Effect of time dependent viscosity and radiation efficacy on a non-Newtonian fluid flow.

In the present article we have studied the radiation effects on the flow of a viscoelastic fluid flow past a spongy plate by considering the viscosity as variable. In order to explore the variable viscosity effects, law of conservation of mass, momentum and energy are flourished. The shooting method is adapted to accomplish the numerical solution of governing equations. The effects of the involved emerging parameters such as Reynolds' model viscosity numbers, Vogel's model viscosity numbers and Prandtl numbers on velocity and temperature profiles are discussed with the help of graphs. The associated physical properties of the flow i.e. the skin friction coefficient and Nusselt numbers are presented graphically for different parameters. The tables for effects of distinct parameters on temperature profile at the wall for Reynolds' model and Vogel's model are given. Impact of various parameters involved on Nusselt number and skin friction are also presented in tables.

Introduction

Fluid Mechanics is the branch of the physical sciences which deals with the behavior of the fluids, whether they are moving or stationary. Fluid Mechanics plays an important role to model the red spot on Jupiter, or predict the behavior of subatomic particles in a betatron, or measure the vorticity in a tornado, or design a transonic wing for a supersonic transport. It provides the base for aerodynamics, propulsion and combustion; meteorology, oceanography, and astronomy; for biofluids, acoustics and particle physics. Materials exhibiting non-Newtonian flow characteristics involve, soap solutions, biological fluids, natural products, polymer melts and solutions, dairy wastes and agricultural wastes, magmas and lavas, personal care products including toiletries and cosmetics, multiphase mixtures, food products, building materials etc. In literature a number of researchers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] studied radiation effects on fluid flow through different geometries and boundary conditions. The downright and coherent results of non-Newtonian fluids with variable viscosity was anatomized by Yursoy and Pakdemirili [21] and Pakdemirili and Yilbas [22].

The model of Walters' B fluid describes the behaviors of different polymeric liquids upon chemical engineering and biotechnology. The boundary layer flow of viscoelastic fluid close to a stagnation point was studied by Walter and Beard [23]. Walters' B equation as a simpler constitutive equation generates nonlinear initial value problems which are not trivial. Chang et al. [24] examined the numerical solution of Walters' B fluid of viscoelastic flow for free convective heat transfer. Nandeppanavar et al. [25] investigated the Walters' B fluid appearace of nonuniform heat source. Such analysis is extended by Hakeem et al. [26] acompany of radiation. Nadeem et al. [27] also analyzed the flow of Walters' B fluid of oblique flow in the existence of nanoparticles and magnetohydrodynamics. The Walters' B flow bounded by an increasing stretching sheet was given by Talla [28]. Ramesh and Devakar [29] discussed the peristalsis of Walters' B fluid through a vertical channel.

Radiation effects on boundary layer flow has much importance in engineering, physics and industries such as furnace aspire, polymer proceeding, space gas cooled nuclear reactors, glass handiwork and way technology. The merge backwash of linear and nonlinear radiation precipitate on stagnancy point over a stretching surface was analyzed by Hussain et al. [30]. The sway of thermal radiation sequel on heat flow athwart a pad sheet was studied by Cortell [31]. He engages the nonlinear Rosseland approach to assimilate the thermal radiation move. Motivated from the consequence of the fluid flow, the effects of radiation on steady flow of Walter's B fluid model with variable viscosity past a porous plate is studied. Numerical solution of the problem is obtained. The tables of temperature on effects of distinct parameters at the wall are given. Impact of various parameters involved on skin friction and Nusselt number are also presented in tables and graphs. Graphs of different emerging parameters are also presented. The problem is divided into two different parts in which first part explicates that the plate has greater temperature than fluid's temperature. Second part describes that plate is insulated. The results are also discussed at the end section.

Model

The cauchy stress tensor of Walter's B fluid is given byHere, represents pressure, identity tensor, is viscosity and is short memory coefficient, which are defined aswhere represents the distribution function along with relaxation time .

In the present problem, we discussed the radiation effects on a viscoelastic Walters' B fluid model past an infinite pervious plate (Fig. 1).

Fig. 1

Schematic figure of the problem.

