Study of Flow and Heat Transfer on a Stretching Surface in a Rotating Casson Fluid

Abstract

The steady, boundary layer flow and heat transfer on a stretching surface in rotating fluid has been examined. Using suitable similarity transformations, the partial differential equations governing the flow and heat transfer phenomenon convert to a system of non-linear ordinary differential equations. The obtained equations are solved by using the shooting technique with fifth order Runge–Kutta–Fehlberg method. The parameters involving in the problem are Casson fluid parameter $\beta$, non-dimensional parameter $\lambda$ that signifies the importance of rotation rate to stretching rate and Prandtl number $Pr$. The effects of these parameters on physical quantities such as velocity and temperature profiles, skin frictions and Nusselt number are inspected with the aid of graphs and tables.

1 Introduction

The boundary layer flow induced by the continous stretching of a surface about a fixed point was initially investigated by Crane [1]. He assumed that the surface is stretching with a velocity proportional to the linear distance from the fixed point. We need heat transfer phenomena in flow over a stretching sheet [2–15].

The rotating fluid flows have applications in industrial processes, astrophysical and geophysical phenomenon, biomechanics, cosmic fluid dynamics, in the design of turbines and turbomechanics and rotating heat exchangers. The migration of underground water and movement of petrol, oil and gas through the reservoirs are some examples of rotating flows. Deka et al. [16] examined the viscous, incompressible rotating fluid flow induced by uniformly accelerated flat surface. Wang [17] investigated the flow over a stretching surface in a rotating viscous fluid and obtained the self-similar solutions. The magnetic field effects on flow and heat transfer characteristics on a stretching sheet in a rotating fluid were examined by Takhar et. al. [18]. The unsteady flow of a viscous fluid over a surface stretching linearly in its plane in a rotating fluid was analyzed by Rajeswari and Nath [19] and Nazar et al. [20]. Abbas et al. [21] extended the Nazar’s problem [20] by considering the effects of magnetic field and obtained the solution of the problem by using the implicit finite difference scheme known as Keller box method. Zaimi et al. [22] studied the fluid flow due to stretching of a surface in rotating viscoelastic fluid and found that viscoelastic parameter has increasing effect on velocity in $x$-direction whereas opposite behavior is observed for velocity in $y$-direction.

The fluids found in nature and industry exhibit non-Newtonian behavior and it is not possible to represent the behavior of these fluids by classical Navier–Stoke’s equations. Different models have been proposed by the rheologists as no single constitutive equation could be found to describe all the properties of non-Newtonian fluids. Of these models, Casson fluid model represents a shear thinning fluid which is assumed to have infinite viscosity at zero rate of shear [23]. The flow and heat transfer characteristics of Casson fluid from different physical and mathematical aspects have been investigated by several researchers [24–32].

The present study focusses on boundary layer flow and heat transfer due to a surface stretching in a rotating Casson fluid. A complete parametric study is presented through graphs and tables.

2 Mathematical Formulation

Consider a steady, incompressible boundary layer flow caused by the stretching of a heated surface in a rotating Casson fluid. The flow is three dimensional due to the presence of Coriolis force. It is assumed that the surface is stretching in the $x$ direction with a velocity proportional to the distance from the origin. Let $(u,v,w)$ denote the velocity components in the $(x,y,z)$ directions with the axes rotating at an angular velocity $\Omega$ in the $z$ direction (see Fig. 1). The temperature of the stretching surface is kept constant at $T_{w}$ and the far away fluid is assumed to have the temperature $T_{\infty }$. The rheological model that characterizes Casson fluid is given by :

$$\begin{aligned} \tau _{ij}=\left\{ \begin{array}{ll} 2\left( \mu _{B}+\frac{p_{y}}{\sqrt{2\pi }}\right) e_{ij},&{}\pi >\pi _{c} \\ 2\left( \mu _{B}+\frac{p_{y}}{\sqrt{2\pi _{c}}}\right) e_{ij},&{}\pi <\pi _{c} \end{array}\right. , \end{aligned}$$

