Navier–Stokes equilibrium solutions of a viscous fluid confined between two infinite parallel plates that can independently stretch or shrink in orthogonal directions are studied. It is assumed that the admissible solutions satisfy spatial self-similarity in the stretching or shrinking perpendicular coordinates. The nonlinear steady boundary-value problem is discretized using a spectral Legendre method, and equilibrium solutions are found and tracked in the two-dimensional parameter space by means of pseudo-arclength continuation Newton–Krylov schemes. Different families of solutions have been identified, some of which are two-dimensional and correspond to the classical Wang and Wu self-similar flows arising in a plane channel with one stretching–shrinking wall [Wang, C.-A. and Wu, T.-C., “Similarity solutions of steady flows in a channel with accelerating walls,” Comput. Math. Appl. 30, 1–16 (1995)]. However, a large variety of three-dimensional solutions have also been found, even for low stretching or shrinking rates. When slightly increasing those rates, some of these solutions disappear at saddle-node bifurcations. By contrast, when both plates are simultaneously stretching or shrinking at higher rates, a wide variety of new families of equilibria are created and annihilated in the neighborhood of cuspidal codimension-2 bifurcation points. This behavior has similarities with the one observed in other planar and cylindrical self-similar flows.

The flow induced by a simultaneously rotating and radially stretching disk is revisited in this paper. The steady solution is given by a power-series of exponentials including the governing parameter of the problem c=Ω∕s, which is the ratio of the angular velocity of rotation Ω and the strain rate of stretching s . Since the series satisfy the far field conditions automatically, the boundary value problem reduces to an initial value problem and this guarantees the uniqueness of the solutions. Based on the exponential series solution, the features of the flow in the stretching dominated the subcritical regime 0ccrit are discussed in detail. Further features of engineering interest, as the pumping performance of the "rotating and stretching viscous pump", as well as its heat transfer characteristics are also considered in the paper.

The hydrodynamic flow generated between a biorthogonally stretched membrane and a steadily rotating flow is shown to be an exact self-similar solution to the three-dimensional Navier-Stokes equations. Identifying such exact solutions is significant because these solutions often give illuminating insight into more complex flows. Dual (and in some cases four) solutions are identified for a given ratio of stretching rates and far-field rotation rate, and the stability of these solutions is examined.

A modification of Hiemenz's two-dimensional outer potential stagnation-point flow of strain rateais obtained by adding periodic radial and azimuthal velocities of the formbrsin2 theta andbrcos2 theta, respectively, wherebis a shear rate. This leads to the discovery of a new family of three-dimensional viscous stagnation-point flows depending on the shear-to-strain-rate ratio gamma=b/athat exist over the range-infinity<gamma<infinity with reflectional symmetry about gamma=0. Numerical solutions for the wall shear stress parameters and the displacement thicknesses are given and compared with their large-gamma asymptotic behaviors. Sample similarity velocity profiles are also presented. It is noted that the results presented here are in many ways similar to the results reported for non-axisymmetric Homann stagnation-point flow.

The frictionally coupled sliding and spinning of a disk on an inclined plane is considered. The main results of the paper are: (i) The exact solution of the problem can be obtained by quadratures (i.e. in terms of integrals). It includes the kinematical steepness S=σ(tanϕ)∕μ of the plane, with ϕ the declination angle, μ the kinetic friction coefficient, with σ=+1 for downward and σ=−1 for upward motion, respectively. (ii) The phenomenon of the simultaneous stopping of the two motions known from the case of horizontal plane (S=0) is only possible for −0.0804632≤S<1∕2. For example, the choice μ=0.3 leads to an upper bound of ϕmax=8.53o and ϕmax=1.38o for the downhill and uphill motions, respectively. (iii) In the range −0.0804632≤S<0 of the kinematical steepness dual solutions for the uphill motions exist. (iv) When the initial state coincides with one of the fixed points of the system (which only exist in the range −0.0804632≤S<1∕2), the exact solution reduces to a simple elementary form which describes the uniformly decelerated sliding and spinning of the disk until the simultaneous stopping of the two motions. This is a universal phenomenon in the sense that it occurs in all the frictionally coupled sliding and spinning motions considered to date in the literature.

A modification of axisymmetric outer potential stagnation-point flow of strain rate a in a porous medium is obtained by adding periodic radial and azimuthal velocities of the form br sin 2 theta and b r cos 2 theta, respectively, where b is a shear rate. This leads to the discovery of a new family of asymmetric stagnation-point flows depending on two parameters, namely the shear-to-strain-rate ratio gamma and the porous media parameter sigma. Numerical solutions for the wall shear stress parameters and the displacement thicknesses are given and compared with their large-gamma asymptotic behaviors. Sample similarity velocity profiles are also presented.

A free convection wall jet flow with a prescribed wall temperature which enables the boundary-layer equations to be reduced to similarity form is treated. It is seen that there is a finite range σ1<σ<σ0 of the Prandtl number σ for which there are solutions. The singular nature of the solution as σ→σ0 and σ→σ1 is examined. An alternative (convective) boundary condition characterized by the parameter λ is also considered with critical values λc of λ dependent on σ being found. An examination of these critical points shows that they can exist only in the range σ1<σ<σ0. The values of λc become singular at these limits and achieve a maximum value within this range giving only a finite range of both σ and λ for solutions to exist.

