Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article
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Applied Mathematics and Statistical Mechanics and their Applications

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Research Article | Open Access

Volume 2021 |Article ID 6641835 |

Muhammad Bilal Riaz, Maryam Asgir, A. A. Zafar, Shaowen Yao, "Combined Effects of Heat and Mass Transfer on MHD Free Convective Flow of Maxwell Fluid with Variable Temperature and Concentration", Mathematical Problems in Engineering, vol. 2021, Article ID 6641835, 36 pages, 2021.

Combined Effects of Heat and Mass Transfer on MHD Free Convective Flow of Maxwell Fluid with Variable Temperature and Concentration

Academic Editor: Fateh Mebarek-Oudina
Received18 Nov 2020
Revised15 Mar 2021
Accepted26 Mar 2021
Published14 Apr 2021


Heat and mass transfer combined effects on MHD natural convection for a viscoelastic fluid flow are investigated. The dynamics of the fluid are controlled by the motion of the plate with arbitrary velocity along with varying temperature and mass diffusion. The non-dimensional forms of the governing equations of the model are developed along with generalized boundary conditions and the resulting forms are solved by the classical integral (Laplace) transform technique/method and closed-form solutions are developed. Obtained generalized results are very important due to their vast applications in the field of engineering and applied sciences; few of them are highlighted here as limiting cases. Moreover, parametric analysis of system parameters is done via graphical simulations.

1. Introduction

In science and in many engineering applications such as in condensation, evaporation, and chemical process, many transport processes are influenced by the combined action of the buoyancy forces from both heat and mass diffusion. Heat and mass transfer combined effects are studied extensively due to their significant role in chemical processing equipment, oceanic circulation, emergency cooling system of advanced nuclear reactors, cooling process of plastic sheets, formation and dissipation of the fog, processing and drying the food, temperature distribution and moisture of agriculture fields, and production of polymer. In recent years, a lot of practical applications attracted many scientists and engineers to pay a considerable amount of focus to learn the heat and mass effects either analytically or numerically [14].In industrial and engineering processes, most fluids are non-Newtonian. Since the non-Newtonian fluids deal more complexities due to the rheological behavior than Newtonian fluids, distinct models were proposed. The influence of heat and mass transfer in the non-Newtonian fluid is an important subject from the theoretical as well as practical point of view due to its abundant applications in industry and engineering. Common examples include polymer extrusion, the emergency cooling system of nuclear reactors, food processing, thermal welding, to name a few. Convective flow is a self-sustained flow with the effect of the temperature gradient. In literature, different theories are made to see the occurrence of heat and mass transfer in convective flows of different fluids. Mebarek-Oudina et al. [5] investigated the natural convective heat transfer phenomenon of water-based hybrid nanofluid in a porous medium along with the magnetic field. Das et al. [6] considered the natural convective flow of the electrically conducting fluid past on vertical plate embedded in a permeable medium and explored the impacts of heat and mass transfer. The heat transfer phenomenon of Casson nanofluid flow is taken into account by Abo-Dahab et al. [7]. They analyzed the problem with the convective boundary conditions and discussed the influence of chemical reaction and heat source. Sajad et al.[8] studied the heat transfer and magnetic effect on hybrid nanofluid. Nazish et al. [9] explored the influence of heat and concentration/mass transform with the existence of fields developed by magnetic in the Maxwell fluid model. Ahmad [10] explored the heat transfer for the Maxwell fluid on the stretching plate with the slip boundary on the velocity. They explored the numerical solutions and showed the heat flux effect using the Nusselt number and the Prandtl number. A computational analysis is performed to study the effects of the transverse magnetic field at the unsteady Poiseuille–Rayleigh–Benard flow by Marzougui et al. [11]. The thermal properties’ effects on the soil temperature are modeled and investigated numerically by Belatrache [12]. To have more insight about heat and mass transfer mechanisms in fluid flow and their applications, readers are referred to review references [1316].On the other hand, many researchers paid a significant amount of attention to the study of MHD free convective flows due to its numerous applications in solar and stellar structure, radio propagation, MHD pumps, MHD bearings, aerodynamics, polymer technology, petroleum industry, crude oil purification, glass fiber drawing, etc. In light of these applications, many researchers such as Rajput [17], Gupta [18], and El Amin [19] studied the MHD flow of different fluids. They found the exact solution for velocity, concentration, and temperature by the Laplace transform method. Heat and mass transfer simultaneous effects on MHD flow of Maxwell fluid have been investigated by Nadeem et al. [20]. Recently, the study of the unsteady boundary layer heat transfer of Maxwell viscoelastic fluid was carried out by Zhao et al. [21]. Ahmad [22] studied MHD viscous, with constant density, electro-conducting fluid in the existence of the radiation, thermal diffusion, free convection, and mass transfer flow. These results motivated Chaudhry et al. [23] and they used classical integral transform to obtain the exact solutions of natural MHD convective flow past on an accelerated surface submerged in a permeable medium. Das [24] developed the closed-form solution of the unsteady MHD natural convection flow on a moving vertical plate accompanied by mass transfer and thermal radiation. Carrying on, Das et al. [25] investigated the time-dependent MHD natural convection flow past a moving vertical plate dipped in a porous medium and studied the different aspects of heat and mass transfer. They discussed the problem with the uniform, oscillating, and impulsive motions of the plate besides considering the constant heat and mass diffusion and implemented the Laplace integral transform to develop the analytic solutions.Motivated by these investigations, the objective of this manuscript is to study the combined effect of heat and mass transfer on MHD Maxwell fluid. Laplace integral transformation is used to obtain the unique solution of temperature, velocity, and concentration under the impact of generalized boundary conditions on temperature, velocity, and concentration. The importance of the problem is highlighted by showing its impact/applications in the field of engineering and applied sciences. The paper is organized into six sections. After the introductory section in Section 2, the dimensionless governing equations are developed. In Section 3, Laplace integral transform is implemented to find the exact solution of the temperature, velocity, and concentration field. In Section 4, some applications in different fields are discussed as limiting cases to justify our results. In Section 5, the effect of physical parameters is analyzed graphically. The concluding observation is listed at the end.