(2.1)

where $\tau _{ij}$ is the Cauchy stress tensor, $\pi =e_{ij}e_{ij}$ is the product of components of deformation rate with itself, $e_{ij}$ is the $(i,j)\rm{th}$ component of deformation rate, $\pi _{c}$ is the critical value of this product based on the non-Newtonian model, $\mu _{B}$ is the plastic dynamic viscosity of the non-Newtonian model and $p_{y}$ is the yield stress of fluid and then the governing equations for the continuity, momentum and energy are:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0,$$

(2.2)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{ \partial u}{\partial z}-2\Omega v=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\nu \left( 1+\frac{1}{\beta }\right) \nabla ^{2}u, $$

(2.3)

$$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{ \partial v}{\partial z}+2\Omega u=-\frac{1}{\rho }\frac{\partial p}{\partial y}+\nu \left( 1+\frac{1}{\beta }\right) \nabla ^{2}v,$$

(2.4)

$$u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{ \partial w}{\partial z}=-\frac{1}{\rho }\frac{\partial p}{\partial z}+\nu \left( 1+\frac{1}{\beta }\right) \nabla ^{2}w,$$

(2.5)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{ \partial T}{\partial z}=\frac{k}{\rho c_{p}}\nabla ^{2}T. $$

(2.6)

The boundary conditions related to the present problem are

$$\begin{aligned} \left. \begin{array}{c} u=u_{w}(x)=ax,v=0,\,w=0,\,T=T_{w}\quad {\text {at}} \quad z=0, \\ u\rightarrow 0,v\rightarrow 0,\,T\rightarrow T_{\infty }\quad {\text {at}} \,z\rightarrow \infty . \end{array}\right\} \end{aligned}$$

(2.7)

Fig. 1

Schematic diagram of the problem

Here $a>0$ is a constant, $\beta =\frac{\mu _{B\sqrt{2\pi _{c}}}}{p_{y}}$ is the Casson fluid parameter, $\nu$ is the kinematic viscosity, $k$ is the thermal conductivity, $c_{p}$ is the specific heat of the fluid at a constant pressure, $\rho$ is the density and $T$ is the temperature of the fluid. Introducing the following similarity variables:

$$ u=axf^{\prime }(\eta ),\,v=axg(\eta ),\,w=-\sqrt{a\nu }f(\eta ), \quad \eta =\sqrt{\frac{a}{\nu }}z,\,\theta =\frac{T-T_{\infty }}{T_{w}-T_{\infty }}. $$

(2.8)

By substituting (2.8) in Eqs. (2.2–2.5), Eq. (2.2) is identically satisfied and Eqs. (2.3, 2.4, 2.6) become

$$\left( 1+\frac{1}{\beta }\right) f^{\prime \prime \prime }+ff^{\prime \prime }-f^{\prime 2}+2\lambda g=0,$$

(2.9)

$$\left( 1+\frac{1}{\beta }\right) g^{\prime \prime }+fg^{\prime }-f^{\prime }g-2\lambda f^{\prime }=0,$$

(2.10)

$$\theta ^{\prime \prime }+Prf\theta ^{\prime }=0,$$

(2.11)

where $\lambda =\Omega /a$ is the ratio of rate of rotation to stretching rate and $Pr=\mu c_{p}/k$ is the Prandtl number. The boundary conditions (2.7) in dimensionless form are

$$\begin{aligned} \left. \begin{array}{c} f^{\prime }(0)=1,\,f(0)=0,\,g(0)=0,\,\theta (0)=1, \\ f^{\prime }(\infty )=0,\,g(\infty )=0,\,\theta (\infty )=0. \end{array}\right\} \end{aligned}$$

(2.12)

The skin friction coefficients along $x$ and $y$ directions, i.e., $C_{fx}$ and $C_{fy}$ are:

$$ C_{fx}=\frac{\tau _{wx}}{\rho u_{w}^{2}},\quad C_{fy}=\frac{\tau _{wy}}{\rho u_{w}^{2}}, $$

(2.13)

where $\tau _{wx}$ is the surface shear stress along the $x$ direction and $\tau _{wy}$ is the surface shear stress along the $y$ direction and are defined as

$$\tau _{wx}=\left( \mu _{B}+\frac{p_{y}}{\sqrt{2\pi }}\right) \left( \frac{ \partial u}{\partial z}\right) _{z=0},\quad \tau _{wy}=\left( \mu _{B}+\frac{p_{y}}{\sqrt{2\pi }}\right) \left( \frac{\partial v}{\partial z} \right) _{z=0}.$$

