MHD stagnation point flow of Williamson and Casson fluids past an extended cylinder: a new heat flux model
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Abstract
The impact of heat transfer in MHD flows along a stretching cylinder is playing a vital role for heat exchangers, fiber coating, transportation, etc. Currently, a lot of theoretical models are accessible for illustrating the thermal transfer impact of nonNewtonian liquid flows over a cylinder. Also, an external magnetic force is spoton to deal with the physical features of the fluids to oversee the nature of thermal and momentum transfer in the system. Considering this fact, we inspect the heat transport behavior of two different nonNewtonian MHD flows due to stretching of a cylinder with heat generation, by taking the advantage of a new heat flux theory conceived by Christov–Cattaneo. The basic PDEs are converted into ODEs with the suitable similarity transformations. These ODEs are solved by fourth order Runge–Kutta based shooting system. Plots are drawn to discern the influence of sundry parameters on the flow fields (velocity and temperature). Along with them the rate of heat transfer and friction factor are bestowed in table. From the results, we notice that the influence of thermal stratification and curvature parameters have a propensity to increase both the velocity and thermal fields. Thermal relaxation parameter effectively lifts the friction factor in the flow of Williamson fluid than that of Casson fluid.
Keywords
MHD Heat transfer NonNewtonian fluid Convection Thermal stratification CylinderList of symbols
 \( B_{0} \)
Applied magnetic field strength
 \( b \)
Radius of the cylinder
 \( C_{f} \)
Friction factor
 \( Gr \)
Grashof number
 \( J_{m} \)
Wall shear stress
 \( J_{w} \)
The measure of the heat transfer
 \( k \)
Thermal conductivity
 \( l_{0} \)
Characteristic length
 \( M \)
Magnetic field parameter
 \( m_{1} ,m_{2} \)
Constants
 \( Nu \)
Nusselt number
 \( \Pr \)
Prandtl number
 \( p_{0}\,\&\,p_{\infty } \)
Velocity components
 \( c_{p} \)
Specific heat
 \( Q_{l} \)
Component of heat source
 Q
Heat source parameter
 S
Thermal stratification parameter
 T
Temperature of the fluid
 \( T_{0} \)
Component of temperature
 \( {\text{V}}_{\text{r}} \)
Velocity ratio parameter
 \( u,v \)
Velocity components in \( x,y  \) directions
 \( u_{e} \)
Free stream velocity
 \( u_{w} \)
Stretching velocity of the cylinder
Greek letters
 \( \beta \)
Thermal relaxation parameter
 \( \upsilon \)
Viscosity (kinematic)
 \( \sigma \)
Electrical conductivity
 \( \beta_{T} \)
Thermal expansion coefficient
 \( \delta \)
Relaxation time of heat flux
 \( \rho \)
Fluid’s density
 \( \varGamma \)
Positive time constant
 \( \gamma \)
Casson fluid parameter
 \( \lambda \)
Williamson fluid parameter
 \( \xi \)
Stream function
 \( \mu \)
Viscosity (dynamic)
 \( \zeta \)
Similarity variable
1 Introduction
In nonNewtonian liquids there is a nonlinear interaction between change in warping and shear stress. The investigation on those fluids has allured so much attention due to their importance in industries, medicine and mechanical engineering. Fluids that do not fulfil the above said nonlinear correspondence are termed as nonNewtonian fluids. Some agreeable nonNewtonian fluids are starch, blood, ketchup and some oils etc. The most commonly known nonNewtonian fluids are Casson and Williamson fluids. Casson fluid is a shear thinning fluid. Honey, soups, thick juices and tomato sauce belong to Casson fluid. The analysis of the boundary layer flows of Williamson liquids have great importance in abundant branches of engineering and science, mainly in geophysics, bioengineering, food processing and drilling operations. Owing to this significance, number of mathematical models can be found in the literature to examine their flow behaviour. Hayat et al. [1] bestowed an exact solution for boundary layer motion of nonNewtonian liquid across a stretched sheet. The impression of drag force on Newtonian fluid motion owing to an exponentially stretched sheet was deliberated by Nadeem et al. [2]. Further, the flows of nonNewtonian type were reported by Mukhopadhyay [3], with the aid of shooting technique. Malik et al. [4] inspected the impression of heat sink on Williamson liquid over a cylinder and conclude that nonNewtonian parameter has a propensity to reduce the velocity field. Recently, Kumar et al. [5, 6] presented dual solutions for MHD shear thinking liquid over a curved geometry.
Convective heat transfer shows an essential task in some physical practices like tumour cures and industrial processes to elucidate engineering problems. Normally, the performing ability of the machines will be contingent on strength of the fluid’s heat transfer rate (water, oils, kerosene etc.). Revankar [7] discussed the impact of free convection on permeable vertical plate with suction or injection. The flow and heat transfer aspects of Newtonian fluid caused by a stretched geometry were reported by Chiam [8]. The similar type of investigation with velocity slip and heat transfer impact on third grade fluid flow was elucidated by Sahoo [9]. Hsiao [10] studied the action of drag force on nonNewtonian fluid owing to permeable wedge with natural convection.
