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

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 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

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

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

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 convection stagnation point flow along a vertical stretching permeable surface with heat source/sink

In this paper, we consider a two-dimensional stagnation point flow and heat transfer due to suction or injection towards a stretching surface. The governing nonlinear boundary layer equations are transformed into the system of nonlinear ordinary differential equation (ODE) using similarity transformation. The equations are then solved numerically by using the bvp4c function in MATLAB software. The effects of the governing parameters, namely velocity ratio parameter, Hartmann number, mixed convection parameter, Prandtl number, heat generation/absorption coefficient and suction/injection parameter are pdiscussed and presented graphically

Magnetohydrodynamics (MHD) Flow And Heat Transfer Of A Doubly Stratified Nanofluid Using Cattaneo-Christov Model

The present study utilized Cattaneo-Christov heat flux model for solving nanofluid flow and heat transfer towards a vertical stretching sheet with the presence of magnetic field and double stratification. Thermal and solutal buoyancy forces are also examined to deal with the double stratification effects. Buongiorno’s model of nanofluid is used to incorporate the effects of Brownian motion and thermophoresis. The boundary layer with non-Fourier energy equations are reduced into a system of nonlinear ordinary (similarity) differential equations using suitable transformations and then numerically solved using bvp4c solver in MATLAB software. The local Nusselt and Sherwood numbers of few limited cases are tabulated and compared with the earlier published works. It showed that a positive agreement was found with the previous study and thus, validated the present method. Numerical solutions are graphically demonstrated for several parameters namely magnetic, thermal relaxation, stratifications (thermal and solutal), thermophoresis and Brownian motion on the velocity, temperature and nanoparticles volume fraction profiles. An upsurge of the heat transfer rate was observed with the imposition of the thermal relaxation parameter (Cattaneo-Christov model) whereas the accretion of thermal and solutal stratification parameters reduced the temperature and nanoparticles concentration profiles, respectively

Hybrid nanofluid stagnation point flow past a slip shrinking Riga plate

Magnetic nanofluids cover many of uses since their characteristics are externally controllable, and their physical properties may vary with the nanoparticle volume fraction and magnetic field strength. Hybrid nanofluid also has been commercialized as the advancement of traditional nanofluid. The preliminary research on hybrid magnetic nanofluids inspired the present study to discover the stagnation-point flow of hybrid magnetite-cobalt ferrite/water nanofluid towards a shrinking Riga plate with the presence of velocity slip. The complex governing model of the flow is simplified by implementing the similarity transformation. A well-established numerical package, namely bvp4c in MATLAB, is used for numerical calculation as well as stability analysis. Two solutions are found due to the opposing flow from the shrinking Riga plate. From the stability analysis, the first solution which fulfills the boundary condition is the physically stable solution. The rising values of electromagnetohydrodynamic (EMHD) parameter and cobalt ferrite concentration augment the skin friction coefficient. Specifically, the critical point is lessened by 3% when the EMHD parameter is augmented from 0.3 to 0.5 and 0.5 to 0.7, which concludes that a suitably higher EMHD parameter could prevent the separation of the boundary layer. The heat transfer progress is actively performed with the enhancement of EMHD and velocity slip parameters which conclusively shows the suitability of these parameters in developing the cooling heat transfer fluid