Stability of hydromagnetic boundary layer flow of non-Newtonian power-law fluid flow over a moving wedge

Abstract

We consider the two-dimensional laminar boundary-layer flow of power-law fluid over a moving wedge in which the uniform magnetic field is applied normally to the flow. The motion of the mainstream and wedge is approximated in terms of the power of the distance from leading boundary-layer edge which helps to reduce the governing partial differential equations to an ordinary differential equation. No analytic solution is feasible due to high nonlinearity; therefore, the numerical simulations are sought by Chebyshev collocation and shooting techniques. These techniques provide a means to assess physical insights into the hydrodynamics of complex flows and particularly the Chebyshev collocation method enables the construction of efficient tools suitable for nonlinear problems. This method often leads to a matrix-based analysis and an iterative technique needs to be employed to solve the problem. The various results on the physical parameters show that the shear-thinning and shear-thickening solutions are demarcated by the Newtonian fluid. It is also noticed that the system yielded a non-unique solution structure for a given set of parameters. However, the non-uniqueness of the solutions disappears for increasing the magnetic field. To assess the nature of the non-unique solutions, it is important to perform the linear stability using large-time asymptotics. This analysis gives as to which of these solutions is practically feasible and models the flow. Eigensolution-based analysis reveals that the first solution is always stable while the other one leading to unstable mode. The results further show that an increase in the magnetic field stabilizes the fluid flow over the moving wedge and promotes the unique solution to the problem. Further, the obtained numerical results are compared by examining the large λ asymptotics. It is observed that there is a qualitative comparison between solutions. The hydrodynamics behind the magnetic field and non-Newtonian fluid are discussed in detail.

Appendix

1.1 Derivation of boundary layer equations

Consider the Navier–Stokes equation in the vector form where the symbols have their usual meaning (as discussed earlier in the formulation Sect. 2.1).

$$\begin{aligned} \dfrac{\partial u }{\partial x} + \dfrac{\partial v}{\partial y}&= 0 \end{aligned}$$

(42)

$$\begin{aligned} \dfrac{\partial u}{\partial t }+ u \dfrac{\partial u }{\partial x } + v \dfrac{\partial u }{\partial y}&= \dfrac{-1}{\rho }\dfrac{\partial p }{\partial x} + \dfrac{\mathcal {K}}{\rho } \left( \dfrac{\partial \tau _{xx} }{\partial x } + \dfrac{\partial \tau _{xy} }{\partial y } \right) - \dfrac{\sigma }{\rho } B_y{^2} u\end{aligned}$$

(43)

$$\begin{aligned} \dfrac{\partial v}{\partial t }+ u \dfrac{\partial v}{\partial x } + v \dfrac{\partial v}{\partial y}&= \dfrac{-1}{\rho }\dfrac{\partial p }{\partial y} + \dfrac{\mathcal {K}}{\rho } \left( \dfrac{\partial \tau _{yx} }{\partial x } + \dfrac{\partial \tau _{yy} }{\partial y } \right) . \end{aligned}$$

(44)

The stress tensor $\tau$ is defined as

$$\begin{aligned} \tau =-\mu (\dot{\gamma })\dot{\gamma },\ \dot{\gamma } =\nabla u + \nabla u^{T}, \end{aligned}$$

(45)

where $\dot{\gamma }$ is the second invariant of strain tensor. We now refer the non-Newtonian viscosity $\mu$ as an apparent viscosity $\mu _{app}$, in which case, we have

$$\begin{aligned} \tau _{ij} = \mu _{app}\bigg ( {\frac{\partial u_i}{\partial x_j}}+ {\frac{\partial u_j}{\partial x_i}}\bigg ). \end{aligned}$$

(46)

Hence,

$$\begin{aligned} (\tau _{x x}, \tau _{y y}) = 2\mu _{app}\bigg ({\frac{\partial u}{\partial x}},{\frac{\partial v}{\partial y}}\bigg ),\ (\tau _{y x}, \tau _{x y}) = \mu _{app}\bigg ({\frac{\partial u}{\partial y}}+ {\frac{\partial v}{\partial x}}\bigg ), \end{aligned}$$

(47)

where

$$\begin{aligned} \mu _{app} = \mathcal {K}\bigg |2\bigg ({\frac{\partial u}{\partial x}}\bigg )^2 + 2\bigg ({\frac{\partial v}{\partial y}}\bigg )^2 + \bigg ({\frac{\partial u}{\partial y}}+{\frac{\partial v}{\partial x}}\bigg )^2\bigg |^{\frac{n-1}{2}}. \end{aligned}$$

