Purpose The purpose of the present study is to analyze the mixed convection water boundary layer flows over moving vertical plate with variable viscosity and Prandtl number. The non-linear partial differential equation governing the flow and thermal fields are presented in non-dimensional form by using appropriate transformation. The quasi-linearization technique in combination with implicit finite difference scheme has been adopted to solve the nonlinear-coupled partial differential equation. The numerical results are displayed graphically to illustrate the influence of various non-dimensional physical parameters on velocity and temperature. Further, the numerical results for local skin-friction coefficient and local Nusselt number are also reported. The present findings are compared with previously reported results, and these comparisons are found to be in excellent agreement. Design/methodology/approach The nonlinear partial differential equations governing the flow and thermal fields have been solved numerically using the implicit finite difference scheme in combination with the quasi-linearization technique. The numerical results are presented in terms of skin friction and heat transfer rate which are useful in determining the surface heat requirements for stabilizing the laminar boundary layer flow over a moving plate in water. Findings The effect of the ratio of free-stream velocity to the composite reference velocity is significant on the velocity profile. Near the wall region, as ratio of free stream velocity to composite reference velocity increases form 0.1 to 0.5, the velocity overshoot gets enhanced from 3 per cent to 41 per cent. The influence of buoyancy parameter and ration of free stream velocity to composite reference velocity on temperature profile is comparatively less than on velocity profiles. The increase in the skin friction coefficient is dependent on the increase in the value of ratio of free stream velocity to composite reference velocity if the buoyancy parameter λ is fixed and vice versa and increases in ΔT results in a decrease in N and Pr. Originality/value The present investigation is to deal with the solution of steady laminar water boundary layer flows over a moving plate with temperature-dependent viscosity and Prandtl number applicable for water using practical data. The fluid considered here is water, as it is one of the most common working fluids found in engineering applications.

The current investigation is based on Williamson's features of a hybrid nanofluid flow over a curved surface made mixture of silver (Ag) and titanium dioxide (TiO2) with engine oil. Under the presumption of a low magnetic Reynolds number, a constant homogenous magnetic field is applied. Consideration is given to the ramping temperature and the time-varying surface concentration. Thermal absorption and first-order consistent chemical reaction are also taken into account. To create a hybrid nanofluid, silver (Ag), and titanium nanoparticles are dispersed in a base fluid made of water and engine oil. Quasi-linearization technique and Finite difference scheme are employed on the nondimensional partial differential equations. All physical parameters of practical importance, such as velocity, temperature, and concentration profile are analyzed and provided in tables and graphs along with the impact of physical parameters. As the Williamson parameter (W) increases, the surface velocity of the steep plate decreases. Also, as the parameter temperature ratio of Nt and Rd increases, the forces opposing the flow field reduce the friction force, and the thermal field increases with temperature effects.