# Momentum and heat transfer of a special case of the unsteady stagnation-point flow

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## Abstract

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer, which is also an exact solution to the unsteady Navier-Stokes (NS) equations. The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions. The wall temperature and heat flux have power dependence on both time and spatial distance. The solution domain, the velocity distribution, the flow field, and the temperature distribution in the fluids are studied for different controlling parameters. These parameters include the Prandtl number, the mass transfer parameter at the wall, the wall moving parameter, the time power index, and the spatial power index. It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems. The heat transfer solutions are given by the confluent hypergeometric function of the first kind, which can be simplified into the incomplete gamma functions for special conditions. The wall heat flux and temperature profiles show very complicated variation behaviors. The wall heat flux can have multiple poles under certain given controlling parameters, and the temperature can have significant oscillations with overshoot and negative values in the boundary layers. The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified. The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed. Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations, which can be used as a benchmark problem for numerical code validation.

## Key words

unsteady stagnation point flow Navier-Stokes (NS) equations analytical solution heat transfer## Nomenclature

*u,v*velocity components, m/s

- T
fluid temperature, K

*ν*kinematic viscosity, m

^{2}/s*p*fluid pressure, Pa

*α*fluid thermal diffusivity, m

^{2}/s*k*fluid thermal conductivity, W/(m·K)

*U*_{∞}free stream velocity, m/s

*U*_{w}wall moving velocity, m/s

*V*_{w}mass transfer velocity, m/s

*T*_{w}wall temperature, K

*T*_{∞}free stream fluid temperature, K

*T*_{ref}reference temperature, K

*q*_{w}wall heat flux constant, W·m

^{−2}·s^{1}/^{2}*θ*non-dimensional temperature

^{[1]}*Q*free stream velocity coefficient

^{[1]}- λ
wall moving parameter

^{[1]}*s*mass transfer parameter

^{[1]}*Pr*Prandtl number

^{[1]}*ψ*stream function

^{[1]}*η*similarity variable

^{[1]}*f*(*η*)similarity function

^{[1]}*F*(*η*)function transformation

^{[1]}*M*confluent hypergeometric function of the first kind

^{[1]}*m,n*real exponents

^{[1]}.

## Chinese Library Classification

O357.1## 2010 Mathematics Subject Classification

76D05## Preview

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