(2.14)

Using Eqs. (2.14) and (2.8), we obtain

$$Re_{x}^{1/2}C_{fx}=\left( 1+\frac{1}{\beta }\right) f^{\prime \prime }(0), \quad Re_{x}^{1/2}C_{fy}=\left( 1+\frac{1}{\beta }\right) g^{\prime }(0). $$

(2.15)

The local Nusselt number $Nu_{x}$ is defined as

$$Nu_{x}=\frac{xq_{w}}{k(T_{w}-T_{\infty })}, $$

(2.16)

where the heat flux $q_{w}$ is given as

$$ q_{w}=\left. -k\frac{\partial T}{\partial z}\right| _{z=0}.$$

(2.17)

Substituting the value of $q_{w}$ into Eq. (2.16), we have the non-dimensional of Nusselt number as

$$ Re_{x}^{-1/2}Nu_{x}=-\theta ^{\prime }(0).$$

(2.18)

3 Numerical Results and Discussion

A numerical study of flow and heat transfer on a stretching sheet in a rotating Casson fluid has been carried out and the non-linear differential equations (2.9–2.11) with the boundary conditions (2.12) are solved with the help of shooting method with fifth order Runge–Kutta–Fehlberg method. The scheme has been carried out in symbolic software $MAPLE$. The boundary conditions defined at infinity are replaced by a sufficiently large value $\eta =\eta _{\max }$. In the present study $\eta _{\max }$ is taken to be 15 for all values of involved physical parameters. The accuracy up to 6 decimal places is chosen to be the convergence criteria and step size is taken as $\Delta \eta =0.001$. In order to check the accuracy and validity of the obtained results, Tables  1, 2 and 3 list the obtained values of $f^{\prime \prime }(0)$, $g^{\prime }(0)$ and $\theta ^{\prime }(0)$ and compared with those of Wang [17] and Nazar [20] under limiting conditions.

Table 1 Comparison of obtained values of $f^{\prime \prime }(0)$ with those of Wang [17] and Nazar et al. [20] when $\beta \rightarrow \infty$

Table 2 Comparison of obtained values of $g^{\prime }(0)$ with those of Wang [17] and Nazar et al. [20] when $\beta \rightarrow \infty$

Table 3 Comparison of obtained values of $\theta ^{\prime }(0)$ with those of Wang [17] when $\beta \rightarrow \infty$ with error defined as Error= |θ′(0)[17]−θ′(0) _Present |

The effects of the Casson fluid parameter $\beta$ on the similarity velocity profiles $f^{\prime }(\eta )$ and $g(\eta )$ in $x$ and $y$ directions are presented in Figs.  2 and 3. It is noticed that the velocity profiles $f^{\prime }(\eta )$ and $g(\eta )$ decrease with an increase in the value of Casson fluid parameter $\beta$. This is due to the fact that the introduction of tensile stress due to elasticity creates resistance in the fluid flow which results in reduction of velocities. For higher values of $\beta$, the momentum boundary layer thickness decreases. On the otherhand, an increase in thermal boundary layer thickness is observed with increase in $\beta$ as shown in Fig.  4.

Fig. 2

Influence of Casson fluid parameter $\beta$ on $f^{\prime }(\eta )$ when $\lambda =0.5$

Fig. 3

Influence of Casson fluid parameter $\beta$ on $g(\eta )$ when $\lambda =0.5$

Fig. 4

Influence of Casson fluid parameter $\beta$ on $\theta (\eta )$ when $\lambda =0.5$

