Unsteady flow and heat transfer over a permeable stretching/shrinking sheet with generalized slip velocity

Purpose: This study aims to investigate the unsteady two-dimensional viscous flow and heat transfer over an unsteady permeable stretching/shrinking sheet (surface) with generalized slip velocity condition. Design/methodology/approach: Similarity transformation is used to reduce the system of partial differential equations into a system of nonlinear ordinary differential equations. The resulting equations are then solved numerically using “bvp4c” function in MATLAB software. Findings: Dual solutions are found for a certain range of the unsteady, suction and stretching/shrinking parameters. Stability analysis is performed, and it is revealed that the first (upper branch) solution is stable and physically realizable, whereas the second (lower branch) solution is unstable. Practical implications: The results obtained can be used to explain the characteristics and applications of the generalized slip in boundary layer flow. Such condition is applied for particulate fluids such as foams, emulsions, polymer solutions and suspensions. Furthermore, the phenomenon of stretching/shrinking sheet can be found on the manufacturing of polymer sheets, rising and shrinking balloon or moving and shrinking polymer film. Originality/value: The present numerical results are original and new for the study of unsteady flow and heat transfer over a permeable stretching/shrinking sheet with generalized slip velocity

Three-dimensional viscous flow over an unsteady permeable stretching/shrinking sheet

In this study, a numerical investigation on the unsteady three-dimensional boundary layer flow of a viscous fluid past a permeable stretching/shrinking sheet is considered. Similarity transformation is employed to reduce the governing system of nonlinear partial differential equations into the ordinary (similarity) differential equations. These equations are then solved numerically by using a shooting method. Both stretching and shrinking cases are considered. Effects of the unsteadiness parameter, stretching/shrinking parameter, mass suction parameter and ratio of the surface velocity gradients along the vertical y- and horizontal x- directions are presented and discussed in detail. The numerical results show that for the shrinking case, the skin friction coefficient and the velocity boundary layer thickness increase with increasing unsteadiness parameter, while the skin friction coefficient decreases and the velocity boundary layer thickness increases with increasing ratio of the surface velocity gradients. The results also show that dual solutions exist for both cases of stretching and shrinking sheet

Thermally Stratified Flow Of Cu-Al2O3/Water Hybrid Nanofluid Past A Permeable Stretching/Shrinking Circular Cylinder

The present study emphasizes the thermally stratified hybrid nanofluid flow due to a permeable stretching/shrinking cylinder. Thermal buoyancy force is also taken into consideration to incorporate with the thermal stratification process. An improved hybrid nanofluid (dual nanoparticles) may offer a better heat transfer performance in many engineering applications. In the present work, the combination of copper (Cu) and alumina (Al2O3) nanoparticles with water as the working fluid is analytically modeled using the extended form of Tiwari and Das nanofluid model. A suitable transformation is adopted to simplify the boundary layer and energy equations into a nonlinear system of ODEs. A boundary value problem solver with fourth order accuracy (bvp4c) in the MATLAB software is utilized to solve the transformed system. The change in velocity and temperature as well as the heat transfer rate and skin friction coefficient are deliberated and graphically manifested for appropriate values of the dimensionless stretching/shrinking, nanoparticles volume fraction, and thermal stratification parameters. The presence of dual solutions is seen on all the profiles within the range of selected parameters

Numerical solutions of boundary layer flow over an exponentially stretching/shrinking sheet with generalized slip velocity

In this paper, the problem of steady laminar boundary layer flow and heat transfer over a permeable exponentially stretching/shrinking sheet with generalized slip velocity is considered. The similarity transformations are used to transform the governing nonlinear partial differential equations to a system of nonlinear ordinary differential equations. The transformed equations are then solved numerically using the bvp4c function in MATLAB. Dual solutions are found for a certain range of the suction and stretching/shrinking parameters. The effects of the suction parameter, stretching/shrinking parameter, velocity slip parameter, critical shear rate and Prandtl number on the skin friction and heat transfer coefficients as well as the velocity and temperature profiles are presented and discussed

Three-dimensional stagnation point viscous flow on a permeable moving surface with anisotropic slip

In this paper, the problem of a steady laminar three-dimensional stagnation point boundary layer flow on a permeable moving surface with anisotropic slip in a viscous fluid is investigated. A similarity transformation reduces the governing system of nonlinear partial differential equations into the ordinary (similarity) differential equations. The resulting equations are then solved numerically by using the bvp4c function in Matlab. The effects of surface mass transfer parameter, slip parameter, ratio of slip factors and moving parameter on the fluid flow characteristics are presented in the forms of tables and figures and are discussed in details

Stagnation Point Flow Of Hybrid Nanofluid Over A Permeable Vertical Stretching/Shrinking Cylinder With Thermal Stratification Effect