Velocity and temperature fields figured out as

As represents temperature, and indicates velocity components. The governing equations which are the conservation of mass, momentum and energy are,where is body force, is fluid density, shows internal energy, heat flux vector, represents velocity gradient and also is signifies radiant heating. For incompressible fluid flow Eq. (5) becomeswhere is suction and represents blowing at the plate. The governing momentum equation with viscosity as variable (in the absenteeism of body forces) for present problem may be written as

Now defining a modified pressure

The Eqs. (10), (11), and (12) can be written in the form

Eq. (14) can be expressed ashere

Now the appropriate boundary conditions are

Since Eq. (17) is third-order equation, so more boundary conditions are required. As fluid is moving with uniform velocity. Far away from the plate so flow in free stream shear stress is zero. So,

We have necessary conditions i. e,and also take another assumption that

Now we concentrate on Eq. (7). Heat flux is

And in Eq. (25). Radiation parameter is expressed as

The energy equation for the under consideration study takes the formWhere is specific heat.

The suitable boundary conditions for Eq. (25) are:

Case 1: Constant wall temperature

Case b: Insulated wall

Solution for constant wall temperature

We define non-dimensional parameterswhereis the characteristic “length” and also

Using above relations Eqs. (17) and (25) becomes

For simplicity, we remove the bars from Eqs. (33) and (34) and getWhere

The suitable non-dimensional boundary conditions areand

Solution for insulated plate

In this case, we have dimensionless parameter of temperaturewhere can be considered the bulk temperature and all other remaining important dimensionless parameters are similar as given above in Eq. (37) The Eckert number is

The suitable non-dimensional boundary conditions are

We have removed the bars and stars for our simplicity, all quantities are non-dimensional.

Skin friction and Nusselt number of fluid flow are expressed as

In above equation skin friction is denoted by and Nusselt number is shown as . Here, represents wall shear stress and indicates wall heat flux that are explained asby using dimensionless transformation, we getHere, represents Reynold number.

Reynolds' model

In this model, the viscosity is expressed in the formwhich by Maclaurin's series can be written as

Using the value of in Eqs. (35) and (36) we obtain

Vogel's model

In this model,

Above equation can be written in the formusing Eq. (53) in Eqs. (35) and (36) we get

Methodology

The governing partial differential equations which are highly non-linear are transfigured to ordinary differential equations by using suitable transformations with shooting technique is used along with Runge Kutta scheme [32,33,34, 35]. Runge-Kutta method is a technique that solves the initial value problems. First of all we covert momentum and energy equations in first order form.

Solution for Reynold's model

In this case, Eqs. (50) and (51) for the above mentioned desired form is

Now, we define new variables to get first order equations from higher order ordinary differential equations.

We get the new system as:

Along with boundary conditions

Solution for Vogel's model

In this procedure, Eqs. (54) and (55) gets the form

As in above case, we havealong with boundary conditions of the problem as described in Eq. (64).

Results & discussion

In the realm of graphical portray, impact of physical parameters on temperature and velocity profiles are canvassed briefly. Fig. 2 renders the effect of on temperature for Reynolds' model. Fig. 3 depicts the behavior of temperature against for Reynold's model. Temperature profile is going down by rising . Fig. 4 delineates the outcome on velocity field for Reynolds' model on . The velocity field decreases with the increment of . Fig. 5 describes the on temperature distribution for Reynolds' model. Fig. 6 shows the effect of on velocity for Reynolds' model. Fig. 7 draws the consequences of for velocity distribution of Vogel's model. The profile enhances by escalating . Fig. 8 gives the temperature profile on for Vogel's model. Fig. 9 shows the impact on velocity for on Vogel's model. Fig. 10 delineates the behavior on for temperature field of Reynolds' model. Fig. 11 describes effects of on velocity profile of Vogel's model. Figs. 12 and 13 indicates effects of and for temperature of Vogel's model. Fig. 14 draws the consequences of on velocity distribution for Vogel's model. Fig. 15 describes the influence of on temperature of Vogel's model. Fig. 16 describes the effects on velocity field of Vogel's model. Fig. 17 depicts the effects of for velocity of Vogel's model. Fig. 18 shows that temperature of gives lower behavior for Vogel's model. Fig. 19 depicts that and decreases skin friction for Vogel's model. Fig. 20 displays that consequences of and on Nusselt number goes down due to accretion in . Fig. 21 explicates the effects of and on Nusselt number grows for Reynolds' model. Figs. 22, 23, and 24 clarify stream lines graphs for different values of for Reynolds' model. Figs. 25, 26, and 27 clarify the three dimensional graphs for different values of for Reynolds' model. Table 1 represents behavior of and on Nusselt number. Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14 shows values of for and for both models. Table 15 explicates the consequences on skin friction coefficient for and which is escalating with the accretion in the values of .