Figure 5 exhibits the impact of rotation parameter $\lambda$ on similarity velocity profile $f^{\prime }(\eta )$ in $x$-direction. An increase in the value of $\lambda$ results in decrease in the velocity $f^{\prime }(\eta )$. For small values of $\lambda$, i.e for $0\le \lambda \le 1$, the stretching rate of surface is more than or equal to the fluid rotation rate and a monotonically exponential decay in $f^{\prime }(\eta )$ is observed. On the otherhand, for large values of $\lambda$, i.e for $\lambda >1$, the fluid rotation rate is significant and an oscillatory decay in velocity $f^{\prime }(\eta )$ is noticed. The effects of variation in rotation parameter $\lambda$ on similarity velocity profile $g(\eta )$ in $y$-direction is shown in Fig.  6. The velocity profile $g(\eta )$ is increased oscillatory for large values of $\lambda$ whereas there is no oscillation in velocity for smaller values. Figure  7 exhibits the effects of rotation parameter $\lambda$ on the temperature profiles $\theta (\eta )$. An increase in thermal boundary layer thickness is noticed with increase in $\lambda$. Thus, increase in rotation parameter results in rise in the fluid temperature. Figure  8 indicates that increasing the Prandtl number $Pr$ results in decrease in the thermal boundary layer thickness.

Fig. 5

Influence of rotation parameter $\lambda$ on $f^{\prime }(\eta )$ when $\beta =2.0$

Fig. 6

Influence of rotation parameter $\lambda$ on $g(\eta )$ when $\beta =2.0$

Fig. 7

Influence of rotation parameter $\lambda$ on $\theta (\eta )$ when $\beta =2.0$

Fig. 8

Influence of Prandtl number $Pr$ on $\theta (\eta )$ when $\beta =2.0$, $\lambda =0.5$

The effects of variation in rotation parameter $\lambda$ and Casson fluid parameter $\beta$ on the skin friction coefficients in the $x$- and $y$-directions, $Re_{x}^{1/2}C_{fx}$ and $Re_{x}^{1/2}C_{fy}$ are presented in Table  4. It can be seen that there is an increase in $Re_{x}^{1/2}C_{fx}$ and $Re_{x}^{1/2}C_{fy}$ as the value of Casson fluid parameter $\beta$ increases. Moreover, as the value of rotation parameter $\lambda$ increases, the values of the skin friction coefficients in $x$ and $y$ directions decrease thus resulting in reduction in the thickness of momentum boundary layer. Table  5 illustrates the effects of rotation parameter $\lambda$, Casson fluid parameter $\beta$ and Prandtl number $Pr$ on local Nusselt number $Re_{x}^{-1/2}Nu_{x}$. A drop in heat transfer rate is observed as the values of the rotation parameter $\lambda$ and Casson fluid parameter $\beta$ increase whereas there is an enhancement in the heat transfer coefficient with increase in the value of Prandtl number.

Table 4 Numerical values of $Re_{x}^{1/2}C_{fx}$ and $Re_{x}^{1/2}C_{fy}$ for different values of rotation parameter $\lambda$ and Casson fluid parameter $\beta$

Table 5 Numerical values of $Re_{x}^{-1/2}Nu_{x}$ for different values of rotation parameter $\lambda$, Casson fluid parameter $\beta$ and Prandtl number $Pr$

4 Conclusions

We investigate the flow and heat transfer phenomenon on a sheet stretching in a rotating Casson fluid. The exact similarity solutions are obtained and the results are analyzed and interpreted. The main observations of the study are:

• (a) The similarity velocity profiles $f^{\prime }(\eta )$ and $g(\eta )$ in $x$ and $y$ directions decrease as the value of Casson fluid parameter $\beta$ increases.

• (b) The velocity profile $f^{\prime }(\eta )$ in $x$ direction decrease with increase in rotation parameter $\lambda$.

• (c) An increase in the thermal boundary layer thickness is observed with increase in Casson fluid parameter $\beta$ and rotation parameter $\lambda$ whereas reduction in thermal boundary layer thickness is notices as the value of Prandtl number $Pr$ increases.

• (d) The skin friction coefficients $Re_{x}^{1/2}C_{fx}$ and $Re_{x}^{1/2}C_{fy}$ in $x$ and $y$ directions decrease with rotation parameter $\lambda$. On the otherhand, an increasing effect on $Re_{x}^{1/2}C_{fx}$ and $Re_{x}^{1/2}C_{fy}$ is observed as the Casson fluid parameter $\beta$ increases.

• (e) The local Nusselt number $Re_{x}^{-1/2}Nu_{x}$ decreases with increase in Casson fluid parameter $\beta$ and rotation parameter $\lambda$ whereas Prandtl number $Pr$ has increasing effect on heat transfer coefficient.