2. Problem Formulation

We studied here the motion of the viscoelastic, in-compressible, electronically conducting Maxwell fluid due to plate motion with arbitrary velocity . The plate is along and is considered normal on the plate. In the first instance, at the plate and fluid are at temperature and concentration . With the time , the plate starts to move in its own axis. Then, the level of temperature and concentration takes up to and where and are piecewise continuous functions that vanish at . Details of different parameters are given in Table 1. Momentum, energy, and concentration equations are formed as follows:


Velocity of fluid
Magnetic field parameter
Laplace transforms parameter
Mass diffusivity
Thermal expansion parameter
Concentration expansion coefficient
Thermal conductivity
Density of fluid
Relaxation time
Electric conductivity coefficient
Dynamic viscosity
Kinematic viscosity
Specific heat
Heat source parameter
Chemical reaction coefficient
Gravitational acceleration
Schmidt number
Parameter due to magnetic field
Prandtl number
Grashof number due to thermal effect
Grashof number due to concentration

The imposed initial and boundary conditions are

For dimensionless problem, we use the following relations:

After non-dimensionalizing, the governing equations becomealong the following initial and boundary conditions:

3. Solution of the Problem

3.1. Concentration

Transforming equation (7) after applying the Laplace integral transform and utilizing the corresponding initial condition, we get

The above differential equation solution is

The solution of equation (11) with the transformed form of boundary conditions becomes

Applying the Laplace inverse on equation (12) and using the with , convolution theorem and equation (A.22), the generalized solution for concentration isand is specified in equation (A.23).

3.2. Temperature Distribution

Implementing the Laplace transform on equation (5) and using the concerned the initial condition, we get

The solution isAfter implementing the boundary conditions, equation (15) becomes

The Laplace inverse on equation (16) and using the with , convolution theorem and equation (A.24), the generalized solution for temperature obtained iswhere is defined in equation (A.25).

3.3. Velocity

Employing the Laplace transform on equation (4) and using the corresponding initial condition on velocity form the following differential equation:

In order to solve equation (18), we use the value of , from equation (12) and equation (16), respectively. With boundary conditions use on velocity, the following solution is obtained:

Further simplification reduces equation (19):where , .

Generalized expression for velocity field is acquired by employing the inverse Laplace transform on equation (20):wherewhere

By using equation (A.20) and equation (A.21), expressions for the and are evaluated as follows:where is given in equation (23) andwhere

Similarly,where is given in equation (23) andThe above results are obtained for generalized time-dependent boundary conditions on velocity, concentration, and temperature. These results have many applications in engineering and applied science. Now, we will consider and discuss its few applications.

4. Applications

4.1. Application 1:

This function value shows the motion of the fluid is because of the motion of an infinite plate in its plane with constant velocity. This function has importance in a lot of engineering problems such as signal waves, driving forces that act for a short time only, and impulsive forces acting for an instance such as a hammer blow.Substituting the value of into equation (12) and applying the Laplace inverse, the expression for concentration iswhere is known as Dirac delta function.