The analysis of MHD deals with the motion of electrically accompanying liquids. This conception has related in numerous processes such as flow meters, power generators, pumps, magnetic drug treating, cooling of reactors, plasma studies and the design of heat exchangers. The Lorentz force impact on Casson fluid due to stretched geometry was scrutinized by Nadeem et al. [11]. Sandeep and Sugunamma [12] considered a time dependent flow problem with heat source. The impact of frictional heat and Lorentz force on Newtonian fluid across a shrinking surface was reported by Kumar et al. [13]. Recently, Kumar et al. [14] and Kalyani et al. [15] discussed the flow and heat transfer characteristics of nonNewtonian fluid over a stretching surface in the presence of heat source.
Stagnation point flow plays a significant role in travelling management, hot rolling, production of paper, some extrusion production and wire drawing etc. Najib et al. [16] described the chemically reacting flow across a cylinder near the stagnation point. Further, this work was extended by Yasin et al. [17] with drag force and thermal transport. The influence of heat transport on nanofluid motion owing to stretched cylinder was deliberated by Omar et al. [18] and Raju et al. [19]. They found that suction constraint has propensity to shrink the thermal fields. Further Kumar et al. [20] deliberated the flow and thermal attributes of Newtonian liquid past a sheet near the neighbourhood of stagnation in the attendance of radiation.
The flows across a cylinder (stretched) has noble significance in manufacturing process, metal industries, plastic sheets extrusion, cooling of a large magnetite plates in a hot tub, making paper plates and spinning of fiber. Initially, in 1961, Rashko [21] designed and developed a model to examine the flow past a cylinder. In continuation of this, Proudman and Johnson [22] reported the stagnation point flow past a cylinder. Later on, Fornberg [23] had been given a numerical treatment to examine the flow owing to cylinder. The impact of magnetic force and constant heat flux on an unsteady flow through a cylinder was discussed by Ganesan and Loganthan [24] and they state that a growth in the magnitude of Prandtl number boosts the heat transfer performance. The characteristics of the flow caused by a circular cylinder were reported by Nobari and Naderan [25]. Recently, Authors [26, 27] inspected the flow of MHD nonNewtonian liquid past a melting surface with the attendance of heat source/sink. It was concluded that heat source/sink parameters plays a crucial role in the heat transfer performance.
The heat flux phenomenon was first initiated by Fourier [28] in 1900 century. Then after, so many researchers employed the Fourier’s law to illustrate several engineering and industrial situations and found some drawbacks in this model. So Cattaneo [29], recommended a new model with thermal relaxation time. Further, Christov [30] had given an advanced heat flux derivative model by extending the earlier Cattaneo’s model. Later on, sundry researchers focused their research on different kind of flows with Christov–Cattaneo heat flux. Han et al. [31] bestowed an analytical treatment for Cattaneo–Christov motion of viscoelastic liquid over a stretched plate. Recently, Mustafa [32] and Waqas et al. [33] used the above said heat flux concept to analyse the thermal transport behaviour and concluded that thermal conductivity parameter helps to inflate the temperature distribution. The impact of variable heat generation or absorption on Cattaneo–Christov flow through a wedge was reported by Kumar et al. [34]. Ramadevi et al. [35] continued this work to scrutinize the influence of chemical response and heat source/sink on the flow past a variable thicked surface. They stated that the action of chemical reaction parameter is decreasing function of concentration field. Very recently, Sandeep and Reddy [36] focused on Cattaneo–Christov flow caused by cone or plate considering frictional heating and concluded that Eckert number escalates the momentum and thermal distributions. The impact of thermal stratification on Jeffery fluid flow due to stretched cylinder with modified heat flux was explained by Hayat et al. [37] and stated that thermal and momentum fields diminish with the action of thermal stratification.
To the best of our knowledge, it is clear that a few researchers concentrated on the stagnation point motion of fluids due to stretched cylinder. Also, to the extent of our belief no researcher examined the Cattaneo–Christov motion of nonNewtonian liquids via stretching cylinder near the stagnation point. Heat source and thermal stratification are accounted. Similarity transmutations are applied to alter the flow governing nonlinear equations into coupled ODE’s. Bvp5c technique is utilized to solve these ODE’s. Impacts of numerous dimensionless parameters on the heat and velocity functions are represented through plots. Along with them measure of thermal transport and skin friction coefficient are presented in a separate table.
2 Formation of the model
 (i)
Casson and Williamson nonNewtonian fluid model.
 (ii)
Ohmic heating is very small.
 (iii)
The coefficient of electrical conductivity is regarded as scalar.
 (iv)
Impact of frictional heat and induced magnetic field are ignored.
 (v)
Cattaneo–Christov heat flux model is adopted.