Hybrid nanofluid is invented to improve the heat transfer performance of traditional working fluids (water, traditional nanofluid) in many engineering applications. The present study highlights the numerical solutions and stability analysis of stagnation point flow using hybrid nanofluid over a permeable stretching/shrinking cylinder. The combination of copper (Cu) and alumina (Al2O3) nanoparticles with water as the base fluid is analytically modeled using the single phase model and modified thermophysical properties. A set of transformation is adopted to reduce the complexity of the governing model and then, numerically computed using the bvp4c solver in Matlab software. Suction parameter is vital to generate dual similarity solutions in shrinking cylinder case while no solution is found if the surface is impermeable. Two solutions are possible for the assisting and opposing flow within a specific value of the buoyancy parameter. For the shrinking cylinder, Al2O3-water nanofluid has the lowest heat transfer rate than Cu-water and hybrid Cu-Al2O3/water nanofluids. A suitable combination of alumina and copper nanoparticles volumetric concentration in hybrid nanofluid can produce higher heat transfer rate than the Cu-water nanofluid. The execution of stability analysis reveals that the first solution is more realistic than second solution. However, the present results are only fixed to the combination of copper and alumina nanoparticles only and the other kind of hybrid nanofluid may have different outcome

Thermal Marangoni flow past a permeable stretching/shrinking sheet in a hybrid Cu-Al2O3/water nanofluid

The present study accentuates the Marangoni convection flow and heat transfer characteristics of a hybrid Cu-Al2O3/water nanofluid past a stretching/shrinking sheet. The presence of surface tension due to an imposed temperature gradient at the wall surface induces the thermal Marangoni convection. A suitable transformation is employed to convert the boundary layer flow and energy equations into a nonlinear set of ordinary (similarity) differential equations. The bvp4c solver in MATLAB software is utilized to solve the transformed system. The change in velocity and temperature, as well as the Nusselt number with the accretion of the dimensionless Marangoni, nanoparticles volume fraction and suction parameters, are discussed and manifested in the graph forms. The presence of two solutions for both stretching and shrinking flow cases are noticeable with the imposition of wall mass suction parameter. The adoption of stability analysis proves that the first solution is the real solution. Meanwhile, the heat transfer rate significantly augments with an upsurge of the Cu volume fraction (shrinking flow case) and Marangoni parameter (stretching flow case). Both Marangoni and Cu volume fraction parameters also can decelerate the boundary layer separation process

MHD Mixed Convective Stagnation Point Flow With Heat Generation Past A Shrinking Sheet

This paper investigates the in uence of magnetohydrodynamics (MHD) mixed convective stagnation point flow over a shrinking sheet with the enhancement of heat generation/source. Using appropriate similarity transformations, the model are transformed into a system of nonlinear equations and then solved using bvp4c built-in-function in Matlab. Numerical results are presented graphically for the distributions of velocity, temperature as well as the skin friction coefficient and local Nusselt number. The findings revealed the dual solutions obtained within a particular range of the mixed convection parameter and shrinking parameter. It is found that the fluid velocity increases with the increasing values of the magnetic and mixed convection parameter while opposite results obtained for the fluid temperature. A stability analysis was performed and it is proven that the first solution is physically realizable and stable whereas the second solution is unstable

Unsteady three-dimensional flow and heat transfer past a permeable stretching/shrinking surface

The unsteady laminar three-dimensional boundary layer viscous flow and heat transfer over a permeable stretching/shrinking surface (sheet) is studied numerically. The system of nonlinear partial differential equations is transformed into ordinary differential equations via the similarity transformation. The resulting governing equations are then solved numerically by using the shooting method. Effects of the unsteadiness parameter, the stretching/shrinking parameter, the mass suction parameter and the Prandtl number on the skin friction coefficients, the local Nusselt number as well as the velocity and temperature profiles are presented and discussed in detail. The results also show that dual solutions exist for both cases of the stretching and shrinking surfaces

Boundary layer flow and heat transfer over a permeable exponentially stretching/shrinking sheet with generalized slip velocity

In this paper, the steady laminar boundary layer flow and heat transfer over a permeable exponentially stretching/shrinking sheet with generalized slip velocity is studied. The flow and heat transfer induced by stretching/shrinking sheets are important in the study of extrusion processes and is a subject of considerable interest in the contemporary literature. Appropriate similarity variables are used to transform the governing nonlinear partial differential equations to a system of nonlinear ordinary (similarity) differential equations. The transformed equations are then solved numerically using the bvp4c function in MATLAB. Dual (upper and lower branch) solutions are found for a certain range of the suction and stretching/shrinking parameters. Stability analysis is performed to determine which solutions are stable and physically realizable and which are not stable. The effects of suction parameter, stretching/shrinking parameter, velocity slip parameter, critical shear rate and Prandtl number on the skin friction and heat transfer coefficients as well as the velocity and temperature profiles are presented and discussed in detail. It is found that the introduction of the generalized slip boundary condition resulted in the reduction of the local skin friction coefficient and local Nusselt number. Finally, it is concluded from the stability analysis that the first (upper branch) solution is stable while the second (lower branch) solution is not stable