches exponentially. The fluid moves through a porous matrix in the presence of a magnetic field that is perpendicular to the flow direction. To achieve the main objective of efficient thermal transportation with increased thermal conductivity, the possible settling of nano entities is avoided with the bioconvection of microorganisms. Furthermore, thermal radiation, heat source dissipation, and activation energy are also considered. The formulation in the form of a partial differential equation is transmuted into an ordinary differential form with the implementation of appropriate similarity variables. Numerical treatment involving Runge–Kutta along with the shooting technique method was chosen to resolve the boundary values problem. To elucidate the physical insights of the problem, computational code was run for suitable ranges of the involved parameters. The fluid temperature directly rose with the buoyancy ratio parameter, Rayleigh number, Brownian motion parameter, and thermophoresis parameter. Thus, thermal transportation enhances with the inclusion of nano entities and the bioconvection of microorganisms. The findings are useful for heat exchangers working in various technological processors. The validation of the obtained results is also assured through comparison with the existing result. The satisfactory concurrence was also observed while comparing the present symmetrical results with the existing literature.

MHD Williamson Nanofluid Flow over a Slender Elastic Sheet of Irregular Thickness in the Presence of Bioconvection

Bioconvection phenomena for MHD Williamson nanofluid flow over an extending sheet of irregular thickness are investigated theoretically, and non-uniform viscosity and thermal conductivity depending on temperature are taken into account. The magnetic field of uniform strength creates a magnetohydrodynamics effect. The basic formulation of the model developed in partial differential equations which are later transmuted into ordinary differential equations by employing similarity variables. To elucidate the influences of controlling parameters on dependent quantities of physical significance, a computational procedure based on the Runge–Kutta method along shooting technique is coded in MATLAB platform. This is a widely used procedure for the solution of such problems because it is efficient with fifth-order accuracy and cost-effectiveness. The enumeration of the results reveals that Williamson fluid parameter λ, variable viscosity parameter Λμ and wall thickness parameter ς impart reciprocally decreasing effect on fluid velocity whereas these parameters directly enhance the fluid temperature. The fluid temperature is also improved with Brownian motion parameter Nb and thermophoresis parameter Nt. The boosted value of Brownian motion Nb and Lewis number Le reduce the concentration of nanoparticles. The higher inputs of Peclet number Pe and bioconvection Lewis number Lb decline the bioconvection distribution. The velocity of non-Newtonian (Williamson nanofluid) is less than the viscous nanofluid but temperature behaves oppositely.

Boundary Layer Flow and Heat Transfer Analysis by Using the Correlation Coefficient and Regression Model Over a Stretching Bullet-Shaped Object

The boundary layer theory is important when fluid flows over a solid surface that is moving or stationary. In presence of the boundary layer, the effective shape of the body may change leading to changes in pressure distribution, as a result, the overall lift and drag forces change. Therefore, the Boundary layer theory helps in designing aerofoil’s, to compute the lift and drag forces for the aerospace and automobile designers, to control the heat transfer rate from the device, etc. So, the present problem will help design the various types of bullet-shaped objects in the field of automobile engineering. Therefore, the current problem has focused on the two-dimensional axisymmetric BL flow over a stretching bullet-shaped object for the effect of magnetic field strength (M), linear stretching parameter (M), and surface thickness parameter (s). Therefore, the main goal of this work is to determine the relation by applying the correlation coefficient among the physical parameters and velocity field, temperature field, shear stress (τw), Nusselt number (Nux). Hence, the novelty of the current paper is to develop the relationship among the dependent and independent parameters by the correlation coefficient and also developed the numerical method to solve these highly nonlinear equations. The numerical results are discussed for the three different values of the stretching ratio parameter and two values of the surface thickness parameter. The velocity and temperature distribution equations are compressed into a system of ODEs with similarity transformations. These ODEs are then solved using a spectral quasi-linearization method (SQLM) by applying Taylor series expansions that can be used to linearize the non-linear terms in the equations. These resulting linearized systems of equations are determined by the spectral collocation method. The convergence of the numerical solutions was performed by using the residual error of the PDEs. The error analysis is established for the validity of the present model. This error norm is applied to establish the validity and convergence of the numerical solution. The outcome of the mentioned dimensionless parameters over the fluid velocity field, temperature field, skin friction coefficient (Cf), and Nusselt number (Nux) are displayed graphically. It is observed that the parameters M and M are positively correlated with fluid velocity distribution within the BL but the surface thickness parameter(s) are negatively correlated. The rate of temperature increases for the parameter M and Pr but decreases for M and s. Therefore, the boundary layer thickness reduces for increasing the values of M and M but increases for increasing the values of s. The velocity of the fluid is about 80% higher in the case of a thinner surface (s = 0.2) than the thicker surface (s = 2.0) and the heat transfer rate is also igher in the case of a thinner surface comparatively thicker surface. The innovation of this present problem lies in the unification of more physical parameters into the governing equations and an attempt to give a thorough analysis of how the flow properties are affected by these parameters.

On Numerical Analysis of Carreau–Yasuda Nanofluid Flow over a Non-Linearly Stretching Sheet under Viscous Dissipation and Chemical Reaction Effects

This work reports the Carreau–Yasuda nanofluid flow over a non-linearly stretching sheet viscous dissipation and chemical reaction effects. The coupled system of non-linear partial differential equations are changed into a system of linear differential equations employing similarity equations. The spectral quasi-linearization method was used to solve the linear differential equations numerically. Error norms were used to authenticate the accuracy and convergence of the numerical method. The effects of some thermophysical parameters of interest in this current study on the non-dimensional fluid velocity, concentration and temperature, the skin friction, local Nusselt and Sherwood numbers are presented graphically. Tables were used to depict the effects of selected parameters on the skin friction and the Nusselt number.