Embedding the value of into equation (16) and taking Laplace inverse make the expression of temperature

The equation of velocitywhere

is obtained asand for (see equation (A.1)).

After substituting the value of into equation (25)

Equation (26) takes the form after employing the value of whereand for (see equation (A.1)).

After substituting the value of into equation (29)

Similarly, equation (30) after substituting the value of andand for (see equation (A.1).

Similar result for concentration was obtained by Nehad Ali Shah [26] (equation (35)). Thus, our result supports the result already present in literature.

4.2. Application 2:

The important concepts of engineering are based around linear functions. They are often used in engineering to explain data and evaluate the lines that best fit the given data sets. It has a lot of applications in engineering and it can be represented in a variety of ways. One of the particular interests is direct variation, which forms many engineering applications such as Hooke’s law and Ohm’s law. To learn about slope, engineers use linear functions to interpret and understand graphs that describe displacement, velocity, and acceleration. They use these functions to analyze data to learn how to design their engineering products more efficiently, reliably, and safely.For the choice of equal to in the appropriate equations and employing the Laplace inverse, the expression of , and then changes into, respectively,wherewhere (see equation (A.1)) and (see equation (A.2)).where (see equation (A.1)) and (see equation (A.3)).where (see equation (A.1)) and (see equation (A.4)).

4.3. Application 3:

The choice of this function shows the fluid motion due to the oscillation of the plate. It has a lot of applications in physics such as wave motion, other oscillatory motions, and engineering. It is used to model the behavior that repeats. Trigonometric functions are used to calculate angles in many engineering problems. In civil and mechanical engineering, trigonometry is used to calculate torque and forces on objects, which help build bridges and girders. In the construction of bridges, we need to consider the forces which keep the bridges at their balance and trigonometry helps us to calculate the static force which keeps the bridges static. In engineering, trigonometry is used to decompose the forces into horizontal and vertical components that can be analyzed.The expression for concentration after putting the value of into equation (12) isthe expression for temperature become after putting the value of into equation (16) isand velocity change after substitute the value of into equation (21) iswherewhere (see equation (A.1)) and (see equation (A.5)).where (see equation (A.1)) and (see equation (A.6)).where (see equation (A.1)) and (see equation (A.7)).

4.4. Application 4:

The exponent functions are used for real-world application as for calculating area, volume, determining growth or decay, and impacts of force. In engineering, it helps them to design, build, and improve the machinery, structure, and equipment. For example, in sound engineering, it is used to calculate sound waves. In basic engineering, it is used to compute the tensile strength, which finds out the amount of stress that a structure can withstand. In aeronautical engineering, it is used to predict how airplanes, rockets, and jets will perform during flight. To determine the kinetic and potential energy, pressure, heat, and airflow of waves behavior, it is very helpful. Nuclear power sources are one of the important things developed by nuclear engineers. They used the exponents to work with extremely small numbers to make the big things happen. Substituting the value of into equation (12), the concentration equation after implementing the Laplace inverse becomesand equation of temperature distribution after putting the value of into equation (16) and applying Laplace inverse

The expression for velocity iswhereand (see equation (A.1)) and (see equation (A.8)).and (see equation (A.1)) and (see equation (A.9)).

Similarly,and (see equation (A.1)) and (see equation (A.10)).

4.5. Application 5:

Inserting the into equation (12) and applying the Laplace inverse, we getand insert the into equation (16) and take Laplace inverse:The takes the form after embedding the where (see equation (A.1)) and (see equation (A.11)).where (see equation (A.1)) and (see equation (A.12)).where (see equation (A.1)) and (see equation (A.13)).

4.6. Application 6:

The choice of the value of makes the expressionwhere (see equation (A.1)) and (see equation (A.14)).where (see equation (A.1)) and (see equation (A.15)).

Similarly,where (see (A.1)) and (see equation (A.16)).

4.7. Application 7:

By putting the value of into equation (12) and employing Laplace inverse,

By putting the value of into equation (16) and Laplace inverse gives the expressionwhereand (see equation (A.1)) and (see equation (A.17)).where (see equation (A.1)) and (see equation (A.18)).and for (see equation (A.1)) and (see equation (A.19)).These are solutions for the choice of same function for , and from the list of functions . We can consider the problem with the different choice of function for , e.g., and find its solution. For the validation of results, if we take with choice of or , in our system of equations (4)–(8), the results obtained are the same as the result obtained by Nehad Ali shah [27] (choosing the