Fig. 2

Influence of λ on Temperature profile for Reynolds' model.

Fig. 3

Effects of γ on Temperature field for Reynolds' model.

Fig. 4

Impress of M for Velocity profile of Reynolds' model.

Fig. 5

Impact of Pr for Temperature outline of Reynolds' model.

Fig. 6

Affect of γ for Velocity silhouette of Reynolds' model.

Fig. 7

Sway of C on Temperature contour for Vogel's model.

Fig. 8

Movement of A on Temperature profile for Vogel's model.

Fig. 9

Influence of R on Temperature field for Vogel's model.

Fig. 10

Effect of M onTemperature outline for Reynolds' model.

Fig. 11

Impact of B on Velocity field for Vegol's model.

Fig. 12

Affect of B on Temperature profile for Vogel's model.

Fig. 13

Influence of C∗ on Temperature field for Vogel's model.

Fig. 14

Influence of C on Velocity profile for Vogel's model.

Fig. 15

Impact of Pr for Temperature outline for Vogel's model.

Fig. 16

Effect of R for Temperature field for Vogel's model.

Fig. 17

Impact of γ on Velocity field for Vogel's model.

Fig. 18

Effect of γ on Temperature outline for Vogel's model.

Fig. 19

Effect of γ and A on skin friction for Vogel's model.

Fig. 20

Influence of γ and A on Nusselt number for Vogel's model.

Fig. 21

Impact of γ and M on Nusselt number for Reynolds' model.

Fig. 22

Stream lines for γ = 0.1 for Reynolds' model.

Fig. 23

Stream lines for γ = 0.3 for Reynolds' model.

Fig. 24

Stream lines for γ = 0.5 for Reynolds' model.

Fig. 25

Three dimensional graph of for γ = 0.1.

Fig. 26

Three dimensional graph of for γ = 0.5.

Fig. 27

Three dimensional graph of for γ = 0.9.

Table 1

Values for Nusselt number of distinct parameters for Vogels' model.

γ	Pr	λ	C	B	C^∗	A	R	−θ'(0)
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675
0.2	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.72213
0.3	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.77681
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675
0.1	7.3	1.5	2.9	1.5	0.9	0.08	0.8	0.62488
0.1	7.4	1.5	2.9	1.5	0.9	0.08	0.8	0.63304
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675
0.1	7.2	1.6	2.9	1.5	0.9	0.08	0.8	0.60731
0.1	7.2	1.7	2.9	1.5	0.9	0.08	0.8	0.59791
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675
0.1	7.2	1.5	3.0	1.5	0.9	0.08	0.8	0.61376
0.1	7.2	1.5	3.1	1.5	0.9	0.08	0.8	0.61077
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675
0.1	7.2	1.5	2.9	1.6	0.9	0.08	0.8	0.62246
0.1	7.2	1.5	2.9	1.7	0.9	0.08	0.8	0.62974
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675
0.1	7.2	1.5	2.9	1.5	0.1	0.08	0.8	0.62974
0.1	7.2	1.5	2.9	1.5	0.2	0.08	0.8	0.59990
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675
0.1	7.2	1.5	2.9	1.5	0.9	0.09	0.8	0.60350
0.1	7.2	1.5	2.9	1.5	0.9	0.10	0.8	0.59456
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.9	0.78562
0.1	7.2	1.5	2.9	1.5	0.9	0.08	1.0	0.85214
0.1	7.2	1.5	2.9	1.5	0.9	0.08	0.8	0.61675

Table 2

Values of at the wall for for Reynolds' model.

λ	R	M	Pr	γ	Results
1.0	0.5	0.3	7.2	0.1	−.1108
3.0	0.5	0.3	7.2	0.1	−1.427
5.0	0.5	0.3	7.2	0.1	−1.732
6.5	0.5	0.3	7.2	0.1	−1.952

Table 3

Values of at the wall for for Reynolds' model.

