Three‐dimensional Hiemenz stagnation‐point flows

Abstract

A modification of Hiemenz's two‐dimensional outer potential stagnation‐point flow of strain rate a is obtained by adding periodic radial and azimuthal velocities of the form brsin2θ and brcos2θ, respectively, where b is 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 γ=b/a that exist over the range −∞<γ<∞ with reflectional symmetry about γ=0. Numerical solutions for the wall shear stress parameters and the displacement thicknesses are given and compared with their large‐γ 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.

On dual solutions of the three‐dimensional Hiemenz flow over a stretching/shrinking sheet

“…Following Weidman [24], we consider the three‐dimensional Hiemenz flow over a stretching/shrinking sheet. Let $$(u_r, u_\phi , w)$$ be the velocity components in the directions of cylindrical coordinates $$(r,\phi ,z)$$, respectively.…”

“…Let $$(u_r, u_\phi , w)$$ be the velocity components in the directions of cylindrical coordinates $$(r,\phi ,z)$$, respectively. We get the following potential flow field [24]: $$\begin{eqnarray} u_r(r,\phi )=\frac{ar}{2}(1+\cos 2\phi )+br \sin 2\phi ,\quad u_\phi (r,\phi )=-\frac{ar}{2}\sin 2\phi + br \cos 2\phi ,\quad w(z)=-az, \end{eqnarray}$$that satisfies the equation of continuity, that is $$\begin{eqnarray} \frac{\partial }{\partial r}(ru_r)+\frac{\partial }{\partial \phi }(u_\phi )+r\frac{\partial }{\partial z}w=0, \end{eqnarray}$$and the corresponding pressure field is given by …”

“…\end{eqnarray}Again following Refs. [5, 24], we formulate the problem along the principle axes and drop the bar notation in Equation (). Thus, the similarity transformations of this problem are considered as follow: $$\begin{align} u(x,y,z)=\Delta _1xF^{\prime }(\eta ),\quad v(x,y,z)=\Delta _2yG^{\prime }(\eta ),\end{align}$$ $$\begin{align} w(z)=-\sqrt {\frac{\nu }{a}}[\Delta _1F(\eta )+\Delta _2G(\eta )], \quad \eta =\sqrt {\frac{a}{\nu }}z.$$…”

“…They have found more than dual solutions of the similarity equations over some particular range of the stretching parameters in all these three problems. Very recently, Weidman [24] studied three‐dimensional Hiemenz flow where the reported results are related to that of Weidman [5].…”

“…The present study is motivated by the work of Weidman [24] and Mahapatra and Sidui [19]. Weidman [24] generalized the Hiemenz's problem by adding two periodic terms $$br\sin 2\phi$$ and $$br\cos 2\phi$$ to the radial and azimuthal velocities, respectively, of Hiemenz's outer potential flow that leads to a new family of three‐dimensional Hiemenz flow governed by the shear‐to‐strain rate parameter. Several applications exist for such flow problems.…”