Dual solutions of a magnetohydrodynamic stagnation point flow of a non-Newtonian fluid over a shrinking sheet and a linear temporal stability analysis

The present study accentuates the magnetohydrodynamic and suction/injection effects on the two-dimensional stagnation point flow and heat transfer of a non-Newtonian fluid over a shrinking sheet. The set of Navier-Stokes equations are converted into a system of highly non-linear ordinary differential equations by employing suitable similarity variables. The obtained self-similar equations are then solved numerically with the aid of shooting technique. The similarity equations exhibit dual solutions over a certain range of the shrinking strength. It is observed that the solution domain increases as the suction/injection parameter, the non-Newtonian parameter and the magnetic parameter increase. Moreover, it is further noticed that these two solution branches show opposite behavior on the velocity and temperature profiles for the combined effects of the several flow parameters. Emphasis has been given to determine the most feasible and physically stable solution branch. Thus a linear temporal stability analysis has been carried out and the stability of the these two branches are tested by the sign of the smallest eigenvalue. The smallest eigenvalues are found numerically which suggest that the upper solution branch is stable and the flow dynamics can be describe by the behavior of the upper solution branch.

The lid-driven right-angled isosceles triangular cavity flow

We employ lattice Boltzmann simulation to numerically investigate the two-dimensional incompressible flow inside a right-angled isosceles triangular enclosure driven by the tangential motion of its hypotenuse. While the base flow, directly evolved from creeping flow at vanishing Reynolds number, remains stationary and stable for flow regimes beyond $Re\gtrsim 13\,400$, chaotic motion is nevertheless observed from as low as $Re\simeq 10\,600$. Chaotic dynamics is shown to arise from the destabilisation, following a variant of the classic Ruelle–Takens route, of a secondary solution branch that emerges at a relatively low $Re\simeq 4908$ and appears to bear no connection to the base state. We analyse the bifurcation sequence that takes the flow from steady to periodic and then quasi-periodic and show that the invariant torus is finally destroyed in a period-doubling cascade of a phase-locked limit cycle. As a result, a strange attractor arises that induces chaotic dynamics.

Stagnation Point Flow with Heat Transfer and Temporal Stability of Ferrofluid Past a Permeable Stretching/Shrinking Sheet

In this paper, the hydromagnetic stagnation point flow and temporal stability of Fe3O4-water ferrofluid over a convectively heated permeable stretching/shrinking sheet is theoretically investigated. The model equations of momentum and energy balance are obtained and transformed into ordinary differential equations using appropriate similarity variable. Using shooting method together with Runge-Kutta-Fehlberg numerical scheme the model nonlinear boundary value problem is tackled numerically. Pertinent results with respect to the basic steady flow velocity, temperature, skin friction and Nusselt number are obtained graphically and in tabular form. It is found that a critical value of shrinking parameter (λc) exists below which no real solution can be found. In addition, dual solutions (upper and lower branch) are observed for a range of shrinking/stretching parameter (λc<λ< 1), while for the stretching case (λ 1), the solution is unique. The obtained steady state solutions are examined for temporal development of small disturbances. The smallest eigenvalues reveal that the upper solution branch is stable and physically reliable while the lower solution branch is unstable and unrealistic. Both suction and magnetic field widen the range of the shrinking parameter for which the solution exists and boost the flow stability while nanoparticles volume fraction lessens it.

Maximal heat transfer between two parallel plates

The divergence-free time-independent velocity field has been determined so as to maximise heat transfer between two parallel plates with a constant temperature difference under the constraint of fixed total enstrophy. The present variational problem is the same as that first formulated by Hassanzadeh et al. (J. Fluid Mech., vol. 751, 2014, pp. 627–662); however, the search range for optimal states has been extended to a three-dimensional velocity field. A scaling of the Nusselt number $Nu$ with the Péclet number $Pe$ (i.e., the square root of the non-dimensionalised enstrophy with thermal diffusion time scale), $Nu\sim Pe^{2/3}$, has been found in the three-dimensional optimal states, corresponding to the asymptotic scaling with the Rayleigh number $Ra$, $Nu\sim Ra^{1/2}$, expected to appear in an ultimate state, and thus to the Taylor energy dissipation law in high-Reynolds-number turbulence. At $Pe\sim 10^{0}$, a two-dimensional array of large-scale convection rolls provides maximal heat transfer. A three-dimensional optimal solution emerges from bifurcation on the two-dimensional solution branch at $Pe\sim 10^{1}$, and the three-dimensional solution branch has been tracked up to $Pe\sim 10^{4}$ (corresponding to $Ra\approx 2.7\times 10^{6}$). At $Pe\gtrsim 10^{3}$, the optimised velocity fields consist of convection cells with hierarchical self-similar vortical structures, and the temperature fields exhibit a logarithmic-like mean profile near the walls.

Branching of stretch histories in biaxially loaded nonlinear viscoelastic fiber-reinforced sheets

This work considers an experiment in which a nonlinear viscoelastic square sheet, subjected to uniformly distributed tensile forces on its edge surfaces, undergoes a homogeneous biaxial extensional creep history. The sheet is fiber-reinforced with the direction of reinforcement either normal to the midplane of the sheet or parallel to one of its edges. The possibility is considered that when the governing equations are solved for the biaxial creep response, there may be a time during the deformation when a second solution branch can form. Thus, if equal biaxial forces are applied to the sheet, its deformed states are squares until some time when they become rectangular. The material is modeled using the Pipkin–Rogers nonlinear single integral constitutive equation for a transversely isotropic material. A condition is derived to determine the time when the solution to the governing equations forms a new branch. Numerical examples are presented for fibers oriented normal to the sheet and in the plane of the sheet.