M	R	λ	Pr	γ	Results
0.5	0.5	1.0	7.2	0.1	−.701
2.5	0.5	1.0	7.2	0.1	−1.284
4.5	0.5	1.0	7.2	0.1	−1.359
7.5	0.5	1.0	7.2	0.1	−1.416

Table 4

Values of at the wall for for Reynolds' model.

Pr	R	λ	M	γ	Results
7.3	0.5	1.0	0.3	0.1	−0.666
8.3	0.5	1.0	0.3	0.1	−0.762
9.3	0.5	1.0	0.3	0.1	−0.860
10.3	0.5	1.0	0.3	0.1	−0.958

Table 5

Values of at the wall for for Reynolds' model.

γ	R	λ	M	Pr	Results
0.1	0.5	1.0	0.3	7.2	−1.111
0.15	0.5	1.0	0.3	7.2	−1.370
0.2	0.5	1.0	0.3	7.2	−1.653
0.25	0.5	1.0	0.3	7.2	−1.954

Table 6

Values of at the wall for for Reynolds' model.

R	γ	λ	M	Pr	Results
0.5	0.1	1.0	0.3	7.2	−1.111
0.6	0.1	1.0	0.3	7.2	−1.126
0.7	0.1	1.0	0.3	7.2	−1.135
0.8	0.1	1.0	0.3	7.2	−1.147

Table 7

Values of at the wall for for Vogels' model.

γ	C	C^∗	A	B	R	λ	Pr	Results
0.23	2.9	0.9	1.3	1.5	0.8	1.0	7.5	−0.932
0.3	2.9	0.9	1.3	1.5	0.8	1.0	7.5	−1.363
0.4	2.9	0.9	1.3	1.5	0.8	1.0	7.5	−2.026
0.5	2.9	0.9	1.3	1.5	0.8	1.0	7.5	−2.724

Table 8

Values of at the wall for for Vogels' model.

Pr	C	C^∗	A	B	R	Λ	γ	Results
7.3	2.9	0.9	1.3	1.5	0.8	1.0	0.23	0.846
10.3	2.9	0.9	1.3	1.5	0.8	1.0	0.23	1.084
13.3	2.9	0.9	1.3	1.5	0.8	1.0	0.23	1.314
16.3	2.9	0.9	1.3	1.5	0.8	1.0	0.23	1.526

Table 9

Values of at the wall for for Vogels' model.

λ	C	C^∗	A	B	R	γ	Pr	Results
1.0	2.9	0.9	1.3	1.5	0.8	0.23	7.5	−0.502
2.0	2.9	0.9	1.3	1.5	0.8	0.23	7.5	−0.281
3.4	2.9	0.9	1.3	1.5	0.8	0.23	7.5	−0.166
4.5	2.9	0.9	1.3	1.5	0.8	0.23	7.5	−0.053

Table 10

Values of at the wall for for Vogels' model.

C^∗	C	λ	A	B	R	γ	Pr	Results
0.8	2.9	1.0	1.3	1.5	0.8	0.23	7.5	−0.146
1.0	2.9	1.0	1.3	1.5	0.8	0.23	7.5	−0.223
1.5	2.9	1.0	1.3	1.5	0.8	0.23	7.5	−1.974
2.0	2.9	1.0	1.3	1.5	0.8	0.23	7.5	−2.394

Table 11

Values of at the wall for for Vogels' model.

C	C^∗	λ	A	B	R	γ	Pr	Results
0.5	0.9	1.0	1.3	1.5	0.8	0.23	7.5	−5.044
1.0	0.9	1.0	1.3	1.5	0.8	0.23	7.5	−4.622
1.5	0.9	1.0	1.3	1.5	0.8	0.23	7.5	−3.566
2.0	0.9	1.0	1.3	1.5	0.8	0.23	7.5	−0.535

Table 12

Values of at the wall for for Vogels' model.

B	C^∗	λ	A	C	R	γ	Pr	Results
1.49	0.9	1.0	1.3	2.9	0.8	0.23	7.5	−0.160
1.6	0.9	1.0	1.3	2.9	0.8	0.23	7.5	−0.206
2.0	0.9	1.0	1.3	2.9	0.8	0.23	7.5	−0.262
15.0	0.9	1.0	1.3	2.9	0.8	0.23	7.5	−0.535

Table 13

Values of at the wall for for Vogels' model.