Planar stagnation-point flow normally impinging a rotating plate is studied. Symmetries show that along a circle of radius r, the radial velocity ur can be computed from results in one quadrant and the angular velocity uθ can be computed from results in two quadrants. The results of this study for various values of the dimensionless angular velocity σ are compared with axisymmetric Homann stagnation-point flow on a rotating plate first studied by Hannah.

The steady laminar flow of two immiscible, uniformly rotating fluid layers is studied and exact similarity solutions of the axisymmetric Navier-Stokes equations in cylindrical polar coordinates are found. The similarity solutions occur with a flat interface at z = 0 under the parameter restriction that sigma(2)rho = 1, where sigma is the ratio of the fluid angular velocities at z = +/-infinity and rho is the density ratio of the two fluids. Under this restriction the problem reduces to one with two independent parameters sigma and mu, the viscosity ratio of the fluids. Numerical results of the resulting system of ordinary differential equations are found for selected values of mu and sigma, and it is shown that similarity solutions exist for sigma(c)(mu) <= sigma <= 1, where sigma(c)(mu) < 0 (i.e., counterrotating flows). For sigma < 0 the lower fluid can become divided into two distinct recirculation regions between which fluid cannot transfer.

A study is made to determine similarity solutions for uniform shear flow along horizontal and vertical heated plates. Only for specified variations of the wall temperature along the plates are these solutions possible. For vertical walls the temperature must decrease as x − 1 ∕ 3 while for horizontal walls it must increase as x 1 ∕ 3 , where x is the streamwise coordinate along each semi-infinite plate.

Received 17 April 2018DOI:https://doi.org/10.1103/PhysRevFluids.3.100001©2018 American Physical Society

The flow of one Howarth stagnation-point flow impinging directly on another Howarth stagnation-point flow is studied, and an exact similarity solution to the Navier–Stokes equations is found. The upper layer fluid has density ρ1 and kinematic viscosity ν1 while the lower layer fluid has density ρ2 and kinematic viscosity ν2 and the two fluids are assumed to be immiscible. This problem has potentially five independent parameters to investigate, but application of the continuity of the normal stresses at the interface imposes restrictions which reduces the problem to one with three independent parameters, namely a ratio σ of strain rates and the fluid parameter ratios ρ=ρ1∕ρ2 and ν=ν1∕ν2. Numerical results are presented for selected values of ρ and ν for a range of σ and show that stable results exist for all values of σ>0, and for a range of negative σ values. Sample stable velocity profiles are also presented.

The combined effects of wall stretching and transpiration on an Airy wall jet is considered. An exact solution dependent on two parameters is found. One parameter fw measures the transpiration through the surface and the other parameter λ measures the extensional motion of the surface. Suction occurs for fw>0 and blowing occurs for fw<0. The surface stretches when λ>0 and shrinks when λ<0. When fw=λ=0 the original result of Weidman et al. (1997) is recovered; when λ=0 the results of Magyari et al. (2003) are recovered; when fw=0 the results of Kubitschek and Weidman (2017) are recovered. Solution of an implicit equation provides the boundary in fw-λ space for the existence of solutions. The region of existence of solutions increases with increasing suction.

The normal impingement of axisymmetric Homann stagnation-point flow on a surface executing perpendicular, planar, biaxial stretching is studied. The flow field generated is an exact solution of the steady, three-dimensional Navier–Stokes equations in the form of a similarity solution. It is shown that two sets of dual solutions exist, forming four different branches of steady solutions. For sufficiently small stretches (including compressions of the surface) the four branches exhibit a multi-branch spiralling behaviour in the surface shear stress parameter plane. The linear stability of the solutions are also examined, identifying only one stable solution for each set of parameters.

A unified formulation for stagnation-point flows and linearly stretching plates is given wherein the two can occur separately or in unison. Reductions to known cases are given. It is noticed that previous work on stretching plates beneath planar and axisymmetric stagnation point flows have respectively aligned planar stretching and axisymmetric stretching. The general formulation reveals other combinations of stretching beneath stagnation-point flows exist and three new cases are studied in detail. The linear stability of dual and multiple solutions are calculated.

The flow induced by an impermeable flat surface executing orthogonal stretching and orthogonal shearing in a rotating fluid system is investigated. Both the stretching and shearing are linear in the coordinates. An exact similarity reduction of the Navier–Stokes equations gives rise to a pair of nonlinearly-coupled ordinary differential equations governed by three parameters. In this study we set one parameter and analyze the problem which leads to flow for an impermeable surface with shearing and stretching due to velocity u along the x-axis of equal strength a while the shearing and stretching due to velocity v along the y-axis of equal strength b. These solutions depend on two parameters—a Coriolis (rotation) parameter \(\sigma = \Omega /a\) and a stretching/shearing ratio \(\lambda =b/a\). A symmetry in solutions is found for \(\lambda = 1\). The exact solution for \(\sigma = 0\) and the asymptotic behavior of solutions for \(|\sigma | \rightarrow \infty\) are determined and compared with numerical results. Oscillatory solutions are found whose strength increases with increasing values of \(|\sigma |\). It is shown that these solutions tend to the well-known Ekman solution as \(|\sigma | \rightarrow \infty\).