Magneto-Nanofluid Dynamics in Convergent-Divergent Channel and its Inherent Irreversibility

The effects of Cu-nanoparticles on the entropy generation of steady magnetohydrodynamic incompressible flow with viscous dissipation and Joule heating through convergent-divergent channel are analysed in this paper. The basic nonlinear partial differential equations are transformed into a system of coupled ordinary differential equations using suitable transformations which are then solved using power series with Hermite- Padé approximation technique. The velocity profiles, temperature distributions, entropy generation rates, Bejan number as well as the rate of heat transfer at the wall are presented in convergent-divergent channels for various values of nanoparticles solid volume fraction, Eckert number, Reynolds number and channel angle. A stability analysis has been performed for the shear stress which signifies that the lower solution branch is stable and physically realizable, whereas the upper solution branch is unstable. It is interesting to remark that the entropy generation of the system increases at the two walls as well as the heat transfer irreversibility is dominant there whereas the fluid friction irreversibility is dominant along the centreline of the channel.

Effects of Second-Order Slip and Viscous Dissipation on the Analysis of the Boundary Layer Flow and Heat Transfer Characteristics of a Casson Fluid

The aim of the present study is to analyze numerically the steady boundary layer flow and heat transfer characteristics of Casson fluid with variable temperature and viscous dissipation past a permeable shrinking sheet with second order slip velocity. Using appropriate similarity transformations, the basic nonlinear partial differential equations have been transformed into ordinary differential equations. These equations have been solved numerically for different values of the governing parameters namely: shrinking parametersuction parameterCasson parameterfirst order slip parametersecond order slip parameter Prandtl number and the Eckert number using the bvp4c function from MATLAB. A stability analysis has also been performed. Numerical results have been obtained for the reduced skin-friction, heat transfer and the velocity and temperature profiles. The results indicate that dual solutions exist for the shrinking surface for certain values of the parameter space. The stability analysis indicates that the lower solution branch is unstable, while the upper solution branch is stable and physically realizable. In addition, it is shown that for a viscous fluida very good agreement exists between the present numerical results and those reported in the open literature. The present results are original and new for the boundary-layer flow and heat transfer past a shrinking sheet in a Casson fluid. Therefore, this study has importance for researchers working in the area of non-Newtonian fluids, in order for them to become familiar with the flow behavior and properties of such fluids.

Extended series solutions and bifurcations of the Dean equations

Steady, incompressible flow down a slowly curving circular pipe is considered. Both real and complex solutions of the Dean equations are found by analytic continuation of a series expansion in the Dean number, $K$. Higher-order Hermite–Padé approximants are used and the results compared with direct computations using a spectral method. The two techniques agree for large, real $K$, indicating that previously reported asymptotic behaviour of the series solution is incorrect, and thus resolving a long-standing paradox. It is further found that a second solution branch, known to exist at high Dean number, does not appear to merge with the main branch at any finite $K$, but appears rather to bifurcate from infinity. The convergence of the series is limited by a square-root singularity on the imaginary $K$-axis. Four complex solutions merge at this point. One corresponds to an extension of the real solution, while the other three are previously unreported. This bifurcation is found to coincide with the breaking of a symmetry property of the flow. On one of the new branches the velocity is unbounded as $K\rightarrow 0$. It follows that the zero-Dean-number flow is formally non-unique, in that there is a second complex solution as $K\rightarrow 0$ for any non-zero $\vert K\vert $. This behaviour is manifested in other flows at zero Reynolds number. The other two complex solutions bear some resemblance to the two solution branches for large real $K$.

Effects of Magnetic Field and Thermal Radiation on Stagnation Flow and Heat Transfer of a Power-Law Fluid over a Shrinking Sheet

An analysis is made on the steady two-dimensional boundary layer magnetohydrodynamic (MHD) stagnation-point flow and radiative heat transfer of an electrically conducting power-law fluid over a shrinking sheet which is shrunk in its own plane with a velocity proportional to the distance from a fixed point. The similarity transformations are used to transform the boundary layer equations into a system of nonlinear ordinary differential equations which are then solved numerically using shooting technique. It is found that multiple solutions exist for a certain range of the ratio of the shrinking velocity to the free stream velocity (i.e., α) which again depends on the magnetic parameter (M) and the power-law index parameter (n). The results pertaining to the present study indicate that as the strength of the magnetic parameter increases, the range of α where similarity solutions exist gradually increases. It is also observed that the temperature at a point decreases with increase in M for the first solution branch, whereas it increases with increase in M for the second solution branch. The reported results are in good agreement with the available published work in the literature.

Bistability of flame propagation in a model with competing exothermic reactions

We investigate a diffusional-thermal model with two-step competitive exothermic reactions for premixed combustion wave propagation in one spatial dimension under adiabatic conditions. A criterion based on the crossover temperature notion was used to qualitatively predict the region in the space of parameters where three travelling combustion wave solutions coexist, which are further studied via numerical means. It is demonstrated that under certain conditions the flame speed is an ‘S’-shaped function of parameters. The fast branch is either stable or is partly stable and exhibits the Andronov–Hopf (AH) bifurcation before the turning point is reached. The mid-branch is completely unstable. The slow solution branch is either unstable or partly stable and exhibits a single or a pair of AH bifurcations. The AH bifurcations are shown to be supercritical giving rise to stable pulsating waves. Bistability and hysteresis phenomena are also demonstrated.