A	C^∗	λ	B	C	R	γ	Pr	Results
0.8	0.9	1.0	1.5	2.9	0.1	0.23	7.5	−0.160
0.9	0.9	1.0	1.5	2.9	0.1	0.23	7.5	−0.206
1.3	0.9	1.0	1.5	2.9	0.1	0.23	7.5	−0.262
1.5	0.9	1.0	1.5	2.9	0.1	0.23	7.5	−0.535

Table 14

Values of at the wall for for Vogels' model.

R	C^∗	λ	B	C	A	γ	Pr	Results
0.8	0.9	1.0	1.5	2.9	0.1	0.23	7.5	−0.160
0.9	0.9	1.0	1.5	2.9	0.1	0.23	7.5	−0.267
1.0	0.9	1.0	1.5	2.9	0.1	0.23	7.5	−0.374
1.1	0.9	1.0	1.5	2.9	0.1	0.23	7.5	−0.481

Table 15

Values of skin friction coefficient for A and γ for Vogels' model.

γ	A	λ	C	B	C^∗	Pr	12C_fRe
0.1	1.3	1.5	3.5	1.7	1.7	7.3	0.42382
0.2	1.3	1.5	3.5	1.7	1.7	7.3	0.63031
0.3	1.3	1.5	3.5	1.7	1.7	7.3	0.78747
0.1	1.3	1.5	3.5	1.7	1.7	7.3	0.42382
0.1	1.4	1.5	3.5	1.7	1.7	7.3	0.48943
0.1	1.5	1.5	3.5	1.7	1.7	7.3	0.54517
0.1	1.3	1.5	3.5	1.7	1.7	7.3	0.42382
0.1	1.3	1.6	3.5	1.7	1.7	7.3	0.43324
0.1	1.3	1.7	3.5	1.7	1.7	7.3	0.42907
0.1	1.3	1.5	3.5	1.7	1.7	7.3	0.42382
0.1	1.3	1.5	3.6	1.7	1.7	7.3	0.40182
0.1	1.3	1.5	3.7	1.7	1.7	7.3	0.38175
0.1	1.3	1.5	3.5	1.7	1.7	7.3	0.42382
0.1	1.3	1.5	3.5	1.8	1.7	7.3	0.30023
0.1	1.3	1.5	3.5	1.9	1.7	7.3	0.17688
0.1	1.3	1.5	3.5	1.6	1.7	7.3	0.42382
0.1	1.3	1.5	3.5	1.7	1.8	7.3	0.43749
0.1	1.3	1.5	3.5	1.7	1.9	7.3	0.46459
0.1	1.3	1.5	3.5	1.7	1.7	7.3	0.42382
0.1	1.3	1.5	3.5	1.7	1.7	7.3	0.43532
0.1	1.3	1.5	3.5	1.7	1.7	7.4	0.543727
0.1	1.3	1.5	3.5	1.7	1.7	7.5	0.42382

Conclusions

An exploration has been implemented to interrogate the study of Reynolds' and Vogel's viscosity models for the flow of a viscoelastic fluid. Using the similarity transformation governing equations are transmuted into ordinary differential equations. The numerical solution is obtained. Consequences of all different parameters on temperature and velocity fields are interrogated to probe the comportment of fluid's flow. Parameter's sway on temperature and velocity are explored with tables and graphs. The significant points of the findings of the present study are as follows:

An apprehensible excrescency is sighted in velocity portray with increasing C and reduction occurs in velocity curve due to increment in curvature parameter , A and M on Vogel's and Reynolds' models.

The temperature of both models decreases due to escalation of and .

Stream lines compresses and 3D graphs crooked due to increase in of Reynold's model.

Skin friction graph is explicating the lower behavior because of increasing A and Nusselt number graph increases by change in A.

Declarations

Author contribution statement

Azad Hussain: Analyzed and interpreted the data; Contributed analysis tools or data.

Sobia Akbar, Lubna Sarwar: Conceived and designed the analysis; Contributed analysis tools or data; Wrote the paper.

Sohail Nadeem, Zaffar Iqbal: Contributed analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Competing interest statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.