Casson fluid flow: \( \varGamma = 0,\,\,\gamma \ne 0 \).

Williamson fluid flow: \( \varGamma \ne 0,\,\gamma \to \infty \).
Here \( \text{Re}_{x} = \frac{{xu_{w} }}{\upsilon } \) (Reynolds number).
3 Solution procedure
The system of nonlinear ODEs (9) and (10) with the restrictions of boundary (11) and (12) have solved by Runge–Kutta fourth order method. Afore that the boundary value problem denoted by Eqs. (9)–(11) are converted to initial value problem by making use of shooting technique. Since the governing dimensionless equations are nonlinear, they cannot be solved by Runge–Kutta method. We solve them by reducing to a set of first order partial differential equations. The procedure is as follows.
4 Expansion of results
The transmuted nonlinear ODE’s (9) and (10) in view of the conditions (11) and (12) are solved by shooting and fourth order R.K. methods. For calculation drive, we assumed sundry parameter values as \( S = 0.5 \), \( \delta = 1 \), \( \beta = 1.2 \), \( Q = 0.5 \), \( M = 0.2 \), \( Gr = 0.1 \), \( \Pr = 0.71 \), \( \xi = 2 \) and \( {\text{V}}_{\text{r}} = 0.1 \). In the entire analysis the above standards are believed as perpetual excluding the variants display in the graphs and table. In graphs dashed and solid line specify the curves of Williamson and Casson liquids correspondingly. The impact of dimensionless sundry constraints on skin friction coefficient, measure of thermal transport, velocity and temperature were reported through tables and plots, which are obtained by bvb5c MATLAB package.
Variation in \( f^{\prime\prime}(0) \) and \(  \,\theta^{\prime}(0) \) for Casson and Williamson fluids
\( \delta \)  \( M \)  \( Gr \)  \( {\text{V}}_{\text{r}} \)  \( \Pr \)  \( \beta \)  \( Q \)  \( S \)  \( f^{\prime\prime}(0) \)  \(  \,\theta^{\prime}(0) \)  

Casson fluid  Williamson fluid  Casson fluid  Williamson fluid  
1.5  1.0167  1.0341  2.7159  2.6797  
3.0  1.3752  1.5255  2.1661  2.1392  
5.0  1.6274  1.9205  1.8349  1.1851  
1.5  − 0.5308  − 0.6696  1.7526  1.7066  
3.0  − 1.0377  − 1.1356  1.6291  1.5930  
5.0  − 1.5813  − 1.6124  1.5105  1.4844  
1.5  0.3158  0.2446  1.7278  1.6863  
3.0  1.4054  1.6257  1.9605  1.9414  
5.0  2.6625  3.4646  2.1837  2.1950  
1.5  − 0.9535  − 1.0250  2.7062  2.6098  
3.0  − 0.9124  − 0.9818  2.8319  2.7572  
5.0  − 0.8539  − 0.9210  2.9676  2.9100  
1.5  0.6233  0.6108  3.8565  3.8056  
3.0  0.2742  0.2172  5.2251  5.1576  
5.0  − 0.0064  − 0.0768  6.5828  6.5074  
1.5  2.2891  2.7735  2.8182  2.8186  
3.0  2.0350  2.3761  3.3145  3.3130  
5.0  1.7869  2.0174  3.8312  3.8247  
1.5  3.3579  4.6893  2.7744  2.7991  
3.0  3.5420  5.1093  2.4460  2.4806  
5.0  3.8508  5.8726  1.9339  1.9919  
1.5  − 0.1330  − 0.2354  1.5496  1.5294  
3.0  0.8616  0.9269  2.7413  2.7112  
5.0  2.1022  2.6424  4.5076  4.4801 
5 Conclusions

Thermal stratification and curvature parameters have a propensity to boost up both velocity and thermal fields where as the result is reversed with the action of Prandtl number.

Lorentz force has a tendency to diminish the friction factor.

Prandtl number displays higher impact on heat transfer rate when compared to the friction factor.

In the attendance of heat source parameter Casson fluid is slightly more affected when compared with the Williamson fluid.

Thermal relaxation parameter has proclivity to diminish the curves of velocity and temperature.

Local heat transfer rate and friction are escalating functions of Grashof number.
Notes
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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