By introducing the following non-dimensional variables

$$\begin{aligned} (x^*,y^*) = \bigg ({\frac{x}{L}},{\frac{y}{\delta L}}\bigg ),\ (u^*,v^*) = \bigg ({\frac{u}{U}},{\frac{v}{\delta U}}\bigg ),\ t^* = {\frac{L}{U}}t ,\ p^* = {\frac{p}{\rho U^2}} \end{aligned}$$

(48)

where L, U are the reference length and velocity, respectively, and $\delta$ is of the order of boundary layer thickness to get the following equations:

$$\begin{aligned} \dfrac{\partial u^* }{\partial x^*} + \dfrac{\partial v^*}{\partial y^*}&= 0 \end{aligned}$$

(49)

$$\begin{aligned} \dfrac{\partial u^*}{\partial t^* }+ u^* \dfrac{\partial u^* }{\partial x^* } + v^* \dfrac{\partial u^* }{\partial y^*}&= -\dfrac{\partial p^* }{\partial x^*} + \dfrac{\mathcal {K}}{\rho } \left( \dfrac{\partial \tau _{x^*x^*} }{\partial x^* } + \dfrac{\partial \tau _{x^*y^*} }{\partial y^* } \right) - \dfrac{\sigma }{\rho } B_y{^2} u^*\end{aligned}$$

(50)

$$\begin{aligned} \dfrac{\partial v^*}{\partial t^* }+ u^* \dfrac{\partial v^*}{\partial x^* } + v^* \dfrac{\partial v^*}{\partial y^*}&= -\dfrac{\partial p^* }{\partial y^*} + \dfrac{\mathcal {K}}{\rho } \left( \dfrac{\partial \tau _{y^*x^*} }{\partial x^* } + \dfrac{\partial \tau _{y^*y^*} }{\partial y^* } \right) .\nonumber \\ (\tau _{x^*x^*}, \tau _{y^*y^*})&= 2\bigg ({\frac{U}{L}}\bigg ) \mu ^*_{app}\bigg ({\frac{\partial u^*}{\partial x^*}},{\frac{\partial v^*}{\partial y^*}}\bigg ),\ (\tau _{y^* x^*}, \tau _{x^* y^*}) \\ &= \bigg ({\frac{U}{L}}\bigg ) \mu ^*_{app}\bigg ({\frac{1}{\delta} }{\frac{\partial u^*}{\partial y^*}}+\delta {\frac{\partial v^*}{\partial x^*}}\bigg )\nonumber \\ \mu ^*_{app}&= \mathcal {K}\bigg ({\frac{U}{L}}\bigg )^{n-1} \bigg |2\bigg ({\frac{\partial u^*}{\partial x^*}}\bigg )^2 + 2\bigg ({\frac{\partial v^*}{\partial y^*}}\bigg )^2 + \bigg ({\frac{1}{\delta} }{\frac{\partial u^*}{\partial y^*}} \\&\quad+ \delta {\frac{\partial v^*}{\partial x^*}}\bigg )^2\bigg |^{\frac{n-1}{2}}, \end{aligned}$$

(51)

and the apparent Reynolds number is

$$\begin{aligned} Re = {\frac{\rho U^{2-n} L^n}{\mathcal {K}}}. \end{aligned}$$

(52)

Neglecting terms of order $O(\delta )$ and small on the bases that Re is large, we obtain the following boundary layer equations:

$$\begin{aligned} \dfrac{\partial u }{\partial x} + \dfrac{\partial v}{\partial y}&= 0\end{aligned}$$

(53)

$$\begin{aligned} \dfrac{\partial u}{\partial t }+ u \dfrac{\partial u }{\partial x } + v \dfrac{\partial u }{\partial y}&= -\dfrac{1}{\rho }\dfrac{\partial p }{\partial x} + \dfrac{\mathcal {K}}{\rho } \left( \bigg |\dfrac{\partial u }{\partial y} \bigg |^{n-1}\dfrac{\partial u}{\partial y} \right) - \dfrac{\sigma }{\rho } B_y{^2} u\end{aligned}$$

(54)

$$\begin{aligned} \dfrac{\partial p }{\partial y}&=0. \end{aligned}$$

(55)

Here $^*$ has been dropped for simplicity and Eq. (55) infers the pressure varies towards the mainstream flow direction alone. Hence we arrive at (3).